Relevant Logics and Their Rivals 1 (Western Philosophy Series)

(A, B) is A B, i.e. 'if A then B', where the connection is that of a non-logical relevant conditional, e.g. a subjunctive conditional. (4) (A, B) iff appropriate weakness;

A -*• B, where L is a genuinely relevant logic of or equivalently (for many such systems),

(5') <J>(A, B) iff there is a route from A to B licensed by natural deduction rules for L; or equivalently again, but less formally, in advance of explication of 'use of premisses', (5") <J>(A, B) iff there is a derivation (according to system L) of B from, or using, A. (5)-(5") take up the traditional characterisation of entailment in terms of which A -»• B iff B can be derived, or inferred, from A by valid rules. The issue is then: what constitute valid procedures? Values (5) - (5") are spelt out in detail in ABE for favoured exportative relevant systems, but similar syntactical methods extend to a variety of other relevant systems. And there are other syntactical explications which fulfil an analogous role and so likewise provide values of (j), e.g. the tableaux and Gentzen methods studied in subsequent chapters. More satisfying perhaps, there are semantical explications, the fundamental one, in terms of which sufficiency as deployed in (4) can be explained, being as follows:(6) <|>(A, B) iff for every (L-)situation or world a, when A holds in a then B holds in a; or, put differently, A is logically sufficient for B iff without exception A situationally guarantees B, or, differently again, iff A on its own without any other assumptions (which would be peeled off by suitable falsifying situations) guarantees B. Such semantical analyses, especially (6) itself, are the main object of subsequent investigations (from chapter 2 on). For all the main relevant systems studied Bennett's requirements are met, entailment ensures that the <}> relation obtains, but <|>(P & ~P» Q) is falsified for (weakly) irrelevant Q. The main charge, on which stricters always fall back, is that the standard relevant logics involve giving up the intuitively valid principle of Disjunc-

24

1.4 THE BURDEN OF PROOF AS REGARDS V1SJUNCTJVE SYLLOGISM

tive Syllogism. This is prima facie extremely implausible ....[Disjunctive Syllogism is] not to be rejected except on the basis of strenuous arguments (Bennett 69, p.205). Disjunctive Syllogism is not intuitively valid; it is easily falsified by inconsistent situations (as we explain subsequently in this chapter and show in detail in 2.9).1 Briefly, A & (~A v B) is not logically sufficient for B because in inconsistent logical situations both A and ~A may hold though B does not. Such logical situations are readily envisaged and described (see 1.7 and 1.8). But the antecedent of a valid entailment has to guarantee the consequent over all logical situations. A & (~A v B) does not so guarantee B. Hence Disjunctive Syllogism is not valid, as this intuitive argument shows. As to whether the argument sketched is strenuous enough for Bennett, we make no conjectures; but we do disapprove mildly of the transparent way in which Bennett, here and elsewhere in his defence of strict implication, has shifted the onus of argument. It seems to us that intuitively correct principles should have sound intuitive supporting arguments: that it is not that classical, or strict, implicational principles should be accepted until they can be knocked down, but that they should not gain accreditation without intuitive supporting considerations. Methodologically a principle of weakness, not the all too familiar principle of strength, should be adopted in assessing the correctness of logical principles (as reason indicates2, and we try to explain in chapter 3). Similar points tell against other attempts to reinforce the status quo that entailment is some brand of strict implication or its metalinguistic reformulation - by shifting the onus of argument to the opponents of strict implication. Although methodologically unsound procedures abound among defences of strict implication, Pollock 66 surely deserves some sort of prize for his effort. Pollock's overall argument takes the following lines:Firstly he tries, on the basis of Lewis's arguments, to shift the burden of proof: i) All the inferences in the derivations of the hard Lewis paradox are just too intuitive and 'obvious to be denied unless some very good reason can be given for their denial' (66, p.183). By i) intuition is supposed to show ii) 'there is a very strong prima facie case for thinking that the paradoxes of ... implication are logical truths' (p.183). iii) No philosophers have produced a reason as required by i), i.e. have undermined the "intuitive" case of ii). Therefore, Pollock concludes, iv) 'the paradoxes are in fact true principles of logic, surprising though they may be' (p.180; also p.195 and p.196); indeed, this is 'the only justifiable conclusion'! 1

We also show in Chapter 2 that Disjunctive Syllogism involves illegitimate suppression. At the same time we deal with Lewy's failed attempt to accurately locate suppression in Disjunctive Syllogism and show where Lewy's stimulating but thoroughly unsatisfactory discussion (in 76) goes astray.

2

And Lewis observed:

see footnote 1 p.36.

25

1.4 POLLOCK'S DEFENCE OF STRICT-IMPLICATION AS ENTAILMENT

Similar, invalid, arguments would have entrenched the Aristotelian theory of motion (before Galileo) and the Newtonian theory (before Einstein), and more generally are of great assistance to the establishment. But the fact that no investigators have so far upset a prima facie case hardly suffices for the conclusion drawn, unless the investigations have been suitably exhaustive. Pollock adduces no evidence that they have, merely claiming, without any requisite argument, what appears false, that he has examined the three most plausible arguments philosophers have located. Not only is the overall argument invalid, the premisses are also defective. If intuition has too commonly supported the argument for the paradoxes, it has also supported the feeling that there is something badly wrong with the paradoxes. In short, intuition commonly generates a conflict about what holds good for implication, and because it does, it does not produce such a very strong prima facie case for thinking that the paradoxes of implication are logical truths. (Compare other paradoxes: does the Liar paradox create a strong prima facie case for thinking that some statements are both true and false? That is not what is usually said.) Since intuition too commonly yields inconsistent findings about the logical truths of implication, on a strict account (consistencizing and applying Antilogism) one perhaps ought to hold also that the Lewis argument provides a very strong prima facie case for thinking that at least one of the premisses in each "intuitively convincing" Lewis argument is incorrect. It is a piece of history that this was the position taken by Nelson 30, Duncan-Jones 34, and other investigators, all of whom found the paradoxes more intuitively outrageous than certain of the premisses of the Lewis arguments. Pollock tries to circumvent this problem by denying that there is - or can be - anything intuitively wrong with the paradoxes. First he appeals to his own case, that he has 'no intuitive reaction to the paradoxes one way or the other*. How much is this case worth? The fact that students brainwashed by orthodox logic courses frequently have no reaction even to the paradoxes of material-implication hardly assists Pollock's case: many students could almost as easily be made flat-earthers. More to the point, for students of relevant logic the principles of Disjunctive Syllogism and Antilogism are not intuitively acceptable, for advocates of connexive logic other Lewis principles are unacceptable, yet Pollock does not relinquish the first premisses of his argument. Pollock is favouring strict intuitions over others: on what basis? Pollock contends, against intuitive rejections of the paradoxes, that 'a philosophical position cannot be refuted by an intuition that is not generally shared' (p.183-4). If so, it would seem that a philosophical position could not (perhaps even less) be so established, thereby undercutting Pollock's overall procedure which begins from intuitions which are "not generally shared". Pollock is sunk unless he can find some clear asymmetry between acceptance and rejection intuitions. He has two suggestions (p.183). One is that those who reject the paradoxical implications have defective intuitions, resulting from confusing implication with some other relation such as a meaning-inclusion one. The move fails to establish the requisite asymmetry because it can equally well be applied against those who accept the Lewis premisses; for example Anderson and Belnap have repeatedly argued that intuitive acceptance of Disjunctive Syllogism rests on a conflation of extensional and intensional disjunction (see especially 62). The second suggestion seems to be - and will have to be if it is to stand a chance of success - that we cannot have intuitions as to invalidity, but only of validity. The assumption is that complex validity can be determined by putting together elementary steps, but that we can never

26

1.4 THE METHOV OF OVERKILL, APPLJEV TO THE PARADOXES

be intuitively certain that by a suitably complex argument that some inference step will not be derivable. Setting partly aside the syntactical-semantical confusion, it is simply false that creatures never have correct intuitions of invalidity. Furthermore validity and invalidity are semantical considerations, often determined, not step by step, but by model-theoretic considerations; and an intuitive countermodel to an implication is enough for intuitive invalidity. There need be no intuition, or argument, about an unlimited number of steps or a tree path leading to the result. Pollock's picture is far too simple to handle evident intuitive data. The second suggestion, like the first, fails. It has become fashionable to follow up emphatic assertion - without any of the necessary supporting argument - of the intuitive validity of all the principles used in the Lewis paradox arguments with statements as to the innocence and sheer desirability of the paradoxes - what might be called the method of overkill. Thus, for example Hughes and Cresswell: ... we are inclined to go further: the "paradoxes" seem to us on reflection not to be tiresome (though harmless) eccentricities which we have to put up with in order to have the disjunctive syllogism, transitivity of entailment and the rest, but sound principles in their own right: a logic of entailment ought, for example, to contain some principle which reflects our inclination to say to someone who has asserted something self-contradictory, 'If one were to accept that, one could prove anything at all' - and the princ iple that (p.~p) entails q expresses this in just the way that a formal system might be expected to (68, pp.338-9).1 In rather similar vein Bennett contends that the hardest Lewis paradox is not 'something unpalatable which must be choked down because there is no convincing way of faulting the Lewis argument which supports it', but that 'the fact that [Lewis's] analysis generates the paradox is part of the case the overwhelming case - for the analysis' (69, p.198). But how this paradox is part of the positive case is never clearly revealed. The most we are 2 explicitly offered depends on a figurative connection of contradictions which conflict with 'the system of move licenses'. The 'positive merit in Lewis' analysis that it implies that each impossible proposition entails every proposition' is supposed to emerge in the following way according to Bennett: 'what the paradox says ... is that if you start with something conflicting with the system of move-licenses there is nothing you cannot arrive at' (69, p.218). When this is cashed out it can be seen to depend upon having already, quite circularly, adopted a strict system of inference rules. It has been unkindly remarked that Hughes and Cresswell were so little interested in alternatives to strict implication that they did not even trouble to get right the axioms of the one alternative to strict implication that they do specifically mention in the text of their appendix on entailment and strict implication, the system E. Axiom El for entailment should be ((p -*• p) -*• q) ->• q, not the special case presented (68, p.299), namely ((p -»• p) p) pWe will take up shortly a different respect in which paradoxes of the form that anything at all entails a necessary truth have been claimed, both by Lewis and by followers like Bennett and more reluctantly Lewy 76, to be essential to deductive reasoning, namely through the suppression of necessary truths they admit. Compare also Popper's use of paradoxes in characterising derivability.

n

1.4 THE PARADOXES ARE NOT INNOCENT OR HARMLESS

The same arguments which show that Disjunctive Syllogism is invalid also show that the paradox A & ~A B is not sound. And it is widely recognised as unsound and as counterintuitive. Correspondingly the inclination to say that someone who has accepted or is committed to some particular contradiction is bound to, or might as well, accept anything at all, is by no means widespread - for good reasons: There are many statements that dialecticians and others who accept certain contradictions will refuse to accept. More generally, we shall argue that a logic for entailment ought not, as a condition of adequacy, to contain paradoxes such as A & ~A B because it thereby precludes the investigation of non-trivial inconsistent theories (1.8). This is part of our case for claiming that the paradoxes are not simply tiresome eccentricities (or Don't Cares), but are, to understate the matter, positively harmful. The larger case (a beginning on which may be found in UL, §4) is that the paradoxes have done untold damage, especially in philosophy, where time after time they have subverted straightforward resolutions of philosophical problems. Thus it is not just false, it is decidedly misleading to make the common claim that the principle A & ~A B 'is at worst an innocent one: it could never lead us astray in practice by taking us from a true premiss to a false conclusion' (Hughes and Cresswell 68, p.338). The paradox is not innocent because of its effects in applications; and it can lead us seriously astray as we shall see: just as we can go wrong with material-implication (1.6) so we can go wrong, seriously wrong,with strict-implication. The semantical arguments which counter Disjunctive Syllogism and the negative paradox A & ~A -*• B also invalidate the positive paradox B A v ~A and the main disputed assumption in Lewis's "independent" argument for this paradox, namely the expansion principle B -*-. (B & A) v (B & ~A). For suppose B holds in an incomplete situation where neither A nor ~A holds. Then A v ~A fails to hold, and likewise (B & A) v (B & ~A) fails to hold; so the antecedents of the implications are not sufficient for their respective consequents; and the implications are falsified. (The details of such intuitive countermodels to the positive paradox principle and to Expansion will be elaborated and clarified in chapter 2.) Stricters commonly defend Expansion on the grounds that A and ~A exhaust the cases in every situation, so that there are 1

Lewis and Langford (32, p.252) make an analogous defence: the logical division of any proposition B with respect to any exhaustive set of other alternatives, A v ~A, does not alter the logical force of the assertion B. But on our account the division does alter the logical content. following in Lewis's footsteps, asks:

Bennett,

When we take a premiss and split it up into jointly exhaustive subpossibilities, are we arguing invalidly or "suppressing" something which needs to be explicitly stated? (69, p.230). The short answer is: Yes, the expansion argument is invalid; the further necessary premiss A v ~A has been suppressed, and without it the subcases are not exhaustive. But the larger issue which emerges, the inadmissibility of necessary premiss suppression, remains to be sorted out. Stricters are, all the same, a little uneasy about proofs using Expansion. The proofs have the air of clumsy legerdemain, not because they are sophistical, but because they are 'so unnecessary' (Lewis and Langford 32 p 252) ' p* (footnote continued on next page)

28

1.4 FAULTING, ANV CORRECTING, EXPANSION PRINCIPLES

no incomplete deductive situations. But all the evidence points to the fact that the evaluation of entailment requires that all situations be taken into account, including incomplete and inconsistent situations, where deduction and reasoning do not cut out. Semantical analyses of systems S2 and S3 reveal, through the situations admitted where entailments fail to hold (the so-called 'non-normal' or 'queer' worlds), that even the framework of strict implication allows to a limited extent for such situations. But of course strict implication does not go nearly far enough in this direction; for although it recognises situations where Identity, A -*• A, does not hold (in correctly rejecting the paradox B ->. A -»• A and such principles as A A A), it fails to observe that there are likewise situations where other laws of thought fail to hold, where in particular Excluded Middle, A v ~A, breaks down. Yet surely if A A can fail in some situations so can A v ~A. We shall find that it is easy to locate situations where Excluded Middle does fail, incomplete ones - and these serve to falsify Expansion. Unlike the negative paradox where the various relevant approaches are divided, several approaches join forces against Lewis's positive paradox argument in rejecting Expansion; relevant logics, conceptivism and connexivism all reject Expansion. Indeed the positive paradox argument - though according to Bennett (69, p.229) simply the contrapositive of the negative argument (it is not) - has not enjoyed anything like the reputation or plausibility of the negative argument. Lewis's independent argument is as follows:B -*•. B & A v. B & ~A (B & A) v (B & ~A) ->-. B & (A v ~A) B & (A v ~A) A v ~A B -»-. A v ~A

Expansion Converse Distribution by Simplification applying Rule Syllogism repeatedly.

The commonplace objection to this argument is that Expansion is false and should be corrected to B & (A v ~A)

B & A v. B & ~A

Corrected Expansion,

i.e. to a form of Distribution. This is a correction upon which conceptivism and connexivism agree with the relevant position; and it reduces the paradox conclusion to the principle B & (A v ~A) A v ~A, to nothing but a form of Simplification. Bennett, in his apologia for strict-implication, has endeavoured to come down heavily upon the commonplace objection. He charges (in 69, p.229) that resulting positions deny the widely accepted principle A & C + B, DC-OA + B

Necessary Premiss Suppression.

This is not true of connexivism. But it is true of relevant logics, which view necessity suppression as a widespread fallacy, as an extremely damaging principle which leads directly to paradoxes. For in virtue of Simplification, •B -* A -»• B at once results. So likewise the Lewis paradox and much more result, shortcircuiting the "independent" argument. Bennett contends, how(footnote 1 continued from previous page) And Bennett, though he first suggests (p.230), without any evidence at all, that Expansion is part of 'the common concept of entailment', then concedes that it may look odd. We suggest that Expansion is not part of the common concept - to the limited extent that there is such a common concept. Our evidence is circumstantial: part is the consensus among a variety of nonclassical positions that Expansion is invalid, and part is the fact Lewis's argument for the positive paradox has never carried the conviction that the argument for the negative paradox has, and that the step in the argument students regularly baulk at is the first expansion step.

29

1.4 DEFEATING THE LEWIS CARROLL ARGUMENT FOR SUPPRESSION OF NECESSARV TRUTHS

ever, that anyone who rejects Necessity Suppression totally is caught in an infinite regress, as Lewis Carroll found (and Lewy 76 makes the same, invalid, point)• For if Suppression is rejected every necessary assumption used in an argument must be stated and conjoined to the given premisses; but to this procedure there is no end. If this picture were right there could be no deduction in relevant logics since these prohibit necessary premiss suppression: but as there are deductions, the picture cannot be right. It is very far from right. The myth that deduction not merely permits, but requires for its very operation, the deletion of (conjoined) premisses when they are logically necessary, is embedded deep in the modal Weltanschauung; it is worth some detail to try to dispell it. Let us begin with the exportation-importation confusion that is one of the muddles underlying the assumption that conjoined necessary antecedents must be sometimes 'deletable without loss of validity' (Bennett 69, p.230). According to Bennett, As Lewis Carroll once showed, we cannot automatically declare Q -*• S to be false just because the derivation of S from Q rests upon, presupposes, or requires a necessary truth which is not a conjunct in Q; for this would commit us to denying every entailment statement. For example, the derivation of (2) [i.e. (Q & £) V (Q & ~P)] from A [i.e. Q & (P v ~p)] rests upon B: A D (2) ; and the derivation of (2) from A and B rests in turn upon C:

(A & B) 3 (2)

and so on backwards and outwards. The argument confuses 'resting upon' a premiss in the sense of being implied by a premiss, with its imported form, being conjoined with a premiss. Consider the example. The derivation of (2) from A applies Modus Ponens, not Material Detachment, and rests upon B' : A

(2) .

The derivation of (2) from B' and A "rests", for what it is worth, on C* : B' -»•. A + (2)

,

i.e. on a case of Identity: A -*• (2) A -»• (2), not on its imported form A & (A (2)) (2). Thus with C*, i.e. B' -*• B', the regress effectively terminates. All steps 'backwards and outwards' are cases of Identity, C' -»• C', C' -»• C' -»-. C' -* C', etc. There is thus no vicious regress, and no basis for conceding that Necessary Premiss Suppression is "sometimes valid". The Carroll argument does not show, what Bennett claims (69, p.230), that someone who denies that (1) -*• (2) [i.e. Expansion, using the commonplace objection] must say that necessary premisses are sometimes but not always deletable without loss of validity. But non-classical logicians will in any case admit that antecedents, whether necessary or not, are sometimes deletable without loss of validity, notably where the resulting implication is independently valid; e.g. B is deletable in A & B -*• A because A A is valid. This admission naturally does nothing to reinstate Expansion.

30

1.4 ADMISSIBLE SUPPRESSION, ACCORDING TO HOOKERS AND TO STRICTERS

Relevant logicians agree with stricters that Rule Exportation is inadmissible, that true (or even proven true1) antecedent conjuncts cannot be omitted from entailments without invalidity. Exactly the same holds for necessarily true components; but this the stricter denies: in stating a strict implication one cannot omit a merely true premise which is one of a set of premises which together give the conclusion; but one can omit a necessarily true premise. The omission of a premise which is a priori or logically undeniable does not affect the validity of a deduction; but the omission of a required premise which is true but not a priori leaves the deduction incomplete and, as it stands, invalid (Lewis and Langford 32, p. 165).2 Such an omission would, prima facie, leave the deduction incomplete, and so invalid, irrespective of the modal status of the omitted antecedent, and irrespective of whether it was known to be necessary or not. There is nothing special about necessarily true premisses that permits their omission. The stricter thinks there is, but much of his case relies on circularly appealing back to what is at issue, the positive paradox and the assumption that all statements entail ones that are necessarily true. Indeed the latter, e.g. DC -fi> D C, together with Disjunctive Syllogism, yields (in the framework of basic relevant logic) Necessary Premiss Suppression, as follows: A & C - > B - * ~ B - > ~(A H C) A & C B, DC -F ~B C, ~B -> ~(A & C) ~B ->. C & ~(A & C) -o ~B -> ~A A -> B

Rule Contraposition Paradox principle Rule Composition Disjunctive Syllogism, Rule Syllogism Rule Contraposition.

The hooker can similarly argue, using the positive material paradox C -*»D -»• C, that true premisses can be suppressed; and the stricter who mistakenly accepts Necessitation, C-^QC, everywhere can -likewise argue in this way to the (exportation) rule A & C - > B , C-fA^-B The rule is defective, and is defeated by any case where A and C jointly entail B but do not do so separately and C happens to be true (e.g. by the switches paradox presented above). So the stricter is right as against the hooker; but he is only part-way right, for what he says against the hooker's omission of merely true premisses can be duplicated for the omission of necessarily true premisses. That is, counterexamples and countermodels invalidating the Necessity Suppression Rule are readily devised. Inversely, what the stricter urges against those who reject the suppression of necessarily true premisses can be recast as castigation of those who reject the suppression of true premisses, whereupon the inadequacy of the case becomes evident, even to some stricters. Consider, for instance, Bennett's assertion (69, p.233) so adapted: Because in S3 and subsystems the Necessitation Rule: A DA is inadmissible. Entailments, though true, are not according to the systems necessarily true. Thus the stricter is committed to one of Quine's two dogmas of empiricism, that the necessary/contingent distinction can be made good. The relevant position need not depend on this distinction at all.

31

1.4 HOW STRICT SYSTEMS ERASE A FUNDAMENTAL DISTINCTION

Someone who thinks that ... validity may be lost by the deletion of - - - true premisses must presumably be muddling validity with educational or heuristic satisfactoriness. That is certainly not a claim Bennett would agree to, though it differs from his remarkable assertion only by deletion of the word 'necessarily' (where the three dashes occur). Furthermore it points out the way in which it would be argued, against Bennett, that in the necessary case there is no muddle: it would take almost precisely the lines along which Bennett should argue that in the truth case there is no muddle. To put it briefly, there is no muddle because validity requires a universal guarantee over all deductive situations: the deletion of a true, or necessarily true, premiss can easily undermine this guarantee (as the semantical analyses of chapters 2 and 4 will make very clear). What Bennett really wants to insist upon is not this warm-up point but that the suppression of necessary truths (and the corresponding negative suppression of logically impossible statements) which the paradoxes facilitate, is an inevitable feature of deductive reasoning, and ipso facto of applications of entailment within philosophy. It is on this sort of ground that the paradoxes are often claimed to be essential, and desirable, outcomes of a correct analysis of entailment (and this very likely covers what Bennett intended, in saying that the paradoxes are part of the case for Lewis's analysis, 69, p.198). The critical issue here, and also underlying Lewis's independent argument for the positive paradoxes, comes down to whether a distinction can be made between two roles for necessary entailment statements, an inference facilitating role and a conjoined premiss role (cf. Bennett 69, p.230 ff). Despite Bennett's doubts, such a distinction of roles can be drawn very easily, in both syntactical and semantical fashion. Syntactically the distinction is that between A -*-. B C and A & B -»• C with A an entailment (i.e. of the form D -*• E) ; and what permits inference is Modus Ponens, not Necessary Premiss Suppression. The requisite distinction is obliterated by all strict systems that have ever been seriously considered as offering analyses of entailment; for even in system S2° A -»•. B •*• C and A & B -*• C are interderivable where A is necessitated. Nonetheless the distinction between exported and imported forms is syntactically quite obvious and is clear from elsewhere, indeed not just from relevant systems but from weaker modal systems such as SO.5 which still maintains Lewis's definition of strict implication. Semantically, the distinction is that between situations conforming to an entailment (or principle of valid inference) and an entailment's holding in situations (see FD and UL). Once again strict systems erase this fundamental distinction in the case of necessary truths (so it is not altogether surprising that Bennett fails to see it); for necessary truths are supposed to hold everywhere.1 We can usefully adapt the distinctions between premiss roles in dealing with attempts that have been made to reinstate the material-conditional as the conditional and strict-implication as entailment. Consider, for example, the material case. Simons argues for intuitive recognition of that version of Exportation he calls 'conditional proof', namely: if A and B entail C then A entails that if B then C, by way of examples like the following:1

In S2 this result is only achieved by not asserting D(A A). That is, A -»• A can fail in some situations because the theory does not require that it is a necessary truth - and really cannot so require.

32

1.4 HOW WHEW EXPORTATION VIELVS A CORRECT RESULT IT IS OTIOSE

From the theory of universal gravitation (G) together with the initial condition that x is unsupported near the earth (U) it follows deductively that x moves towards the centre of the earth (C). This is to say (29) G, U therefore C is a valid argument. But "if U then C", the singular consequence of the lawlike generalisation is thereby established as a consequence. But if so, it seems to be established because (30) G, therefore if U then C is taken to be valid if (29) is valid (Simons 65, p.83). The mistake in this argument can be pinpointed in the word 'thereby1; for the point at issue is thereby assumed. Less obliquely, (30) is not established from (29), except incorrectly; but (30) may be established along with (29). The argument, which illustrates a method of avoiding "conditional proof" of wide applicability, namely an Instantiation method, is as follows:Firstly, G entails that, for every y, if y is unsupported near the earth then y moves towards the centre of the earth. Then (30) follows by Instantiation and Syllogism. Simon's more general argument for conditional proof in the sciences, from which he argues to the result that the conditional of the sciences is truth-functional, may be dealt with in the same way. The correct argument is not: T (Theory), A (antecedent) .*. C (consequent), so by "conditional proof": T if A then C; but T so (lawlike) generalisation: always if Ax then Cx, and so T if A then C. The incorrect exportation step is otiose. The Simons example also illustrates in a very elementary way more general features of the way in which relevant theories, such as relevant arithmetic and set theory, can operate without suppression of necessary truths; namely it is very often possible to prove relevantly what classically or modally is proved by suppression means or using paradoxes. For example, in standard relevant logics and many relevant theories it can be shown that Rule y of Material Detachment is admissible, that whenever both A and ~A v B are provable then B is also relevantly provable. Modally, of course, B results immediately by detachment from Disjunctive Syllogism, but this illegitimate step does not furnish a relevant proof of B. A relevant proof can however always be obtained by different, and legitimate means. The situation is familiar from mathematics and the theory of constructive proofs: often a theorem proved by indirect means can be proved (in a less controversial way) constructively. For instance, the whole of classical Peano arithmetic can be reworked intuitionistically. Relevant logics require not that valid arguments be constructive1 but that for an argument to be valid it conform to relevant standards. An argument which proceeds by suppression of necessary premisses will be admissible only if it can in principle be reconstructed relevantly or replaced by a relevant argument. A great many classical arguments are so reconstructible or replaceable (cf. investigations of relevant arithmetic). This is more 1

Constructive relevant logics where proofs conform both to relevant and to constructive standards can, of course, be designed: sentential parts of such logics are studied in chapter 6.

33

1.4 BENNETT'S CASE FOR THE POSITIVE LEWIS PARADOX IS UNCONVWCINGLV

CIRCULAR

than enough to refute the view that necessary premiss suppression is an inevitable feature of deductive reasoning. The strategy of replacing the Necessary Premiss Suppression, A & C

B, DC-0A->B

by the case of Modus Ponens, C ->•. A

B, C - P A + B

,

is just a very simple example of one way in which replacements can sometimes be effected. Bennett has, he believes, a case for the positive paradox 'which rests neither on Lewis's analysis nor on the second Lewis argument* (69, p.232). It is based on the feature of derivation and proof that we may at any stage in a proof introduce a previously proved statement, or, in the parallel informal case, that 'a necessary proposition is entitled to crop up anywhere in an argument without being explicitly related to anything that has gone before' (p.233). But contrary to what Bennett assumes, this is not a 'stronger truth' than 'that a necessary proposition is entailed by every proposition': it does not imply the paradox at all without further assumptions, such as that the procedure can be extended to hypothetical proof and a deduction theorem applied. Derivation and arguments of the line-by-line type introducing established statements are relevantly acceptable and do not lead to paradox (as chapter 4 will make plain). It is only if classically designed inference rules are erroneously reconstrued as representing valid arguments - they do not - that anything like Lewis's positive paradox is produced: the derived rule B-» A v ~A

,

of relevant logic E, for example, does not however represent valid argument in any sense. There are derivation rules, especially derived rules, which correspond to valid argument and derivation rules that do not. The logic of entailment distinguishes the cases, but there are other ways of doing so, e.g. natural deduction explications of proof from hypotheses, and explications of formal deducibility as distinct from derivation. Now Bennett does realise that it can be objected that the introducibility of a necessary proposition in an argument without loss of validity is not the same as the deducibility of a necessary proposition from any premisses (69, p.234); but he does nothing effective about meeting the objection. Instead, to erase the introducibility/ deducibility distinction, he appeals back to the strict analysis he is supposed to be defending: accordingly the "case" for the positive paradox reduces to an unconvincing circular one. The case for the strict paradoxes is, despite initial protestations, the strict theory itself. For Bennett simply accepts the Lewis thesis that arguments purporting to prove necessary propositions cannot be sorted out into valid and invalid simply on the basis of whether each nonpremise line is entailed by earlier lines (p.234). And this is simply to accept from the outset, what was to be defended, that necessary propositions are entailed by every proposition. There is no argument, but instead another piece of tournament philosophy: no argument, but another challenge. Bennett invites those who do not accept the amazing Lewis thesis to defend a rival position. But each different theory of entailment can furnish such a sorting of the arguments in question into valid and invalid, at least for sufficient range of cases, e.g. by use of canonical systems the challenge can readily be met (later chapters show how in detail).

34

7.4 THE MISTAKEN SUBSTITUTION PRINCIPLES STRICT-IMPLICATION SUPPLIES

Just as the framework of the material-conditional, classical logic, is inadequate to define a decent conditional, so, to turn now to the attack, the framework of strict-implication systems is inadequate to entailment. The basic trouble with strict-implication can be located in the substitution conditions it admits. (The key to the logical behaviour of a connective lies in its substitutivity conditions.) The equivalence which guarantees intersubstitutivity within strict implication sentence frames is strict equivalence, defined as strict coimplication thus: A 6-4 B =

(A -» B) & (B

A) .

For systems which include Feys' system S2° or Lewis's system S2 - the only strict systems ever seriously considered as offering an analysis of entailment - coimplication can be redefined: A M B = D f d(A = B). The upshot is that provable material equivalents are everywhere intersubstitutible in these systems, so that a great many undesirable implicational theses are immediately derivable. In particular, no discrimination in deductive power can be made between theorems, since any two have exactly the same set of consequences - something that is obviously false. Similarly no discrimination can be made between logically false statements: they all have the same set of consequences, namely everything. These disastrous results are magnified in theories based on entailment, such as theories of logical content and semantic information (as FD and UL explain in detail), e.g. all necessary truths have the same logical content, namely none, and convey the same information, again none. The worst manifestations, in purely entailmental terms, appear in the paradoxes - that a contradiction entails every statement and that any statement whatsoever entails each necessary truth. For any contradiction is strictly equivalent to B & ~B, and B & ~B entails B for arbitrary B; hence, by substitutivity, any contradiction entails B. Similarly using substitutivity, B entails, on the strict account, every necessary truth. These results can be traced back of course, to the fact that strict implication, as an analysis of entailment, amounts to necessary material implication. As in the case of material implication, however, it is not just that the analysis admits these paradoxes. There are several other principles admitted which (though analytic of course for strict implication) fail to hold when -3 is construed as entailment. The stronger the strict implicational system the more of these unwarranted and unwanted principles there are as theses. Because of these significant differences it is important that stricters tell us which system of strict implication it is that really captures entailment. But this most stricters are extraordinarily reluctant to do, even when they are informationally equipped to make a choice1 - perhaps for good, if rarely divulged, reasons. 1

This applies in part even to Lewis, who wrote in 1932, at a stage when he was still informationally unequipped to distinguish S2 and S3: Prevailing good use in logical inference - the practice in mathematical deductions, for example - is not sufficiently precise ... to determine clearly which of these five systems expresses the acceptable principles of deduction. (Lewis and Langford 32, pp. 501-2). (footnote continued on next page)

35

1.4 IMPLAUSIBLE "ENTAILMENT" PRINCIPLES OF SB

The few who do choose often plump for S51 which, despite its merits as an analysis of modal notions such as logical necessity, is hopeless as an analysis of entailment, containing some singularly undesirable principles. For example, Peirce*s "law" ((A ^ B) o A) => A, rightly rejected by all logics (such as intuitionistic logic) built on positive logic as an unsatisfactory implicational principle, reappears in an unconvincing modalised form in system S5 (but not in S4 or weaker systems) as ((A -f B) H A) A, where A is strict, i.e. of the form C -3 D. In fact S5 is distinguished as a strict implicational system from its rival S4 by the modalised Peirce "law" or one of its complications such as:(((A-?B)-JC)-JB)-i. A -J B (((A -J B) -3 C) B) B-iD-J. A W D (((A-3 B) C) B) -5. B -i D -?. E -i. A—? D (((AH ( ( (A

B ) - I C ) - J D ) -4. B D -3. A -3 D B) C)T3 D ) B - » D -3T. E . A-* D

Although none of these principles is evident2 or indeed at all plausible as an (footnote 1 continued from previous page) Apart from the limitation to the five systems S1-S5, this is a claim with which we have a good deal of sympathy (cf.chapter 3). Lewis's indecision, such as it was, was not to last. Already in Lewis and Langford (32, p.178) it is said: It is this System 2 which we desire to indicate as the System of Strict Implication; and by 1960 Lewis wanted to insist: I wish the system S2 ... to be regarded as the definitive form of Strict Implication (Preface to Dover Edition of Lewis 18). l The choice is often made on such insufficient grounds as strength, mathematical elegance and simplicity: cf. Lewis's astute observation (32. p.502): Those interested in the merely mathematical properties of such systems of symbolic logic tend to prefer the more comprehensive and less 'strict' systems, such as S5 and Material Implication. The interests of logical study would probably be best" served by an exactly opposite tendency. Occasionally however the choice is based, more satisfactorily, on the semantical structure of S5 and its logical theory of modalities. This points up the unfortunate historical conflation of entailment with modality, and how a very likely account of logical modality can lead to a most unlikely account of logical implication. 2

AS Prior (62, p.49) remarks of Peirce, p -»• q -*• p -»• p, itself: Its truth is not evident at a glance, but we might look at it this way: it means If p does not imply q without being true, then it i:s true. That is, it means Either p implies q without being true, or it is true. If p is true the second of these alternatives holds. And if p is false, then it does imply q (for COq = 1, whatever q may be) and of course does so without being true, so in this case the first alternative holds. (footnote continued on next page)

36

1.4 LEWIS SYSTEM S2 AMD OTHER MODAL AFFIXING SYSTEMS

like the system S3 that Lewis once favoured, has to be made. It is easily demonstrated (using matrices or a semantical analysis of non-normal modal systems, e.g. Kripke 65) that S3 has no theses of the form A -5 B. The focussing by Lewis of the choice of a correct (or as he would have said, acceptable) logic of deducibility on the systems S3 and S2 undoubtedly represented a vast improvement over most of what had gone before him, especially in the direction of the formalisation of the logic of entailment. Since S2 is, according to its author, 'to be regarded as the definitive form of Strict Implication' (Lewis 60) and since it provides the locus of much subsequent discussion of entailment, we exhibit the system in explicitly entailmental form, not in the usual modal form. Connectives -*• (symbolising Lewis entailment, i.e. Strict Implication), &, and ~ are taken as primitive, and further extensional and modal connectives are defined thus: A v B ~(~A & ~B); A a B - d £ ~A v B; A = B « D f (A = B) & (B => A) ; A B » D f (A B) & (B -*• A); DA ~A A; 0A » D f ~0~A. (A detailed discussion of the morphological structure of sentential entailmental systems may be found in 4.1.) The postulates of S2 in Lewis entailmental form are these:- A •* A (Identity) (A -*• B) & (B -*• C) A -»• C (Conjunctive Syllogism) A&B+A A&B+B (Conjunctive Simplification) (A -*• B) & (A C) A -»» B & C (Composition) A & (B v c ) (A & B) v C (Distribution: the scheme is not independent in S2) ~~A -»• A (Double Negation) A •*• ~B •*•. B -»• ~A (Contraposition) (A ~A) ~A (Reductio) A&B-+C->-. A&~C->-~B (Antilogism) A, A -*• B B (Modus Fonens) A, B A & B (Adjunction) A -*• B, C D-»B C •*•. A D (Affixing, or Becker rules). Other strict implicational systems of entailmental interest are these (the classification anticipates that we subsequently adopt in 3 for entailmental systems):Other Affixing Systems: These result by varying the axiom schemes for S2; e.g. Feys' system S2° results by deleting Reductio, or what perhaps surprisingly (and wrongly in the larger relevant context) is axiomatically equivalent: A & (A B) -»• A, (Conjunctive) Assertion. Another example is the relevant logic DL (of DCL, formulated without sentential constants) which results from S2 by deleting Antilogism; and yet another example is the relevant logic DK (of UL) which results from S2° by deleting Antilogism, i.e. from S2 by deleting both Antilogism and Reductio. The formulation of S2 given thus brings out sharply the key separation point between strict and relevant accounts of entailment, a point to which we will repeatedly allude (especially 2.6 ff), namely the principle of Antilogism and its outcome, Disjunctive Syllogism. It was this one major obstacle that kept Lewis forever out of the relevant paradise. (But really the unquestioned loyalty to Disjunctive Syllogism and its mates, such as Rule Antilogism, is part of the larger classical-modal psychosis which we subsequently analyse.) Hitherto the main concentration in relevant logic studies has been on analogues of the exportative affixing system S3, obtained by addition to S2 of one form of Exported Syllogism, i.e. A -*• B C -*• A C -*• B (Prefixing) , or A B B -»• C A -»• C (Suffixing), or of one of its modal equivalents, e.g. A-*B-*-. DA-*- []B (Necessity Distribution), or A -»• B -»-. OA -*• 0B (Possibility Distribution). Plainly both the Affixing rule and Conjunctive Syllogism are otiose in such exportative affixing systems. Other Exportative Affixing Systems. These include S30 - S2° + Exported Syllogism; S4 which adds to S3 any of the following schemes: B A -*• A, •(A -»• A), DA -*• CDA, OCA
38

1.4 A CLASSIFICATION OF MODAL LOGICS OF ENTAILMENTAL INTEREST

which adds to DK both forms of Exported Syllogism and A -*• (A -*• B) -»-. A -»• B (Absorption). (On formulations of modal systems, see Feys 65; for formulations of exportative affixing systems, see ABE p.340 ff.) The very close connection of Anderson and Belnap system E with Lewis system S3 can be brought out by a slight reformulation of S3. Take the formulation of S2 given, and replace Conjunctive Syllogism by its exported form, Suffixing, i.e. A B -»•. B -»• C A •* C, and Affixing by N-necessitation, i.e. A-fcA -*• B •*• B; so results S3. Now E is obtained simply by deleting Antilogism; i.e. there are formulations of E and S3 which are separated just by the principle of Antilogism.1 In the quest for a correct entailment system or group of such systems, it is important to look inside S2 and DK as well as beyond. So result two further classes of systems, both non-affixing systems:Replacement Systems. These systems result from affixing systems by replacing the Affixing rule by a Replacement rule, i.e. a rule of intersubstitutivity of provable coimplicants: A **• B, C(A)

C(B)

(Replacement or Substitutivity),

where C(A) is some wff containing zero or more occurrences of A as a well formed part and C(B) results by replacing zero or more of these occurrences of A by B. The affixing systems introduced are all replacement systems in that Replacement is a derived rule. The new class of systems of interest comprises replacement systems which are not affixing systems. Best known of these in the modal arena are the systems SI and Sl° which result from S2 and S2° respectively simply by replacing Affixing in the initial formulations given by Replacement. Relevant analogues of those systems result in a similar way from DL and DK. Non-replacement Systems; These systems dispense with replacement rules, and so characteristically lack--affixing rules also. Examples are provided, in the modal case, by systems in the vicinity of SO.5 and, in the relevant case, by Arruda-da-Costa P systems. The issues raised by these various different classes of systems, in particular the correctness or satisfactoriness of Replacement, of Affixing, and of Exported Syllogism, are taken up in chapter 3. The classes of systeas differ not only syntactically, but the difference is reflected in a deeper way semantically: the various semantical analyses are the main concerns of chapter 4 and later chapters. The criticisms already made and now to be made of modal affixing systems, and especially of preferred Lewis systems S3 and S2, extend, for the most part, to non-affixing systems. Although systems S3 and S2 avoid some of the worst of the paradoxes of normal strict implicational systems, e.g. Ackermann fallacies,2 they hardly offer a panacea. We set out criticism of these systems serriatim. 1

The analogues of a system obtained in this way are far from unique. For example, corresponding to S2 is not only DL but a system ES2 obtained from reformulating S2 with N-necessitation as an added rule by deleting Antilogism. ES2 retains the restricted Commutation principles of E while abandoning Exported syllogism principles.

2

Ackermann fallacies are inevitable in normal modal systems. The argument which proves this relies on the standard definition of normal for these systems, and takes the following semantical lines:Suppose where A B is a theorem, that p A -»• B is not valid. Then, presupposing details (footnote continued on next page)

39

1.4 RELEVANT CRITICISM OF SVSTEMS S2 AND S3

1) The systems still contain a great many undesirable principles as theses, the worst of which are of course the much advertised paradoxes of strict implication which occur in S2 in the forms: OB A B and A -3 B (in SI there are equally damaging T-forms of these paradoxes). Since every truthfunctional tautology is provably necessary and every truth-functional contradiction demonstrably impossible in S2 (and in SI) the familiar "independently provable" paradoxes already discussed emerge, and also typical irrelevant entailmental principles such as C ~t D A ~f A and C D A B where A -3 B is any theorem. But the systems also contain several principles independent of the paradoxes which, if not objectionable, are least very questionable. (We will question them, and reject them in chapter 3 where we present our case for concentrating on different entailment logics from the usually favoured systems E and NR.) For example, S3 and stronger systems share with relevant system E such theses as A -*• (B C) -*• D -»-. B C A D (Restricted Commutation) and (A -*• B) -»-. (A •*• B) C C (Restricted Assertion) . In S2 these counter-intuitive principles appear in rule form, and in both cases they lead to such defective theses as ((A A) -»• B) -»• B (Suppression) and, more generally, (D B) -»• B where D is a thesis, i.e. to the Anderson and Belnap rule of necessitation A NA where NA = ^ (A -*• A) -*• A. In these entailmental respects 53, like its relevant mate E, is too strong: but in modal respects S3 is too weak. 2) The elimination of the worst of the paradoxes of implication by non-normal modal logics is far from costless. For it is achieved only with the simultaneous elimination of apparent implicational truths. In particular, S2 and S3 lack the expected principles 0(A A) and D(A & B-^A); more generally they lack D(A -3 B) for each tautology A ^ B. The addition of any one of these principles takes S2 into modal system T (von Wright's M) and S3 back into 54, and thus leads to Ackermann fallacies. Yet surely the entailment A -*• A is necessarily true, and a satisfactory modal system should assert its necessity. 1 So emerges a serious dilemma for the stricter trying to choose the modal system that really captures deducibility, a dilemma which is a direct outcome of the erroneous strict conflation of entailment and modality (a conflation that is retained in systems like E). The rejection of incorrect entailment principles, e.g. the avoidance of Ackermann fallacies, limits the choice of acceptable modal systems for entailment to non-normal systems, whereas the retention of correct modal principles for logical necessity, such as D(A -> A) and DA -> QUA, forces the choice back to normal modal systems. Relevant logics escape this dilemma, most simply by observing a proper separation of entailment and logical modalities, which modal logics confuse. (footnote

2

continued from previous page)

of the semantics to be developed in chapters 3 and 6, for some normal world a, I(p,a) = 1 and I (A •*• B) = 1, whence for some normal world b, for which Rab, I(A,b) = 1 ^ I(B,b). But as A -»• B is a theorem and so valid, for every normal world if I(A,b) = 1 then I(B,b) » 1, which is impossible. Thus p -*•. A -»- B is valid, and so a theorem. 1

In this respect E appears to represent a substantial improvement upon S3, since D(A -*• A) (with • equated with N) is a thesis of E. However, as E c S3, N(A A) is also a thesis of S3. One of the troubles with S3 is, to put the point under discussion differently, that S3 admits of two nonequivalent accounts of necessity, • and N. In a satisfactory modal theory these accounts would coincide; however from the relevant viewpoint to be advanced neither account is satisfactory (see further, chapter 10). On the normalising effect of [D(A" => B), where A => B is a tautology, in SI, S2 and S3, see Feys 65, pp.99 and 127.

40

1.4 FURTHER FALLACIES TO AVOW

IN THEORIES Of ENTAILMENT ANP

COmTlONALlTV

3) The modal weakness of S2 and S3 can be exposed in another, equally damaging, way - through their Hallden incompleteness. A sentential system is Hallden reasonable iff, whenever A v B is a theorem with A and B wff which share no sentential parameter, either A is a theorem or B is a theorem. Hallden showed (in 51) that both S2 and S3 (and indeed a range of modal systems from SI through S3) are unreasonable in this sense, and argued that this revealed a serious semantic incompleteness in these systems, namely that there are no normal interpretations with respect to which the systems are complete. But strengthening S2 and S3 to remove this incompleteness leads back yet again, if the modalities are construed as genuine logical modalities, to normal modal system S4 (alternatively, under conventional construals of modalities, removal leads to S6 and S7). Yet again this problem is avoided by relevant analogues of the modal systems, which are (as we show in chapter 5) Hallden reasonable. In sum, there are severe problems with the non-normal strict implicational systems favoured as analyses of entailment, problems which are in each case overcome by analogous relevant systems. The most notable problems with strict implication, the validation of a great many undesirable principles, are an inevitable outcome of the fact the strict implication is a modal account of implication, which tries to account for two-place entailment connections in terms of a one-place modal connection in tandem with truth functions. Such a modal reduction cannot, of course, succeed with entailment; nor can it succeed for conditionals. Determination of adequate theories of entailment - and, so it will turn out, of conditionality - requires, to summarise then, avoidance of the following further fallacies:5 . The modal dependence fallacy. The fallacy is that of supposing"that whether A -*• B holds true sometimes depends just on the modal value (e.g. necessity, impossibility) of one of the components A and B - independently of any connection between A and B. Neither the implications a statement has, nor what implies a statement, are ever functions of the statement's modal values, of its impossibility, contingency or necessity. The Stalnaker and Lewis theories of conditionals commit the modal dependence fallacy as well as the truth copulation fallacy. A special case of the modal dependence fallacy is 6. The necessity suppression fallacy. The fallacy is that of taking A -> B as true where B is necessary. 7. The modal reduction fallacy. The fallacy is that of defining A -»• B as i(f>(A,B) where i is a (one-place) modal connective and 4>(A,B) is a truthfunction of A and B. More generally, neither entailment nor conditionality are modal. Any adequate account will meet this strong intensionality requirement; and this is the basic reason why the investigation of implication generally has to be in the area of the intensional beyond the modal, why - to state a central thesis soon to be advanced - it has to be ultramodal. § 5. Conditionals: the theory sought in contrast to extensional and modal attempts. The orthodox attempts at gaining an analysis of entailment - the metalinguist ic shift to avoid overt systemic introduction of modal and intensional notions, and the strict implication account in terms of modality - leave one with no satisfactory account of non-logical implication. Thus, given that our case against material-implication stands up, the problems of a natural implication, and of conditionals, are not satisfactorily resolved at all within the usual metalinguistic and strict implicational frameworks. Yet a great many

41

7.5 'IF' STATEMENTS, AMD WHV SOME NEW CLASSIFICATION OF THEM IS ESSENTIAL

implicational statements are not analytic, for instance the lawlike statements of science and everyday conditionals, and many conditionals are, as we have noticed, not implications: but such statements are commonly deployed in reasoning, e.g. in the empirical sciences, and in legal and ordinary arguments, and they have a logic, distinct from material-implication, which it should be the business of a full theory of implication to capture. In an effort to fill these lacunae there have recently appeared on the market various theories of conditionals, which are supposed to supplement the modal view and thereby overcome its acknowledged weaknesses on the conditionals front. Logical space is made for these theories, not by abandonment of the thesis that entailment is necessary implication but by the distinction of nonnecessary implication from the conditionality relation. The conditional connections these theories are intended to capture are supposed to be neither sufficiency relations nor extensional connections such as material-implication (cf. Stevenson 70, pp.28-29). So much is not in dispute: There certainly is a class of non-sufficiency conditionals, in common usage in natural languages, whose intensional analysis is an important logical matter. Although some have hoped that these conditionals could be defined enthymematically in terms of a satisfactory sufficiency relation, there is nothing to stop, and reason to encourage, their independent logical investigation. For, among other things, if enthymematic reducibility is to be established as distinct from just claimed, an independent account of -the class to be reduced needs to be given. There is not only a strong case for independent logical investigation of (non-sufficiency) conditionals - and conditionals are after all central in much everyday reasoning - but there is also a good case for jettisoning many former divisions and exclusions built into or presupposed by theories of conditionals. For example, there are semantical reasons for not trying to make the artificial and never satisfactorily determined or vindicated division of English conditionals into "counterfactuals" and "noncounterfactuals", where a "counterfactual" usage is not simply (a counterfactual) one where the antecedent is false, but 'one in which its antecedent is uttered in the knowledge, belief, or temporary supposition that it is false' (Cooper 68, p.291). Part of the argument is that there is no evidence that 'if' differs in semantical meaning when contextual conditions are so varied, and indeed evidence that it does not - just as there is evidence that the correctness of an implication cannot be determined from the truth- or modal-value of its consequence. Likewise there is a prima facie case for not separating out counterlogical or counterphysical conditionals, i.e. those with logically or physically impossible antecedents. Nonetheless it seems that any non-vacuous logical theory of conditionals is bound to exclude some natural language occurrences of if ; that is, it seems that there is no non-vacuous way of encompassing all occurrences of 'if' within a single uniform analysis. The argument, such as it is, is that any all-encompassing theory would be vacuous because any proposed thesis it contained could be defeated by a counterexample of sorts. Remember that in English 'if' can do duty for such connectives as 'whether', 'even if* and 'as if', as well as for 'if ... then'. The claim is that the intersection of the logical properties of these diverse connectives is null. Consider, to illustrate the inconclusive case-by-case method of confirming the claim, a few of the more likely or desired principles of an i^ of argumentative value. Firstly, consider detachment, i.e. Modus Ponens. This appears to fail in cases where 'if' can be replaced by 'whether', e.g. from "you see if you can turn the wheel" and "you can turn the wheel" we cannot validly infer "you see". Next, consider Identity, and let 'if' do duty for 'even if* in the intended

42

1.5 GENUINE CONDITIONALS SURVIVE THEN-TRANSFORMATION

sense where this makes a contrast, as in 'he is intelligent if crazy*. Then "he is intelligent (even) if he is intelligent" is not just weird but not true. Differently "I am wondering if 1 am wondering" is not always true. In the same way, by taking examples where 'whether* or 'even if' can be substituted for 'if', countercases to such basic principles as Conjunctive Simplification can be devised. No principles, it would seem, survive this sort of procedure. The procedure does indicate, however, the first kind of condition that should be used to restrict the class of genuine conditionals, namely that uses of 'if' where 'if' does duty for 'whether', 'even if' or 'as if' should be excluded (at least from the core theory). There is a simple linguistic test which shakes out many of the undesired cases, namely reordering and then insertion. The test, which we call then-transformation, is this: given 'A if B' reorder to 'if B, A' and insert then to obtain 'if B, then A'; finally ask whether the result makes sense or has the same sense as the original. Examples: 'if you can turn the wheel then you see' does not have the same sense (as qualified salva veritate tests will show) as 'you see if you can turn the wheel', and 'he is sound if unimaginative' does not have the same sense as 'if he is unimaginative then he is sound'. A necessary condition on a genuine conditional is that it should survive then-transformation. Genuine conditionals do appear to have a logic, even if the pure conditional logic itself is not of vast logical interest, consisting, as it seems to at bottom of Modus Ponens and Identity (such a logic is reviewed in 2.1 and subsequently in chapter 5: the minimal B conditional logic is in fact the pure implicational fragment of system B of chapter 4, and all its theorems are instances of Identity). But, rather like the parallel logic of deducibility, the interest and difficulty of a general conditional logic quickens when further connectives, such as those of the orthodox set {&, v, are introduced. Many principles that appear correct for an implicational logic fail for a general conditional logic - something to be expected since the I-C transformation (of 1.3) fails for many genuine conditionals. For example Contraposition which appears correct at least in some forms for implication (e.g. in the form: when A ->- B then ~B -»• ~A) fails for conditionals taken generally. Counterexamples to Contraposition abound: while "if you are tired then we will sit down" may be true "if we won't sit down then you are not tired" is commonly false; "if it rains then I'll take my umbrella" rarely ensures "if 1 don't take my umbrella then it doesn't rain"; and so on. The failure of Contraposition (especially for "temporal" if-thens) is indicative of the fact that conditionals do not in general meet sufficiency requirements: that the antecedent A of a true (or acceptable) conditional A •*• B is commonly not logically or naturally sufficient for the consequent B. When A is not sufficient for B, further background assumptions are involved in getting B from A, so that when ~B holds it may not be A that is negated but some of the further assumptions that lie in the background.1 But not all negation principles fail: 1

Syntactical enthymematic representation of the conditional A 3 B in the relevant framework as A & t -»• B, where t represents some background class of truths, shows up a technical reason for Contraposition failure. ~B D ~A would require ~B & t -»• ~A, that is it would depend on the rejected Antilogism. But the syntactical representation is of only limited value, since it sanctions irrelevance; it also happens to validate Augmentation.

43

7.5 THE PROBLEM OF THE WATERSHED PRINCIPLE OF AUGMENTATION

Double Negation, for example, appears to hold in both directions. Contraposition is not the only implication principle that is faulted when ordinary conditionals which do not pretend to offer sufficient conditions are countenanced. With conditionals, as distinct perhaps from implications and entailments, very many commonly accepted logical principles are up for reconsideration, and rejection. A critical issue in determining the general logic of conditionals is whether the rule Affixing holds; for the rule is a watershed principle in semantical analyses of implicational-conditional logics (as will become evident in subsequent chapters). An important test case for Affixing is provided by the principle A -*• C -*•. A & B -*• C

((Antecedent) Augmentation)

in combination with A & B -*• A; for were Affixing correct, Augmentation would result from Conjunctive Simplification. But according to recent logical wisdom Augmentation fails for the conditional relation. The case against Augmentation is two-fold; it tends to be based on intuitive counterexamples, these being backed up by various, incompatible, theoretical explanations drawn from the different theories that have emerged. A familiar counterexample (Stalnaker 68, p.106; Stevenson 70, p.27) runs as follows: there are cases where it is true that if you strike that match it will light without it being true that if you wet that match and strike it it will light.1 Another example is this: "If I walked on the ice, it would remain firm" could be true, while "if I walked on the ice and I wore 60 lb. boots, it would remain firm" is not (cf. Bennett 74, p.384). There are ways of explaining away examples like these; but there are, so Lewis, Bennett and others contend, too many examples for the dismissals to continue to look convincing. As almost always, the arguments for this kind of claim are not decisive. Bennett's argument, for example, makes the assumption that moves to explain away apparent counterexamples to Augmentation must treat conditionals as elliptical (or enthymematic), to be expanded, in determining the logic of conditionals, by further clauses conjoined to the antecedents. And it is true that there are then well-known difficulties about ever completing such clauses satisfactorily, since the background conditions to be incorporated are open-textured in several ways which defy complete linguistic description. But the strategems aimed at explaining away the counterexamples do not have to be of this sort; they can legitimately refuse to take up background or contextual conditions syntactically in antecedents.2 It is enough, presumably, to meet the "counterexamples" to contend that the conditions which falsify the conclusion of a deduction of A & B C from A C (to take an entailmental form of Augmentation), that is typically which guarantee A & B but falsify C, undermine the basis for, or are incompatible with, or background conditions for, A -*• C. Thus in Bennett's example, my wearing 60 lb. boots undermines (or in Adams' revealing terms, is incompatible with) the background or contextual conditions under which the conditional "If I walked on the ice, it would remain firm" is held true. This strategem, which we call the A-stratagem (after Adams 65), is of very general application, defeating, in its fashion not merely all counterexamples to Augmentation but also those to Conjunctive Syllogism and to other conditional principles that initial intuitive considerations might seem to reject (cf. p.10 above). 1

It is enough that there be appropriate cases to rebut an entailment or conditional. The fact that the Greenlite matches that Routley often has to use at Plumwood Mountain will light after being dipped in water does not count against familiar cases where non-waterproof matches are used. The fact does help expose however one of the background assumptions - that that match is non-waterproof - thereby emphasizing the non-sufficiency character of the connection. (footnote 2 on next page) 44

7.5 THE SEPARATION OF RULES FROM THESES IN THE LOGIC OF CONDITIONALS

On the other hand, there are, rather obviously, various difficulties with this sort of stratagem. For one, it is no longer possible to assess on their own untreated conditionals, conditionals that have been subject to no analysis or investigation1: to know whether a conditional holds one needs to know, so to speak, what is supposed to follow from it. And this makes conditionals virtually unassessible, and a great many ordinarily acceptable conditionals false - because otherwise, were they true, we could, by defeating background conditions, counter Augmentation. Given that Augmentation is to be rejected for untreated conditionals, and also for important subclasses of conditionals such as counterfactuals, given indeed that Augmentation is a "fallacy" (as D. Lewis puts it), an important question for formalisation arises: What forms of the principle are fallacious? The same difficult question arises also for other rejected principles, Contraposition and Syllogism. Granted that both conditional and entailmfental forms of Augmentation fail for conditionally, i.e. neither Antecedent Augmentation as formulated above nor its entailmental reformulation, A -* C A & B C (where distinguishes entailment), are correct, what of the rule form: A C —t>A & B -*• C? The rule is evidently separable from theses forms, and does not succumb to the counterexamples given to theses forms. It is noteworthy, for instance, that, despite Stalnaker's emphasis on the breakdown of Augmentation, the formal system Stalnaker arrives at admits the rule form. Similarly, though Hunter and Graves 73 consider Augmentation to be a principle which differentiates conditionality from entailment, they adopt principles (including Conjunctive Syllogism) which lead to the rule form of Augmentation.2 Thus an easy answer to the question about the rule form is that we can leave the matter open for the present: what fallacies there are, if any, are avoided by avoiding the theses forms. The distinction of rules and theses is important not merely with respect to the axiomatisation of the logic of conditionals, but also as regards the common claim that the failure of Transitivity (in conjunctive form) follows from the failure of Augmentation (Stalnaker 68, p.106; Lewis 73, p.32ff; Bennett 74, p.385). Unless entailment is erroneously identified with strict implication, this common claim is false. The assumption is that Simplification, A & B ->• A, being logically necessary, can be suppressed; so given Conjunctive Syllogism in the form (A & B A) & (A C) A & B C, Augmentation results (and similarly where the main connective is strengthened to an entailment). But such suppression of conjoined necessary premisses is inadmissible, and fallacious. All that the failure of Augmentation establishes against Transitivity is the failure of Suffixing (in thesis and rule forms), not the failure of Conjunctive Syllogism. Only the failure of the rule form (footnote 2 from previous page) Bennett has, it may be alleged, made the similar error - in assuming that background conditions can be taken up as antecedent conjuncts - to that he made in his "case" for the general suppression of necessary truths. 1

Thus heavy use of the A-strategem would violate our earlier methodological strategy.

2

The collapse of the original Hunter-Graves system does not detract from this point. For the point is correct for subsystems of the HunterGraves one.

45

7.5 THE FURTHER CASE AGAINST CONJUNCTIVE SYLLOGISM, AND ITS DISSOLUTION

of Augmentation would undermine Conjunctive Syllogism, as it would undermine even Rule Syllogism, A B, B C A -»• C. Whether Rule Syllogism holds or not makes a substantial difference to the semantics of a logic (as will become evident in chapter 13); so the determination of exactly what is supposed to be rejected in conditional and counterfactual logics is not a merely idle matter. From this perspective the way the common rejection of transitivity is stated is rather sloppy, not to say misleading; thus, for instance, Stalnaker writes: '... the conditional corner is a non-transitive connective. That is, from A > B and B > C, one cannot infer A > C* (68. p.106). But in the usual way in which inference rules are stated (e.g. in Church 56) one is entitled to infer, in Stalnaker*s system C2, A > C from A > B and B > C (for when A > B and B > C are valid so is A > C). The claim intended is that A > B and B > C do not entail A > C; that there is, in this sense, no valid inference from the premisses to conclusion. The remaining case, directed against Conjunctive Syllogism is again twofold; arguments based on direct counterexamples, and more theoretical arguments from the analyses of conditionals proposed. We have already glanced at the sorts of direct counterexamples offered, and observed that there are conventionalist strategems by which they can be disposed of; and one stratagem will do, the A-strategem. Consider how this disposes of a counterexample to Syllogism from Bennett: If there were snow on the valley floor, I would be skiing along it; and if there were an avalanche just here, there would be snow on the valley floor; but it is false that if there were an avalanche I would be skiing on the valley floor — (74, p.385). According to the stratagem, the antecedent of the conclusion, that there be an avalanche, is incompatible with the truth of the initial premiss, because if there were snow from an avalanche I (or Bennett) would not be skiing along it. Insufficiency implications, such as conditionals are, are contextually-bound: they are true at best under present conditions, and when the conditions are changed, as by an avalanche, they fail. The apparent counterexample emerges only, so the A-strategem complains, by changing the contextual conditions as we go from premisses to conclusion. As we have observed, however, in the case of Augmentation, the A-strategem, may do too much; and there is, moreover, a good methodological case for a theory of (relatively) untreated conditionals. Furthermore if a general theory of untreated conditionals can be devised, we can then check the adequacy of reductions and likewise of strategems rigorously, and we can also investigate the particular logics of conditionals which satisfy given classes of conditions, e.g. contraposable conditionals, transitive conditionals, augmentable conditionals, to begin on a classification by logical principles of conditionals (cf. Rennie's procedure in the case of adverbial modifiers, in 74).1 The idea is that the logic of augmentable, contraposable conditionals, for example, will emerge as a special case of a more general conditional logic, in something the same way that the' theory of commutative groupoids comes out as a special case of the theory of groupoids. Similarly the logic of counterfactual conditionals, as a case of the logic of lawlike conditionals where the antecedent is false, will be a subcase of the logic of l The classification of conditionals by way of logical principles they satisfy is only one, among many, of the classifications that might be tried. Other classifications, most of them less than fully satisfactory, already occur in the literature, e.g. Goodman's classification in 55.

46

7.5 FAILINGS OF RECENT (IRRELEVANT) THEORIES OF CONDITIONALS

transitive augmentable contraposable conditionals, assuming (as we shall argue in chapter 8) that these properties hold for lawlike connections. The acceptance of Conjunctive Simplification separates main modal theories of conditionality from connexivism, a position with which, because of the rejection of Augmentation, they otherwise have a good deal in common. For the deletion and defeating features of the added antecedent in Augmentation is exactly what lies behind the connexivist critique of Augmentation, a critique which extends automatically to apply against Simplification (see chapter 2). It is not too surprising, then, that modern theories of conditionality are split by Conjunctive Simplification into two factions, a connexivist group (which includes Cooper 68, Stevenson 70, Downing, e.g. 61) and a non-connexivist group (which includes not only resemblance theorists such as Woods 67, Stalnaker 68 and Lewis 73 but many others, e.g. Hunter and Graves 73, Xqvist 71, Gabbay 73). Many of the recently proposed theories of untreated conditionals, whether connexivist or not, have the alleged virtue that they do not retain the faulty principles of Augmentation, Contraposition and Transitivity. But most of these theories are nonetheless defective, in that they incorporate principles which do not hold for conditionality, in particular relevance-violating principles. Thus they succumb, exactly like the theories they reject, to intuitive counterexamples. For example, the most publicised systems (American ones of course), those of Stalnaker and Thomason 70 and of D. Lewis 73 are badly irrelevant. Both the Stalnaker-Thomason system and Lewis's official logic of counterfactuals have as theses (and in rule form) the paradoxes: (~A -*• A) (B A), i.e. anything implies a conditionally necessary statement, and A & B A -»• B, thus any truth conditionally implies any other truth.1 In short, the theories provide no requisite connection between true antecedents and true consequents in a conditional or between any arbitrary antecedent and any necessary consequent: in such cases the accounts are every bit as fallacious as material-implication and strict implication, committing truth-copulation and modal-copulation fallacies. These systems also contain several other bizarre or undesirable principles - arising in part from the fact that they were obtained principally by working back from a possible world modelling to an axiomatic basis. For example, Stalnaker's system (investigated formally by Stalnaker and Thomason) includes the principle (A B) v (A -»• ~B) already criticised, the paradoxes (A -*• ~A) A -*• B (a principle which also rules out proper treatment of counterlogical conditionals) and A = (B =. A •*• B), and the dubious principle ~(A -»• ~A) A -*• B =. ~(A -* ~B) . In rejecting these theories we also reject the particular semantical and philosophical analyses in terms of which the acceptances and rejections of logical principles are justified. The worst faults of the semantics, the modal framework, we shall criticise, but the question of what can be salvaged from the extraordinary theories of neighbouring worlds and similar worlds which are written into the intended construal of the semantics, we postpone until we elaborate our own (ultramodal) theory of conditionals (at which stage resemblance theories are severely criticised). 1

~A -»• A B -»- A holds in all three Lewis systems, and A, in two systems.

47

A ->• B

7.5 THE INADEQUACY OF MODAL THEORIES

Other recently proposed theories of conditionals, especially those that are properly equipped with semantical modellings, are equally defective, both in terms of the principles they contain and as regards the underlying philosophical analysis. For example, according to Aqvist's 71 system a principle of connectivity: (A -*• B) v (B -*• A) , where B is of the form C •> D, holds, while according to Gabbay's 73 system, if A s B is a theorem, so is A •*• B, whereupon the full force of the paradoxes is immediate. One trouble with all these theories of conditionals is that they take as entailment, as the underlying logical implication, strict-implication - something we have already seen to be inadequate. But the basic trouble with these theories of conditionals - of which the adoption of strict-implication as logical implication is symptomatic - is that, like strict-implication they operate essentially within the possible worlds framework of modal logics, and so are subject to the inevitable deficiencies of such approaches, namely modal substitutivity conditions and ensuing paradoxes and sundry other unpleasantness. As customary, we say that an n-place connective $ is modal in the ith place iff whenever A B B is true (or A = B is a theorem) $(... A ...) iff $(... B ...). A functor is modal iff it is modal in each place, and a sentential language is modal iff all its connectives are modal.1 Possible world modellings are only adequate for modal languages. Modal notions have been raised to a position of excessive prominence in recent philosophy, partly due to the mistaken idea that the entailment connection, propositional identity, and conditionality are modal matters, and partly owing to technical limitations, that known semantical analyses - possible world semantics, and their metalinguistic correlates involving state descriptions succeeded at best for modal notions. This elevation of modal notions has done an enormous amount of philosophical damage, not merely in the case of entailment and conditionality, but with a range of other notions of major philosophical importance and interest such as evidence, confirmation, belief, knowledge, perception and obligation (see UL). It has even fostered the grossly mistaken, but widespread, idea that all intensional notions are modal or, if not, merely sentential (admitting replacement of no logical equivalents of any sort but only of identical sentences). The fact is that a great many notions fundamental in philosophy (including all those mentioned above and many more) are of more than modal strength: they are ultramodal intensional notions, which cannot be forced into the modal straitjacket without grave distortion and the generation of many gratuitous philosophical problems (as UL and BP explain). Most important here, none of the main notions we are seeking to explicate, namely entailment or logically necessary implication, law-like implication, and conditionality, nor any of the associated notions, such as propositional identity, are modal. All are ultramodal. Take implication. It is not a modal matter; for the substitutivity conditions required are stronger than modal, i.e. formally the truth (or provability) of A H B guarantees neither A -»• C iff B C nor C -*• A iff C -*• B. The ground we have already traversed shows this, and more. More generally, there can be no reinstatement, by patch-work repairs and qualifications from strict-implication and material-implication and similar purely modal and extensional notions such as the modern probability relation, of strict-implication or material-implication as satisfactory accounts of entailment and conditionality, and similarly there can be no constructions of satisfactory accounts from these elements. For such accounts always satisfy modal substitutivity conditions. This disposes of many proposed analyses (e.g. those of Adams 65, Stevenson 70, Geach 74). 1

These technical senses take up one (the main) traditional sense of 'modal'.

48

1.5 CHARACTER OF THE SOUGHT THEORY OF CONDITIONALS

So results a further condition of adequacy on theories of entailment and conditionality, the strong intensionality, or ultramodal, requirement, that adequate theories must be ultramodal. Just as strict-implication is insufficiently intensional to provide a satisfactory account of entailment, so, as we have now seen, modal accounts of conditionality, such as modal similarity theories, are likewise insufficiently intensional. Actually, as compared with implication and conditionality, which are fundamental notions in argumentation and reasoning and so central in philosophy and also in such more practical matters as the law, modal notions are of relatively slight importance. Through modal muddlement, however, which has taken entailment and conditionality as modal notions, the importance of modal notions and modal logics has been much exaggerated - though not of course to the extent that the importance of classical logic has. Modal befuddlement is not the only factor at work in modal-style analyses of entailment and conditionality. There is also the older, associated, assumption to be found in C. I. Lewis and in Burks' analysis of causal implication 51, but reappearing in Stalnaker and Thomason 70 and D. Lewis 73, that conditionality can, like implication, be correctly defined in terms of a oneplace connective (characteristically but not necessarily, modal), such as physical necessity or conditionally-defined necessity, in combination with truth functional connectives. Meyer has put the kibosh on this assumption (in 74b; adapted in ABE under the misleading title 'Relevance is not reducible to modality', since neither relevance nor modality are to the point). There is no need to appeal to the outcome of modal substitutivity conditions to show that genuinely two-place connectives such as entailment and conditional relations cannot be defined in terms of some one-place connective, modal or~ not, applied to a combination of truth functional connectives. That is, A ->• B cannot be defined as $$(A,B), where $ is a one-place connective satisfying a fairly minimal set of conditions and \|»(A,B) is some truth-functional combination of A and B.1 To begin, then, from the assumption that conditionality can be captured by simply adding a one-place connective to the logic of truth-functions, is to make a false start. Some of the newer systems of conditionals have implicitly recognised this limitation, that conditionality will have to employ genuinely two-place connectives. However they have persisted with the assumption that conditionals are nonetheless modal. This is false, as relevance considerations alone reveal in the case of natural language conditionals. Assembling the conclusions reached sets the shape of the general theory of conditionality sought. The theory will contain an irreducibly two-place conditionality connective which is ultramodal. The theory will be relevant, and ideally will be coupled with a relevant theory of entailment - certainly not with a strict one. The theory will, in its most general form (of 13), fail such principles as Contraposition, Augmentation and Transitivity, but validate such principles as Simplification. The theory should be underpinned by a semantical analysis preferably of a suitable philosophical cast. There are several hitches to this ambitious undertaking, to some of which we now turn, the main intellectual obstacle being the mistaken idea that any satisfactory theory must include a classical sublogic. 1

For the conditions required, see ABE, p.466, p.471. Note that system E can be replaced by any subsystem of RM3 which includes the basic logic of chapter 13; and accordingly that the argument applies both to the desired system D of deducibility, whatever its precise final form, and to that sought for conditionality.

49

7.6 DISTINGUISHING FORMS OF THE CLASSICAL DISEASE

§6. The classical hang-up; how one can go wrong with material implication, and why no classically-based logic can be adequate. The trouble with all the attempts, so far criticised, to produce satisfactory theories of conditionals and logical implication is that they are far too classical in orientation features which emerge, firstly, in the way they all proceed by adding to classical logic and aim to incorporate classical logic as an integral part,1 and, secondly, in the semantics where only classical-like possible worlds are admitted. For no such classical-type account is going to provide a tight enough connection of antecedents and consequents in conditionals or therefore to preserve relevance. Strict-implication and irrelevant conditionals, are both classical in a worlds way, in that each world used in their semantical analysis conforms to classical canons. It is from this classicalness that their main inadequacies stem. Classicalness is a persistent and widespread disease (which affects even relevant logicians who like the logical ease and "smoothness" of classical semantical corditions which involve no new or less familiar functions). It has resulted in a vast overemphasis of modal notions, and underlies attempts to provide modal analyses of a great many - sometimes, in extreme cases, all - intensional notions, though it is evident that many of these notions are bound to resist modal, or extended-classical, analyses. But in contrast with implication - which is the basic argument relation* and hence fundamental in philosophy - modality is much less important; it has assumed an exaggerated importance largely through defective accounts of implication and conditionals, which reduce them to modal notions and reduce coimplicational substitutivity conditions to modal interchange conditions. A much weaker claim is sometimes advanced in favour of classical logic or modal logic, than that they are adequate. It is claimed that even if material-implication [strict-implication] is defective as an implication [entailment] and perhaps needs some augmentation, classical logic is nonetheless correct and classical logic has to be included in any adequate logic of implication. Moreover, since one can not go wrong using classical logic, it provides a minimal working base, which there is a good case for using in controversial areas. It is, so to speak, a core logic with which one can comfortably rest. This is all too familiar bunk, which we will try to expose. The core-correctness claim may be weakened still further to: one can't go wrong with material-implication so long as simply figures as the main connective in inferences - which is all it is ever needed for. Even this latter assertion is of pretty doubtful correctness unless only very limited purposes are envisaged - certainly not formalisation of much of discourse as it figures in commerce and in newspapers and occurs on television, nor formalisation of the intensionally less rich mathematical discourse, where nonetheless second degree arguments and inferences with iterated implications occur. We will examine these stronger and weaker forms of the claim that one can't go wrong with hook together. First of all, we do not of course dispute the truism that classical logic is correct in classical contexts, i.e. 1

Modal logic is incorrect for the same reason, because it incorporates classical logic as an integral part of its structure, whereas a correct modal logic would be built on a relevant base. Similarly for modal conditional logic.

50

7.6 WAV'S OF GOING WONG

WITH CLASSICAL LOGIC

in those contexts where it is correct. But these contexts provide only a quite proper subclass of the contexts where some logic is needed, and where classical logic is supposed and intended to apply. There are then logical contexts which are not classical, and within these classical logic is incorrect (see also UL). Moreover while we agree that the & - v — theorems of classical logic though not doxastically compulsory - are true (for the extensionally-refined connectives of natural language that they consider), and accordingly should be theorems of any adequate logic of implication, it does not follow that classical sentential logic is correct, or should be included in every theory, or can never go wrong. Classical logic is only correct if it is coccoonised, if it does not assert too much more than the & - v — tautologies. It goes wrong all too quickly under needed and intended applications (as the counterexamples of 1.2 reveal). One can go wrong with classical logic in two main sorts of ways. One can go wrong interpretationally, by interpreting => as amounting to some sort of implication rather than as simply short-hand for an extensional combination (e.g., not-or)1, and one can go wrong in a deeper way by making the consistency and completeness assumptions of classical logic and their worlds. We deal with these ways of going wrong in turn.2 A common view - summarised in the slogan 'You can't go wrong with material-implication' - is that because => is truth-preserving it is safe to use for valid argument. The common view is well-represented in Lemmon (65, p.60): While admitting that this discrepancy [between ^ and if-then] exists, we may continue safely to adopt 'A => B' as a rendering of 'if A then B' serviceable for reasoning purposes, since, as will emerge ... our rules at least have the property that they will never lead from true assumptions to a false conclusion. Indeed how can any truth-falsity counterexample (i.e., where the antecedent is true and conclusion false) to any of these laws be possible, when these laws are laws of material-implication, which is, as everyone knows, truthpreserving? The truth-falsity counterexamples are possible firstly because A ^ B is at best a necessary, and is not a sufficient, condition for A -*• B. The truth-preserving properties of material-implication mean that one cannot find truth-falsity counterexamples - of an extensionally-admissible kind (i.e. one world examples) - to A -* B when A ^ B holds; but if A ^ B is only a necessary condition for A -*• B, one can find a truth-falsity counterexample to the main connective in A => (B •* C), when A = (B => C) holds. If B C is only a necessary, and not a sufficient, condition for B -*• C, B C may be false when B a C is true. Hence one has only to find one of these cases to obtain a higher-degree truth-falsity counterexample to the main connective in A 3 (B -*• C) when A is true and A ^ (B ^ C) holds. (Exportation and Commutation may be countered in exactly this fashion: cf. 3.7 where counterexamples to Commutation are presented.) 1

The counterexamples of earlier sections all rely on intended interpretations of connectives, on the interpretation of extensional connectives in terms of natural language surrogates, and of and -3 as 'if-then* and 'entails' respectively.

2

In discussion of the first way we borrow heavily from V. Routley 6 7.

37

1.6 WHV HOOK IS MOT SAFE TO USE IN REPRESENTING CORRECT ARGUMENT

The second reason why counterexamples are possible - though not counterexamples of a one-world truth-falsity variety - is because, though A => B is true, A B is rendered false by a situation, distinct from the actual one which determines truth, where A holds but B fails to hold. Modal counterexamples to A -*• B, where A and B are both contingently true, are of this sort; a possible world is envisaged where A => B fails to hold. And the specific countercases to Disjunctive Syllogism (given in 2.9) are of this kind; though A & (~A v B) a B holds in the actual situation, deductive situations can be designed where A & (~A v B) holds but B does not. In the light of this, the common view, that because is truth-preserving it is safe to use for valid argument, or as a surrogate for correct conditionals, must be defective. First, the truth-preservation property of => does not show that it is safe unless it is assumed quite circularly that the only way an implication can be wrong is by having its premiss true and conclusion false in the actual situation, the real world, T. Second, o is not truth-preserving in the right sense to guarantee safety for valid argument. For it is not truth-preserving in the sense that if we substitute 'there is a valid argument from ... to ...' or is deducible from ...' or even 'only if' for all occurrences of '=>' in the laws of =>, what we obtain will continue to be true where the original => law was true. Similarly, once such a substitution is made, => can lead from true premisses to a false conclusion, as counterexamples show. Admittedly all the false conclusions are themselves about valid argument (or conditionals), but failure of this sort can not be discounted given the uses of deducibility in assessing responsibility. Nor is it an adequate defence against these counterexamples to claim that the supposedly false conclusions are always about valid argument and that since truth-preservation, as represented by =>, is all that is required for deducibility, they must not be false at all, but true, even if surprising. Such a defence is quite circular, for we can only decide whether a particular connective, say ij^, as claimed, adequate for valid argument, by substituting 'there is a valid argument from ... to — ' for '=>' and seeing whether the results continue to hold; hence we could not without circularity decide these results on the basis of what holds for the connective itself. (A similar objection applies to the claim that the material-implication paradoxes are not paradoxical for deducibility because they follow from the truth-table for =>.) A connective is only safe to use for valid argument if upon substitution of 'there is a valid argument from ... to ...' for all occurrences of the connective in its laws, the result continues to be true, i.e. if every law of the connective is also a correct law of valid argument. It must licence no methods which are not also methods of valid argument; but many of the principles discussed, which are => laws, do license methods which are unacceptable as methods of proof or valid argument.1 And thereby one is licensed to go wrong. 1

These points also destroy van Fraassen's curious claim that classical logic is correct for mathematical English (71, p.4). Insofar as mathematics relies on valid argument, its proper formalisation is not in terms of classical logic. While it is true that much of classical mathematics can be reformulated using classical logic, such a formalisation is hardly unique, and alternative formalisations using strict systems or even relevant systems can undoubtedly be devised. In fact classical mathematics uses only comparatively weak logical inference principles, few highly nested implications and nothing like the power of classical logic or stronger Lewis systems. Finally what 'the correct logic for a language' is remains a somewhat obscure matter; but what seems clear is that a language may have a variety of logics associated with it (see Routley 75, for further discussions of this issue).

52

1.6 THE PEEPER TROUBLE WITH CLASSICAL THEORIES: MATERIAL DETACHMENT

Even if the interpretation of ^ as any sort of implication or validargument indicator or conditional is given away entirely and = construed as no more than an extensional abbreviation (e.g. for not-or) classical logic and classically-based logics remain in real trouble. For there is a deeper way in which classical logic is wrong, and in which exponents can, and do, go wrong with classical logic. Even though all extensional tautologies are true, one can still go seriously wrong through assuming the unqualified correctness of the principle of Jfeterial Detachment: Y- Where A and A => B are theses, so is B, or, in application to theory c, through assuming its theory analogue: Yc. Where A and A => B hold in c, so does B. Classically, of course, yc is assumed not only where c is the class of all truths T but also for every theory (i.e. y is assumed to hold not just for the actual world but for mathematical and scientific theories). This is part of the (mistaken) assumption that classical logic is universally applicable, to every (scientific) theory, and that it is fundamental in reasoning. It has too often escaped attention that principle y is a fundamental component of classical logic and of all its patch-ups. Classical sentential logic, as ordinarily presented and understood, consists not just of and-ornot schemes, or some batch of these taken as axioms, but also of the rule of material detachment. The deeper trouble with classical systems lies in the unrestricted assumption of this rule - which would be correct were => indeed a sufficiency connective - in the assumption, that is, that y holds for any and every application of the logic, for every theory. Consider what happens when classical logic is applied as the underlying logic to a theory which is simply inconsistent but not trivial, in that not every proposition holds in it. Such theories are not just important, they are common: philosophical theories, logical systems, dialectical positions, and belief systems furnish a wealth of examples (detailed cases are set out in DCL and in Routley2 75). But the presence of material detachment excludes them, and trivialises every simply inconsistent theory, thus:- Since a simply inconsistent theory contains B and ~B for some sentence B, it also contains ~B v C, i.e. B C, for arbitrary sentence C. Hence by Material Detachment it contains C, and so it contains every sentence.1 That is, it is trivial. Exactly the same problem also arises from use of any implication, such as strict implication, which satisfies the principle of Disjunctive Syllogism, DS . A & (~A v B)

B.

For this principle yields, by Modus Ponens closure, y. Or, more directly, since ~B v C and B belong to the simply consistent theory, so, by DS and implication closure, does C. The limited applicability of the rule of Material Detachment emerges from the conditions necessary and sufficient for its correctness. We have already observed that simple consistency is a necessary condition for its correctness. For sufficiency, let c be a theory which is prime, i.e. whenever A v B e c either A e c or B e c, and simply (i.e. negation) consistent, i.e. for no A do both A and ~A belong to c. Then y holds at c, i.e. yc is correct. For suppose A e c and ~A v B e c. As A e c, ~A / c, by negation consistency. But, as c is prime, either ~A e c or B e c; so B e c.2 1

It is assumed that the theory meets certain quite minimal normality conditions - which will happen in main cases of interest. (footnote 2 on next page)

53

1.6 TYPES Of OBJECTIONS TO MATERIAL VETACHMENT

The first class of objections to Material Detachment, and derivatively to its thesis analogue Disjunctive Syllogism, centres, then, on the point that they obliterate a class of theories of considerable philosophical and logical interest. A second class of objections arises from this, namely that many other important theories may, for all we know, belong to this class (cf. Lukasiewicz 70). In particular, it may well be that some important mathematical theories fit into the category of simply inconsistent but non-trivial theories. Naive set theory is a prime candidate. It is certainly simply inconsistent, but it is only obvious that it is rendered trivial by paradoxes like Russell's when it is classically formalised. We now know that, on the contrary, naive set theory is non-trivial when given a suitable relevant formalisation - a far more appropriate formalisation of the intuitive theory than conventional classical ones (as EMJB explains; see also 4.2).1 (footnote 2 from previous page) The sufficiency conditions also indicate the line of a main strategy for proving that y is admissible for a given logic L, namely by showing that each L-theory c which keeps out D can be replaced by a prime simply consistent L-theory c' which keeps out D. The feat is accomplished, where it can be, by first expanding c, by Lindenbaum methods, to a prime L-theory c + . But c + will commonly be negation inconsistent, so the technique is to select a subtheory c' of c + with the desired properties. This can be done, in some cases, by splitting c + , or, what accomplishes the same, by metavaluation methods (as in Meyer 72). *Russell observed long ago (06) that any way of avoiding the logical paradoxes, as classically formulated, involves changing the logic (of set theory), and accordingly he went on to consider three important ways of amending the logic of sets. But what he did not consider was the obvious move of changing the underlying classical logic. Yet this was the change of logic that really should have been considered. The option opened up, of retaining all of highly appealing intuitive theories by adjusting the underlying, usually classical, logic presupposed for formalisation, raises some awkward questions for the dedicated pragmatist; for example as to whether to trade off little used sentential principles for the powerful unqualified abstraction thesis in a formalisation of set theory. But the classical hang-up most pragmatists have has blocked investigation of this attractive alternative. The abandonment of the unwarranted classical assumption that every simply inconsistent theory is trivial also raises other severe problems for pragmatism. For the pragmatist conception of logic which is supposed - in opposition to absolutist views on logic - to make the choice of underlying logic a pragmatic matter, in fact in the presentations of its leading exponents, assumes that classical consistency conditions must be met - otherwise revision in the light of recalcitrant experience could not play its assumed role. In short, the pragmatist view presupposes part of the absolutist position it is supposed to be rejecting. It is facile to try to meet this objection as Haack (74, p.36) and others do by trying to rule out simply inconsistent logics. Dialectical logics are formally viable, and are coming to be part of the logical scene: they cannot simply be ruled out of court as not 'logics'. Nor can they be dismissed on the grounds that they fail to discriminate valid arguments since an inconsistency entails everything. For such paradoxes of deducibility any worthwhile dialectical logic would repudiate.

54

1.6 GROUNDS FOR SCEPTICISM ABOUT MARKETED SET THEORIES

Set theory of adequate power may well be inconsistent. (No doubt consistent theories may be obtained by sufficient mutilation - simple type theory without an infinity axiom provides a familiar example - but so far these theories are far too underpowered to haul the classical mathematical train.1) Most important, any classical set theory that approximates closely enough to naive set theory to undertake its work is likely to be simply inconsistent. Marketed set theories don't approximate too well at all, and there is a chronic need for better set theories than the inferior goods we have so far been offered in the way of formalisations.2 The fact that Quine's theories appear in some respects better than most others is indicative of the situation. Marketed set theories don't however have to approximate very well for inconsistency to threaten. Indeed it seems to us that some marketed set theories - Quine's system ML is the most conspicuous example are quite likely inconsistent. So scepticism about set theory and set theoretical foundations for mathematics has some warrant. A trivial foundation for mathematics - which is all that classically formulated inconsistent set theory would provide - is hardly of much merit, and one does not have to go through the laborious circuit of set theory to provide it. The conventional classical reply is that if a theory such as set theory is simply inconsistent it might as well be trivial; simple inconsistency is hardly any better than triviality. This supposes that a theory which is simply inconsistent but not trivial is of no logical interest, an assumption we have already seen to be false. Abstract set theory, with abstract sets characterised in terms of the abstraction axiom, contains, as Cantor saw, inconsistent sets, and accordingly it provides a good prima facie3 candidate for a simply inconsistent theory; but that's no good reason for concluding that the theory is trivial. Thus universal use of classical logic in set theory is without justification. The situation with regard to various other parts of mathematics is scarcely better than the set theoretic situation (cf. DLS).U Other historical mathematical theories have also proved to be simply inconsistent, most conspicuously the theory of infinitesimals and the theories of calculus and analysis that this theory supported. It was not evident - it is still not obvious - that these theories were thereby trivialised - and under a dialectical formalisation the theories may well turn out to be viable, as historically they were thought to be. 1

With only a small increase in capacity, however, they are powerful enough to haul the main semantical train of subsequent chapters.

2

The classical story of what mathematicians do when they use intuitive set theory, of what they used to do before the paradoxes were discovered and continued doing afterwards, is largely mythology - there is the parable of the consistent subtheory, of the restricted domain of discourse where sets are well-behaved, and so on.

3

The case so provided is only prime facie because there can of course be consistent theories of inconsistent objects. Other considerations lead, however, in the case of set theory to the conclusion that the theory is simply inconsistent, especially the intuitive arguments for the logical paradoxes.

u

Furthermore, through classical set theoretic formalisation, much of the rest of modern mathematics has been contaminated by set theory. But really classical set-theoretic formalisation is too rough a device to capture subtler parts of mathematical argumentation and inference, and even at quite (footnote continued on next page)

55

1.6 LIVING WITH CLASSICAL LOGIC IS LIVING DANGEROUSLY AND IRRATIONALLY

Worse still, the consistency of a sufficiently rich classically formulated mathematical theory can never be classically assured, and so conventional use of Y can never be justified. As a result of the classical limitative theorems, and especially Godel's consistency theorem, classically based theories are forced into a completely untenable position. For let MT be any sufficiently rich classical theory, i.e. any classical theory in which every recursive function is representable. Most mathematical theories satisfy this requirement by virtue of including arithmetic. Then, by Godel's theorem, the consistency of MT cannot be proved using just methods of MT; any attempt to prove the consistency of MT in a non-question-begging way will lead to an endless regress through larger and larger, and increasingly suspect, theories. In short, a non-circular proof of the simple consistency of MT is classically impossible. But vindication of the use of y in MT requires a non-circular proof of the simple consistency of MT. Thus use of y can never be classically vindicated in the case of any sufficiently rich mathematical theory. Accordingly, too, use of classical logic in formulating such theories cannot be justified. There is indeed far too much complacency about formalisations, especially classical formalisations, of set theory and other substantial regions of the foundations of mathematics. The fact is that they are in dreadful philosophical and logical shape. A lot of glib and supposedly reassuring tripe is trotted out about how everything is alright - like the economic experts on an obviously ailing economy. But really the crisis in logic is far from over. It is simply that, like hyperinflation in some economies, people have come to live with it. The fact that one can live with crises, can live dangerously with classical logic (as van Fraassen is quite explicitly prepared to do: 71, p.5), is hardly a good reason for continuing to do so when the risks can be reduced as they can be giving up material detachment everywhere except where consistency can be established, that is in most places. There is no need to live in the shadow of possible disaster in the fashion of classical set theorists, and it is mostly irrational to do so. Nor can there be any doubt that allowing for the possibility that a set theory is a simply inconsistent non-trivial theory substantially reduces risks.1 Naturally where a version of Godel's theorem applies, risk cannot be entirely eliminated; but firstly, Godel's result has not been established for theories which are not classically based (see UL), and, secondly, there is good prospect that for such theories a non-Godelian strategy can succeed (for example, dialectical theories automatically escape the main impact of limitative theorems, since they admit the paradoxes, whereas the limitative theorems, and the scepticism they inject, are consequences of getting burnt by the paradoxes after their main fire has allegedly been extinguished). (footnote u continued from previous page) elementary levels creates distortion. (Consider, for example, the settheoretic representation of groups and semi-groups discussed in van Fraassen 71.) Too many logicians are prepared to tolerate this level of distortion, to accept isomorphisms for identities, etc. Probability considerations support this claim. Just as, to use a geometrical analogy, it is probable that isolated singularities remain isolated and do not connect with every point, so there is a considerable probability that isolated inconsistencies do not spread everywhere.

56

1.6 THE NEED FOR REASSESSMENT OF THE PLACE OF CLASSICAL LOGIC

In view of all this, and especially the severe limitations on the reliable applicability of the core classical rule of Material Detachment, and (as we shall see) the defensibility of dialectical theories, we believe that a reassessment of the place of classical logic in theory formulation is wanted. Classical logic was the first formalised logic of much scope on the scene and it remains by far the best developed. Practically all the extant formalisations of mathematical theories have been carried out on a classical basis not because classical logic was necessarily or obviously the best medium for all - or any - of these formalisations, but in part because of historical accident and vagaries and of conservatism of the educational process and in part because it alone was there. Nowadays classical logic, which has many powerful adherents, has consolidated its position and become entrenched,1 and much mathematical theory is being rewritten in terms of it. A comparison with the rewriting of history in terms that suit a new dominant ideology is not inappropriate. For it appears that the informal reasoning used in classical mathematics at least until the mid-century rarely2 or never deployed the excessive power of, or, for that matter, the paradoxical features of, material implication, and it also appears that most of the reasoning can be reset on alternative logical bases including relevant ones. It is our view that the importance of classical logic - which captured the market because first into it and because of its childish simplicity - has been grossly exaggerated. Given time and relevant effort it is our hope that history will in due course bear out our claim on this matter. What importance classical logic does have is in providing a simple model, and testing ground, for arguments that can then be re-engineered to extend to more adequate alternative systems.3 1

Thus rival logics are derisively dismissed as 'deviant' in (much of) U.S.A. and its intellectual satellites: elsewhere they are 'non-classical'.

2

If there were many such cases, which we doubt, they could be catered for in a relevant formalisation by additional mathematical postulates - the classical postulational bases having in any case a measure of indeterminacy.

3

We have already, impetuously, proposed relevant logics as candidates for the job from which we are arguing classical logic should be dismissed (see, e.g. UL). As a matter of history, relevant logics were not introduced merely for recreational or academic advancement, or demotion, purposes, but were intended as serious alternatives to classical logic and to such improvements on classical logic as modal logics. We certainly believe that relevant logics should, because of their unusual scope and range of other virtues, be very seriously considered as alternative logics. So it pained us to encounter Haack 74 - a whole book on alternative logics in which relevant logics get no mention. Of course recognition of relevant logics would have destroyed the organisational framework and several of the arguments of Haack's book. To begin with, relevant logics do not fit into S. Haack's oversimple classification (in 74, p.4ff) of alternative logics. For although all theorems of classical (sentential) logic are theorems of the main relevant logics studied (those including system G below), these relevant logics do not in general simply extend classical logic, but lack (except as an admissible extra in many cases) the characteristic classical rule of Material Detachment. Moreover non-trivial inconsistent extensions of the best relevant logics - the dialectical systems - no longer allow material detachment as an admissible addition. Thus relevant logics - with -*• considered, like the strict implication and necessity of modal logics, as a further connective beyond the standard set {&, v, ~} of classical logic - though they typically include classical logic as admissible, are not essential extensions of that logic. Doesn't that make relevant logics quasi-deviant logics on Haack's classifi(footnote continued on next page)

1.6 THE RATIONAL LOGICAL POLICY IS TO REDUCE RELIANCE ON CLASSICAL LOGIC

To sum up the main drift thus far:- Not even the residual claim made for classical logic - that classical logic will have to be part of any adequate logical system - is correct. On the contrary the residual claim is drastically mistaken. Admittedly the tautologies of classical logic are analytically true (though, as noticed, not compulsory, in that one doesn't have to believe them). Material Detachment is however an integral part of the classical scene, and this rule is formally unsatisfactory and becomes positively undesirable when investigating the consequences of philosophically and mathematically important theories which may be inconsistent. Then one can go seriously wrong using classical logic. One can, in short, go wrong with classical logic if one applies it in the wrong situations. And if one applies it to discourse concerning a range of worlds that are not classical such as inconsistent and paradoxical situations, then one is quite certain to go wrong: and so of course for any arbitrary situation there's a chance of going wrong. Furthermore for most theories of interest the requisite conditions for the application of classical logic cannot be classically established. Standard logic is not, ther, a device that one can apply with real confidence anywhere much beyond decidable theories. Only the completely unfounded, and classically unsupportable, presumption of correctness props up the universal application of classical logic, and a rapid - but impermissible - retreat towards pure logic is often attempted by exponents in the face of criticism. Rational applications of logical theories should include logical caution, which means dropping classical logic as a universal, or even extensively used, tool especially in investigations of consistency. For we do not check situations carefully to make sure they are not bugged by contradictions somewhere before applying logical reasoning - yet that is what the safe application of classical logic would appear to require. Moreover, even if, as is classically assumed, the world is consistent it is not generally safe to apply classical logic in investigating theories (which may diverge from the world). The rational course would seem to be to reduce risks and adopt a relevant logic (a thesis developed in detail in DCL). § 7. Dialectical logic, and the repudiation of the dogma that the world is consistent. So far we have been arguing against classically-based logics primarily on the grounds that there are non-trivial inconsistent theories which just cannot be discounted, and that, moreover, a great many of our other intuitive theories may, for all we know, be of this sort (and that classical theory prevents us from ever knowing otherwise). (footnote

continued from previous page)

cation? No, for the theorems of the common vocabulary with connectives &, v, ~ do not differ. But the matter of the choice of connectives of alternative logics and the comparison made with the connectives of classical logic is important, and, unfortunately for Haack's classification, the status of a logic is not invariant under choice of its formulation. For example, Lewis's systems are classed by Haack as extended logics. But should we consider Lewis's systems of strict-implication, as formulated in a quite standard way with connective set {&, v, =>} with = as a strict implication, as an alternative to classical logic (the way Lewis and others have thought of these logics), then Lewis systems are deviant logics on the Haack classification - not extended logics as Haack has it. Similarly relevant and intuitionist logics so considered are deviant logics, and so likewise are the very different connexive logics which contain decidedly non-classical theses and which should be separately classified. Accordingly the Haack classification is neither suitably stable under systemic formulation nor sufficiently revealing, and it is neither exhaustive or appropriately exclusive.

58

1.7 PARACONSISTENT LOGICS, ANV THE DIALECTICAL CRITICISM OF CLASSICAL LOGIC

The dialectical criticism of classical logic is much harsher. For according to it, there are true theories that are inconsistent, and thus the world T (considered as everything that is case, as the class of truths) is simply inconsistent. If so, classical logic cannot be reliably applied, even in its home (extensional) territory, to true theories: it is incorrect in a quite drastic way. The new dialectical criticism of classical logic and repudiation of the consistency of T is based on the development of paraconsistent logics. The new logical case reinforces the older intuitive case, in a quite remarkable way (as we shall see in 6.5), and furnishes many of its main theses. A paraconsistent logic is, at bottom, a logic which admits arbitrary contradictions without thereby being trivialised, i.e. a necessary condition for a paraconsistent logic is that it does not contain (as a derived principle) the spread rule: A, ~A -frB.1 Let us take this necessary condition to characterise weakly paraconsistent logics, so leaving it open what additional conditions have to be met by paraconsistent logics proper2 - just as we have characterised weakly relevant logics in terms of satisfaction of Belnap's weak relevance requirement and so far left open what additional conditions must be met by relevant logics (but this question is closed in chapter 3, where relevant logics are characterised as logics which are weakly relevant and conservatively extend distributive lattice logic). Since most positive logics would qualify as paraconsistent under the account we also require that the logics concerned should be sentential and contain as well as •*• a full stock of extensional connectives, typically &, v and 1

A less sweeping requirement that merits some investigation is that under which a logic admits some contradictory pair A and ~A of further theses without being trivialised.

2

Da Costa in a number of publications, particularly 74, has imposed further conditions on paraconsistent logics, and similarly Jaskowski has required that his discussive logics - which are paraconsistent logics - should meet further conditions. But some of these conditions are not formally tractable (and are too strong insofar as they are), e.g. that a paraconsistent logic should have an intuitive sense or that is should be sufficiently rich to formalise the bulk of useful sentential reasoning (Jaskowski 69, D'Ottaviano and da Costa 70), and others are undesirable, e.g. the condition that a paraconsistent logic should not contain the thesis ~(A & ~A). Some important dialectical logics have ~(A & ~A) as a thesis though for some B, B & ~B is also a thesis, e.g. the system DL of 6.5. Though the modern logical theory of formal inconsistent systems originated in Poland with Jaskowski - its roots go back to Lukasiewicz 70 - extensive development of theory has occurred in Brasil, and is to be found in the work of da Costa, Arruda and coinvestigators. The theory of formal inconsistent systems, or as they are now called, paraconsistent systems (the term 'paraconsistent* being coined by Quesada), has become very much a Latin American institution, and distinguishes Latin American logic in much the way that the identity theory of mind used to distinguish Australian philosophy. Important work on paraconsistent systems has also been done by Asenjo, Apostel, Priest, Routley and others (see DLS).

59

1.1 THE CLASSICAL FAITH (IN CH) CONTRASTED WITH DIALECTICAL HARDHEADEDNESS

It is evident that the classes of logics distinguished, weakly relevant and weakly paraconsistent logics, properly overlap. The relevant affixing systems (of chapter 3) generally fall into the overlap, but most of the systems studied in depth by da Costa are irrelevant (the P systems are an exception), while the original rigorous implication systems of Ackermann 56 are relevant but not paraconsistent since they contain the rule y.1 The usual irrelevant logics which all contain Disjunctive Syllogism are neither relevant nor paraconsistent. A dialectical logic is a paraconsistent logic that realises the potential that a paraconsistent logic allows for, that is, a dialectical logic is simply inconsistent as well as non-trivial, i.e. it contains contradictory theses.2 Thus dialectical logics are a subclass of paraconsistent logics, and relevant dialectical logics - those of main interest to us - of relevant paraconsistent logics. Let c be a situation that conforms to a dialectical logic. The semantics of relevant logic will reveal that such non-classical situations are easily supplied. Situation c upsets an application of y, namely yc. So much we have already seen. The additional dialectical thesis is that T is such a situation, and that so also are certain subsituations, such as c, of T. Hence, by the negation of the consistency hypothesis, yT is upset. There are three positions that can be taken with respect to the consistency hypothesis, CH, that the world T is consistent, and so three positions with respect to yT - positions which correspond to the three standard positions with respect to the existence of God. There are, firstly, theists or believers who accept CH, mostly as an article of faith. Such is the classical position, such also was Ackermann's position, and, so it appears, that of Anderson and Belnap. Secondly, there is the agnostic position, argued for as the rational position in DCL. Finally, there are atheists or infidels: such are thoroughgoing dialecticians (and the same dialectical position is argued for in UL and in DLS).3 l It is assumed in characterising paraconsistent logics that the extensions considered are closed under the rules of the logic: otherwise even classical logic could be paraconsistent. 2

0ur use of the term 'dialectical' has encountered a fair measure of criticism, partly because 'dialectical' and 'dialectic' have another important (related) sense, and partly because the links of dialectical logic with the enterprises of Hegel and the dialecticians have not been made sufficiently clear. But, firstly, our use of 'dialectic' falls squarely under the second part of the dictionary listings for the terms (see OED): dialectic, n (often in pi.) Art of investigating the truth of opinions, testing of truth by discussion, logical disputation; (Mod. Philos; not in pi.) criticism dealing with metaphysical contradictions 6 their solutions. For we are certainly concerned with the application of dialectical logic in resolving metaphysical contradictions (see below): this is the central reason for investigating them. (Jaskowski's discussive logics and paraconsistent logics are intended to tie in also with dialectics in the Greek meaning, i.e. as under the first part of the dictionary listing.) Secondly, Hegel and his followers do not have a monopoly on the use of the term 'dialectic' even if (footnote 2 and 3 continued on next page)

60

1.7 WHAT IS AT STAKE IS THE QUESTION OF THE CONSISTENCY OF THE WORLD

For present purposes we do not need to settle the issues between consistency agnostics and consistency atheists, even if we could; it is enough to knock out the classical, theistic, position. What we shall do is to set out the seemingly powerful, and pretty impregnable, case that the agnostic has (adapting the argument from DCL) and in the course of presentation, indicate how it can be biased in favour of the dialectical position. We have seen that what is at stake is nothing less than the question of the consistency of the world T, that if the world is simply consistent (and also, as against the intuitionists, complete) then the classical position is correct, whereas if the world is inconsistent then the dialectical position is correct.1 Whichever is the case, however, the relevance position does not go wrong. This provides a major reason for claiming that the relevance position is more rational than the other positions, should it turn out that the matter of the consistency of the world cannot be definitively settled. But the question of the consistency of the world cannot be conclusively settled in a non-question-begging way - so at least the agnostic argues hence the keeping-options-open strategy of (non-classical non-dialectical) relevant logic is the rational one from a decision-making viewpoint. The rival positions of course contend that the matter can be settled, in their way - not perhaps definitively, but in the way that other high level theoretical hypotheses are settled. Here the dialectician has a point, as we shall see. What is at stake is not the absolute consistency or non-triviality of the world, or the logic or theory whose truth it reflects. All the positions can agree that the world is absolutely consistent - which is as well for them, since the absolute consistency of the world can be empirically verified. The statement, q , "R. Routley is cutting firewood at Plumwood Mountain on the (footnotes

2

and

3

continued from previous page)

they were responsible for the modern usage; and though it would be very interesting to apply logical and semantical methods to the explication of Hegel's philosophy, we are under no obligation to do so. We do believe, however, that the dialectical logics we have so far investigated have a direct and important application in clarifying the logic of Soviet dialectical philosophy (see DCL). 3

0ne of the fringe benefits of going dialectic is that one can no longer be condemned quite so easily for the occasional inconsistency in one's work. But of course we can make the classical plea that our position has changed - improved we should like to say - over time.

x

The dialectician can admit with the intuitionist that the world is incomplete, but the intuitionist cannot admit with the dialectician that the world is inconsistent; for intuitionistically, since A -*• B, A -+A •*•. A -*• B, whence —iA A -*• B and A & —|A -*• B, i.e. intuitionistic inconsistency implies collapse into triviality. Of course a minimalist who rightly rejects A -»• B can go some way with the dialectician. In general, minimalism is a relevantly rather more congenial position than intuitionism: if only however "minimal" logic had been based on a relevant positive logic instead of on Hilbert's positive logic (the so-called "absolute calculus").

61

1.7 THE CHARACTER OF THE CONSISTENCY HYPOTHESIS

afternoon of August 1, 1976", for example, can be empirically falsified by observing Routley in Brasil at that date. Hence q Q is false and q^ does not belong T. Therefore, T is absolutely consistent. It is often supposed that given the absolute consistency of the world, the (simple) consistency of the world is also proved, and the classical position thereby established. However the "proof" of simple consistency must depend on the special rule n.

A, ~A -o B

which transforms simple inconsistency into absolute. But this paraconsistencyexcluding rule is tantamount to rule y, even in weak sublogics of classical logic (e.g. in DML of 2.8), and is semantically equivalent to the assumption that T is simply consistent (given absolute consistency or non-degeneracy of the world). Hence the question is begged by rule n> since precisely what is at issue is rule y and the consistency of T. The question of the consistency of the world is not, it seems, empirically decidable. Nor i~ it a high level scientific hypothesis. It is a metaphysical thesis, but a perfectly significant one. Since it involves a universal claim to the effect, as canonical semantical modellings will show, that for every statement A exactly one of A and ~A belongs to T, i.e. holds true - it would appear that the dialectician is in a much stronger position to establish his claim than the classicist, since universal claims can be falsified in principle by a single counterexample. To establish his thesis however the classicist has to establish a claim that extends over all true theories, whether scientific, mathematical, evaluative, or whatever. Should a contradiction lurk deep in some still unpenetrated mathematical or other theory then the classicist is undone. Only a priori "evidence" that such a contradiction or inconsistency cannot occur would seem to exclude the possibility: thus the classical position if true is not empirically so in any simple way, no endless search is contemplated, nor would its results be relied upon. The dialectician, on the other hand, has the chance of falsifying the consistency hypothesis by a solitary counterexample. But, as any dialectician soon finds out, a simple falsification of the consistency hypothesis which would satisfy the opposition is not so easily achieved, as each alleged contradiction can be avoided, though often none too convincingly when the wider theoretical framework is remembered, by theoretical shifts (coupled with conventionalist strategems). In this way too it quickly becomes apparent that the consistency hypothesis is a very highly theoretical claim. Characteristically dialecticians have appealed to paradoxes to establish the inconsistency thesis. Hegel was, it seems, convinced both by the Kantian antinomies and by Zeno's paradoxes of motion; Soviet philosophers are more impressed by Zeno's paradoxes and invest heavily in the contradictions they observe in motion (which generalise to development). None of these cases are however decisive, and even if the classicists do not have really convincing solutions, say to some of Zeno's paradoxes, they do have alternative classical resolutions which so far get by. Much more convincing examples may be drawn from the logical and semantical paradoxes (this is the basis of the argument in UL against CH), but dialecticians have not generally made much appeal to such examples. Our case however is based, in the first place, squarely upon the logical and semantical antinomies and their like (self-referential arguments, including those of the limitative theorems). The arguments for the Kantian antinomies are conspicuously fallacious, and Zeno's arguments generally fail to convince. The antinomies we depend on have a different character: the arguments are convincing and are not conspicuously fallacious, if falla-

62

7.7 THE ARGUMENT TO INCONSISTENCY FROM PARADOXES

cious at all. The treatment of these antinomies within the framework of classical logic leaves an enormous amount to be desired, philosophically, linguistically, and mathematically. The artificial hierarchies of languages or types that classical semanticists appeared forced into, to avoid the catastrophic effect of semantical antinomies in combination with classical logic, are frankly unbelievable.1 One basic problem is that classical logic has to exclude deductive reasoning within and surrounding the antinomies, though it undoubtedly occurs (cf. the Prior-Mackie exchange, especially the initial paper, Prior 61). What has seemed especially puzzling is that paradoxical situations seem to be perfectly possible, in the sense that they could quite easily occur, and perhaps sometimes do occur. And this would make T inconsistent - which is one reason why attempts to dispose of the antinomies have exercised philosophers and logicians so extensively. Consider, for example, the policeman-prisoner situation, where the prisoner states only that everything the policeman says is true while the policeman, whose statements are otherwise unquestionably true asserts that whatever the prisoner asserts is false. Nothing stops such a situation from occurring indeed it could be arranged in a real-life courtroom situation.2 But no catastrophical breakdown would occur: legal reasoning could go on as before. And in fact such a paradoxical situation could go unnoticed. Paradoxes just do not spread and affect everything else: paradoxical situations do not have the logical features that classical analyses ascribe to them. The case for dialectical logic and the inconsistency of the world is not confined to that based upon "paradoxes" of one sort or another, of logic and semantics, of mathematics and physics. There are inconsistent principles in many legal codes which are valid (even if they are not said to hold true), and within the shadows of which legal reasoning takes place. To put it bluntly and without the detailed argument really called for: the prevailing law is sometimes inconsistent, yet deductive reasoning, incorporating the principles of the law as postulates, continues both within and outside the courts without being trivialised. Legal logic would accordingly appear to be dialectical logic. The argument we have outlined (more detail appears in UL and DLS) does not pretend to be entirely conclusive against the opposition, though the case seems to us very damaging. The classicist will insist of any such theory which contains a contradiction that it is not true, that it is (if perchance absolutely consistent) a non-standard theory distinct from the actual one which must be consistent. Thus counterexamples lead to a stalemate: the consistency hypothesis is not straightforwardly falsifiable or, the relevance position tries to insist, falsifiable at all. A useful example of the theory-saving methods of the classicist in the face of apparent inconsistency in nature is provided by Rescher 73, who, confronted by the incompatible actual states assumed by the Everett-Wheeler 1

The manifold deficiencies of the hierarchial approaches are well enough known: they are brought together, e.g. in Routley 66. The hierarchical approaches are not of course completely unavoidable classically; for every consistent, classically-based set theory will, in principle, yield an analogous protothetic; but these will typically be even less satisfactory than the set theory from which they derive.

2

How it is described is another matter, of course.

63

1.7 REASONS FOR REJECTING CLASSICAL THEORY-SAVING STRATAGEMS

theory of wave packet reduction in quantum physics, casts around for a difference of respect between the incompatible states in order to save the law of non-contradiction, and "locates", i.e. postulates, such a suitable distinguishing respect in a further time dimension.1 Rescher's difference-of-respect procedure (a method that goes back to the Socratic dialogues) thus offers further confirmation for the popular claim that a logical or mathematical theory2 can always be saved - at varying costs - by making sufficient changes or complications in scientific theories, since logical principles rarely confront empirical data in isolation and generally only do so rather indirectly in combination with other theoretical assumptions (as pragmatists have long pointed out). But though a non-empirical principle, such as the consistency hypothesis, never directly encounters the hard empirical data and can always be saved in one way or another, with greater or less cost, by changes elsewhere, the costs may be too high, and it may be better to give up the principle. A convincingly microphysical theory based on dialectical logic might provide such a reason. For success with theories based on classical logic has always been at the finite macro-level (as Br^uwer would have said): there is, in principle at any rate, no reason why classical logic should not go the way of Euclidean geometry satisfactory locally in regions of consistency, but globally defective. It should now be obvious that such a change would be perfectly possible; and persuasive evidence supporting the call for a conceptual revolution in logic, which queries the consistency hypothesis, is fast accumulating. If the consistency hypothesis is not straightforwardly falsifiable, it is far less readily verifiable. There is no easy or obvious way of surveying or sampling the class of true theories, particularly those yet or never to be discovered. Moreover by virtue of a by-product of the logical paradoxes, the situation appears to look still blacker for the classical position. Consistency is a logical matter, and if it can be established it will presumably be by logical means. Suppose, with a view to proving consistency, that the theory of T is formalised as far as it can be classically, and suppose that the truths of T included provide the principles of Peano arithmetic: then, according to Godel, the consistency of T is not incontrovertibly provable classically except by means at least as demanding as the resources of T (nor can they exceed the resources of T without appealing to falsehoods?). As we have observed (in §6), the same problem arises each time a reformalisation is attempted 1

To what extent the Everett-Wheeler theory can be satisfactorily constructed on the basis of a dialectical logic is an interesting question that takes us beyond the scope of the present venture. The same goes for questions as to whether such discredited theories as the classical theory of infinitesimals can be "satisfactorily" reformulated in the framework of dialectical logic, and the connected question of the extent to which the admission of inconsistencies at the infinitesimal level would help to resolve Zeno's paradoxes. For each of these theories, each of which leads to at least isolated inconsistencies at micro-levels, would demand a substantial amount of technical development before a worthwhile discussion could get off the ground.

2

An analogous, equally unsatisfactory, method in legal theory is to insist that there is an implicit priority ranking on laws, which, should a contradiction become manifest, the courts will explicitly determine. But what such post-hoc determinations really indicate is that before the consistencizing step, legal reasoning had been operating with inconsistent premisses.

64

1.7 THE MODAL STATUS OF THE CONSISTENCY HYPOTHESIS

aimed at incorporating outstanding truths syntactically and thus approximating syntactically to T. In short, the classicist can never conclusively establish his position; for, any formal proof of his hypothesis will use means at least as powerful and accordingly as questionable as his claim. Although we can be sure that the world is locally consistent for some regions we work in, like locally Euclidean (more exactly, there are consistent subtheories, like pure quantification theory, that we commonly use),1 globally we can have no such confidence, thanks to the fine dose of scepticism Godel's classical results inject (cf. 1.6). Therefore the question of the consistency of the world cannot be conclusively resolved classically in favour of the classical position, which was one of the theses to be argued. This in turn suggests the uncharitable proposition that belief in the consistency of the world is a mere act of faith, a part, so to speak, of the classical religion. While the belief may be an act pf faith for many people, it doesn't have to be quite so obviously one; for, the consistency hypothesis may be represented as part of a correct scientific theory, and overarching principles of sufficiently comprehensive scientific theories are in somewhat the same kind of epistemological plight as the consistency hypothesis. There can be good evidence for such theories even when they cannot be conclusively verified. The uncharitable proposition then rests on a false dichotomy between conclusive resolution or else a mere act of faith. Even so the comparison of CH with an overarching scientific principle does not, as already observed, stand up to too much examination: rather the hypothesis functions classically as an unfalsifiable metaphysical hypothesis. This takes us straight into one remaining puzzle, namely the modal status of the consistency hypothesis: is it a contingent statement, or a purely logical one, or something else again? For example, what seems to us more likely, synthetic non-empirical? It certainly does not appear to be empirical in the way that the deeper assumptions of physics such as invariance principles are. The alternative, on the simplistic classification that positivism so obligingly imposes, seems even less satisfactory, namely that the hypothesis is logical, and so if true analytic and if false logically false. The alternative does however have the virtue of grouping the consistency hypothesis with a range of other principles which raise similar epistemological problems, in particular the axiom of choice and axioms of infinity. The consensus that seems to have been reached regarding these principles is that we reason as far as possible without them, note carefully where they are used, try to establish whether or not they are essential where they are used, mark out those cases where they are essential, and so on. And the consensus view reflects a rational course of action, namely to minimize questionable logical assumptions. The same method would incline one to the relevance position to eschew the consistency hypothesis and to work, where consistency cannot be established, i.e. in most classical places, with a relevant logic or, at worst (as in 3.3), with some restriction of CH as a further explicit assumption - since this seems the rational course of action. But there are further factors that are not to be forgotten, in particular that, in contrast to the axiom of choice, there is a substantial dialectical case against CH. In fact 1

The official Soviet view is not dissimiliar; e.g., classical logic does apply in the case of simple, relatively stable objects and relations: see the discussion of Kedrov's position in Kline 53 p.84, and compare and contrast the intuitionist position on the areas where classical logic is correct and may be safely applied.

65

7.7 COROLLARIES

OF REJECTION

OF THE CLASSICAL

FAITH

the comparison of CH with the axiom of choice is in some ways misleading, and a better comparison is that (emerging from Rescher 73) with the principle of the uniformity of nature or that, hinted at above, with the assumption of the existence of God. Admittedly there are differences between these cases but the similarities are striking. In each case the assumption underpins the application of an extensive theory, classical logic, theories of induction and religions respectively; in each case the assumption is not, in its standard senses at least, an analytic one1, and would fail in its intended role if it were; in each case, however, it is extremely doubtful that the assumption is an empirical one - the assumptions being impervious to falsifying empirical evidence - yet in each case these are considerations, metaphysical arguments for instance, which incline many people to conclude that they are false; and yet in each case the assumptions can be maintained, despite all counter-considerations, and they can be bolstered or supported by a variety of expedients, by faith, by metaphysical arguments (invariably of a questionbegging cast), by the purchase of Kantian glasses (the wearing of which makes the assumptions part of one's perceptual scheme), or by other familiar political means, e.g. exhortation, persuasion, propaganda, repression, force, and so on. But we don't have the classical faith, we are not persuaded by the questionbegging classical arguments for CH, we don't wear Kantian glasses and, in any case consider the whole Kantian business fraudulent, especially as false assumptions can be enforced in this fashion, and we hope we can avoid classical thuggery, for lesser political means are not going to move us. The viability of dialectical logic has more corollaries than just the unseating of the consistency dogma, and curbing of classical logical imperialism. Several other classical projects and accounts are also upset. Firstly, for example, the classical idea, deriving from Hilbert, of absolute, i.e. nonrelative, simple consistency proofs by way of a modelling in the world has to be adjusted. Non-triviality may be so established, but to establish negation consistency it has also to be shown that a consistent subsituation of the world is selected. (Alternatively such proofs could be reconstrued as relative consistency proofs, showing simple consistency relative to T, for what that is worth.) Secondly, the familiar account of rationality which makes simple consistency a necessary condition of rationality in beliefs, theories, or actions has to be discarded. There is nothing in principle to prevent a dialectician's beliefs from being perfectly rational and reasoned, and a dialectical theory may be completely logical. The two issues, of rational inconsistent theories and rationally-held inconsistent beliefs are of course intricately connected: 2 for (so Routley 75 argues) an animal's beliefs constitute its theory as to how things are, i.e. as to part of T. The "framework of rational inquiry" thus extends beyond the consistent to dialectical theories and practices. Consistency imposes no Kantian (or Strawsonian) bound on rational inquiries. The same holds for intelligibility: intelligibility likewise is not bounded by consistency (see further 5.6). The rational intelligent person is not one who adheres to the consistency hypothesis but one who allows for the possibility at least that the world, though hopefully rational and intelligible 1

Forcing CH to be analytic takes us around the meaning-of-negation circuit that we shall go around in 2.8 and 6.5. Briefly, however, natural language negation is not so restricted in sense as to render CH analytic, and does not conform to classical exclusion principles.

66

1.7 THE PARACONSISTENT REQUIREMENT ON AN AVEQUATE THEORY Of ENTAILMENT

but possibly neither, is inconsistent, and who realises that inconsistency implies neither irrationality nor unintelligibility nor total disorganisation. To cope with the wide range of dialectical reasoning, both everyday and technical, an adequate theory of entailment, and likewise an adequate theory of conditionality, should be paraconsistent. Let us call this the paraconsistency requirement. It is a corollary that adequate theories cannot be simply extensions of classical logic, e.g. modal logics; and it will emerge that the requirement wipes out various apparent alternatives to relevant logics, e.g. certain conceptivist and non-transitivist positions. Since classical and modal logics are rejected as inadequate, the question is: what are the new logics of entailment and conditionality to be like?

67

2.0 TAKING THE ROAV TO SIMPLE DEDUCIBILITY ANV TO FIRST DEGREE ENTAILMENT

CHAPTER 2. DERIVABILITY> DEDUCIBILITY, AND TEE CORE OF ENTAILMENT. In this chapter we begin logically at the beginning, with the notion of deducibility in the absence of any sentential connectives, and then gradually add on connectives and complicate the class of formulae admitted until we arrive at the theory of simple deducibility (tautological entailment) and (early in chapter 3) at the first degree theory of entailment. We hope that in this way the reader will be led, ineluctably, to the first degree theory of entailment, the common core of all the good entailment logics studied in later chapters. The first degree is remarkably stable; it is the same for a surprising span of systems, and it leaves, so we shall argue, but little room for choice of rival candidates. That it to say, while there is, as we shall see in detail in chapter 3, a great deal- of room for debate as to the specific postulates of the higher degree of entailment, there is no such room for debate, in our opinion, at the first degree. We will nonetheless take time off to display some of the rival lower degree systems, to present semantics for them and to aid their case or, occasionally, to pour scorn on them. The chapter puts together in a self-contained and very elementary way results about deducibility and entailment many of which are known to the cognoscenti but which are not readily accessible or assembled elsewhere. §2. Semantical analysis of the deducibility relation j-f. In the absence of connectives there is a remarkably simple account of deducibility that wins almost universal acceptance among competing theories of implication: it is in fact precisely the classical theory of deducibility, which goes wrong only in its analysis of connectives and in its misguided attempt to reduce logical truth to deducibility (from null premisses). In order to facilitate comparison with the classical theory of deducibility, and for simplicity, we concentrate upon the case where the deducibility relation [•} is singular on the right. That is we, we consider primarily the form A 14^®, where A is a non-null sequence of wff of the logic L and B is a wff of L.

ff^ is

the deducibility relation with respect to logic L. For the most part the background logic L will be taken for granted, and ' |4j/ written simply ' [4 '. 'A (4 B' reads 'B is deducible from sequence (subsequently, set of wff) A' or 'A entails B'. Where A is a finite sequence, (A^,...,An) say, A f| B is equated with A^,...,An |4 B, and this sequence is read 'A^ and ... and A q entail Bf (cf. Smiley 59, p.235). So long as A is non-null the account of relevant deducibility - in the absence of specific sentential connectives - need not diverge at all from the classical (modal) account (see LC). That is, A |4 B is true in model M = <..., K,...,I>1 with K a set of worlds or situations and I a two-valued interpretation on K, iff whenever I(A,a) = 1 for every wff A in A then I(B,a) = 1 for every a in K. Then A |4 B iff A (4 B is true in every model. Thus the relevant account is exactly the classical account. 1

Allowance is made for additional items to go in M.

69

2.1 LOGICAL TRUTH DISTINGUISHED, AND THE PROPERTIES OF VEVUCIBIL1TY

It is evident that [4 can easily be extended to handle multiple expressions on the right - and we shall take advantage on occasions of the symmetrical form A |4 T. Then A ff T is true in M iff whenever I(A,a) = 1 for every wff A in A then I(B,a) = 1 for some wff B in F, for every a e K. Furthermore the definition is easily recast in terms of satisfaction upon using the following interconnection: an assignment of values at worlds satisfies A at world a iff it assigns value 1 to A at a. Then, as classically, A^,..., A^ |f B iff every assignment of values at each world a which satisfies each of A,,...,A at a also satisfies B at a. 1 n Where A is however null, the account begins to diverge from the classical account. Classically |4 B, i.e. B is a logical truth or logically valid, is equated with A f f B, i.e. B is true at every world in every model. But this account leads either to paradox or error according as the class of worlds considered is restricted to the logically regular ones (i.e. to "possible worlds" in one sense) or not. If not logical truths may fail to get verified, whence error ensues. A superior approach is to equate ff B with t ff B, where t is the truth constant, already mentioned, which satisfies the following interpretation rule: I(t,a) = 1 iff a is regular. And this approach, which will be exploited later, then reveals that t is avoidable. ff B is true in M - with M now of the form <•..0,K,...I>, where the set of regular worlds 0 is a subset of K - iff I(B,a) = 1 for every a e 0. A separate definition, in place of the classical reduction of ff B to A [4 B, is enough, then, to avoid paradoxes of deducibility such as that anything entails a logical truth.1 Finally, f f B iff f | B is true in every model. Several properties of the relevant deducibility relation are now derivable from the semantical account quite independently of whether any particular connectives occur or not (Smiley's list of properties, 59, p.236, is varied): Order properties: Reflexivity:

A |-f A

Transitivity:

If A |4 B and B, T ff C then A , T f4 C.

Substitutivity:

If A f f B and B f-f A then when A, A |-f C also B, A |-f C.

The property of substitutivity is derivable from transitivity. The order properties of |-f are really those of a partial order, since when both A |4 B and B |4 A, A and B are deductively identical. Hence too the more explicit formalisation of deducibility in algebra using the partial order < in place of |4But the cost of the explicitness is that a theory of identity is thereby presupposed. Set properties: Permutation:

If A , A, B, T |4 C then A, B, A, T ff C.

Repetition Omission:

If A, A, A f f B then A, A |4 B

Repetition Introduction: If A, A ff B then A, A, A f f B

1

And the simple avoidance of these paradoxes is of course highly desirable. The whole issue is taken up again in chapter 12 with the further discussion of the consequence relation - a basic relation for the algebras of logics.

7Q

2.1 AUGMENTATION v-6 SUPPRESSION;

ANV THE REPUDIATION OF SUPPRESSION

Premiss addition and deletion properties: Augmentation: Modus Ponens:

Iff A |j- B then C, A |j- B.1 If If- B and B ||- C then |f- C.

The latter property is the relevant cut-down of the usual classical property of suppression of logical truths. Thus although Augmentation or the addition of irrelevant premisses is validated, its classical mate, the suppression of logically true premisses, is not. That is, the following ridiculous classical property gets repudiated:(Positive) Suppression:

If |f- B and B, A |J- C then A |}- C.

For even if, in a given model, B holds in all regular worlds, still in other worlds C may only be deducible from A given B, i.e. B is essential even though logically true. In multiple formulation Positive Suppression, expressed more generally as: If if- B and B, A If- V then A if- T, is matched by its dual Negative Suppression: If C If- and A |f- T, C then A \\- T, where C |f- if C is a logical falsehood, i.e. C fails to hold at every regular world (i.e. classically ~C is logically true, i.e. holds at every regular world). The rejection of Suppression marks out clearly which of the two evident main classes of routes has been taken in avoiding a classical deducibility paradox, with an arbitrary B entailing a logical truth A, derived as follows (cf. Smiley again 59, p.246):A |J- A A, B If- A B ||- A

Identity Augmentation (and Permutation) Suppression

The mechanism of the negative paradox is exactly parallel but requires multiple formulation for its expression: D |f- D D ||- B, D D If- B

Identity Augmentation Negative Suppression,

where D is a logical falsehood. The other main option to the rejection of Suppression, within the framework of the deducibility theory offered, is the connexivist position (criticised in §4 and discussed again in §7), that of abandoning Augmentation. However the route of rejecting Suppression is far from uniquely determined: it does not, for example, distinguish a relevant logic approach from a Parry analytic containment approach. The differentiation of these approaches will turn on the semantical rules given for specific connectives, especially &, v and There remains a variant way of blocking the deducibility paradoxes which lies outside the framework of the classical theory since it involves changing the semantics for If- , the method of rejecting not merely Suppression but more generally Transitivity (of which Suppression can be seen as a degenerate case). This is one of the important preliminaries to investigate before semantical rules for specific connectives are introduced. The other, to which we turn first, is the matter of the axiomatic characterisation of the semantic relation of deducibility. We consider these in turn:*In the computing science literature this principle is called Monotonicity. While deducibility is certainly monotonic, there are inferential relations (e.g. those of inductive, analogic and probable reasoning) which are not, where inferences may break down when further premisses are added (e.g. because initial premisses are undermined). Such is the proper basis for nonmonotonic logics, which include connexive and various conditional logics.

71

2.2 AXIOMATISATION OF THE DEDUCIBILITY RELATION

§2. The axiomatisation of the relation \\- by a derivability relation [- , and the objection from Non-transitivism. Let L be a formal logic with denumerably many wff A, B, C Let A, T etc. be non-null sets of wff of L: in short the set properties of the intended relation will be built into the initial formulation. But a Gentzen-style formulation in which set properties required are explicitly listed could easily be designed. Each set A is countable. The sole formation rule of logic ML based on L and with the vocabulary cited, together with the symbol is: Where A is a wff and A a set of wff of L then I^A and A f^A are wff of ML. Much as before the subscript L is usually omitted, and where A = {B^,....B^}, A (- A is written B1,...,Bn f- A. A is derivable from A. Axiom scheme: Rules:

'A

A' is read:

The postulates of ML are as follows:

A {- A

From A (- B and B, T |- C derive A, T (- C (Transitivity, Cut). From A |- B and A £ T derive T |- B (Augmentation, Weakening). From

B and B (- C derive

C

(Modus Fonens).

Theoremhood of ML is defined in the usual sequential way. It is perhaps worth observing that f- A is a theorem of ML amounts to l-^. (fr A) and that ML L A B is a theorem of ML amounts to f ^ (A which corresponds to

More important, we do not need to see the formal structure in the object language-metalanguage way the system notation was chosen to suggest. Alternatively one might contend that one really has just one system ML with further unspecified formal objects (called obs) A, B, C, etc. and sets of these. Then the inner logic L drops out of the picture. (This type of approach, due to Curry 63, is adopted in the formalisation of distributive lattice logic below). Lemma 2.1 D € V iff T (- D. Proof. If D « r , then as Df-D, T, Df-D by augmentation, and so T |- D. For the converse it is shown by induction on proofs that if A [- D then D e A. Where the theorem is an axiom A = {D} so D e A. Moreover the rules that apply (Modus Ponens does not) preserve the property. That Augmentation does is immediate. For Cut, if A, T |- D then by one premiss D e T or D = B whence by the other premiss if D = B D e A. Hence D e A or D e T as required. Corollary ML is decidable. Proof. To decide whether T (- D is a theorem or not, enumerate T and begin checking through the sequences to determine in which D occurs. finitely many steps D will appear in one or other of T and T. Then r a theorem or not according as D occurs in F or in T.

_ T, and After [• D is

An ML model M, to be more explicit, is a structure & = <0,K,I> where K is a set of worlds, 0 is a subset of K, the regular worlds, and I is a bivalent interpretation function which assigns to each wff A at each world a in K just one of {1,0}, i.e. I(A,a) = 1 or = 0. A |- A is true in M iff whenever I(B,a) = 1 for B e A, I(A,a) = 1, for every a e K; and |-A iff I(A,a) = 1 for every a e 0. Finally, A f- A is valid iff A |- A is true in every model, and |- A is valid iff |- A is true in every model. Moreover, to connect with the semantical notions, A ff A iff A (- A is valid and ||A iff |-A is valid.

72

2.2 THE FIRST ADEQUACY THEOREM - FOR PURE VERJVABTLJTY AS IN ML

Adequacy Theorem 2.1. A |- A Is a theorem of ML Iff A |-f A, and |-A is a theorem iff |-f A. Proof. Soundness is established by induction over the length of proofs in ML. A f-A, the only axiom, is plainly valid, since if I(A,a) = 1 then I(A,a) 38 1 always; and direct verification shows that the rules preserve validity. Completeness is proved by the routine method of providing a canonical model. Let a theory c be any class of wff closed under (- , i.e. if A c c and A |- B then B e e . A theory c is normal iff whenever |-B, B e c. Define a canonical model M = <0 ,K ,I> as follows:~c c c K is the class of all theories; 0 is the class of all normal theories: and c ' c I(B,c) = 1 iff B e e for each wff B and each theory c. Suppose A is not derivable from A, i.e. A J-A. In order to show A is not deducible from A, i.e. ~(A |4 A), a simple Lindenbaum construction is first used. Let Aq,A^,...,An,... be an enumeration of the wff of L. Define a sequence of sets Aq, A^,...An,... inductively as follows:-

A^ ® A; if A |~Aq then

=

A u {A }, and otherwise A .. = A . Let a = u A . Then n n n+1 n n <w n (i) if D e a then A f- D, for each wff D. This is almost immediate from the construction of a;

it may be established by induction.

by the axiom scheme and augmentation. for D to belong to

If D e A^, then A f- D

Otherwise D = A^, for some i and in order

A |- D.

(ii) If a |- D then A |- D. (i) A |- D.

For if a |- D then by a lemma D e a, and so by

(iii): a is a theory. Let T be some subset of a and D some wff such that T |- D. D will be some element of the enumeration, say A^. It has to be shown that A

for then A^ e a as required.

Since T £ a and T |- A^, by augmentation,

a (- A^, for then A^ e a as required. Hence by (ii) A |- A.. Since A

A, by (i) A f a though A c a with a e K . As I (A,a) + 1 A |- A

is not true in M ;

~c

and so ~ ( A

14 A).

''

Suppose, for the other case, |-B is not a theorem. It suffices to find a normal theory a to which B does not belong. For then I(B,a) ^ 1, and so ~ |-| B. Let c be the class of all wff B such that |- B is a theorem. (in the degenerate case we are concerned with here c is null). Then c is a (degenerate) normal theory to which B does not belong. Corollary. ML is completely axiomatised by the sole axiom scheme and the rule Augmentation. For of the rules, only Augmentation is used in the completeness proof. In fact, in the narrow framework of ML, Modus Ponens is vacuous, and Cut is a derived rule, obtained thus:Given A |- B and B, T |- C, then by the first premiss B e A , i.e. {b} c_ A and so {B} u T c A u T , whence by Augmentation and the second premiss A, T |- C. Though there is very extensive agreement among the implicational camps as to the correctness - if classical incompleteness - of such a core theory of deducibility as ML captures, the agreement is not universal. For Augmentation is found objectionable by connexivists (for reasons we will outline first of all when conjunction Is introduced, since connexivist objections to Augmentation are put in terms of the conjunction of drone antecedents); and

73

2.2 THE ATTEMPT TO FAIL TRANSITIVITY OF ENTAILMENT

Transitivity is rejected under the von Wright-Geach method of diverting the paradoxes of implication. According to von Wright (57, p.181) A entails B, if and only if, by means of logic, it is possible to come to know the truth of A => B without coming to know the falsehood of A or the truth of B. The account1 was taken by its exponents to rule out transitivity of entailment; for example,to admit the premisses A & B -»• A, A & (~A v B) •*• B, etc., of the classic Lewis paradox argument but to reject the conclusion A & ~A -»• B. But critics were to point out both the many alternative construals of coming to know, in particular how one might come to know A & ~A B by the paradox argument using transitivity of implication, and also that both informal and technical accounts by von Wright and Geach were defective and failed to block not merely the Lewis paradox arguments but other equally damaging paradoxes (see ABE, p.215ff., Strawson 58, Bennett 59 and especially Lewy 76, pp.141-8). There are, then, formal explications of the von Wright-Geach account (none of them bearing a very good resemblance to the original, since epistemological features drop out) some of which defectively bring out transitivity and others of which fail transitivity.2 The account does not settle the issue of transitivity, then, and in a good sense the question of the correctness of transitivity is something to be settled prior to such an account - as condition of adequacy on a notion of entailment, as distinct from other notions such as obvious entailment, immediate inference, etc., 1

The account has antecedents in Russell, Johnson and Broad (as Lewy 76 observes in the two latter cases). What Johnson has to say on the paradoxes of implication is especially relevant to our subsequent criticism of Lewy, that Lewy has illegitimately converted what is satisfactory as ? necessary condition into a sufficient condition. According to Johnson (21, p.47): I believe that they [the paradoxes of implication] can all be resolved by the consideration that we cannot without qualification apply a composite and (in particular) an implicative proposition to the further process of inference. Such application is possible only when the composite has been reached irrespectively of an assertion of the truth or falsity of its components. In other words, it is a necessary condition for further inference that the components should really have been entertained hypothetically when asserting that composite.

Although Johnson is right about the necessary condition, the condition does not on its own provide, or effectively delineate, a resolution of the paradoxes: for example, it leaves the choice of ways of defeating the negative Lewis paradox almost entirely open. 2

There are serious formal difficulties in the main accounts so far presented. For example, that of Geach 74, an adaption of Smiley 59, depends on the decidability of the underlying classical logic on which the account is parasitic, and so will fail to extend from sentential logic even to quantificational logic. For other defects in Geach's theory see Dale 73 and Kielkopf 75a (the latter shows that the theory violates the paraconsistency requirement), Kielkopf 77, Briskman 75 (which shows that Geach is committed to Lewis-style paradoxes), and Iseminger 78.

74

1.1 THE OVERWHELMING CASE FOR TRANSITIVITY OF ENTAILMENT

It is evident that there are non-transitive restrictions of entailment, such as obvious entailment, and also that given such non-transitive notions a transitive notion can be obtained by taking the transizive closure. That is, let B- be some immediate deducibility-type relation which however is not transitive; then a transitive deducibility relation ff can be formed as the ancestral of B- . But it is this closure relation, not the immediate relation, which is what is ordinarily understood by deducibility, and it is the closure relation that we are interested in explicating. Moreover without the ordinary closure relation ('ultimate deducibility1 as Parry 76 calls it), much of the power of deducibility, and of the point of logic in applications, is lost. As Smiley (59, p.242) put it, in an emphatic statement: .... the whole point of logic as an instrument, and the way in which it brings us new knowledge, lies in the contrast between the transitivity of 'entails' and the non-transitivity of 'obviously entails' and all this is lost if transitivity cannot be relied on. To put the central point in terms of 'proof' or 'valid argument': a very conspicuous feature of proof is that a proof may extend over a substantial sequence of steps. Proof is characteristically carried by an implication or analogous (derivability) relation. Without the transitivity of this relation central, and immensely important, features of proof and deductive argument would be lost. The overwhelming case for the transitivity of entailment derives, then, both from the sense of 'entails' and from the properties of the cluster of notions analytically tied to entailment. By entailment is meant the converse of deducibility: since deducibility is transitive, so is entailment (for when R is transitive so is RC). One argument for the transitivity of deducibility is as follows: B is deducible from A iff there is, in virtue of the meaning of deducibility, a (valid) deductive argument from A to B, i.e. a sequence of elements beginning with A and ending with B each element of which is validly obtained from predecessors; in short there is a valid proof sequence beginning with A and ending with B. But such a notion of deductive argument or proof is automatically transitive: for if there is an appropriate sequence beginning with A and ending with B and another beginning with B and ending with C then there is of course one beginning with A and ending with C. The analysis of deducibility in terms of logical consequence yields the same result; then B is deducible from A iff B logically follows from A, i.e. iff, syntactically, there is a valid derivation of B from A, and iff, semantically, B holds in every deductive situation in which A holds. Both syntactical and semantical analyses ensure transitivity (as has been shown by way of ML). The same applies to every analysis of entailment based on inclusion; e.g. A entails B iff the logical content of B is included in that of A, i.e. c(B) _£ c(A). But inclusion is transitive; hence entailment is also. Given that logically necessary conditionals are entailments, the arguments for transitivity vindicate Conjunctive Syllogism, (A B) & (B C) -»•. A -*• C: they do not vindicate, however, the exported forms of this principle which feature prominently in the system E of "entailment". Arguments along similar lines, that is arguments from the sense and analytic connections of entailment, may be likewise used to vindicate other entailment principles, e.g. Identity and Modus Ponens. Analyses of entailment through inclusion, for instance, at once yield Identity.1 This is not to say that there are not relations closely related to entailfootnote continued on next page)

75

2.2 "THE CAMBRIDGE CASE AGAINST UNRESTRICTED TRANSITIVITY "

Marshalling these points in favour of transitivity serves to emphasize that no explication of entailment that rejects transitivity - in the form: if A B and B ->• C then A C, as a matter of logical necessity - can be adequate to capture the preanalytic notion. Transitivity is, in short, a requirement of adequacy on analyses of entailment. There is no sense of entailment - as distinct from restrictions of entailment, and as distinct from conditionality - in which entailment is not transitive, and accordingly there is no resolution of paradoxes of entailment by replacement of entailment by some non-transitive relation. As it has been supposed that there is, especially by recent Cambridge logicians, it is perhaps worth glancing at the source of this illusion: ... it is this. We want our knowledge that A entails B not to rely on any antecedent knowledge of the propositions involved. The relation between A and B we think we have in mind when we say that A entails B must not be based on the modal value of A or that of B. ... But what the paradoxes show is that entailment which tries to capture this idea, if it is really to work, will lack unrestricted transitivity (Lewy 76, pp. 124-5; cf. also p.128 bottom). The premisses have been granted (in the fallacy of modal dependence, 1.5); but Lewy's conclusion should be resisted: The paradoxes do not show any such thing (unless 'really to work' is seriously overworked). Every analysis of entailment which is paradox-free in a sense avoids the modal dependence fallacy; in particular relevant analyses do. So transitivity need not be sacrificed. There are two reasons why Lewy supposes that such an entailment relation 'must involve a restriction on transitivity'. The first is that all the other moves in the proofs we have discussed, or at least all the other moves in Lewis's first proof, are entirely valid (76, p.130). This is a large claim, which happens to be false (see especially 2.9). The second reason is that Lewy is thinking of proposals like those of von Wright, Geach and Smiley which do not merely offer avoidance of modal dependence as a necessary condition for entailment, but present it as a necessary and sufficient condition. Thus Lewy goes on at once to say (p.125) that Smiley's criterion attempts - unsuccessfully it soon turns out (p.130) - to formalise 'the basic idea' he is talking about. Smiley's proposal (in 59, to which Smiley is not committed) does restrict transitivity. It is as follows, where represents entailment in the sense of Smiley: A^,...An H" g® i^f A^ & ... & A^ z> B is a substitution instance of a classical tautology A^' & ... & A r ' => B' such that neither B' nor ~(A ' & ... & A ') is a classical tautology. (A definition 1 n which goes beyond the confines of sentential logic results upon replacing (footnote

1

continued from previous page)

ment which fail Identity. Non-reflexive (or non-stuttering) entailment, defined through entailment and non-equivalence (as A + B & ~(A B)), and corresponding to proper inclusion, fails Identity. It has turned out that there is a real point in studying implicational-type relations which fail Identity (always), namely in solving technical problems such as the P - W problem (see Martin and Meyer 80), and also in the analysis of progressive reasoning (cf. 11.2).

76

2.2 SOME TROUBLES WITH SMILEY "ENTAILMENT"

classical tautologousness by logical necessity, as Lewy observes.) The Smiley proposal, while (like Lewy's premisses) unexceptional as a necessary condition on entailment, is quite unacceptable as a sufficient condition. But when only its acceptable, necessary half is taken, transitivity can be retained. The argument for these claims is, in outline, as follows:Firstly, where A -»• B is a first degree entailment (cf. system FD belov^ then A HgB. (For proof details, cf. ABE, p.219.) This shows that Smiley entailment , [f „, taken as a necessary condition, is unexceptional from a relevant viewpoint and need lead to no violation of transitivity. Taken as a sufficient, as well as necessary, condition for entailment, however, Smiley entailment underwrites many defective principles: where A and B are contingent, Disjunctive Syllogism is readily established for such an "entailment"; though A & ~A B is out, (A & ~A) v (A & B) -*• B is not; and though A -*- B v ~B is put, A -»•. A & (B v ~B) is not, yet surely the latter is just as paradoxical, and generally undesirable, as the positive paradox. Classical logic is no base from which to start in the quest for a satisfactory account of entailment; and patch-ups like Smiley entailment, which work at best for first degree entailments (where entailment is the main connective, and so never iterated), and which treat only the most obvious symptoms of the paradoxes, retain too much that is classical - thus many of the implicational counterexamples of 1.2 ff. apply against Smiley entailment - and diverge from the classical only at the wrong places, on matters such as transitivity which entailment itself must satisfy. These points provide evidence that Smiley entailment does not capture an intuitive notion of implication. And later (p.137) Lewy 'submits that the criterion "captures" no intuitive concept', though his evidence is hardly 1 impressive. Lewy contends that the considerations that there is something very compelling about Smiley's criterion and that yet there seems to be no intuitive concept which the criterion captures - lend additional weight to [the] claim that the intuitive concept of entailment is inconsistent (p.137). This is all decidedly dubious. Given a little reflection Smiley's criterion is (as we have tried to show) not at all compelling; and it is just false that 'we all have the feeling that it does "capture" some intuitive concept'. Many relevant logicians do not have this feeling (cf. ABE, p.215 ff.). What the criterion does is to approximate, as a necessary condition, to a condition that is intuitively appealing, namely the avoidance of modal dependence fallacies. There would have to be something especially compelling and satisfactory about Smiley's criterion as a sufficient condition for entailment for additional weight to be lent in this way to Lewy's inconsistency thesis; for only in this way would Smiley's criterion support the intuition that entailment is not transitive, in contradiction to the intuition, for which there is good evidential support, that entailment is transitive. *The fact that it does not answer to one intuitive notion that has been suggested, namely obvious entailment, is hardly evidence for the general claim. Lewy's very nice demolition of the suggestion (made, e.g., by Bennett 69) that Smiley entailment reconstructs the idea of obvious entailment is however worth drawing attention to: it is in 76, p.136.

77

2,2 "THE INTUITIVE CONCEPT OF ENTAILMENT IS NOT CONSISTENT"

In Slim, Levy's "additional" case for the inconsistency of the intuitive concept of entailment does not stand up to examination. One of his main arguments for inconsistency hardly differs, and accordingly fares no better: it is (76, p.133): ... consider the following three propositions (1) S-necessitation [i.e. Smiley entailment] is a sufficient condition for "A entails B"; (2) Entailment is not unrestrictedly transitive; (3) It is not the case that (A & ~A) entails B. This triad is patently inconsistent; yet we feel a strong inclination to accept all the three propositions. ... each proposition brings out an intuition we have in the area, but these intuitions are inconsistent. We feel no inclination at all to accept (1); nor could Lewy's argument for (1) persuade us otherwise, unless we granted him - what is fallacious - the conversion of an A-proposition. Furthermore (1) has considerably less claim than (2) and (3) to be intuitively based; indeed we suspect that for very many philosophers (1) is not intuitively assessable at all, in view of the character of Smiley entailment and the way all substitution-instances of complex entailments have to be grasped in assessing (1). Thus (1) does not bring out an intuition most of us have concerning entailment: if, however, 'sufficient' in (1) were replaced by 'necessary' it would yield a proposition that many of us would be lead, by a proof, to accept. (The rest of Lewy's case for his thesis that 'our intuitive, that is pre-formal, concept of entailment is inconsistent' will be examined in 2.9.) §3. Adding conjunction: semi-lattice logic and holim. Fundamental though a derivability relation is, its separate logic is hardly exciting. To obtain more interesting logics where distinctive theses really occur and where rules like Transitivity actually do some work, some operations subject to axiomatic constraints will have to figure in L. In the absence of specific connectives in the sentential case for instance, the powerful deducibility machine merely idles. We begin adding, one at a time, extensional connectives to ML. In the case of logics which lack the undesirable portation principles (which combine importation and exportation), namely in implicational form A

(B -*• C)

iff

A & B -f C

(Portation),

there is definitely - for reasons that will slowly emerge - a preferred order in which standard extensional connectives can be added to the basic theory of implication or deducibility: namely conjunction comes first, disjunction second and negation last. This order also coincides, to a fair extent, with the way in which rival accounts of implication and entailment diverge from one another. Thus there is substantial agreement about the logical behaviour of conjunction within the restricted (lower degree) framework at which we are isolating issues; there is less agreement over disjunction principles, and less agreement still over negation, and moreover some of the disagreement over disjunction principles can be put down to disagreement over negation principles. Indeed negation is the watershed connective.

78

2.3 SEMI-LATTICE LOGIC AMP ITS SEMANTICAL ANALYSIS

Semi-Lattice Logic (SLL) adds to ML the following postulates for the conjunction operation: Axiom schemes:

A & 6 |- A

A & B |- B

A, B (- A & B

(Simplification) (Adjunction)

Alternatively Adjunction may be replaced by the derived rule of &-Composition: r, A j- B, r, A h c-f r, A |- B & C, i.e. if r , A B and r , A |- C (are theses) then T, A |- B & C (is a thesis); otherwise read: from I", A |- B and T, A (- C to infer T, A |- B & C, etc. (The rendition of rules and the theory of inference rules will be issues taken up again later, in 4 and 15.) The rule of &-Composition follows from Adjunction as follows:1. Hence 2. 3.

T, A, B |- A & B

By Adjunction, Augmentation.

If T |- A then r , B |- A & B,

by Transitivity and 1.

If T |- A then T |- B then T |- A & B, by 2 and Transitivity.

Conversely Adjunction follows using &-Composition from Identity and Augmentation (see Tl, §5). The semantical analysis of SLL reveals very clearly the intended sense of the connective &. The analysis is obtained simply by adding to that for ML the normal conjunction rule: I(A & B,a) = 1 iff I(A,a) = 1

and I(B,a) = 1.

A rule stated in this form is intended to hold generally, i.e. for every situation a in K and all obs A and B. Truth, validity and so on are defined on the same plan as for ML. Adequacy Theorem for SLL 2.2. A |- A is a theorem of SLL iff A H A, i.e. iff A |- A is SLL valid. Proof builds on the analogous result for ML. Thus soundness is simply a matter of verification of the new postulates, which is done by applying the &-rule. Suppose, for Simplification, that for a in I(A & B,a) = 1. Then, by the &-rule, I(A,a) = 1 and I(B,a) = 1. Hence both A & B |- A and A & B |- B are SLL valid. Suppose, for Adjunction, that I(A,a) = 1 and I(B,a) = 1, for a e K; then I(A & B,a) = 1, validating A, B |- A & B. For completeness it is enough to show (given adequacy of ML) that for a e K c , I(A & B,a) = 1 iff A & B e a. And this is shown as follows:I(A & B,a) = 1 iff I(A,a) = 1 = I(B,a), by the &-rule iff A e a and B e a, by induction hypothesis iff A & B e a, as the following arguments show:Suppose A e a and B e a. Then as {A, B} c a and A, B |- A & B, by closure A & B e a. Suppose A & B e a. Then as Ta & B} £ a and A & B J- B, by closure of a, A e a and B e a. Completeness proofs for more comprehensive systems containing conjunction (e.g. the systems of chapter 3) require that the theories used be closed under adjunction as well as |- , i.e. that whenever A e a and B e a then A & B e a. This requirement is satisfied by the theories of SLL, but in general closure under entailment (i.e. -*•) does not ensure closure under adjunction. The semantical analysis opens the way for various arguments for the conjunction postulates of SLL, all of them aimed at vindicating the normal

79

2.3 ARGUMENTS FOR ANV AGAINST THE NORMAL CONJUNCTION «/LE

conjunction rule for all logical situations. Most of these arguments begin from the assumption, usually enforced as analytic, that logically to say that A & B holds in a situation just is to say, or means the same as, that A holds and B holds.1 (Such an argument is developed in Routley2 75.) Alternatively the conjunction postulates of SLL may be argued for on the basis of intended interpretations of entailment. Take the logical sufficiency interpretation for example: as A is sufficient for A (i.e. A -*• A) , A and anything else B must also be sufficient for A (i.e. A & B •*• A), for the addition of further information in the shape of B cannot undermine A's sufficiency. (This assumes that the addition of B does not interfere with A and undercut its sufficiency, an assumption - written into the ordinary use of 'and' - that the smart connexivist will dispute.) Similarly when A is sufficient for B and also sufficient for C then it is surely sufficient for both B and C, i.e. for B & C. In an analogous way inclusion interpretations can be used to argue for the conjunction postulates; for example, as the logical content of B is part of the logical content of A & B, A & B + B . Such interpretations also underwrite such characteristic conjunction principles as Commutation (A & B •*• B & A) and Associativity (e.g. A & (B & C) (A & B) & C). But none of these arguments is without assumptions, and accordingly none is completely conclusive. Like everything else, each of the conjunction postulates can be disputed.2 Before entering the dispute, let us record agreements and disagreements between the implication camps over the very ordinary and basic conjunction postulates of SLL. They are accepted by classical, intuitionist, relevant, orthological, non-transitivist and conceptivist positions. The axioms are however rejected by connexivism, for much the same reason that the principle of augmentation of premisses gets rejected, for example that a consequent only follows from antecedents if the joint force of the. antecedents is used, or if there is some cogent way of using all the premisses, certainly not if an added premiss is irrelevant to the derivation of a conclusion, or if an added antecedent somehow undermines the initial antecedent. Tiius connexivism rejects both forms of Simp but retains Adjunction. Connexivism is not alone in the rejection of normal conjunction postulates: Adjunction has also been questioned, but on quite different grounds, by non-adjunctivists. Consider some of reasons that have been given for rejecting Adjunction: there are at least two of importance. Firstly counterexamples to Commutation are easily devised, and these reflect back, not on Simplification, but on Adjunction, which yields A, B ]- B & A. The counterexamples derive from cases where 'and' clearly functions as 'and then', e.g. "he pulled out to pass and he hit an oncoming car", which does not imply "he hit an oncoming car and [then] he pulled out to pass" in the temporal sense of 'and', though it does imply both "he pulled out to pass" and also "he hit on oncoming car". And this already begins to reveal that the temporal conjunction determinable 'and then1, 1

As the English 'and' used in the explanation is supposed to conform to analogues of the semi-lattice postulates, there is an element of circularity in the semantical rule. It is this, undamaging circularity, that makes the analytic entrenchment possible.

2

Just as anything can be believed or disbelieved, so anything, including the correct laws of logic, can be disputed. An argument for this, premissed on relevant logic, is given in Routley2 75; and see further EMJB. Nonadjunctive systems are investigated in chapter 13, when "conjunction" rules are varied.

80

2.3 REASONS THAT ARE GIVEN TOR REJECTING ADJUNCTION

which certainly occurs in English, is a restriction (and/then, so to speak) of the more general, and logically important, commutative conjunction. Thus temporal conjunction is a notion that can be captured later, in the stage of logical analysis when temporal operations and tensing are taken account of (e.g. in 8.1). But the initial, and characteristically logical, objective concerns the logical behaviour of conjunction unrestricted by temporal ordering requirements. A second reason of importance for rejecting Adjunction also succeeds in bringing out significant features of the normal logical conjunction which are commonly overlooked. According to holism, which does not dispute Commutation, the whole is sometimes more than the mere sum or assemblage of the parts: applied to conjunction the message is that A & B may say more than A says and B says, that, more precisely, the logical content of A & B may exceed the logical content of A together with that of B, i.e. in symbols c(A & B) £ c(A)u c(B). The systemic whole may exceed the sum of the parts of a system because, so it is said, of the interrelations of parts that emerge on assemblage and hence because of emergent properties of the system. Whether this is so or not (it depends on how the whole and parts are characterised and whether systemic relations are appropriately incorporated in the characterisation of parts), it is a little fanciful to link conjunctions of statements with systemic wholes. A whole is a certain totality of things and their interrelations, whereas a conjunction is a mere result of an operation on statements. Explication of the notion of a whole is a logically important matter, especially in such disputes as those concerning methodological individualism, but it is not something to be accomplished by displacement of the general notion of conjunction. Rather it is a different and subsequent endeavour, which really requires elaboration of the theory of intensionality, including entailment, in order to proceed in a logically satisfactory fashion. Likewise intensional features, supplied semantically through the theory of worlds, are required to explicate the notion of conjunction which does result from holistic criticism of extensional conjunction a criticism which can stand even when the analogy of wholes with conjunctions is duly weakened. For according to the holistic criticism, and indeed to all the criticisms of the normal conjunction rule, whether A & B holds depends not just on whether A holds and B holds but also on the relation between A and B, and it is this relation that the extensional (intraworld) account leaves out.2 Granted there are intensional conjunctions which do take account of interrelations of components and which merit study (see, e.g. 5.2 for analyses of certain of these); the normal English conjunction is not, when unrestricted, one of these, and it is important for a variety of reasons to include the normal conjunction in a comprehensive theory of entailment and implication (the reasons are essentially those set out in SL, chapter 5, for the inclusion of classical connectives in significance logics). It may also be desirable for various purposes to include other conjunctions as well: it is enough for our argument that the normal conjunction be included, not that other conjunctions be excluded. And there is nothing to prevent the subsequent addition of a holistic conjunction to the connectives admitted 1

Thus too there are ways of recovering the more general operation from the restricted operations.

2

Interrelations of components come down in the semantic analysis to evaluations over related situations. Semantical rules for non-normal conjunctions will be introduced in subsequent chapters, in particular chapter 8.

SJ

2.4 THE LEAVING THESES OF CONNEXIVISM

(in this crucial respect classical negation turns out to differ: it cannot be added without inducing system distortion of an undesirable character). Our main point against holism is then that an English conjunction, the normal one it seems to us, does conform to &-Composition, and that such a connective should be included early on in logical investigations, especially as regards comparisons with going systems. A third ground for non-adjunctionism - really a special case of the second - points to the feature that someone may accept A and B separately where they are encountered, e.g. B is ~A, but not be prepared to accept the conjunction A & B, e.g. the explicit contradiction A & ~A. This ground affects Adjunction not as an entailmental principle however, but the question of the logical behaviour of 'and' when covered by such functors as 'It is accepted that', 'Kick believes that', etc. The latter issue is taken up again in chapter 8. §4. The varieties of connexivism. In contrast to holism, according to which a conjunction may say more than its conjuncts, connexivism maintains that a conjunction nay say less than its conjuncts, i.e. in more precise form c(A & B) ^ c(A) u c(B). Connexivism, that is, rejects Simplification but retains Adjunction. Addition, A A v B, is generally rejected along with Simplification, since Contraposition and De Morgan principles are generally accepted. Similarly Augmentation is rejected. The features do not however fully characterise connexivism1 which has the following two leading theses:1. Simplification (A & B -*• A, A & B B) fails to hold, and its use (and that of its outcome, Addition) is what is responsible for the paradoxes of implication, not the ancient and correct principles of Disjunctive Syllogism or Antilogism. 2. Every statement is self-consistent,2 symbolically A * A, where the relation of consistency with, symbolised o , is interconnected with implication in the standard fashion: A -o B ~(A -»• ~B). The doctrine of universal consistency appears in stronger and weaker forms: the stronger form, which reappears in modern times in Nelson's 1930 system, may be represented sententially by the principle A B ->•. A • B (called Boethius), and this implies the weaker form, ~(A •*• ~A) (i.e. Aristotle) using Identity. (In stronger systems which contain Exported Syllogism the principles are interderivable as we shall see.)

Evidently thesis 2 guarantees thesis 1 - on pain of negation inconsistency otherwise. For suppose Simplification were generally correct. Then, as in the negative paradox arguments, (A & ~A) -»• A and (A & ~A) -*• ~A; but accordingly, by strong consistency, ~((A & ~A) -*• A), contradicting (A & ~A) -»• A. A similar effect is had using A • A when Contraposition is applied. For A -*• ~(A & ~A) , whence (A & ~A) •*• ~(A & ~A), i.e. ~((A & ~A) • (A & ~A)). Thus connexivism avoids the usual counterexamples to Aristotle's principle by relinquishing Simplification. Put differently, Simplification plays a key role in establishing, what is the current orthodoxy, that contradictions are not selfconsistent. These arguments also show the decidedly non-classical cast of principles such as Boethius and Aristotle: their adoption renders inconsistent and so trivialises classical and modal logics. But the interconnections between theses 1 and 2 run much deeper than this; for 1 and 2 may be unified, as we shall subsequently try to show, through an underlying (so far unformalised) semantical view, a view which lies behind and informs much traditional logical thinking. (Footnotes

1

and

2

on the next page)

82

2.4 EXPLAINING THE HISTORICAL PERSISTENCE OF CONNEXIVISM

Connexivism has a much more ancient lineage than any of our other candidates for weakly relevant logics, most of which appear to be1 relative newcomers on the historical scene. Connexivism appears to go back at least to the Stoic dispute over implication and indeed, there is a good case for claiming, to Aristotle (see Lukasiewicz 51, pp.49-50).2 It was accepted, it appears, in the early middle ages by Boethius. In modern times its distinctive non-classical principles keep reappearing in quite diverse writers (some of them not informed by the logical literature on the topic), e.g. Strawson 52, Downing 61, Cooper 68, Stevenson 70, as well as Nelson 30, Angell 62, McCall 66. The persistence of connexivism is something that needs explaining, and which we shall try to explain. It is important to say, though not enough to explain the persistence of connexivism, that it represents one of the main options open to anyone who wants to reject the paradoxes of implication but not to reject Disjunctive Syllogism and Antilogism, and also to have a more or less coherent case for what one is doing. Connexivism may be a relatively minor doctrine but its elucidation and criticism is not just of isolated interest, but is important for the much larger issue of linguistically correct theories of negation and consistency. There are four main classes of objections to connexivism, the last of which ties up with our explanation of the persistence of connexivism; 1. Objections to the connexivist attempt to solve the implicational paradoxes by rejecting Conjunctive Simplification, and to the motivation for this rejection; 2. Objections based on the serious effects of these rejections, e.g. the way argument is hamstrung; 3. The difficulties engendered by these rejections, especially for basic models of entailment such as inclusion and containment; 4. Objections to the underlying motivation for the rejections, namely the theories of incompatibility, negation and conjunction presupposed. The way connexivism avoids the paradoxes of implication is, as we have seen, by throwing out Simplification, or (what is equivalent in the framework of SSL, but commonly thought to be more general) rejecting Augmentation, the addition of premisses. One paradox demonstration (much favoured by Lewis, e.g. 18, and subsequently by Nelson, e.g. 30) which emphasizes how it is that connexivism is a major option in resolving the paradoxes proceeds from just two premisses:(Footnotes from the previous page) 1

We have already remarked in 1.4 that Chrysippus's requirement that implication there be a connexion between antecedent and consequent is, in a sense, satisfied by any weakly relevant logic, but not by connexive logics, e.g. those of McCall - is no basis for calling a connexive in the modern sense.

2

On another familiar account of consistency A & ~A is not self-consistent but inconsistent; for it entails p & ~p for some p. The connexivist (e.g. Nelson 30) has split the usually coincident accounts of consistency, not entailing a contradiction, and not entailing its own negation.

in an - which some logic

Since most Stoic documents have been lost, including all those elaborating the view that implication requires a connection, there can be no certainty that the idea of a relevant implication is not an ancient one. 2

The case would be conclusive were it certain that the parameters Aristotle uses, 'A is white' and 'B is great', are completely general and not contingently restricted.

83

2.4 ATTEMPTS TO SOFTEN THE WHOLESALE CONNEXIVIST REJECTION OF SIMPLIFICATION

A & B |- A

Simplification

A&B

Rule Antilogism

|- C —• A & ~C |- ~B

Then as A & B |- A by the first, A & ~A j- B by the second. Thus either Simplification or rule Antilogism has to go: this is a fundamental choice. Connexivism alone chooses the first route. (It is a matter of history that all who choose the first route and try to justify their choice are tempted by the characteristic connexivist principles, for reasons we shall try to educe when we discuss the deeper motives for connexivism.) Other paradoxical derivations - indeed these and much more - are destroyed by the single expedient of dropping Simplification. Nonetheless the choice may be the wrong choice. For if a combination of two factors - premiss addition and suppression - lead to trouble, and one, suppression, is independently objectionable and the other is not, the choice of which to get rid of is obvious. Surely the Simplification of premisses is sometimes admissible? Surely the addition of irrelevant antecedents is not always objectionable? The wholesale rejection of Simplification - which is, as we shall see, and obviously anyway, too drastic - is not usually what is proposed by connexivists in their criticisms of the paradox arguments. There are three ways in which standard connexivists have endeavoured to qualify their outright rejection of Simplification, namely by admitting that Simplification is correct provided first that the added premiss B (in A & B -*• A) is relevant to A, or provided secondly that augmented premisses are actually used in the derivation (a move which strictly construed rules out all instances of Simplification1), or provided thirdly that the added premiss B is consistent or compatible with A. If the second move means 'has to be used in the derivation' then it is too strong, and no connectives of systems so far formulated satisfy it. Nor is it likely to be satisfied in an adequate system with substitution on variables, since other principles will lead to instances of Simplification. For example, Conjunctive Syllogism (A -»• B) & (B -»• C) -»-. A C has as a special case (A -*• A) & (A -*• A) -*•. A -»• A which violates the requirement. The restriction, which is like a requirement of minimal premisses, would be inoperable in other ways as well. On the other hand, if the second move simply comes down to 'can (or may) be used in a derivation' then the restriction is too permissive, and lets through versions of the paradoxes. The first restriction on Simplification proposed, the relevance restriction, seems to be based on the mistaken assumption that unless A and B are relevant or related they cannot have joint consequences (i.e. the first point boils down to the third).2 This is to assume that consequences result from 1

This is the conclusion that Nelson 30 in fact reaches, though invalidly. Nelson argues that A & B -»- A is not assertible because cases of it together with the correct principle of Antilogism lead to paradox. Then he goes on to claim (30, p.448): In view of the fact that p & q -*• p is not assertible, we must restate the Principle of the Syllogism (p -»• q & q -*• r -»•. p r) this way: p ^ q ^ r p-+-q&q->-r-»-. p + r ... Otherwise, if we substitute, for example q for r, we get p-»-q&q-*q-»-. p q, which is of the form p & q -»• p. But to say that A & B -»• A is not assertible is only to say that it fails in some cases - as the argument for its non-assertibility shows - it does not follow that there are no instances where it holds. ~ |- A & B •»• A is like ~(A)(B) A & B -»• A, i.e. (PA) (PB) ~(A & B -»• A), not. like (A) (B) ~(A & B -»- A) , which corresponds to |- ~(A & B -*• A). Put differently, Nelson's case rests on a scope confusion between ~ |- D and |- ~D.

Footnote 2 on the next page.

u

2 A THE RESTRICTIONS ON SIMPLIFICATION WOULV SABOTAGE VEVUCT1VE REASONING

the intersection, or interaction, of two premisses rather than by inclusion (cf. again Nelson 30). The connexivist seems to be endeavouring to replace the inclusion model of logical consequences by some sort of intersection account, where in order that premisses have consequences all propositions must be used in or are essential in the deduction. If the idea that all premisses must be used in order that the argument be valid, that all conjuncts of Simplification be applied, is carried out consistently it leads again to a demand for minimal premisses. For precisely the argument that counted against Simplification would show that any unused subpart or subconjunct of a used statement concerning conjoint antecedents would render the argument invalid; the conclusion would not follow unless the assumed subpart were removed and the antecedents pruned to the minimum. Thus we would count no argument as valid unless we could guarantee that the premiss set could not be further reduced. It is hardly necessary to emphasise the devastating effect such a requirement would have on the practice of reasoning. For in very few cases can we be satisfied that such a requirement is met and that there does not lurk within one of our statements some subconjunct which is unnecessary to obtain the consequent. Deductive reasoning would become unworkable. Consider, more generally, the effect of connexive constraints on the business of assessing someone's written work, or belief set, or theory. We could not extract subparts of any of these with the clear assurance that they were entailed by the whole set. Thus the practice of analysis would be destroyed, for we could not assess parts of the theories or texts.1 After all, not all theories are consistent, not all parts of texts are consistent with one another, and some authors are repetitious or redundant or insert irrelevant or otiose material.2 Even if the effect of such requirements in sabotaging deductive reasoning is overlooked, the requirement that premisses be constrained, e.g. as minimal, appears in any case quite misconceived and contrary to basic deductive practice. For why should anyone suppose that an argument the premisses of which include some unused information is thereby rendered invalid. If such otiose premisses or details are included, inadvertently or otherwise, no one would say that such an argument was invalid, certainly not in the same way as arguments such as affirming the consequent. Does the teacher send back the student's exercise in which an otiose premiss has been inadvertently included, marked 'Invalid', in the same way as if it included a faulty derivation, for example conversion of an A-proposition. And do we say that an axiom set, one axiom (Footnote 2 from the previous page) 2

The relevance restriction also runs into various technical difficulties. If relevance were all that was required such principles as A & A A should be connexively correct, since A is relevant to A. But then by Rule Antilogism A & ~A ~A, and similarly A & ~A A; and so, by an earlier argument negation inconsistency ensues. Other qualifications of Simplification, likewise suggested by relevance controls also lead to difficulties, e.g. A & (A & B) ->-. A & B.

1

Though this would hamstring proper discussion of authors' views and positions, this would no doubt suit some holists, who maintain that no assertion of a historical figure can be assessed in isolation from everything else he wrote. Connexivism could then provide (unintentional) logical assistance and sustenance for obscurantism.

2

Classical practice is not without its problems as regards the assessment of texts either. For classically a text is tantamount to the conjunction of all the assertions in it, but we do not readily concede that a text is false because just one (perhaps minor) statement in it is.

85

2.4 CONNEXIVISM IS BASED ON AN INADEQUATE DIAGNOSIS OF IUPLICATIONAL PARADOXES

of which is subsequently shown not to be independent, had after all no nontrivial consequences, and that the impression that it had was surely an illusion? If we did say this what sense would be given to independence and how would we establish it? Insofar as standard connexivism is an attempt to solve the implicational paradoxes, it is based on an inadequate explanation of how the paradoxes are caused and what is objectionable. It takes irrelevance as caused by the addition of irrelevant premisses as epitomised in A & B -*• A. But this is an incorrect analysis of what has gone wrong in the paradox arguments; it is not our freedom to introduce irrelevant or unused further premisses which is at fault (and objectionable), but our ability to suppress necessary truths, for example through Disjunctive Syllogism or Antilogism (or, what proves the semantical equivalent, the exclusion of worlds which admit inconsistent assumptions). It is possible to show independently of their roles in the paradox arguments how these principles warrant the suppression of necessary truths. It is also possible to show how the suppression of necessary truths leads to such paradoxes as that everything implies a necessary truth through intuitive models for deducibility such as containment. Irrelevance is simply a symptom of this more basic disease of suppression, which is at least as damaging to the entailment notion as irrelevance itself. For the suppression of logical truths and falsehoods is greatly at odds with the ordinary (nonmathematical ly corrupt) practice of deducibility. Thus the rejection of Simplification may be seen as an attempt to maintain this objectionable feature, the suppression of necessary premisses, but by a shallow formal trick to avoid the manifestation of its most obvious symptom, namely irrelevance. The trick is worked by an unwarranted and misconceived restriction on our freedom to form and augment premiss sets. Thus as a solution to the paradoxes, standard connexivism is quite unsatisfactory; it removes the more obvious symptoms but does nothing about the real disease. What is in effect being proposed by those who would solve the paradoxes by abandoning Simplification, is that in order to maintain one traditionally accepted argument, allegedly endorsed by Aristotle, which is however peripheral for many deductive purposes, we should abandon much of the basic application of deducibility in traditional deductive practice. Unlike their relevant rivals (for which (y) is provable) standard connexive logics do face the charge that they are too weak, and that crucial principles needed for deductive proof have in fact been abandoned. For the relevant logics can in fact show that strict and classical logics are enthymematically recoverable, but the argument depends crucially on the presence of Simplification. (The point and importance of enthymematic recovery will be explained subsequently). Deducibility is not simply a notion that floats free and which can be eliminated in any way that one chooses. Deducibility has to answer back to requirements imposed by associated models for deducibility, and its properties are in part constrained by such models, for example, such models for deducibility as sufficiency, inclusion and containment models. A fundamental feature of all these models is that they allow the addition of further premisses without sacrifice of deducibility, that is they validate Augmentation and Simplification. Consider the sufficiency picture for instance. If A is logically sufficient for B then the addition of a premiss C to A is not going to render it insufficient for B - unless C should somehow remove or weaken A: but how can it do that? For similar reasons Simplification must be part of any inclusion model, for if C is contained in A it must be contained in any augmentation of A. These important models and connections are lost if Simplification is abandoned. Such traditional models can, however, be maintained and improved by relevant logics which keep Simplification but reject Antilogism and Disjunctive Syllogism (see further 3.3). A conjunction for which Simplifi-

S6

2.4 DISTINGUISHING C0NNEXJV1SMS: STANDARD AD HOC AND INTUITIVE C0NNEXIV1SMS

cation holds good is an essential component of any adequate theory of deducibility (but this fact does not of course exclude other conjunction-like connectives for which Simplification does not hold good generally, any more than the necessary inclusion of a normal negation in an adequate theory excludes other negation-like connectives such as more intuitionistic negations). The standard connexivist view accordingly appears mistaken: adding to the antecedent of a conditional may interfere with consistency, as adding A to ~A does. But it does not affect the logical power of what it is added to or delete from its class of consequences - unless C somehow removes or deletes some of the content of A, something that the mere addition of C to a situation where A holds never does. But according to some varieties of connexivism it does: the addition of C may delete some of the content of A. It is apparent then that connexivism either violates Intuitive models such as sufficiency or else is bound to take a different view - the uncharitable would say a curious view - of either conjunction or negation or, as it will turn out, both. For, as to the first point, the sufficiency model just is that which is explicated (at the first degree) by way of a universal guarantee in all situations. If a connexivist accepts this, then he is obliged to vary his conjunction evaluation rule from the normal. It is important, at this stage, to distinguish two forms of connexivism ad hoc connexivism, and intuitive connexivism. Ad hoc connexivism gives no intuitive basis deriving from the character of conjunction for its rejection of Simplification; either it offers no basis for its rejection of Simplification or else argues for the rejection of Simplification simply on the basis that its incorporation would lead to inconsistency or paradoxes. The formal connexivist theories so far developed (those of Nelson 30, Angell 62, McCall 66) are all of the ad hoc variety. Nelson, for example, argues for the rejection of Simplification on the ground that it together with Antilogism leads to paradox (since Rule Antilogism takes A & ~B -*• A into A & ~A -*• — B ) . Ad hoc connexivisms are subject to the criticism - additional to those already made of connexivism which continue to hold - that they lack a proper motivational basis. Intuitive connexivism, in contrast to merely formalistic connexivism, tries to give some intuitive basis, in terms of the meaning of negation and conjunction, for its rejection of Simplification.1 According to this view of conjunction and negation, conjoining an antecedent as in Simplification is not, as is commonly thought, introducing a further premiss which holds in the antecedent situation: it is introducing an assumption which can interfere with what already does hold. In particular, it can do this if it negates or wipes out part of what is supposed already to. hold. Thus "conjunction" so construed - combined with negation, so construed as to reflect wipe-out - can undermine cases of Simplification, and this presumably is what happens, according to the intuitive version of connexivism, in some of the classic paradox arguments. In the Lewis argument for A & ~A -»• B, Simplification appears in the forms A & ~A -*• ~A and A & ~A •*• A. The connexivist is presumably bound to say that although A entails A the addition of ~A as an antecedent undercuts A so that the consequent A no longer ensues. Unless we can guarantee that a B added to A does not interfere with the deductive force of A we cannot assert A & B + A, a s A & B will not be sufficient for A. Since for an arbitrary B 1

The formalism of ad hoc connexivism can of course be underpinned by an intuitive basis. Attempts to offer semantics for Hilbert-style systems of connexive logic are characteristically attempts to do just this. On the other hand the intuitive bases so arrived at may show the need for changes in the formalism.

87

2.4 EQUIVALENCI ZING CONJUNCTION ANV VELET10NAL1Z1NG NEGATION

we have no such assurance, Simplification is connexively invalid. But the invalidity so claimed is only as good as the accounts of conjunction and negation underlying it. The connexivist owes us an account of these notions, because they are considerably removed from the orthodox notions. The character of the connexivist "conjunction" may be brought out through semantical modellings for connexive logics. Thus Routley and Montgomery 78 show that a number of connexive logics, including McCall's system CC1 which axiomatises the Angell's four-valued matrices for connexive logics, interpret conjunction as an equivalence relation in some situations. But a conjunction that sometimes turns into an equivalence is only an Alice-through-the-Looking-Glass sort of conjunction. Such a modelling is, however, only one among possible semantical analyses of connexive conjunction. The conjunction may be - and in the case of more satisfactory connexive systems apparently has to be - analysed in less damaging fashion through an intensional conjunction rule (see e.g. chapter 5), which does not - at least for satisfactory systems - reduce in some worlds to an equivalential rule. Unless implication is characterised using conjunction (either directly, or indirectly through consistency construed as joint possibility) negation as well as conjunction has to be given a non-normal interpretation. For just as conjunction has to be equivalencised in some worlds to invalidate A & B -»• A, so negation has to be "deletionised" to validate Aristotle's principle, ~(A -*• ~A) , in which conjunction does not explicitly enter. Sometimes connexivists have reduced the problems to one connective. For example, McCall 66 and Angell (in some unpublished writings) define '->•' in Lewis fashion with A B = D~(A & ~B). In this way the question of the interpretation of connexivism may be transferred to the question of the interpretation of conjunction alone (cf. Routley and Montgomery 78, where the semantics for connexive logics given differs from modal logic semantics in the evaluation rule for conjunction only, not on rules for necessity and negation). Alternatively the interpretational issues might be locatable in the interpretation of negation alone. Such reductions of the interpretational problem depend however on such mistaken accounts of entailment as the Lewis account, and tend to lead to paradoxes. In general, semantics of paradox-free connexivism is going to require revision in the interpretation of both conjunction and negation, and also of such interdependent notions as compatibility and inconsistency. There is a variety of connexivism, then, which is based on a formalistic and opportunistic resolution of the Lewis paradoxes and which offers a mistaken account of entailment in the context of truth functional connections. Typically this sort of connexivism rejects or restricts Simplification principles in an ad hoc way because it cannot see how Antilogism and other suppression principles can possibly be rejected. There is however another approach which is connexivist in character since it restricts Simplification and accepts Aristotle, which does have a sounder intuitive base and which can offer an important insight into problems of relevance and sufficiency. This sort of connexivism qualifies Simplification not because it in combination with Antilogism yields paradoxes, because it is «jased on a subtraction account of negation and incompatibility which is non-classical and which forces qualification of Simplification and related principles. This line of connexivistic development, though historically important and intuitively well motivated, has however as yet not received any clear formal expression (and there may be some difficulties in obtaining a formal analysis for it, for reasons we will come to).

88

2.4 NEGATION AS CANCELLATION, DELETION, ERASURE, SUBTRACTION

Subtraction or cancellation negation, if not, as Strawson (52, p.8), rashly asserts, 'the standard and primary use of not', is at least a basic and important negation in natural language. It takes up the basic notion of negation as a deletion or removal of a condition. Thus it is often pictured in simple and natural terms such as the following: p as running up a flag [which states the condition given in p], ~p as running it down again; p as writing something on a board, ~p as rubbing it out again, or putting a line through it, cancelling it out,1 p as recording a message, ~p as erasing it; p as stating something, ~p as withdrawing it. These sorts of pictures for the deletion notion, however appealing, are nonetheless defective, since they confuse statings with statements, the condition stated by p with the performance of asserting, in some mode, p. Thus the examples above are all examples of undoing or deleting the performance of asserting p, not the undoing or unmaking of the condition p itself. Thus for example, if the flag we run up the flag-pole states that the fox is in the hen-house, the lowering of the flag has only undone the performance of asserting this and has done nothing whatever about undoing the condition stated by the flag, that is done nothing about removing the fox from the henhouse . The notion of the negation of p as the deletion undoing or removal of the state or condition that p, is better viewed as a propositional analogue of the familiar operation of subtraction in mathematics. It is because the minus operation, of quantity subtraction, in mathematics has had such a powerful influence in logical thinking about negation that we have called connexive negation subtraction negation. Just as the negative integer -a is obtained by the subtraction operation from positive integer a, so on the subtraction view of negation the negative proposition ~A is obtained by negation from the positive proposition A, and similarly where B is negative (i.e. ultimately of the form ~C) the positive proposition ~B (ultimately C) is its negation. And just as a positive number added to its negative counterpart gives zero, so a positive proposition A conjoined with its negative ~A is seen as giving zero (A-)content. Thus the subtraction account contrasts sharply with the classical complementation view of negation according to which a proposition conjoined with its negation has total content, i.e. says everything. (The classical negation picture is explained in more detail in 2.9.) These results follow for subtraction negation because in A & ~A, ~A is not seen as adding a further condition to A but rather as the removal, deletion or subtraction of the condition first given, so one is left with nothing. Thus the whole A & ~A is, on the subtraction view less than the sum of each part since each part takes something from, and indeed may quite nullify, the other part. Accordingly 1

Thus Strawson (52, p.3): Contradicting oneself is like writing something down and then erasing it, or putting a line through it. A contradiction cancels itself and leaves nothing. The linkage of contradiction with negation is, according to Strawson (p.79), as follows:a standard and primary use of 'not' in a sentence is to assert the contradictory of the statement which would be made by the use, in the same context, of the same sentence without the word 'not'. A beginning is made in NC on tracing the long history of cancellation accounts of negation.

89

2.4 HOW THE SUBTRACTION ACCOUNT YIELDS DISTINCTIVE CONNEXIVE PRINCIPLES

Simplification, in the forms A & ~A ->• A and A & ~A ~A, fails. Since in A & ~A each of the conditions deletes the other the whole A & ~A contains neither of its apparent parts; thus A & ~A does not imply A, and A & ~A does not imply ~A. Thus also A & ~A does not imply A v ~A, i.e. using de Morgan principles A & ~A does not imply ~(A & -A). It is not difficult to see how such a subtraction notion of negation gives rise to Aristotle's law, ~(A ~A) . Suppose, on the contrary, A did entail ~A. Then ~A would be part (of the content of) A. But then, as in the case of A & ~A ~A, ~A would cancel A out, so that ~A could not be implied. Hence A does not imply its negation, and similarly ~A does not imply A. Another argument for Aristotle likewise starts from the assumption that A does entail ~A. Given this assumption, A must be impossible. Since A is impossible, A is equivalent to q & ~q for some q, i.e. A is equivalent to some contradiction. But then by replacement in ~(q & ~q -*• ~(q & ~q)), A + ~A, contradicting the assumption. Hence the principle A -»• ~A fails in the only case in which it could hold, namely where A is an inconsistent statement. The argument may be reformulated in a direct fashion. Either A is possible, in which case A ~A, or A is impossible, in which case by the argument given A -f- ~A; so A / ~A. Given Aristotle's principle, Strawson's principle,1 A -*• B ~(A -*• ~B), i s readily established, by essentially the argument Aristotle himself gave, namely:{A -*• B) & (A

-fi) -»-. (A •*• B) & (B

~A), using Contraposition

-»-. A •* ~A

, applying Syllogism.

Hence, contraposing, ~(A •*• B &.A -»• ~B). Strengthening Strawson's principle to an entailment, i.e. to Boethius's law, A -*• B -»-. ~(A -*• is less straightforward, though given certain principles correct for S4 for example, Aristotle does yield Boethius. The assumptions required are that provable statements (or negated entailments) are necessary, that necessary statements can be commuted out, and that syllogism holds in exported form. (A sufficient set of these principles holds in entailment system E). First, by Exported Syllogism and Contraposition, A -»• B -*-. A -»• ~B -»-. A

~A.

Hence, contraposing again,

A -»• B -*. ~(A ->• ~A)~(A-»- ~B). Since ~(A it can be commuted out, whence A -*• B -*• ~(A

~A) is necessary, since provable, ~B).

Boethius can also be argued for informally, and independently of Aristotle, on the subtraction account of negation. For it follows from the subtraction picture that a statement cannot include an inconsistent part. For using the principles given above, an inconsistent part would delete what it was inconsistent with, and the inconsistent parts would be apparent parts only. But precisely what Boethius asserts is that B's being part of A is logically sufficient for B's being consistent with A, i.e. A -»• B -*. A • B. For A ^ B, i.e. A is consistent with B, iff ~(A -»• ~B). (A similar argument is outlined in Nelson 30, p.447). To state the same argument a little differently:Suppose A entails B. Then as B is contained in A, B must, as a matter of 1

The attribution is based on the following passage (Strawson 52, p.87)5 As an example of a law which holds for 'if', but not for we may give the analytic formula '~[(if p, then q) & (if p then not ~q)].

2.4 THE CONTINUING IMPORTANCE Of DELETION ACCOUNTS OF NEGATION

logic, be consistent with A; for an inconsistent part would have deleted the argument in A that did the entailing.1 Another informal argument for Boethius appeals to the commonly accepted (but as we will see questionable) logical interconnection of A -*• B and A & B A. Now suppose B is incompatible with A, i.e. A ~B. Then, as a matter of logic, A & B does not imply A. For, as B incompatible with A, B subtracts from A, deletes part of A. That is, A ~B -*• ~(A & B ->A), whence Boethius results upon contraposing, and using A B A & B ->• A. (To make the argument connexively acceptable, the last implication should be replaced, e.g. by A -*• B A & B -*• B & B; the argument then relies on the fact that A/B implies that A & B does not imply B & B, because A subtracts from B,) The subtraction or deletion picture of negation and associated accounts of incompatibility have been extremely important historically in thinking on implication. Indeed it, subtraction negation, rather than classical negation, appears to be the notion of negation that underlies traditional logic. For it enables not only fuller and more coherent explanations of the intuitions of many traditional thinkers on implication principles, Aristotle in particular, but also of the central traditional theory of the syllogism. Thus the whole theory of the syllogism comes out on direct translation in connexive quantification theory (cf. McCall 67); whereas classical quantificational theory destroys syllogistic theory unless some very ad hoc assumptions are imposed - assumptions which furthermore destroy, without any warrant, the generality of the theory, which is not restricted in the terms it admits, e.g. to those which carry existential import. The subtraction account of negation continues to influence thinking about entailment in modern times. For example, it finds (as already noted) a modern expression in P.F. Strawson's Introduction to Logical Theory where , the notion of the negation of A as deleting or erasing A and A & ~A as yielding zero content is explicitly stated, without however its being clearly realised that the notion so expounded is quite distinct from classical negation.2 The fact that it has not been appreciated how distinct subtraction negation is from classical negation, helps to explain why a clear formal analysis of the subtraction picture of negation and the resulting, historically important, connexivist theory has never been properly worked out, and 1

2

This may suggest that A & B -»• A is correct subject to the qualification A O B. But this is a qualification that needs to be handled with care, as we have seen. For example, the qualification cannot be satisfactorily rendered as A • B A & B A where it is also conceded that A & B -*• A O B and that Contraction is correct. The question of the respective necessary and sufficient conditions under which Simplification holds is an important open question for connexivism. For these (and other, for example the account given of logical reasoning is neither classical nor connexivist but Peripatetic) reasons Strawson's logical theory is incoherent. It is worth noting that connexivist assumptions appear elsewhere in Strawson's text 52, for example in the proposals (p.24) for avoiding paradoxes of implication, that the inconsistency of a conjunctive assertion is not assured by the inconsistency of a conjunct. In short, in one way or another the principle ->• ~ ^(A & B) is to be avoided; and so, given distribution principles, ~A -*• ~(A & B) is to be rejected.

91

2.4 TROUBLES WITH NAIVE CONNEXIVISM, ANV INTERACTION CONNEXIVISM

also why contemporary efforts to formalise historical connexivism have fallen so far short. Thus, for example, McCall's attempt (in ABE, p.434 ff.) to formalise the first degree part of (an algebraic formulation of) connexive logic is based on, of all things, a Boolean algebra, which, of course, contains classical, not connexive, negation. The confusion between connexive negation and classical negation has, of course, been fostered by the classical logicians' insistence that classical negation is the unique intuitive account of negation found in natural language, an approach which confuses not only the connexivist, but also the classical logician, since the latter is often found providing a subtraction model of negation in the mistaken belief that it is an intuitive model of classical negation. But although the deletion model certainly seems to be coherent and is in many ways intuitively appealing (which helps explain why connexive principles keep appearing anew and being "rediscovered" by new workers on implication), it is not, we believe, the way negation must be explicated - or a correct account of the way negation must operate if we are to understand and to explain satisfactorily linguistic data and the functioning of a large part of language. It is only part of a larger story, and a part that needs to be told more carefully than connexivists have so far told it. Among intuitively based connexivisms, naive connexivisms should be distinguished from a more sophisticated position which we call interaction connexivism. Naive connexivisms are distinguished through their insistence on one, or usually more, of the following claims:- Firstly, as with classical logic, an absolutism about connectives is assumed: there is just one ordinary language negation, not the classical one, the connexivist one, and similarly just one conjunction. Furthermore this negation is the subtraction one modelled by the deletion account. Even so classical sentential logic should hold and does hold for connexive negation and conjunction (i.e. given ~ and & as primitives classical logic is forthcoming, a result written in, e.g., to Angell 62, by the addition of further & — axioms to ensure the full classical apparatus). Interactionist connexivism repudiates every one of these claims. According to interactionism there is, to begin with, more than one negation and more than one conjunction; but among the negation and conjunction determinates of natural language are those of connexivism. But connexive negation is not that given by the simple deletion account. The naive deletion account of negation if taken seriously would have a quite damaging effect on any theory of implication it was coupled with; for it would destroy relevance. If ~A really erased A without residue then A & ~A would have no content, say nothing. So it would not differ from B & ~B as regards content! That is, the deletion account would prevent us distinguishing distinct contradictions. As A & ~A = B & ~B, since they have the same content, namely none, A & ~A B & ~B, contravening relevance, and yielding paradoxes. The naive view of contradictions is hardly better than the classical and modal view. On the modal view inconsistent statements occur, though never in modal worlds and always in a degenerate way, but as far as naive connexivism is concerned there are none such. Such simple models can be powerful and persuasive but they do not accord with the facts of language. The naive deletion account of negation has accordingly to be abandoned, or rather adjusted, and with it the simple arithmetical model. The account can be adjusted, and the arithmetic generalised. The adjustment is by way of the relativisation already hinted at: ~A deletes A content, and A and ~A together say nothing A-wise, i.e. A & ~A has no A content. But to say A & ~A have no A content is not to say it has no B content, so the unfortunate outcome of Strawson's account that all contradictions say the same, namely

92

2.4 STANDARD CONNEXIVE LOGICS VIOLATE PARACONSISTENCY REQUIREMENTS

nothing, is avoided, and with it connexive paradoxes. Correpondingly the arithmetical model for connexivism is to be sought not in ordinary arithmetic with its single line through zero, but in generalised arithmetic (as outlined in Kleene 52) with many zeros - one for each distinct statement and many sequences - one by positive (and negative) extension from each zero. Connexive logics so modelled will presumably be weakly relevant: they would certainly be unsatisfactory if they were not, but relevance is as always not enough. Lastly, interaction connexivism will not incorporate classical logic; for not all classical tautologies are connexively valid, on pain of violation of the paraconsistency requirement (cf.1.7). The connexive logics of Angell 62 and McCall 66, for example, both violate the paraconsistency requirement. Since Disjunctive Syllogism is (claimed to be) connexively correct, the rule Y of Material Detachment holds. But also A & B ^ A , A s . A v B , etc., Hence all principles required to derive the hard Levis paradox in rule form are available, and A & ~A-*B results. Since too A =>. B =>. A & B, paraconsistency is violated. A similar criticism can be made of Nelson 30; for he concedes the principle: A & B-*A. The interactionist strategy for avoiding the problem should be obvious from the intuitive basis for connexivism already sketched: namely A & B —*A requires qualification in much the way, and for just the reasons, that A & B -*• A requires qualification. Thus, for instance, if ~A interacts vith A, so that A & ~A -+• A fails, then presumably ~A interacts with A so that A cannot be derived from A & ~A. Yet surely Angell and McCall are right in thinking that there is a conjunction - in ordinary use - for which all classical tautologies are correct. As this conjunction cannot, given the preceding argument, be connexive conjunction, interaction connexivism has, strictly, to carry two conjunctions, connexive conjunction and (something like) a normal conjunction. And this points the way to a reconciliation of relevant logic and interaction connexivism, with a theory which adds to the underlying common account of deducibility, both normal and connexivist connectives. The question as to how the connexive notions, and in particular connexive negation, are to be clarified formally and semantically is a matter that is best postponed until the larger relevant theory has been elaborated. There are several reasons for this procedure. One is that key connexivist theses such as Aristotle's law cannot be expressed within the logical framework - of first degree entailment - so far elaborated, even when negation is introduced. Negated deducibility statements such as ~(A ff ~A) are not well formed, at least at this stage in the development. (But, as we shall argue, any theory, such as Zinov'ev's in 73, which claims that such expressions are pimply not well formed at all, at any stage, is bound to be inadequate.) The second reason is more important, and more interesting. Whereas the usual views concerning the addition of connectives to the basic theory of implication or deducibility is what can, suggestively, be called separatist (or atomistic), the connexivist view tends to be an interactionist (or holist) one. Connectives interact with one another in a way that defeats natural separation of the role of connectives. The separatist ideal is best exhibited in formulations of systems, such as those of Gentzen, where each connective has its own rules without the appearance of other connectives. It is exhibited to a lesser extent in Hilbert-style formulations of systems where each connective - apart from implication which now assumes a fundamental position - has its own postulates, e.g. separate postulates for &, separate postulates for

2.4 INTERACTION CONNEXIVISM VERSUS THE SEPARATIST

WEAL

separate postulates for v, and for **.1 Connexivism questions, or repudiate this separatism. Connectives interact in a critical way on the connexivist view, partly because the polarity of statements (their positivity and negativity) is written in at the bottom. Thus negation interacts with everything else, with implication as the distinctive connexivist theses show, and in parti cular with conjunction, as the case against Simplification makes evident. The interaction of connectives not only makes the isolation of parts of connexive logic difficult (though so far artificial separation is not excluded): it also makes the determination of the first degree part of connexive logic difficult, and not a particularly natural starting point for the investigation of connexive logic (despite McCallfs proposal in ABE). For these reasons, then we postpone axiomatisation and semantical studies of connexive logic until much later (Part II). The interactionist features of connexivism have one other important outcome; namely that the semantical method (which is the main method of analysis in this text) is not a particularly appropriate method for investigating connexive logic: for the semantical method, at least in its pure forms, presupposes separatism. ^ " Interaction connexivism does not, in a good sense, offer a competing account of deducibility or a competing solution to the paradoxes to that provided by relevant logic. For the interaction connexivist can be seen as one who accepts essentially the same account of entailment as that of relevant logic, but introduces different conjunction and negation connectives, the difference appearing especially in the area where conjunction and negation interact. There need be no conflict unless it is claimed that either theory has captured the One True conjunction or negation of natural language. In our view any such claim by either party would be extremely rash. The accounts are in fact complementary rather than competing, since interaction connexivism has an important insight into the problems of sufficiency and relevance that complements that of relevant logic. Interaction connexivism diagnoses another important way in which loss of sufficiency of premisses can cause irrelevance. According to the theory already sketched in 2.1, principles which allow suppression, together with augmentation principles, may interact to produce irrelevance. The mechanism at the first degree is this. Augmentation princi pies allow the adding of an irrelevant assumption, either as a conjunct to the antecedent or as a disjunct to the consequent. So far sufficiency of antecedent for consequent is maintained. Suppression principles then however allow the dropping of the relevance-carrying part of the antecedent or consequent (depending on whether positive or negative suppression is involved), thus creating irrelevance of antecedent and consequent. Irrelevance is produced because suppression principles enable the destruction of the sufficiency of antecedent for consequent. The solution to irrelevance, then, lies not in rejecting the blameless augmentation principles which maintain sufficiency, but in rejecting the suppression principles which enable the destruction of sufficiency and the dropping of the relevance-carrying parts of antecedents or consequents. 1

Relevant logics do not quite conform to the separatist ideal since, as will be seen, in Hilbert formalisation, v is rather inextricably intertwined with &.

2.4 HOW SUBTRACTION WAV DESTROY SUFFICIENCY AND THE WAY TO SYNTHESIS

The interaction connexivist may appear to be opting for the first and wrong alternative, that of rejecting augmentation principles. But this is not so. What interaction connexivism has seen is that suppression is not the only way to destroy sufficiency. This can be done if we can apparently add to a premiss set a condition which in fact deletes some part of the original e.g. if starting from A ff B we proceed to A, C ff B where C deletes or subtracts a part of A which is necessary for B. This becomes possible when subtraction negation is used, and under these conditions unrestricted augmentation principles will enable the destruction of sufficiency of antecedent for consequent. Under certain conditions the destruction of sufficiency in this way will also produce irrelevance. Augmentation principles maintain sufficiency only provided that what is added is always a further condition which does not interact with and leaves unchanged the original antecedent or consequent, as it does with the conjunction and negation of relevant logic; but where such interaction can occur, as with the conjunction and negation of interaction connexivism, sufficiency can be destroyed by the addition of further conditions, and the interaction connxivist is right to reject augmentation principles. So though at first sight interaction connexivism and relevant logic may appear to be in disagreement, a more careful examination reveals the same insight at the source of each, that it is loss of sufficiency which is damaging, and productive of irrelevance. That connexive connectives are not the only connectives in ordinary use is brought out by ordinary counterexamples to connexive principles. Consider, e.g.., Hunter's counterexample to Strawson's principle. Let A symbolise "he caught the boat" and B "he caught the 2.15". Thus both A and B are contingent (so the example will also serve to counter theses of Stalnaker and Thomason and other proposed logics of conditionals, such as qualified Boethius; A...3. A -*• B = ~(A -*• ~B)). Then according to Hunter (and we agree) one can conceive circumstances in which it would be true both that if he didn't catch the 2.15 he didn't catch the boat, i.e. ~B -»• ~A, and if he did catch the 2.15 he didn't catch the boat, i.e. ~B -> Aj Contraposing, both A -*• B and A ~B, contradicting Strawson's principle. Interaction connexivism and relevant logic have then each discerned basic and important senses of negation and conjunction connectives. Explicitly introducing each sort of connective not only enables the development of a comprehensive logic in which all these basic notions are captured and their differences allowed for and treated, but thus helps avoidance of confusion between them. Failure to distinguish clearly between these different varieties of connectives, especially negation, is responsible for much confusion about which implicational laws hold, both for entailment and more especially in the case of physical sufficiency, causality and counterfactuals (as is argued in chapter 8). Both sorts of theories are essential to any full account of sufficiency, whether logical sufficiency or physical sufficiency. §5. Adding disjunction: distributive lattice logic and conoeptivism. The real issues concerning disjunction turn out to derive from issues concerning the analysis of negation and consistency. These issues are magnified in the case of extensional disjunction which is characteristically connected with conjunction through negation, by the coentailment: A v B •**• ~(~A & ~B). But new issues also enter to complicate matters, in particular 1) the question of constructive proof and truth, which emerges at the first degree sentential level with the introduction of disjunction, and the question 1

The train-then-boat example can be elaborated to remove the charge that one of the ifs involved is really an even if. 95

2.5 NEW ISSUES WITH DISJUNCTION, ANV THE C0NCEPTIV1ST THEME

as to whether A v B should be provable, or constructively true, in cases when neither A nor B is; 2) the question of the distribution of conjunction over disjunction - in such principles as A & (B v C) (A & B) v C - which orthologics and quantum logics reject; and 3) the question of the correctness of Addition (A -»• A v B, B A v B) , in the light of the criticism, especially by Parry, of this principle. The first of these new issues can be postponed (it will reappear with negation) since relevant logics coincide with constructive-intuitionistic logics on positive matters, at least at the first degree. Indeed distributive lattice theory, which we will investigate first, supplies just the sorts of rules for connectives & and v that positive logics ought, according to constructive insights, to have. The second of the new issues also involves negation - because the correctness of distribution is an issue as to the correctness of A & ~(~B & ~C) -> ~ (~(A & B) & ~C) - and we will postpone discussion of it until we have introduced negation. But, unlike constructivism, it provides a direct challenge to the account of disjunction we develop. For according to the proponents of quantum logics (that repudiate classical logic) distributive lattice logic is empirically incorrect. Such a logic no more applies to the quantum world than Euclidean geometry applies to the relativistic universe (see, e.g., Putnam 74). That distributive lattice logic is empirically incorrect is a matter we dispute (see UL, §13); and whether it would matter if it were so incorrect is something we should be prepared to take up if we were to lose the previous round (the issue is dealt with in Lewis 29). The third issue, as to the correctness of Addition, we will take up at once. Much of what we have argued against standard connexivism also applies here. For given a contraposable negation, which ties conjunction and disjunction, repudiation of Addition implies repudiation of Simplification. Parry, in his theory of analytic implication, breaks the link with connexivism by abandoning Contraposition, even in rule form, and so is able to retain Simplification while abandoning Addition. According to Parry, it is Addition that is at fault in the hard Lewis argument, there is nothing at all wrong with Disjunctive Syllogism. A similar position appears in various subsequent writers on implication, some of them unaware of Parry's pioneering work. Parry's position - for which we have coined the ugly term conceptivism is that no implication A -»• B is correct where B contains concepts which do not occur in A. Plainly this makes A -*• A v B incorrect since B may well, in an obvious sense, "contain concepts" not in A. Similarly Rule Contraposition is vitiated since even if A -*• B is provable, so that the concepts of B are indeed contained in those of A, the concepts of ~A will not in general be contained in those of ~B; that is, ~B ~A will be incorrect.1 Rule Antilogism is similarly rejected, and so presumably much of traditional logic theory that connexivism retains. 1

A two-way conceptivism, according to which the concepts of A must coincide with those of B, would restore Rule Contraposition and would render Simplification incorrect. Thus double conceptivism would lead to (weak) subsystems of connexive logic, and again to an equivalential-like "implication". But quite apart from having little to recommend it - double conceptivism would do nothing to motivate characteristic connexivist theses (like ~(A -*• ~A)) unless it also invalidated such theses as A -*• ~~A. Alternative concept and variable sharing requirements will be examined below.

96

2.5 CRITICAL APPRAISAL OF PARRY'S ARGUMENTS FOR CONCEPTIVISM

What is the appeal of conceptivism? Why have some logicians been tempted to say that when A entails B it is not enough that A and B are connected in meaning, that moreover B should be contained conceptwise in A, though A may outstrip B as to concepts contained? Parry's argument against the principle of Addition indicates part of the answer. The argument, presented in his thesis (31, p.119) and reproduced in recent revival work (68, 74, 76), is as follows:- To determine the correct principles of deducibility we should consider concretely a formal deductive system such as Euclidean geometry in a standard formulation. Then one f inds... that a system might contain the proposition "Two points determine a straight line", and yet not contain the proposition "Either two points determine a straight line or some angels have red wings". In fact, a mathematician would rightly consider it, not only ridiculous, but utterly erroneous, to infer the latter proposition from the former (74, p.5; the latter paragraph also appears in 31, p.119). But this argument, were it to succeed, would count equally against a number of principles, such as A i B + A and (A •*• B)&(B -*• C) -*•. A C, that Parry is concerned to retain, as against Addition. For if the system does not contain the proposition "Either two points determine a straight line or some angels have red wings" then it will not (unless extraordinarily formulated) contain such propositions as "Both two points determine a straight line and some angels have red wings" either. And presumably a mathematician's reaction (for what it is worth) to the use of the conjunction in Euclidean geometry would be similar to his reaction to the inferential use of Parry's disjunction. In short, if extra systematic propositions are to be excluded, they should be excluded systematically, in antecedents as well as consequents. But, of course, there is no reason for such exclusions from the point of view of deducibility. For deducibility is not and has never been system confined in the way Parry envisages. Even if it would be strange in a given context to assert A & B because B concerns notions remote from the topic of discourse, the validity of A & B + A is not impugned. The same considerations undercut Parry's other slightly different way of putting his case (68, p.151): The starting point [for analytic implication] is the view expressed in class by H.M. Sheffer: that the conjunction pq really implies p, but that not always does p imply p or q. What is wrong.with the latter? If a system contains the assertion that two points determine a straight line, does the theorem necessarily follow that either two points determine a straight line or the moon is made of green cheese? No, for the system may contain no terms from which 'moon', etc., can be defined. Again, if this is the reason that p -*• p v q fails, that q introduces new terms, then p & q p also fails. But one can see what Parry is getting at: one may want a systemic implication, such that for a system with a given stock C of terms one never derives a theorem containing a term not in C. And the rule A-»A v B would enable this, but A & B d o e s not, since the

97

2.5 REPUDIATION Of THE CONCEPTIVIST ARGUMENT AGAINST ADDITION

confinement of the terms of B to C must have already been established in the premiss A & B.1 A somewhat similar systemic restriction is often argued for in the logic of belief - that though x believes A he may not believe A v B since B introduces new concepts which x does not have. (This no-new-concept position, the initial appeal of which begins to evaporate as the logic of the position is examined, is discussed and criticised in R. and V. Routley 75.) But the systemic restriction so envisaged really applies to rules, not to implications where new unproved hypotheses can always be introduced, e.g. through Simplification. Moreover there is no need to repudiate Addition in order to design a systemic logic. Such a logic can be obtained more satisfactorily either through the formation rules of the system or by appropriate use of restricted variables. Consider the case of a systemic geometry: such constant subjects and predicates as 'the moon' and 'is made of green cheese' simply would not appear, any more than 'is happy' appears in Peano arithmetic. Thus 'the moon is made of green cheese' would not be a wellformed expression of such a systemic geometry and the implication Parry objects to could not be formulated in the system though Addition is maintained.2 Conceptivism, whatever its inital appeal then in the logical analysis of more highly intensional notions such as belief and propositional identity does not impugn Addition as an entailment principle relating statements. For deducibility, as modelled though sufficiency or inclusion of logical content, does not require concept preservation. Parry has claimed (in reply to criticism from Anderson and Belnap, criticism reproduced in ABE, p.432) that containment does require concept preservation, that A cannot contain A v B where B is totally irrelevant to A. On the familiar, and intuitive, account of containment in terms content inclusion, Parry's claim is mistaken. For the content of A v B, written c(A v B), just is c(A) ^ c(B) which is included in c(A), i.e. c(A v B) £ c(A).3 Moreover a cursory examination of paradigmatic entailments, such as "This is red entails that this is coloured" and "Ralph is a brother entails that Ralph is a sibling", would seem to confirm the point, that concept preservation is not required. But Parry wants to separate off such counterexamples to his position through a distinction of logical and structural entailments. Structural entailments lead back however to logical ones. Consider such examples as "Blue Blaze won the 4.30 entails that Blue Blaze was a place-getter" (said, e.g., in explaining to some new chum that he could collect on his bet); this is based on, and expands to, the entailment: Blue Blaze came in first in the 4.30 entails that Blue Blaze came in first or came in second or came in third. That is, the structural entailment is supported by a logical entailment - and surely the antecedent is sufficient for the disjunctive consequent - which conceptivism has to repudiate. Similarly in significance logic, A & B -pA is admissible because A & B, when proved, is significant; but A v B may well be non-significant when A is significant, so A -P A v B is not admissible. Significance issues can however be postponed by restricting consideration to sentences that yield statements: for the criticisms of Parry and others have not been based for the next part, on the significance difficulties A -*• A v B can cause. The question of satisfactory entailment logics with significance is very much an open one: for some of the problems see GR and Woodruff 74. Less charitably, Parry can be seen as forced into criticising Addition by the desire to remove the paradoxes but retain Disjunctive Syllogism. In view of his rejection of Antilogism, in even rule form, his case in favour of Disjunctive Syllogism is decidedly weak (the traditional case that Lewis relies on, for example, is sacrificed). ^ _ 3 ^ . fc (footnote on next page)

U

2.5 EMBARRASING ARBITRARINESS IN CONCEPTIVIST CONTAINMENT REQUIREMENTS

Conceptivism itself, as presented by Parry and others, satisfies only a limited containment of concepts requirement. For the systems proposed include such principles as A -*• ~~A and A -*(A & A) which introduce new concepts in the consequent, negation and conjunction respectively, which may not be contained in the antecedent, A liberal construal of terms or concepts would rule out many of Parry's axioms for analytic implication, e.g. A -*• — A , (A C) & (B -*• C) -»-, (A v B) C, A B ->-. ~A v B, A -»-. A A, each of which may introduce new logical notions in the consequent. Such a liberal construal which took conceptivism seriously and counted all concepts, would indeed cripple logic as an explicative device. Conceptivism depends then, for any plausibility, on a narrow, and rather arbitrary, construal of concepts: only those locked up in variables are allowed to count. This is not just unsatisfactory: it also eliminates conceptivism as a plausible way of formalising the no-new-concept position on belief. For the position rejects such implications as x believes p x believes ~~~~~~p on the ground that x may not have or (like Griss) not acknowledge the concept of negation. The arbitrariness of the conceptivist construal becomes embarrassingly obvious when the question is raised as to what conceptivist quantification logic looks like. Is universal instantiation, (x)A(x) -»• A(t), to be rejected or qualified because t is a new term not in the antecedent, or is it to be admitted because the antecedent is really an infinite "conjunction" and so contains A(t) as a conjunct or because universal quantification says something about everything and so about t? If we say the latter we are on the way to the claim (contemplated by Prior and others) that every statement, and certainly every inconsistent statement, says something about everything, in which case A -»• A v B, should be readmitted, certainly when A is inconsistent. Qualifying instantiation is perhaps the more appealing alternative, and a qualification is suggested by the way Addition and other principles get patched up in Parry's system,2 namely by tacking on newly appearing variables in antecedents, Instantiation would correspondingly be dressed up as (x)A(x) & B(t) A(t>, or given a predicate T (read, say 'is a term') in free logic style as (x)A(x) & T(t) -* A(t). The resemblance to free logic is however superficial: for free logic correctly admits Contraposition and also incorrectly Antilogism, and thereby arrives at a qualified form of existential generalisation, of the form A(t) & T(t) •*• (3x)A(x) . But nothing stops conceptivism from accepting unqualified generalisation - for A(t) -*• (3x)A(x) is like A — A isn't it, both just introducing new logical operators - unless of course (3x)A(x) is construed as an implicit disjunction, in which case existential generalisation will po the way of Addition. But if generalisation is at fault, can't it be (footnote 3 from previous page) Since A ->- B is true iff c(B) £ c(A), A -»-. A v B is true. The content analysis provides one direct defence of Addition. Other intuitive modellings for entailment can likewise be put to work to vindicate Addition. 1

Parry is aware of this, and is tempted by a stronger containment of concepts requirement according to which 'entailment should be treated like empirical concepts, which may not appear in the consequent unless they (or empirical concepts from which they may be derived) are in the antecedent'. The principle of counterexample, A & ~B ~(A B) illustrates the point. 'From my point of view it is perhaps objectionable as introducing the concept of entailment without antecedent, and perhaps I should have this rather in such a form as p & q & +(r s) -*• -(p ->- q) ' (Parry 76, p.25).

2

Parry's Proscriptive Principle, that there is no free variable in the consequent of a (correct) entailment that is not also in the antecedent, requires qualification of Instantiation, and rejection of the picture of a universal quantifier as an infinite conjunction.

99

2.5 THE RELEVANCE

QUALIFIED

PARADOXES

Of PARRY

SYSTEMS

repaired as follows1:A(t) & (3x)B(x) -*• (3x)A(x)? Similar arbitrariness appears also in such questions as to whether change of bound variable is to be admitted or qualified, as to the connections of free variables with bound variables, and so on. In application to formal logic concept preservation is reduced - none too satisfactorily as we have seen - to variable inclusion. According to Parry's so-called "Proscriptive Principle" 'no formula with analytic implication as main relation holds universally if it has a free variable occurring in the consequent but not the antecedent'. The Proscriptive Principle is an attempt to reduce to syntactical form an essentially semantical matter, interrelation of concepts, and suffers from most of the difficulties of such attempts (e.g. that concepts are presented in various linguistic guises, that for interrelations of concepts such as inclusion a notion of entailment is presupposed, so that there is an endemic circularity in the proposal that what entails what should depend on what concepts are included). Systems which conform to the Proscriptive Principle we call Parry systems. The sentential conceptivist, or analytic implicational, systems that have actually been presented - those of Parry, Dunn 72, Urquhart, and Fine 74b - all conform to the principle, as a matrix argument will show (see below)> but they all leave a great deal to be desired. Parry's system, which is included in all these other systems has as an axiom the near paradox A v (B & ~B) -*• A, which permits the suppression of impossible disjuncts. It is only because the addition principle B & ~B A v (B & ~B) is excluded that the systems escape outright paradox. Parry's system also has as an axiom a generalised form of Mingle, namely the scheme .. .A.. .-*•. A -*• A, i.e. any wff containing A entails the law of identity in A. This is but another relevance qualified paradox. This feature is even more blatant in the next weakest of these systems, that of Fine (which Parry now seems inclined to accept), which has a theorem scheme (A & ~A) & ~B -*• B, i.e. the negation of B guarantees the connection. More generally, paradoxes are in order provided variable containment is ensured, e.g. where V(A) represents the variables of A, if V(A) £ V(B) then A - o B -*• A and ~B—frB -*• A. Fine's system also contains a range of other implausible theorems not directly tied to the Lewis paradoxes> e.g. DA & (A -*• B) ~A •*• ~B, in effect a fallacy of conversion. For a counterexample, let A represent 'Some objects have a definite shape' and B 'Some objects have size' (and in case the necessity of A is in doubt construe 'object' in Meinong's fashion). Since definite shape logically guarantees size the.antecedent is true, but that no objects have a definite shape does not entail that no objects have size, since, for example, objects without a definite shape can have a large size. In Dunn's demodalisation of the Fine system (obtained effectively by adding the scheme A -»• OA) , the patently false conversion principle A & (A -»• B) ~A -»• ~B of course results, and a great many other oddities also follow. The semantical analysis of Fine's and Dunn's systems reveals that they are effectively strict and material systems, with a variable or contents inclusion requirement thrown on top. The oddities emerging help to show that the trouble with strict and material systems, is not merely, but only incidentally, their variable-sharing failure. The real troubles go deeper and are not repaired simply by throwing on a variable-inclusion filter - any more than by just ruling out paradoxical cases. These fallacies, near paradoxes, and oddities are not, however, inevitable features of conceptivist systems. It is not too difficult to design other Parry logics which escape these features by amending the axiomatisation of Parry's system (see Part II). But some peculiarities remain, e.g. prefixing principles are in order but suffixing ones are not. For example, l And correspondingly A

— A as:

A & ~C -»• ~~A.

100

2.5 PARRV LOGICS ALSO FAIL PARACONSISTENCY

REQUIREMENTS

while C -*- A & B ->•. C -»• B is derivable in Parry's system its suffixing mate, B -»• C -»•. A & B + C is not. Yet these principles should stand or fall together. (What we have is a further asymmetry induced through the unwarranted failure of Contraposition.) Moreover the general objections made to conceptivism apply equally to "relevancised" Parry systems. So too does the telling objection made by Kielkopf 75 to Parry's system, that it has as a derived rule the spread principle A, ~A-frB. The argument, a variation of the hard Lewis argument, is as follows:A & ~A -*(B v ~B) & (A & ~A),

adjoining the theorem B v ~B*

—p(A & ~A) & ~B v (A & ~A) & B, -*A v B

distributing

, by Simplification and v-Composition.

That is, rulewise the effect of Addition can be obtained indirectly. as A & ~A -»• ~A, by Rule Composition, A & ~A

Hence,

~A & (A v B) -»B;

by Disjunctive Syllogism.

Thus Parry logics fail to meet the paraconsistency condition of adequacy (cf.1.7). The matter is not rectified by abandoning - what is at fault Disjunctive Syllogism. For it is a central tenet of conceptivism that it is Addition that is the faulty move in the hard Lewis argument, and that Disjunctive Syllogism, which meets the variable inclusion requirement, is perfectly in order. Were Disjunctive Syllogism rejected, then - as Antilogism is rejected by conceptivism because it, like Contraposition, fails on variable inclusion - conceptivism would be able to resolve the paradoxes in the relevance fashion and there would be no need to reject Addition and Contraposition. To sum up. The conceptivist objections to Addition - a principle which is validated by underlying informal accounts of entailment - are, when carefully examined, unimpressive, and if they succeeded they would similarly condemn other quite acceptable principles. Moreover the conceptivist objections do not rest on a solid base, but on a narrow and arbitrary assumption as to what counts as a concept or term. Finally, conceptivist accounts fail on important requirements of adequacy, in particular on the paraconsistency test. For all these reasons we reject conceptivism as offering a viable account of entailment.2 But before we turn back to logics not yet out of the running in the entailment and implication stakes, let us glance at other proposals which, like Dunn's demodalisation of Parry's system, essentially characterise entailment by the addition of variable sharing requirements to classical logic (or, equivalently at the first degree stage, modal logic). The proposals take the form: A |- B (or A B) iff A 3 B is a classical tautology and A and B satisfy the proposed variable sharing requirement. Where A and B are zero-degree wff, i.e. contain no occurrences of intensional connectives but only of connectives &, v and there are several possibilities for the interrelations of their variables that 1

The corresponding, and fairly harmless positive .paradoxes - A B v ~B and A C where C is a thesis - are immediate from the classical character of the rule theory. Since Contraposition is not correct for rule theory the positive rule paradoxes do not directly yield the damaging negative paradoxes. The design of non-classical rule theories will be taken up again in chapter 15. 2 Subsequently we shall present however, semantical analysis of certain conceptivist systems. But such analyses do not of course, without much further ado, show the satisfactoriness of an account for a given philosophical purpose.

101

2.5 VARIABLE

INCLUSION,

ZINOV'EV'S

SYSTEM,

A W PARRY PROBLEMS

REPEATEV

have been thought to merit investigation, as follows on the classical provability of A => B, namely, where V(A) represents the class of variables of.A; i) V(B) } C V(A), ii) V(B) = V(A), and iii) V(A) n V(B) 4 { }. The system which emerges from the first proposal, which we call ZV after Zinov'ev, is axiomatised in Zinov'ev 73 (in systems Sj and S 1 : the adequacy of the axiomatisation is proved, in the same text, by Smirnov1). A B is a theorem of ZV iff A B is a classical tautology and V(B) c V(A). ZV is of course decidable, variable relevant, and almost as simple-minded as classical logic. It is a first degree implicational subsystem of all the Parry systems mentioned, since it is a sublogic of the weakest of these, Parry's original system, as derivation of ZV (as formulated in 73, p.79) in Parry's system shows. Indeed ZV is the first degree implicational part of certain of the Parry systems, e.g. of Dunn's system (as the semantics of Dunn 72 will show). Furthermore the following Parry matrices, used to show that the Proscriptive Property holds for Parry Systems, are characteristic for ZV:

*1 *2 3 4

1

2

3

4

1 2 1 2

4 2 4 2

3 4 1 2

4 4 4 2

3 4 1 2

&

1

2

3

4

1 2 3 4

1 2 3 4

2 2 4 4

3 4 3 4

4 4 4 4

That is to say, A (- B is a theorem of ZV iff A j- B takes either value 1 or value 2 for each assignment of values from {1, 2, 3, 4} to its initial obs. (This will be demonstrated, and the Parry matrices analysed subsequently, when normal negation has been introduced, and matrices for normal systems semantically analysed.) The above Parry matrices also satisfy the other Parry systems introduced, that is to say each axiom takes designated values and the rules preserve designated values. Accordingly the Parry matrices yield a simple proof that these systems are indeed Parry systems. For the following assignments show that any wff A (- B (or A -»• B) in which B contains some obs (or variables) not in A takes an undesignated value:- assign to each ob in B but not in A value 4 and to all other obs value 3. Then, by induction, A takes the value 1 or 3 and B takes the value 2 or 4, and hence A + B has value 4; so A ->• B is not provable. The first degree implication system ZV is unsatisfactory in much the way that other Parry logics are: it contains relevancised paradoxes e.g. (A & ~A) &}6(B)

B and C&}S(B) B v ^B, where 6(B) is any extensional funcv v tion of B, and A & ~A &}6(p i ,. . . ,p n ) -»• B and C&}6(p 1 ,.. . ,p^) B v ~B, where 6(p,,...,p ) is any extentional function of p,,...,p and all the variables l n i n of B are included in p ,...,p ; it excludes, without - so far as we can see substantial basis, the principles of Rule Contraposition and Additon which are correct for a sufficiency implication and for such associated notions as inclusion of logical content; and it fails requirements of adequacy, such as the paraconsistency requirement. How is it then that Zinov'ev manages to *It should be a straightforward matter to find less cumbrous axiomatisations than those of Zinov'ev. ^Since this chapter was drafted the system satisfying the third proposal has achieved some celebrity - though it is none the better for that - as the leading example of a relatedness logic (see especially Philosophical Studies 36, 1979). Other relatedness logics vary the implication A ^ B to which the relatedness (or relevance) requirement R(A,B) - generalising iii) - is applied, the connectives to which it is applied, and the conditions on R in addition to reflexivity and usually symmetry.

102

2.5 THE INADEQUACY

OF ZINOV'EV'S

ACCOUNTS

OF LOGICAL ENTAILMENT

AND OF LOGIC

arrive at the conclusion that with system ZV 'the problem of logical entailment can be considered solved in principle' (pp.85-61)? In a thoroughly unsatisfactory and question-begging way is the short answer. The reason Zinov'ev offers is this (p.86): 'there are no other apriori criteria of meaningful sentential relations, which would be as developed and defined as relations of sets of variables occurring in premisses and conclusions'. This already assumes that the correct account of entailment is to be obtained by imposing a (relevance) sieve on classical logic - a seriously mistaken assumption (as we have seen). Even the sieve Zinov'ev mentions is far from uniquely determined, since it would be required that the variable sets concerned are related in other ways, e.g. they are identical or they merely overlap. And many other criteria, both syntactical and semantical, can be just as well developed and just as sharply defined. That is, such criteria do not delineate ZV. Nor does Zinov'ev's other procedure, that of imposing - in a quite unsupported way - conditions of adequacy, uniquely determine ZV. According to Zinov'ev an axiomatisation of the general theory of logical entailment ought to satisfy the conditions: 1) in axiomatic construction all the above assertions should be provable; the construction ought to correspond to the intuitive understanding of logical entailment; 2) in axiomatic construction not just any assertions should be provable (outside of those introduced above) but only those which satisfy the requirement I). For one thing intuitive understandings differ. On ordinary understanding at least, truth functions of entailment statements make sense; so a correct theory should be at least first degree. But on Zinov'ev's theory even (A |- B) v ~(A |- B) is ill-formed, and in fact meaningless. Furthermore if higher degree statements are admitted - as they should be, again on intuitive grounds - then even given variable inclusion there is a wide variety of Parry logics to choose among, and many proposed theses about which intuitions differ. Actually Zinov'ev has the audacity to write down, without discussion, as intuitively acceptable, principles such as Disjunctive Syllogism and Conjunctive Simplification about which there has been considerable discussion and which are far from intuitively acceptable to some thinkers. On the other side implicational paradoxes are now intuitively acceptable to many, so the conditions of adequacy do not strictly rule out material and strict implication, or at least their common first degree implicational part, as providing axiomatisations. The idea of a unique solution, such as that offered by ZV, to the problem of entailment is, moreover, incompatible with Zinov'ev's general relativism about logic (see especially 73, chapter 21). At the outset of his discussion of the problem of logical entailment (p.69 ff) he rejects the 'prejudice that there is some single, unchanging, "natural", "basic", etc. ["intuitive"?], logical entailment and all logic has to do is to find the most accurate and complete (adequate) description of it'. ... there is no single, perfect, "natural", etc. logical entailment which simply has not been adequately described up to now. ... Logic has in fact come to recognise different forms of logical entailment. No one of them can be considered any more "basic" than another. In a certain sense they are 1

See also (on p.86) such overstatements as 'the solution of the problem is basically achieved'. It is amusing to note that 'the systems of Ackermann, Anderson, Belnap and others' are said to be 'in a sense, "improper" ("deformed" etc.) systems', perhaps even "revisionist". 103

2.5 SIMILAR OBJECTIONS

APPLY AGAINST HINTIKKA'S

ACCOUNT ANV RELATIONAL

LOGICS

all equal. The problem of the most adequate description of logical entailment is no longer a matter of finding some logical system as the final and unique theory of logical enailment but of constructing different types of logical systems ...1 Ergo, it is not a matter of finding ZV and its variants. simply incoherent.

Zinov'ev's theory is

Among the alternatives to variable inclusion that have been looked at in the accumulating literature on implication are variable overlap and variable identity. Kielkopf has studied and discarded the proposal that implication requires as well as classical provability variable sharing, i.e. A -+ B iff A ^ B is a tautology and V(A) n v(B) ^ { }. The "implication" defined fails transitivity, which is enough to discredit it, and also replacement of coimplications; and it lets through such relevance qualified paradoxes as (A & ~A) & C -»•. B & ~C.2 But, unlike both Zinov'ev's and Hintikka's accounts, it does admit Contraposition and Addition. Hintikka has defined an implication which, at the sentential level, is as follows: A B iff A = B is a classical tautology and V(A) « V(B); and A -> B iff A •** A & B. Thus Hintikka's theory is but an earlier formulation of Zinov'ev's. For A -*• B is a Hintikka thesis iff A = B & B is a tautology and V(A) = V(A & B), i.e. iff A => B is a tautology and V(B) c V(A), i.e. iff A B is a theorem of ZV. Thus the objections already lodged against ZV apply to Hintikka's account. So concludes our detour into Parry logics: we return at last to logics not yet out of the running in the implication stakes. DLL (Distributive Lattice Logic) is set within a simplified ML: f- in prefix form is omitted (since no formal objects of DLL are theses) and F , A , etc. are (inessentially) restricted to finite sets of formal objects. DLL is built up from the following components (vocabulary from one point of view): Initial formal objects (there is no need to specify what these are). Operations and auxiliaries: &, v, (,); Relation: f- . Formal objects (obs) are defined as follows: 1. Initial formal objects are formal objects. 2. If A and B are formal objects so are (A & B) and (A v B). Statements of DLL are defined thus: If r is a finite (non-null) set of formal objects and C is a formal object then r b B is a statement (wff). The postulates of DLL are these: Axioms: A A Rules: A A r r, J

A (- A & B f- A A & B (• B (- A v B B (- A v B (- B; B, F |- C -h> A , r | - C (- B; A c r r(-B I- B; r I- C r|-B&C A |- C; r , B|- C - » r , A V B

(• C

(Identity) (Simplification) (Addition) (Transitivity, Cut) (Augmentation) (&-Composition) (v-Composition)

The question of the extent to which entailment can be uniquely characterised will concers us much in what follows. The important issue of logical relativism, versus substitutionism, is taken up in detail in chapter 15.

Relational logics - which continue the nontransitivity tradition (discussed above, p.74ff.) - not only wrongly validate basic substitutivity principles, but also validate several principles already or subsequently rejected, e.g. A -»-. A ->• B -»• B and A -*• (A •*• B) -*•. A B (and also A -*• ~A -»• ~A) , and hence fail further important conditions of adequacy for implication in its central roles. Some of these points, especially paraconsistency failure, are elaborated in the disucssion of relational logics in PL.

104

2.5 FEATURES OF SYSTEM DLL OF VISTRIBUTIVE

LATTICE

LOGIC

Logical versions of all the results of distributive lattice theory now follow, but a few sample results suffice here since simple semantical arguments can be used once the adequacy theorem is proved to verify theorems and derived rules. T1 (Adjunction). Pf. B, C |- B

A, B [• A & B

B, C |- C

B, C |- B & C

, by Identity and Augmentation. , by &-Composition.

Unlike the axioms this thesis, which could replace the rule of &-Composition, has multiple antecedents. DR1.

A | - C, B |- C - » A v B I- C

Pf.

A f- C, B

C-frA v B, A |- C, A v B, B (• C,

by Augmentation (invoking an obvious enlargement of the rule notation) —P A v B, A v B (- C , by v-Composition — • A v B |- C

, by set properties.

T2 (&-Commutation) A & B f- B & A Pf. By Simplification and fit-Composition, T3

(v-Commutation). A v B f- B v A

DR2. A, B | - C - * A & B | - C Pf.

A, B |- C, A & B |- A - * A & B, B A & B, B

C, A & B

C,

by Transitivity

B - * A & B, A & B (- C, by Transitivity.

Hence the result using set properties and rule features. T4

(Distribution).

Pf.

A & (B v C) f- (A & B) v C.

A, B |- (A & B) v C, by Tl, Addition, Transitivity.

Since C (- (A & B) v C, by Addition, T3, Transitivity, A, C (- (A & B) v C, by Augmentation. Hence A, B v C (A & B) v c , by v-Composition. Distribution then follows by DR2. As is well-known from lattice theory, multiple antecedents in v-Composition are essential for this proof. It is tempting to suggest that it is only defective formulations of earlier entailment logics that have made distribution an independent and apparently questionable axiom. Orthogonal lattice logic, OLL, discussed below results simply by limiting v-Composition to: A |- C, B

C-»A V B

|- C.

But sufficiency and its offshoot, relevance, demand no such restriction. Weak Relevance Lemma 2.2. If F j- B, then some ob is a component both of B and of some element of T. Proof cannot quite be by induction over proofs in DLL. For though it is a matter of inspection that each axiom satisfies the requirement, and is readily shown that each rule, except Cut, preserves the property, the inductive argument breaks down (in a very characteristic way) on Cut unless Cut is reformulated in a way (such as putting the desired premiss B into the premisses of the conclusion) that no longer cuts out a connecting ob (namely B). But such a reformulation would appear to weaken the scope of the logic:1 unrestricted 1

The argument set out below shows that the restriction would not in fact reduce the class of theorems. Accordingly a more satisfactory, relevant, (footnote continued on the next page)

105

2.5 RELEVANCE

SV THE WAJSBERG MATRICES,

WHICH ARE CHARACTERISTIC

FOR VLL

transitivity is, to repeat this important point, essential to a good entailment and for applications of the logic. The failed argument suggests reformulating DLL in a Gentzen fashion and proving an appropriate Cut-elimination theorem (cf. Kleene 52, p.422 ff.). Such a strategy will indeed succeed (as we shortly indicate). However a simple matrix argument will show that Cut does indeed preserve relevance; and since the argument does all the other cases of our proposed induction for us, we can forget the induction and begin again. We show, by a different induction, that every theorem of DLL takes only designated values for every assignment to initial obs under the following Wajsberg matrices (these matrices are semantically derived, and thereby explained, in 2.6):

h *1

1

4

1 | 1

2

1

1

3 |

3

1

4

1 J ~~3

3

4

1

1

2 I

4

1 1

2

3

The matrices are just the Lewis and Langford Group III matrices (32, p.493), but 1 is taken as sole designated value. To evaluate wff of DLL we construe commas on the left of f- as conjunctions (and commas on the right of }- in multiple formulation as disjunctions), i.e. Ai,...,An |- B is evaluated as Aj&.-.&A^ |- B.

Then, by induction over proofs of DLL,

(1) Every theorem of DLL is satisfied by the matrices. (2) Where T and B share no ob, T |- B takes an undesignated value. To establish (2), assign every initial ob occurring in B one of the values 3 or 4 and every initial ob occurring in an element of T one of the values 1 or 2. Such an assignment is possible because T and B share no ob. A simple inductive argument (confirming inspection of matrix blocks) shows that B takes one of the values 3 and 4 and that T takes one of the values 1 and 2, whence it follows that T B takes value 4. Finally, the theorem follows from (1) and (2). The Wajsberg matrices not merely satisfy DLL; they are characteristic for DLL, i.e. a wff of DLL is a theorem iff it takes a designated value (i.e. value 1) for every assignment of values to initial obs. This will follow from the semantics for DLL, to which we turn after one further preliminary. DR3.

Bi & A bCi, B 2 h c 2 v A

?f.

Bi & A f-Ci, B 2 h C 2 v A

•Bi & B 2 h c i

v C

J

B 2 ) & A |-
(Bi Bi

B2

h

(Ci v C 2 )

c2), v A

But, by Identity and &-Composition, (footnote from the previous page) set of rules can be obtained by adding to the Cut Rule the restriction: provided that in the conclusion, A,T (- C, C and Aur share a component. A similar restriction was proposed by Smirnov for a system of Zinov'ev (73, p.82). Such relevance-restricted rules are important in obtaining a satisfactory, non-classical, rule structure for relevant logics: see further chapter 15. 106

2.5 SEMANTICAL

ANALYSIS

Bi & B 2 b (Ci v C 2 ) v A —frBi & B 2 -•Bi

& B2

OF VJST1UBUT1VE

LATTICE

LOGIC

Bj & B 2 & ((Ci v C 2 ) v A) (®i & B 2 ) & A v (Ci v c 2 ) by T 4 .

Also, by Identity and v-Composition, (Bi & B 2 ) & A |-Ci v C 2 ->((Bi & B 2 ) & A) v (Ci v C 2 ) |- C i v C 2 . The result follows by Transitivity. The rule DR3 plays a key role in the semantical completeness argument. Semantical analysis of DLL simply takes over ML models, deleting 0, and extends the interpretation function I to non-initial obs by classical valuation rules. To be explicit:A DLL model M is a ML model , with I assigning to each initial ob at each a in K just one of {1,0}. I is extended from initial obs to all obs by the following rules, for each a e K. I(A & B,a) = 1

iff I(A,a) = 1 = I(B,a);

I(A v B,a) - 1

iff I(A,a) = 1 or I(B,a) = 1.

and

Truth, validity, etc., are then defined exactly as for ML. Adequacy Theorem for DLL 2.3. A \- A is a theorem of DLL iff A |- A is DLL valid, i.e. true in every DLL model , i.e. extending the deducibility notation, iff A ft A. Proof: Soundness is established by the usual inductive argument. Completeness however requires some complications in the central notion of a theory. A DLL-theory c is a class of wff of DLL closed under |- and also under Adjunction, i.e. if A, B e c then (A & B) € c. A prime DLL-theory c is a DLL-theory such that whenever (A v B) e c then A e c or B e c. The canonical DLL model M

= has K as the class of all prime r c c c DLL-theories, and I defined for all initial obs by: I(B,d) = 1 iff B e d , for d e K c . It is immediate that M £ is a DLL-model. And completeness soon follows given the following Extension Lemma 2.3. If A J- B then there is a prime DLL theory a such that A £ a and B i a. Proof of the lemma is again by a Lindenbaum construction which considers an enumeration AqjA^-.-.A^ of the obs of DLL. In order to ensure primeness of the constructed set a it simplifies matters not just to put obs into a but to keep track of obs that are kept out. To symmetrise acceptance and rejection, we go multiple on the right, and define A }- T as holding iff, for some A>,...,A e A and B ,...,B e T, A &...&A 1i- B V...VB , i.e. some disjunction J i' m i* ' n * l m l n' of elements of T is derivable from some conjunction of elements of A. Where T is a singleton, {B} say, i f A |- T , A |- B i n DLL. (The converse compactness feature also holds, as an inductive argument will show.) Hence A^ J- r where A n = A and IV, = {B}. 0 0 follows:

NOW define A

n+1

and T

n+1

in terms of A

n

and T

n

If A U {A ..} 1!- r , set A . = A , Y , = V U {A ..}, n n+1 n n+1 n n+1 n n+1 i.e. gets rejected. If A U (A , -} 1 L p - = A u {A set A n n+1 n' n+1 n+1 i.e. is accepted. Finally i.e. A v 3 set a = U An

r = r , n+1 n and d = U

Tn .

That a has

the further required features follows from these points:(i)

a |-d.

If not there is some smallest n>0 such that A

107

|- T , but

as

2.5 PROOF OF THE ADEQUACY

J"

Suppose A^ was rejected.

A a = A i _ 1 and (b)

OF THE WORLDS SEMANTICS

= u

h

A^, that (c)

u {A±>. {A^}.

It:

\-

FOR DLL

Then (a) A^ ^ u {A^} fHence, as A

±

and

J- r ,

will follow from (a) and (b), upon eliminating

contradicting the non-derivability.

By (a) there

is some conjunction Bj and disjunction Cj of elements of A^ ^ and respectively such that B^ & A B

and disjunction C , B J 2 2 2'

B^fieB

Cj.

Similarly by (b) for some conjunction

C v A.. 2 1

Hence by DR3 B, & B„ I-C v C . 1 2 ' 1 2

is a conjunction of elements of

elements of

f

^ and C x v C 2 a disjunction of

whence (c).

Thus A must have been accepted. n At =

But

u {A ± } and Y ± = r ^ .

Thus (d) A. , u {A.} 1t- r, ,. i-1 i i-1'

Hence, siape A ± j- r , (e) A

u { A j |- r

,

contradicting (d). (ii) Every ob belongs to just one of a and d. Thus d = a C . By the construction every ob belongs to either a or d. Suppose, to complete the argument, some ob D belongs to both a and d. Then a d, contradicting (i). (iii) A £ a but B it a. (ii), B I a.

By the construction A £ a and B e d , and so, by

(iv) a is closed under (- . Suppose © c a and © (- D. Then a (• D, by Augmentation; and so D e a. For if D e d, a d, contradicting (i). (v) a is closed under Adjunction. Suppose otherwise B, C I a, but B & C i a. Then B & C e d, and so, as B, C |- B & C, a (-d, contradicting (i). (vi) a is prime. whence by B v C

Suppose otherwise B v C e a, B I a, C I a. B v C, a f-d, contradicting (i) again.

Then B, C e d,

Suppose r f- B. Then by the Extension Lemma there is a prime DLL theory a with B I a. To show that T |- B is not valid it suffices to show that I(A,a) = 1 for each A e T and that I(B,a) ^ 1. But all this follows using the equivalence: I(C,a) = 1 iff C e a, for every ob C, which is readily established by induction. The induction basis is a matter of the canonical definition of I, and the induction steps for & and v follow directly from simple properties of prime DLL-theories. The adequacy theorem may be applied to establish a great many properties of DLL and thereby of distributive lattice theory. For example, filtration methods readily establish the decidability of DLL. More directly, the adequacy theorem leads directly to a finite (two-world) canonical model, and so to decidability, and also to the 4-valued Wajsberg matrices. These results follow at once from analogous results for the system DML (of 2.6). Both system DLL and the adequacy proof can be simplified by moving to a multiple formulation. Statements of the system, which we call DLLM, are defined thus: If T and A are finite non-null sets of formal objects then T (- A is a statement (wff). The axiom schemes of DLLM add to those of DLL the following: A, B |- A & B A v B 1- A, B (&- and v-Adjunction) Rules: A |- A, B B, T |- © -o A, T (- A, 6 (Cut) A r , A C A 1 , r C r 1 -o A 1 |- r (Augmentation). The name Cut Is taken from that of the analogous rule in Gentzen-sequents systems. Earlier formulations of Transitivity are special cases obtained 108

2.5 ADEQUACY

RESULTS FOR THE MULTIPLE

FORMULATION

DISTRIBUTIVE

LATTICE LOGIC VLLM

by taking A as null (and T as null in one case) and 0 as {c}. Note that the formation rules allow that in Cut A can be null so long as 0 is not, although A cannot. The main new feature is the replacement of the rule A |- C, B f- C - t A v B (- C b y the more compact and appealing principle A v B f- A, B. The argument for interderivability is, in outline, as follows:- As A |- A,B and B (- A,B, so A v B A,B. Conversely, suppose A f- C, B |- C. From A v B |- A, B and A (- C, A v B |- B, C which together with B f- C yields A v B |- C, C. The semantics for DLLM is precisely the same as that for DLL, but truth is defined as follows: T |- A is (DLLMtrue in M = iff whenever I(A,c) - 1 for every A e T, I(B,c) = 1 for some B e A, for every c e K. Adequacy Theorem for DLLM 2 . 4 < T A is a theorem of DLLM iff T ff A i.e. iff T 1- A is DLLM valid. Proof: Soundness enlarges upon earlier inductive arguments. Consider v-Adjunction. Suppose, for arbitrary a in K, I(A v B, a) = 1. Then I(A, a) = 1 or I(B, a) = 1, so verifying A v B (- A, B. To verify Cut, it suffices to show that when A || A v B and B & Y [f © then A & T || A v 6. Suppose then I(A, c) = 1 for every A in A u T; to show I(D, c) • 1 for some D in A u 0 . Since I(A, c) = 1 for every A in A, by the first premiss either I(D, c) • 1, so finishing the argument, or I(B,c) » 1. In the latter case since I(A, c) = 1 for every A e T, by the second premiss, I(D, c) * 1 for some D C 0 , completing the argument. The shift to multiple consequents simplifies the completeness proof; for a simple elaboration of ML theories can be applied. That is, there is no need to require that theories be prime and closed under adjunction. A theory c is* as for ML, a class of wff closed under J- , that is to say, in the case of multiple formulations, if every element of A is in c and A J— T then some element of T is in c, for every A and T. To establish completeness apply the Lindenbaum construction used for DLL; otherwise however the argument is as for ML and SLL. So it remains to establish the evaluation rule for disjunction, and for this it suffices to show, for arbitrary d in K , and arbitrary obs A and B, A e d or B e d iff A v B e d. Suppose A e d; then as {A} C_ d and A ( - A v B , A v B , by closure A V B e d Similarly when B e d . Suppose, conversely, A v B e d. Then {A v B} e d, but A v B j- A, B; so, by closure, either A e d or B e d. Corollary: T |- B is a theorem of DLLM iff it is a theorem of DLL. For r )- B is DLLM valid iff T |- B is DLL valid, applying definitons. Completeness results then yield the corollary. To formulate the |- v sublogics of DLL and DLLM simply delete the postulates for conjunction in each case. Both systems have the features, for which the Gentzen-sequents systems they resemble are much favoured, of separating the roles of the connectives in distinct postulates. Firstly, DLLM and DLL are readily recast as Gentzen-style systems. Gentzen reformulation GDLLM of DLLM is as follows:Axiom: A (- A Rules: B , r I- 0 r 0, A r h e, B A, r I 0 &

r f- ©,

v

R J-

A & B

Q, A

r|-©,

A V B

A &

r r

B, 0,

r (-

0

B

1- 9 , A V B

109

A & B,

A,

r

A V B ,

\-

r f0

A

0

B,

r |-

r |0

0

2.5 BOOLEAN

Augmentation

Cut

LOGIC, BL, ANV VE MORGAN LATTICE LOGIC, VML

r 1I- e r e, A

»

A,

H

e

r B,

B

>

B

R 1- A,

B

e

U© r (- e

1

>

r

h

6

Because sets are used in the formulation rather than sequences, structural rules other than Augmentation can be dispensed with. Also, as with Gentzen's systems, it can be shown that Cut is eliminable, i.e. a Cut elimination theorem is provable by the usual double induction. (Note that elimination does not follow directly from the semantics, since Cut is needed in proving DR3.) With Cut eliminated weak relevance of DLLM does follow by a simple inductive argument (the formalisation of inspection), since DLLM and GDLLM are equivalent systems in the sense of having the same theorems. §6'. Negation: De Morgan lattice logic and the parting of classical and relevant ways. Thus far classical, modal and relevant positions on deducibility ride along side by side. Other positions such as conceptivism and standard ccmnexivism have departed on their separate ways, but not intuitionist logic and not relevant logic. For DLL i^s a relevant logic. Not so Boolean logic that classicists would have us buy - and that includes all the motley band of modal logicians, for their differences from the classical position and among themselves only emerge when the first degree, involving truth-functions of deducibility, and the higher degree, involving iterations of deducibility, are reached,1 if they are. BL (Boolean Logic) results from DLL by the addition of a one place operation, subject to the following postulates:Axioms: A |- — A ; — A j- A (Double Negation) Rule; r , A f - B - * T , ~B (Rule Antilogism) It is easy to see that BL violates relevance: A, B {- A by, e.g. Tl, T2, Identity; A, ~A (- ~B by Rule Antilogism; and it is easy to trace the source of this violation: Rule Antilogism. The negation axioms do not interfere with the extension of the relevance lemma to encompass negation, but Antilogism does. For suppose, to extend the pseudo-inductive argument for DLL, T, A (- B has the sharing property. It does not follow that T, ~B |- ~A preserves it, for the sharing may be done by B and T with A irrelevant to B and T. Then ~A will be irrelevant to T u {~B}. The argument also reveals that it is not difficult to restore relevance.2 Delete parameter T; that is, restrict Rule Antilogism to: A |- B ~B (- ~A (Rule Contraposition) . Then if A and B share an ob, so do ~B and ~A, so relevance is preserved. DML (De Morgan Lattice Logic) results simply by replacing the classical rule of Antilogism by the relevance-preserving rule of Contraposition, i.e. Truth functional combinations of deducibility statements, such as the erroneous principle (A (| B) v (B [| A), are enough to separate modal (and more generally intensional) positions from the classical one. But the separation of the main modal positions only occurs with the introduction of higher degree statements. Relevance may be restored in more than one way. A different way from that adopted in the text - a way with some appeal - is the following: T, A |- B —* T, ~B |- ~A, provided A and B share a component.

110

2.6 SEMANTICS

FOR VE MORGAN LATTICE LOGIC VML: THE STAR

OPERATION

DML adds to DLL just Double Negation and Rule Contraposition. For ease in logical investigation of DLM, it pays to expose a little of its De Morgan character. T5. ~(A & B) ~A v ~B Pf.

~(~A v ~B)

—A,

by Addition, Rule Contraposition

~(~A v ~B)

A,

using Double Negation, Transitivity.

Similarly ~(~A v ~B) (- B Hence

~(~A v ~B) j- A & B,

by & - Composition.

The result then follows by Contraposition, Double Negation and Transitivity. T6.

~A & ~B (- ~(A v B).

Double Negation

Pf.

A j- ~(~A & ~B),

by Simplification, Contraposition, Double Negation.

Similarly B |- ~(~A & ~B) Hence A v B (- ~(~A & ~B), by v - Composition. The result then follows by Contraposition, etc. The converses of T5 and T6 of course hold also, as semantical adequacy will quickly show. Semantical analysis of DML, unlike that of BL, requires enrichment of ML models, in order that a correct negation rule can be semantically supplied. A DML model M is a structure M • where K and I are as before, i.e. respectively a set of worlds and an interpretation function, and * is an involutary operation on K, i.e. a* e K where and only where a e K, and a** • a. Though it is quite unnecessary to furnish further interpretational details for *, it can in fact be modelled in various ways, e.g. as a reversal operation, topologically as an inside-out operation, and so forth..1 To the rules extending I from initial obs to all obs, the following normal negation rule is added: I(~A, a) = 1 iff I(A, a*) * 1. A f- A is DML valid iff A |- A is true in every DML model, i.e. for every DML model M = and for every a e K, if I(B, a) = 1 for every B e A then I(A, a) = 1. Adequacy Theorem for DML 2.5. A |- A is a theorem of DML iff A |- A is valid. Proof extends the adequacy theorem for DDL. That the Double Negation axioms do not upset soundness follows from the modelling condition a** = a. ad Rule Contraposition. Suppose, for arbitrary DML model M, A |- B is true in M, i.e., for every a e K, if I(A, a) = 1 then I(B, a) = 1. Then, as * is an operation, for every a e K, if I(B, a*) ^ 1 then I(A, a*) ^ 1, that is, for every a e K, if I(~B, a) = 1 then I(~A, a) = 1, which is to say that ~B f- ~A is true in M. Completeness is also a matter of adding the occasional detail to the proof for DLL. In the canonical DML model M » , I is as before for elements of K , K is given a practical adjustment, and for a e K^, a* = {A: ~A i a}. The practical adjustment to the class K c of prime DML theories is that closure under |- is redefined to allow for the non-parametric form of Contraposition (as opposed to Antilogism). A theory c is closed under j- , in the new sense, iff whenever A e c and A (- B then B e e . (In the presence of Adjunction and Simplification the new sense is equivalent to the old, since compactness holds.) Then a* e K iff a e K c , since a* is a prime theory when and only when a is. For (i) a* is closed under |- . Suppose A e a* and A |- B. Then ~A I a and x

The interpretation and derivation of the star operation are much discussed in subsequent sections: see also NC. 111

2.6 AVEOUACV

FOR 8L; M L AS CONSERVATIVELY

EXTENDING

TAUTOLOGICAL

ENTAILMENT

~B |- ~A. Hence since a is closed under |- , ~B I a, i.e. B e a* as required. (ii) a* is closed under Adjunction, Suppose otherwise A, B e a* but (A & B) i a*. Then ~A I a, ~B i a but ~(A & B) e a. Since a is closed under f- and ~(A & B) (- ~A v ~B, by T5, ~A v ~B e a. But a is prime, so either ~A e a or ~B e b, which is impossible. (iii) a* is prime. Suppose otherwise A v B e a*, A e a*, B e a*. Then ~A e a, ~B e a, but ~(A v B) / a . Since a is closed under Adjunction, ~A & ~B e a, whence as ~A & ~B |—(AvB), by T6, ~(AVB) e a, which is impossible. Now suppose r B. There is, as before, by the Extension Lemma, a prime DML theory a extending T such that B { a. Hence since I(D,a) = 1 iff D e a for every a, I(D,a) = 1 for every D e T but I(B,a) i 1, that is f B is not valid. One new inductive step is required in the reduction of holding at a to membership, namely that for negation: I(~C,a) = 1

iff

I(C,a*) ^ 1, by the rule for ~

iff

C i a*, by induction hypothesis

iff ~C e a, by definition of a* Corollary 1. (Adequacy for Boolean Logic). A j- A is a theorem of BL iff A (- A is true in every DML model M = for which a* = a for a e K; hence the operation * can be eliminated, and the normal negation collapses into the classical one, I(~A,a) « 1 iff I(A,a) t 1 for a e K. Proof. Given a* = a, Antilogism is verified. For completeness observe that the proof of adequacy for DML works just as well if only non-degenerate (n.d.) prime theories are considered in the canonical model, that is prime theories that are neither null nor universal (i.e. contain every ob). Use of n.d. theories together with the following paradoxical BL theorems enables proof that a* = a, and so completes the corollary: T7.

A £. ~A j- B

T8.

B (• A v ~A

Pf.

B j- ~(A & ~A) B

~A v — A

~A v — A

|- A v ~A

,

from Rule Antilogism

,

by T7, Antilogism.

,

by T5, Transitivity.

,

from Double Negation,

v-Composition.

ad a £ a*. Suppose otherwise A e a and A d a * . Then ~A e a, so A & ~A whence, by T7, B e a for every ob B, contradicting the non-universality ad a* £ a. Since a is non-null some ob D e a. Hence, by T8, A v ~A e Thus, by primeness, A e a or ~A e a, i.e. A t a*. Since this holds for ob A, a* c a.

e a, of a. a. every

De Morgan Lattice Logic is nothing but the logic of tautological, i.e. first degree, entailment (of ABE, chapter 3) in another guise. Rewrite B, T |- C as B & G C where G is the conjunction of elements of T, and the logic of tautological entailment is forthcoming. Formation rules for system FDE of first degree entailment are obtained from those for DML by deleting the rule for |- , and substituting the following rule for -»•: If A and B are formal objects then A -»• B is a statement (wff). Alternatively FDE has as wff those wff of the full sentential logic of implication (of chapter 3) which are of the form A -»• B where A and B are zerodegree wff, i.e. contain no occurrences of connective i.e. have as their only connectives the extensional connectives &, v and The postulates of FDE can be chosen as follows: Axioms: A A A&B-+A A & B

+ B& A

A +

A V B

A V B -*• B v A

—A

A 112

2.6 THE INTERCONNECTIONS

Rules:

A

B, B

A

B, A -*• C -f A

D 6. A

C

BETWEEN WORLV MODELLINGS

A

AND MATRICES

ANV

LATTICES

C B & C

C, D & B - » - C - * D & (A v B)

C

A -*• ~ B - * B -*• ~A Equivalence Theorem 2.6. A -> B is a theorem of FDE iff A j- B is a theorem of DML. Hence too the semantical analysis of DML may be transferred to FDE'. In fact the semantical analysis given readily furnishes many of the main results concerning tautological entailments. To illustrate the power of the analysis decidability of DML and the Smiley matrices, which are characteristic for DML, are derived. Corollary 2. A f- A is a theorem of DML iff A f- A is true in every two-world DML model M • <{d,d*},*,I>, i.e. K - {d,d*} in every case (and worlds are accordingly dispensible). Proof. Soundness is automatic since truth in every model implies truth in every two-world model. For the converse observe that the extension lemma is only applied once in the completeness argument, thereby delivering d which then, together with its image d*, carries the remainder of the argument; that is, a canonical model M with K = {d,d*} suffices for the argument. Corollary 3. DML is decidable.C For corollary 2 has delivered an effectively specified finite model in which each non-theorem can be refuted. The class of interpretations on a two-world model structure <{d,d*},*>, or {d,d*} for short, can be reexpressed as a class of assignments in terms of four-valued matrices. To avoid repetition in later sections, we first set out, in a more general and systematic way than is required merely for system DML, the interconnections between world modellings on the one hand, and matrices and lattices on the other. A model M can be resolved into a pair <S,I> (or <S,v>) where S which comprises every component of M except I is a model structure (m.s.) and I (or v) is an interpretation (or valuation) on m.s. S. In the case of B models (of 4.4) S = with R a 3-place relation, provides the model structure; in the case of I models (of 7) S = , with R a 2 place relation and N (like 0) a subset of K, is the m.s.; in the case of DML models or, as we sometimes prefer to say since the introduction of base T makes no important difference, is the relevant model structure. The characteristic m.s. for DML, and DLL, <{d,d*},*>, supplied by corollary 2 is called the standard DML m.s. Consider now the class CM of models obtained by taking all two-valued valuations I on model structure S on set K. Following Kripke (63a,p.92), define a matrix value as a mapping whose domain is K and whose range is the set 2 = {1,0}, i.e. as a function in 2^. Accordingly a matrix-value is tantamount to a subset of K; for each matrix value determines that unique subset of K which it maps into 1 and conversely each subset of K specifies a mapping. Furthermore valuation I maps sentential variables into matrix values; for, given variable p, I assigns as matrix-value {a:I(p,a) = l}. Thus the class CM of models provides a matrix. Where p and O are matrix values, define matrix operations 'v' and as follows: (~p)(a) = 1

iff

p(a*) + 1;

(p&a)(a) = 1

iff

p(a) = 1 = a(a);

(pva)

iff

p(a) = 1 or a(a) = 1;

(a)

= 1

and given a B model, (p-KJ) (a) • 1 iff for every b and c in K, if Rabc and p(b) = 1 then a(c) = 1, while given an I model, (p-*0) (a) = 1 iff a e N and,

113

2.6 THE SMILEV MATRICES ARE CHARACTERISTIC FOR DML AND FPE

for every a', if aRa* and p(a') = 1 then a(a') = 1. For DML and first degree models this latter condition reduces, for requisite a, to (p |- 0)(a) = 1 iff, for every a 1 , p(a') | 1 or a(a') = 1. It follows: ~p = {a:p(a*) = 0} and p&O = p 1 ^, and pvct = pUcr, where n and U are set or lattice operations. Finally, in models where truth is evaluated in T, p is a designated value iff p(T) = 1. For standard DML models p is designated iff p(d) = 1 = p(d*). Where S is an m.s. the matrix (or algebra) S + = <M,D,~,&,v,-*> on S is defined as follows: M is the class of matrix values, D is the class of designated matrix values, and &, v, are the class-closing matrix operations defined above. Then S + is a matrix in the standard sense. Lemma 2.4. Where S + is the matrix on m.s. S, and where truth is evaluated at T. S + satisfies A iff, for every I on S, I(A,T) = 1, for every wff A. Where S + is defined on standard DML model S, S + satisfies P iff, for every I on S, I(P,d) = 1 = l(P,d*), for every statement P, i.e. P is of the form r f- A, (Here P, i.e. T (- A,is validated by I at d, written I(P,d) = 1, iff, whenever each element of T holds on I at d, A also holds on I at d.) Proof. Let A + (p + ,... ,p + ) be the matrix function of p"!",...,p+ corresponding to i "n i n wff A(p j,.. p ) under the replacement of connectives by corresponding matrix operations, where I maps p

to p

Let A + be the matrix value of A. tives (or operations),

...,pn to p*. Thus p* = {a: I (p.,,a) « l}. n 'i " 'i] Then, it follows by induction over connec-

(*) a e A + iff I(A,a) = 1. Applying (*), I(A,T) = 1 iff T € A + , i.e. iff A + is designated, i.e. iff for the given assignment S + satisfies A. Generalising, S + satisfies A iff for every assignment I on S,I(A,T) = 1. Similarly I(P,d) = 1 = I(P,d*) iff d e P + & d* e P + , i.e. iff P + is designated; and hence S + satisfies P for DML statements iff for every I, I(P,d) = 1 = I(P,d*). Theorem 2.7. T (- A is a theorem of DML iff the Smiley matrices,

*1

1

2

*1

2

2

2

2

3

3 4

4 4

3 4

4

4 4 1

4 4 4

1

1

satisfy T j- A; i.e. these matrices are characteristic for DML and for first degree entailments. Proof. Let P abbreviate r (- A. P is provable iff P is DML valid. Define matrix values as follows: 1 = {d,d*} = K; 2 - {d}; 3 = {d*}; 4 = A. Then, with matrix operations defined as before for DML models, the Smiley matrices result. The argument is by cases for each operation, ad As matrix value 1 maps both d and d* to 1, ~1 maps neither to 1, i.e. ~1 = 4 ; similarly as 4 maps neither to 1, ~ 4 maps both d* and d = d** to 1, i.e. ~ 4 = 1 . As 2 maps just d to 1, ~2 maps just d* to 0 , i.e. ~2 maps just d to 1, so ~2 = 2; similarly ~3 = 3. ad &. As p&0 = fAJ, the matrix values in the & matrix are given by the intersection of the values of p and a . Hence 4 & 0 = a&4 • 4 , since the intersection of A with any set is A . Hence also lHa = ani = a. Finally 3^2 = 2H3 = {d}n{d*} = A , so 2&3 = 3&2 = 4 ; and ana = a. so 2&2 = 2 and 3&3 = 3. ad |- . By the evaluation rule, (pj-cr)(d) = 1 iff both p(d) = 0 or 0(d) « 1 and p ( d * ) = 0 or a(d) = 1; similarly for (p|- = (p|-a)(d*), p | - 0 = l o r = 4 ; for d and d* are always treated the same. For this reason too it is enough to look at mappings at d. If p » o 114

2.6 THE RELEVANCE

LEMMA TOR DML, AND ITS

COROLLARIES

then the right hand side of the evaluation rule is a tautology, so (p|-a)(d) ® 1. Hence when p = a , p -»• a = 1. Similarly when p = 4 or a = 1, p -»- a = 1; for, to illustrate, if p = 4 then p(d) = p(d*) = 0 so the right hand side is tautologous. In every other case, p-KJ = 4. The argument is cases-ridden. If a = 2, a(d) = 1 and a(d*) 4 1: so ( p f - a ) (d) = 1 iff p(d*) 4 1But when and p = 1 or = 3, p(d*) = 1, so (p|-a)(d) 4 » by earlier argument, p ( - a = 4. The case is analogous when O = 3, giving pf-CT = 4 when p = 1 or p = 2. Finally when O = 4, a(d) + 1 + c*(d*), so (p[-a)(d) - 1 iff p(d) 4 1 Xt also 4 p(d*) , i.e. iff p = 4. Thus for p 4 4, p b a = follows that 1 is the only designated value. Applying the previous lemma, P is DML valid iff it holds at d and d* for every valuation function on K = {d,d*}, i.e. iff the derived Smiley matrices satisfy P. Corollary 1. The disjunction matrix for DML, determined through the definitional connection A v B ~(~A & ~B), is just the Wajsberg matrix for disjunction (of §5). Proof is by direct computation. Alternatively the matrix can be simply derived, like that for &, using the equation pva = pUa. Thus the Smiley matrices, characteristic for DML, are the Wajsberg matrices for DLL together with a distinct matrix for negation. (In the Group III matrices negation is given by the Boolean operation 1 2 3 4 .)

Corollary 2 (Relevance Lemma). T |- B is a theorem of DML only if T and B share an initial ob. Proof, (i) r b B is a theorem iff its normal form C b is a theorem, where C and B' are obs, which may contain disjunction and which contain negation only as applied at most once to initial obs, and C and B' result respectively from the conjunction of obs of T and from B' by repeated application of the following operations (which drive negations down to initial obs): replace ~(C&D) by ~C v ~D, ~(C v D) by ~C & ~D, — C by C. The deductive equivalence of T b B a n d C b B ' will follow either using syntactical results already established, or from the semantics given. For the latter argument, observe that I(~(C & D), a) = I(~C v ~D,a), so the replacements can be made without affecting validity. (ii) T and B share an initial ob iff C and B' share an initial ob. For reduction to normal form neither introduces nor eliminates initial obs. (A detailed proof is by induction.) By (i), (ii) and the theorem it suffices to show that, where C and B' do not share an initial ob, C b takes an undesignated value. To show this make assignments to initial obs - each of which is either negated or unnegated - as follows:- Assign every unnegated ob in B' one of the values 3 or 4 and every negated ob in B' one of the values 1 and 2 and assign every unnegated ob in C one of the values 1 or 2 and every negated ob in C one of the values 3 or 4. Because C and B' share no initial obs such an assignment is legitimate. Then, by induction, B* takes one of the values 3 and 4 and C takes either value 1 or 2; accordingly C' b ® undesignated value 4. The relevance lemma can also be established by purely semantical arguments (for details see FD). Two minor corollaries of the corollary are worth recording. Corollary 3. Antilogism is not an admissible (derived) rule of DML. Corollary 4. The Cut rule of DML may be equivalently replaced by the Relevant Cut rule: A b B B, r b C - • A , r b C, provided that in the conclusion C and AUr share an initial ob.

115

1.6 SEMANTICAL DERIVATION

OF THE FUNDAMENTAL DE MORGAN LATTICE DQ

The semantics not only provides an interpretation of the Smiley matrices; the standard DML m.s. {d,d*} also generates the fundamental De Morgan lattice with diagram, x

and with N defined: N1 • A, N4 - 1, N2 - 2 and N3 - 3. As well as a matrix t m m . s . S a De Morgan lattice L(S) - <M»ri,u»N,<> may be defined on S - and L(K) * L(S) defined on K - as follows: M « - <M,N,<>. Theorem 2.8. Where S is a standard DML m.s. K = {d, d*}, L(S) = L(K) = D Q . Proof. The lattice diagram of L(S) is immediate from inclusion relations aaong the elements of M = {l, 2, 3, 4}; and the properties of N follow issmediately from the definition of N in L(S). The De Morgan lattice L(S) on S may, of course, be read off the matrix S + 5 by replacing mappings by their extensions, the subclass of K they map to operations &, v on mappings by operations N, U on their extensions. Conversely matrix operations &, v may be read off (or identified with) corresponding De Morgan lattice operations. The relevance lemma emerges again using D^:

for, where T and B share no

ot, simply assign each ob in T element 2 and each ob in B 3.

Then T J1- B.

As in the case of DLL both systemic formulation and many results are simplified by moving to a multiple formulation. Multiple system DMLM adds to DLLM Double Negation axioms and Rule Contraposition. Once again the completeness argument for the multiple system avoids some of the complexities of primeness.1 Normal and classical negations are by no means the only negations, i.e. the only connectives that count as negations. But before we turn, firstly, to an examination of other negation notions, and, secondly, to the issue as to whether, and if so how, there can be so many different negations, we make a diversion to link the logics we have developed, especially BL and DML, with familiar algebras. Though the multiple formulation of Boolean logic BLM is readily recast in Gentzen form - simply add to GDLLM the negation rules

~

r j- e,

A,

A

r |- e

~A, r 1- e r (- © , ~A obtaining a comparable formulation of DMLM is not nearly so straightforward. It can be done however by introducing paired rules, so that to each rule corresponds another starred rule: compare the method of tableaux introduced in chapters 3 and 7.

116

=

2.7 EXPLODING

MYTHS CONCERNING

THE FORMALISATION

OF ALGEBRA

§7. On the axioma.tisatt.on and semantic analysis of algebra. So far several familiar algebras have been presented as logics, specifically as formal systems in their own right, and these logics have been furnished with semantical analyses within the framework of the deducibility relation. The formulation of Boolean logic and DML may look rather different, however, from what is ordinarily understood by Boolean algebra and by De Morgan lattice theory. Taking Boolean logic as our working example, we shall try to show that there are formally no substantial differences between Boolean logic as formulated above and Boolean algebra cast as a formal system. As part of the enterprise we shall explode some commonly held views about Boolean algebra and its relation with two-valued classical logic, in particular the thesis that Boolean algebra cannot be presented as a formal or deductive system in its own right. The assumption is that Boolean algebra has to take for granted in its development 'the logic of propositions', that it cannot be developed independently. For example C.I. Lewis (60, p.281) argues that to start with Boolean algebra commits us to the paradox that the system takes the laws of the logical relations of propositions for granted in order to prove them. In similar vein Whitehead and Russell claim (PM, p,90) ... in the theory of classes we deduce one proposition from another by means of principles belonging to the theory of propositions, whereas in the theory of propositions we nowhere require the theory of classes. Hence, in a deductive system, the theory of propositions necessarily precedes the theory cf classes. This position, commonly advanced in the early decades of the century, is not simply some archaic meanderings that modern moderns have seen their way around. It is reiterated in recent literature, McCall, for instance, writing on the completeness of Boolean algebra, asserts (67, p.367): In order to obtain completeness results, however, the theory of deduction implicit here must be fully formalised, and we shall take as our basis the two-valued system of propositional logic (rather than, say, intuitionist logic). While it is true that a theory of deduction, or derivation, of some sort is required for the formalisation of Boolean logic, none of the connectives of standard logics are required. The reason is that Boolean logic only involves the first-degree entailment fragment of these standard logics, only expressions of the form T |- B. And these can be assembled in a system without any assistance from standard logics, as we have seen by way of the Boolean logic already presented. To clinch the point, let us formalise Boolean algebra as conventionally presented. To do so it is enough to formalise the underlying equational theory and add the Boolean equations. As has been our practice we shall not specify the objects: they can be such items as the familiar x, y, z, etc. As before, if A and B are obs so is their intersection (A & B) and complement ~A. The union (A v B) of A and B is defined: A v B = ~(~A & ~B). Where A and B are obs, their equation A 5 B is a statement (wrr). The equation A « B may be read: A is (as regards content) the same ob as B, A is interderivable with B, and so forth. That specifies completely the morphology

117

2.7 THE EQUAT10NAL

FORMALISATJON

OF BOOLEAN AND OTHER

ALGEBRAS

of system BA of Boolean algebra. In setting up the primitive frame of BA the so-called initial postulates, introduced by Huntington 04 in his classical exposition and followed by later mathematicians, have been replaced, by what appears far more adequate since set-theoretic notions are avoided, formation rules. Thus instead of postulating a class K of elements x, y, z ..., relations u, n, etc., such that (1) If x and y are elements of K then x u y is an element of K, and so on, the more modern, exact, method of formal systems has been adopted. New symbols (or objects) are introduced as primitive symbols (or initial objects) and the postulates are replaced by formation (or structural) rules which define 'term' (or 'ob') and 'wff* (or 'statement'). For a rather similar improvement compare modern presentations of propositional logic with presentations up to and including the time of Principia Mathematica. BA has the following postulates undesirable): Axiom schemes: Bl. A & ~(B & ~B) s A B3. A & (B & C) S (A & B) & C B5. A 3 —A Rules: A s B , C s B - * > A s C A-sB-fC&AsB&C As B ~A 5 ~B

(the first two of which are thoroughly

B2. B4.

A & ~A s B & ~B A & (B v C) 5 (A & B) v (A & C)

(Transitivity, i.e. Rule Syllogism) (Rule Factor) (Rule Contraposition)

In virtue of B2, a (minimum) element 0 can be legitimately defined: A & ~A.* Correspondingly a (maximal) element 1 is defined 1 = ~0.

Df The connection with the earlier Boolean logic is made through the definitions: A |- B = , A & B a A, and, in the other direction, A s B = f A (- B and B |- A. Using the definitions Bl and B2 become respectively the more familiar A & 1 9 A and A & ~A s 0. In the development of enough of the logical theory to link BA with other formulations of Boolean algebra and with Boolean logic, let us begin with a derived rule: DR1. A « B D(A) s D(B) , where D(A) is any statement containing A and D(B) results from D(A) by replacing A by B in one or more places (Substitutivity, or Replacement). Proof is by induction on the structure of D(A), and uses the rule structure of BA, and relies on the next four results:A ^ A Tl. Proof A s — A , A s — A , so by Transitivity A s A DR2. A a? B -o B s A A s A, C 5 A -c A s C, by Transitivity. A o C ; then reletter, Hence C e A By DR2 using Transitivity. DR3. A s B , B s C - f r A s C . By Tl and Factor. B & A s A & B. T2. A v 0 5 A T3. Bl. ~A & ~0 S ~A Proof by Contraposition, ~(~A & ~0) s ~ by B5, Transitivity. A v 0 5 A A v ~A 5 1 T4. B2, definition of 0. Proof: A & ~A 3 0 ~A & — A s 0 relettering. Such a definition is admissible only because the role of A & ~A is appropriately independent of A, a point guaranteed by B2 in combination with DRl below. Introduction of 0 provides, in effect, a conservative extension of BA by a new constant; but use of this new constant is always eliminable.

118

2.7 THE EQUIVALENCE

OF FORMALISEV

BOOLEAN ALGEBRA AND SINGULAR

BOOLEAN

LOGIC

Contraposition, ~A & — A ) 5 ~0 v ~A 5 1 by definitions. & A s A T5. & A & 1 by BI. Proof: 5 A & (A v ~A) by T4, DR1. s (A & A) v (A & ~A) by B4, Transitivity. s (A & A) v o definiton of 0. s A & A by T3, Transitivity. Then use DR2. T6. 6 0s 0 Proof, & 0 5 A & (A & ~A) S (A & A) & ~A by B3, using Transitivity. by T5, DR1. 5 A & ~A 3 0 DR4. & ~B s <> A & B & A Proof ~B s 0 -> (A & ~B) v (A & B) s 0 v (A & B), by Tl, DR1 -p-A & (~B v B) 6 A & B, by B4, BI -t> A 6. 1 s A & B, by T4, DR1 -c A 3 A & B, by BI, Transitivity. Conversely, A & B e A -t> (A & B) & ~B 5 A & ~B, by DR1 -t>A & (B & ~B) 5 A & ~B, by B3 -p A & 0 s A & ~B, by definition of 0 -o 0 5 A & ~B, by T6, Transitivity. DR4, together with earlier results, namely T2 and B3, and the identity theory, gives a standard formulation of Boolean algebra (see, e.g. Mendelson 64, p.10). The only difference - and this is the sole strategy needed for the independent formalisation of algebra - is that conditionals are replaced by derivability relations, and biconditionals (iff) by interderivability relations. DR4 also gives a simple key to the syntactical proof that Boolean logic (with singular antecedents) is contained in BA. For, by DR4, A |- B<w>A & ~B 5 0. Hence T7. A [• A T8. A & B 1- A, etc. All the remaining axiom schemes of singular Boolean logic are likewise easily forthcoming, and so, to complete the inductive argument for containment, are the rules. The case of Antilogism will illustrate: DR5. A & B |- C -o A & ~C |- ~B Proof: A & B f- C -o (A & B) & ~C s 0, by DR4 —o (A & ~C) & — B 5 0, by B5, DR1 -c-A & ~C (- ~B, Conversely it is readily shown syntactically using the definition of s in terms of |- , that every theorem of BA is a theorem of Boolean logic. So theoremwise formalised Boolean algebra just is singular Boolean logic. It is a consequence, applying results from the previous section,that BA has a situational semantical analysis and that BA is decidable. The situational semantics is not of course the only semantics that BA has. In a sense every exact representation theorem for a formalised algebra delivers a semantics for the algebra. For example, the familiar representation of Boolean algebra in the theory of classes leads to such a semantics. A Stone model M for BA is a structure <Si, v> where ft is some non-null class1 and v is an assignment function which assigns to each initial object a subset of ft, i.e. in effect v(A) e 2^ for initial A. Assignment v is extended (footnote

1

on next page)

119

2.7 ALTERNATIVE SEMANTICS

FOR BA, ANV EXTENDING

THE DEGREE OF BA

to all obs by the rules: 1. v(A & B) = v(A) n v(B) 2. v(~A) = -v(A), i.e. the completement of the value of A. Hence v(l) = ft and v(0) = A • -fl Further A s B is true in M iff v(A) = v(B), and A g B i s Stone-valid iff it is true in all Stone models. An inductive argument shows that every theorem of BA is Stone-valid, and Stone's representation theorem or the situational semantics can be applied to show that non-theorems are not valid. Narrow Boolean algebra BA is a rather limited logic with quite limited expressive power. There are two ways of breaking out of its confines, both tried, which combined in the orthodox fashion result in classical two-valued logic S. The first way is to extend BA to the first degree by adding the connectives and, or, and not. These may be the operations &, v, ~ of BA assigned an extended role (as in §10), or they may be different operations (for if v were union, say, it would not make sense to assert A v B • 1 v. A v B = 0). The second way (implicit in Lewis's discussion in 60) is to permit the iteration of e, i.e. to extend to the higher degree, with such expressions as (A € A) e l . Such an extension naturally restricts the interpretation of terms to one of a sentential or propositional order, and leads really to a sentential logic with an equivalential base. It is a straightforward matter to apply the axiomatic and semantical methods adopted in revamping Boolean algebra to other algebras, e.g. to lattice theories, topological algebra, and so on. Rather than doing so, we return from our diversion to the development, and to the issue of apparently rival negations. §8. Systems with non-normal negations and disjunctions: orthologics and relevant quantum logicss rival negations and rival connectives and the determinable theory of connectives such as negation. Take a system with a nortnal conjunction, i.e. a system including SLL, add a normal negation and a notmal disjunction will result, given the standard definition of disjunction: A v B ~(~A & ~B). For semantically (there is a parallel syntactical argument): I
1

iff I(~(~A & ~B),a) = 1 iff I(~A & ~B, a*) t 1

The restriction to a non-null domain is desirable in order that set operations such as complement are well-defined. And the restriction in a fair sense rules out nothing, since 1-valued Boolean algebra is trivial; every equation is a theorem. In 1-valued Boolean algebra ft s A , i.e. v(l) = v(0); thus syntactically 1-valued Boolean algebra results by adding the axiom: 0 s 1. Then firstly, A s 1 for every A. For A s A v 0 , by T3 S A v 1 , by 0 = 1 and DR1 o 1 , from T6, Contraposition Hence, secondly A © B, for arbitrary A and B; for A © 1 s B. Note that BA does not provide the apparatus to state that it is not 1-valued, or for that matter that is has 2 or some other number of values. The usual 2-valued assumption A s 0 v A 6 1 is a first degree statement not expressible in BA.

120

2.8 ALTERNATIVE

INTUITIONISTIC

iff iff iff

AND DIALECTICAL

NEGATION

THEORIES

AND

NEGATION

I(~A, a*) i 1 or I(~B, a*) * 1 I(A, a**) » 1 or I(B, a**) - 1 I(A,a) = 1 or I(B,a) - 1.

Hence a non-normal disjunction implies (on weak assumptions) a non-normal negation. This was our excuse for postponing the treatment of non-normal disjunctions until negation was introduced. The converse, that a normal disjunction leads to a normal negation, would require much stronger assumptions - assumptions moreover which are countered by minimal and intuitionistic style logics which combine normal disjunctions with non-normal negations (in part by the expedient of dropping the definitional connection A v B ~(~A & ~B)). There are systems of topicality or interest which separately break each postulate on normal negation: here too nothing is sacred. Observe that when DML is obtained by adding negation directly to SLL, i.e. the route through disjunction is short circuited, the postulates on normal negation are these: A f- — A

(Intuitionistic DN)

—A

A (- B

~B |- ~A

(Rule Contraposition)

A & (B v C) j- (A & B) v C

j- A

(Dialectic DN)

(Distribution, with v defined as usual)

By using stretched models, which (as explained in §9) include a partial order < on worlds, and simply reducing the conditions on * a first group of non-normal negation systems can be captured semantically: 1. Intuitionistic System K weakens the involution requirement on *, namely a = a**, to a < a**. System K, which is related to the intuitionistic and minimal systems IK and JK studied below (§6.3), retains all the normal negation postulates except Dialectical Double Negation; in particular it retains A |- — A . 2. Dialectical system M weakens involution to a** < a. M, unlike, K, invalidates A j - ~ ~ A ; i . e . it keeps Dialectical DN, but rejects Intuitionistic DN. 3. A non-cotttraposible logic which results upon dropping * closure, i.e. a* e K when a e K. That is, * is an operation not on K but on a superset of K. The logic deletes just Rule Contraposition among the normal negation postulates. (Essentially this method, of introducing a perhaps new set of starred worlds to which implicational evaluation does not stretch was exploited in Routley 74 in furnishing semantics for Fitch and Nelson systems.) But the better promoted non-normal negation systems are not so readily obtainable merely by variation of the * conditions, or indeed so obtainable at all. For example, a negation rule of the normal form, I(~A,a) = 1 iff I(A,a*) ^ 1, cannot succeed to model systems which reject Distribution whatever conditions are imposed on *. For within the framework of SLL semantics the rule is bound to validate Distribution. Suppose otherwise that, in some model, I(A,a) - 1 = I(~(~B & ~C),a) but I(~(~(A & B) & ~C),a) t 1. I(~(A & B),a*) = 1 - I(~C,a*), so I(A & B,a) = 0 = I(C,a). Hence I(B,a) - 0. I(~B & ~C,a*) = 0, so I(~B,a*) = 0 or I(~C,a*) - 0, i.e. I(B,a) = 1 or I(C,a) = 1 , which is impossible. However many systems which satisfy Rule Contraposition - indeed every such system if sufficient semantical trickery is permitted (as ER reveals) - can be captured semantically through the relational negation rule now familiar from intuitionistic logic, namely:-

Alternatively, given the models of §9, these logics can be modelled by struct' ures which reject principle m

2.S BASIC CONTRAPOSITION

LOGIC, BCL:

SYNTAX, SEMANTICS,

AND

EXTENSIONS

I(nA,a) = 1 iff, for every b such that Rab, I(A,b) j 1 (weak negation rule: rule). Different systems result upon imposing different conditions on the two-place relation R on K. Among the systems includes are the deducibility parts of intuition and minimal systems, orthologic and Birkhoff-von Neumann quantum logic. Excluded are systems such as Parry logics which reject Rule Contraposition. Basic Contraposition Logic (BCL) adds to SLL - reformulated to include a negation operation -> - the sole postulate of Rule Contraposition. A BCL model is a structure , where K is a non-null set, R is a 2-place relation on K, and I is an interpretation function which satisfies the normal & rule and the -i rule. Thus operation * of structures is generalised to (what it could have been all along) an accessibility (or barrier) relation R.1 A BCL model imposes no conditions on R. Adequacy Theorem for BCL. F j- A in BCL iff T j- A is BCL valid. Proof extends the corresponding theorem for SLL. Ad Rule Contraposition. Suppose A |- B is valid, and I(-iB,a) = 1 ; to show I(-iA,a) = 1. As I(-iB,c) = 1 for arbitrary b e K, such that Rab, I(B,b) ^ 1. Since A \- B is valid, I(A,b) # 1. Hence I(-*A,a) = 1. For completeness, canonical relation R c is defined as follows for a, b e K £ :

R

ab c

f°r every ob A, if TA e a then A i b.

To extend the

argument for SLL it suffices to show that I(-iA,a) = 1 iff TA e a, and for this it is enough to prove, given the inductive hypothesis: iA e a iff, for every b, if R ab then materially A i b. If -iA e a then it is immediate, using the definition of R c , that A i b. the converse theory (i.e. and since |Identity A £

For

suppose -iA I a. Define b = {C: A \- C}. Then b is a BCL an SLL theory). Since |- is transitive, b is closed under |-; satisfies &-Composition, b is closed under Adjunction. By b. It remains to show R c ab. Suppose otherwise that for some

C, iC e a and C e b. Since C e b, A |- C, so by Contraposition iC |- -»A. But then as a is a theory and -«C e a, -»A e a, contradicting nA I a. To obtain the harder-promoted irrelevant non-normal negation systems, we begin on (what will become) the familiar business of ringing the changes on the basic system by adding postulates and determining corresponding modelling conditions. Some of the variations are recorded in the following table (Bad Eggs are marked with a dagger: completeness arguments for them require further restrictions on theories - to rule out analytic, i.e. degenerate, theories). Axiom scheme Corresponding modelling condition 1. t A & -»A j- B R is reflexive. The good scheme corresponding to this Bad Egg requires the introduction of t (of §2), and the corresponding addition of 0 to the model structure. The good scheme is: 2. t -.(A & -nA) If Oa & Rab then Rbb. 3. A }- -v-»A R is symmetric 1

Though relation R is now commonly viewed, at least in explanations of model logic semantics, as an accessibility relation, facilitating qualified access of one sort or another from one world to another (cf. Hughes and Cresswell 68), it may be alternatively linked with other interconnections or disconnections of or within worlds. One such interpretation, of interest in connection with semantic tableaux (as Michael McRobbie pointed out to us) is of R as tied with barriers which impede progress from one part of a world to another: as a simple example, there appears to be a truth barrier between assertions in USA and in USSR. IZZ

2.S PROOFS OF ADEQUACY

4. n A

FOR EXTENSIONS

OF BCL, SUCH AS VARIOUS

|- A

ORTHOLOGICS

For every subset Y £ K, if a e Y then for some b, Rab and for every c such that Rbc, c e Y.

An adequacy theorem covers these extensions1 of BCL. Proofs of adequacy are as follows:ad A & -iA |- B. Suppose I(A,a) = 1 = I(-»A,a). Then since Raa, I(A,a) ^ 1 , contradicting I(A,a) = 1 . Thus I(B,a) = 1 strictly follows, by a paradox. ad Raa. It has to be assumed that situations are not universal. Let B be some ob not in a. Suppose, contrary to Raa that for some ob A, iA e a and A e a. Then A & -iA e a, whence B e a, contradicting assumptions, ad t |- -j(A & nA). Suppose I(t,a) = 1 but I(-«(A & -»A) ,a) j 1. Then Oa and for some b, Rab and I(A,b) = 1 = I(-»A,b). By the last since Oa & Rab implies Rbb, I(A,b) 4 1, a contradiction. ad Oa & Rab materially implies Rbb. Suppose Oa, Rab and iA e b; to show A I b. Since t e a , -I(A & -iA) e a. Hence by Rab, A & TA I b; so TA I b or A I b. By elimination A i b. ad A \- i-»A. Suppose I(-»-»A,a) ^ 1; to show I (A, a) ^ 1. Then for some a, Rab and I(-iB,b) = 1. By symmetry, Rba, so I(B,a) i 1. ad material symmetry, i.e. that Rab materially implies Rba. Suppose Rab, -iA e b and A e a. Then by A |- -i-iA, -nA e a; hence by Rab, TA I b, a contradiction. ad -I-IA j- A. The modelling condition: (DDN) I(B,a) i 1 > (Pb)(Rab & (c)(Rbc > I(B,c) ^ 1) for every ob B and every a £ K, is simply contrived to handle the axiom. One can jump through the detailing hoops as follows:- Suppose I(A,a) ^ 1 = I(-NA, a). Then, as for some b, Rab, etc., by I(A,a) + 1, so by I(-I-IA, a) = 1, I(-»A,b) i 1. But this implies that for some c, Rbc and I step: If I(-iC,a) ^ 1 then, by the interpretation rule for for some b such that Rab, I(C,b) = 1. Now suppose, for arbitrary c, Rbc. By symmetry Rcb, and hence, as I(C,b) = 1, by the -i rule again I(-iC,c) ^ 1, as required. (DNN) may be converted into an R-closure principle (DNN+) For Y £ K, a e Y >. (Pb) (Rab & (c) (Rbc > c e Y)), using the facts that a f {b: I(B,b) ^ 1} iff I(B,a) ^ 1 and {b: I(B,a) ^ 1} £ K. Through the adequacy theorem semantical analyses of various orthologics and minimal logics and intuitionistic logics are furnished; and the results can be extended - though in a less than satisfactory fashion, so it appears to modern quantum logics such as that of Birkhoff and von Neumann (for one such way of extending irrelevant orthologic, see Goldblatt 74). Since all 1

Evidently other extensions could be included. There are however several gaps in our knowledge of extensions; for example:i) What postulates, if any, given the syntactical paucity of BCL, correspond to such conditions as transitivity of R? ii) What "natural" modelling conditions, if any, correspond to such quantum mechanical schemes as t A & (-iA v (A & B)) |- B C )- A - * A & (B v C) |- (A & B) v C, with A v B -i(-iA & -»B)?

m

2.8 ORTHOLOGIC,

OL: ITS SYNTAX,

VARIOUS SEMANTICS,

ANV CONSPICUOUS

DEFECTS

these logics are commonly regarded as competitors to relevant logics, there is point in exhibiting the more important of these logics and exposing some of their defects as explications of implication. Orthologic, OL, is a first degree deducibility logic which has the following postulates: A

A

A |- B, B j- C - * A |- C

A & B |- A

A & B

B

A, B )- A & C

-i-.A |- A

A |- T *

A j- B - » -IB |- -TK

t A & TA (- B (or equivalently:

B

Here &,

n(A & nA))

and j*- are taken as primitive and v defined:

A v B = ^ -I(TA & -»B) .

(Alternatively v may be taken as primitive; the postulates differ from those given for DLL only in the requirement of singularity on the left.) An OL model is a BCL model such that R is reflexive and symmetric1 and conforms to the R-closure condition, i.e. (DNN^-) . By the adequacy theorem, OL theoremhood coincides with validity in all OM models. A pleasing linkage with the jargon of quantum mechanics can be forged by supplanting R by its complement, the orthogonality relation, x on K (i.e. x x y iff Ryx), subject to the derived conditions of irreflexivity and symmetry. In the terminology of Goldblatt 74, which we now follow, S = is called an orthoframe, or orthogonality space, with K the carrier of S. R-closure is replaced by x-closure: a subset Y of K is l-closed iff for every a e K, if a i Y then, for some B e K, b 1 Y but not x x y, where b l Y iff, for every c e Y, b x c. Then an orthomodel is a structure such that is an orthoframe and I' is an interpretation function such that for every initial ob A, I'(A) is a x-closed subset of K. Then, as a matter of definition, I(A,a) = 1 iff a e I'(A). I is extended recursively to all obs as before except that x supplants R. Alternatively I' can be neatly extended using the clauses: I*(A & B) = I'(A)n I'(B) and I'(TA) = {a e K: a x I'(A)}, or these connections derived upon setting I'(A) = {a: I(A,a) = l}. It follows by induction on obs that I'(A) is x-closed. It is a corollary of the adequacy theorem that T |- A in OL iff r j- A is true in all orthomodels. Orthologic OL has two conspicuous defects; firstly, it violates relevance requirements through the inclusion of the final postulate (marked with a Bad Egg sign); and secondly it excludes Distribution, in any form. A relevant orthologic ROL is easily designed, simply by dropping the final postulate: but rectifying the resulting systems so that Distribution is included leads back to system DML. A relevant orthologic which corresponds more closely to ordinary irrelevant orthologic replaces the final postulate which engenders irrelevance by the postulate t (- A v -|A (i.e. extra scheme 2 above), thereby exploiting the sentential constant t already introduced. l It can be seen from these conditions that OL model structures (i.e. models less the interpretation function) are a special case of the model structures which model the Brouwerian system. For details of an exact translation between the systems, see Goldblatt 74.

2.8 QUANTUM

LOGIC AND MODELS, AND RELEVANT

QUANTUM

LOGIC

MODELLED

In the quantum logic of Birkhoff and von Neumann the need for some form of distribution is recognised, and what they regard as the quantum mechanically correct form of the principle, the orthomodular form, is added to orthologic, namely C | - A - * A & ( B v C ) |- (A & B) v C. Within the framework of irrelevant orthologic OL this qualified distribution principle reduces to the slightly more tractable principle t A & ^ A v (A & B)) B (Quantum Disjunctive Syllogism). To derive the latter, observe that the strengthening of B to (A & B) in the antecedent allows distribution to be applied as A & B j- A, whereas of course B |- A commonly fails. Then, distributing, A & (nA v (A & B)) A & TA v. A & B, whence B follows by + A & -iA B, Simplification and v-Composition. The quantum logic QL results by adding A & (TA V ( A & B ) ) |- B to orthologic OL: thus QL is, like OL, irrelevant. One way of obtaining a semantics for QL (a way which however transcends first order conditions) is, as in Goldblatt 74, to add to OL models a further element V, where V is a certain set of ±-closed sets. Specifically, a QL, or quantum, model is a structure where is an orthoframe; V is a non-null set of l-closed subsets of K such that i) V is closed under set intersection and the operation * defined by Y^ = (x e K: x i Y} and ii) if Y, Z e V and Y c Z then Y is l-closed in Z; and I' is a function assigning to each initial ob an element of V. Where Y, Z £ K, Y is i-closed in Z iff for every a e Z a I Y then for some b e Z, b l Y but not a l b . If complexity of this order is the price of a semantics for quantum logic, one might almost as well, it seems, adjust the rule for negation to (the second order form) I(-^,a) = 1, iff Sa [A], i.e. iff£ a {b € K: I(A,b) - l}, and BCL models to structures with S a relation on worlds and sets of worlds, i.e. in KXP(K), and begin again on modelling negation postulates. The (cheating?) method we now illustrate for a weak relevant quantum logic will work (with variations attended to in later chapters) for any "negation" postulates one cares to add to SLL (including the null-class of postulates). Consider the logic RQL obtained by adding to ROL the orthomodular distribution principle. The modelling conditions, rewriting negation postulates, are these:For a, 6, Y £ P(K), a e a iff Sa {a: Saa}, for DN; if a £ 8 then {a: Sag} £ {a: Saa}, for Rule Contraposition; and where y £ a if a e a and Sa{a: Sag} n {a: Say} then Sa{a:^Sa ang} n {a: Say}, for qualified Distribution. That is, an RQL model is a structure which satisfiesthese conditions. Proof ^f soundness for RQL is* rather trivial given that for SLL, for the mode 11 ingtf6ntfitionshave be^n selected to reproduce the negation postulates; and probf o£,€Gmpietettefe|' iS%ltBilarly rather: trivial, pne£ it is established that in the canonical model I^^A.a) = 1, i.e. -iA e a, if Sa[A], i.e. Sa |A| where |A| = {b: A e b}. And this is a matter of defining S suitably, if it can be done. It can, using the following shortcut: let # be an operation on P(K) such that whenever a = |A|, a^ = }-IA| (otherwise # can be arbitrarily determined, e.g. a^ = a when for no A does a = |A|). It needs to be shown that # is indeed an operation. Suppose a = B and, firstly that neither a nor B is of the form |A|; then = a = 3 = If however a = |A then 6 = a = |A|; SO suppose, more generally, £ = |B|. Since |A| = BJ, for every a e K, A e a iff B e a; hence A |- B and B (- A. Thus, by Rule Contraposition TA I- - 3 and -IB j- -A, whence TA e a iff TB e a generally, i.e. |TA| « |-IB|, and a& = B". Now define Saa as (PB) (B e a & |B| C a^), i.e. for some wff B , B e a and |B| C afi. To show -iA e a iff

125

2.S MAKING noAWTUM LOGIC ANV INTUITIONISM

RELEVANT

Sa|A|, suppose first -«A e a. Then iA e a and |-iA| £ |-*A| = |A Suppose conversely Sa|A|. Then for some wff Bj, Bi e a and |Bi so TA £ a, as required.

so SalA|. |a|*~- hAl

To model QL in this fashion, delete the final condition for Distribution, and impose instead for a, 6 £ P(K),a n {a: Saa} £ B,and when a e a and Saa n {a: S c aa n g} then a e 3, i.e. modelling conditions for Bad Eggs. Quantum logics, like other logics, have to be rendered relevant before they can begin to be considered as offering satisfactory accounts of implication: there is no special logical dispensation for corresponding to operations on Hilbert spaces or for consorting with the microphysical world. But once quantum logics are made relevant, along the simple lines sketched, the main case presented for rejecting Distribution is entirely dissolved. For the case rests not merely on the distribution principles of classical (or Boolean) logic but also on paradoxical principles, on irrelevance (as UL explains in detail). Once the paradoxes are removed there is nothing, then, to stop and much to recommend, since there are direct arguments for Distribution based on such accounts of entailment as sufficiency and content inclusion - a return to De Morgan lattice logic, and little to stop its construal not just as the logic of first degree entailment but as furnishing (for all we know at present) a logic for quantum theory as well. First degree intuitionistic implication is likewise irrelevant, containing as it does A & -iA )- B. Indeed it identifies (as far as entailment goes) all (explicit) contradictions: the common object that results is the absurd, or the impossible, A. Intuitionism can be relevancized at this first degree stage, as for orthologic, by abandoning A & -tA |- B, i.e. A j- B. Thus relevancised intuitionism is at this stage like minimal logic, which, very correctly, does not include the thesis that absurdity entails triviality. It appears moreover to be something of an historical accident that Heyting wrote the negative paradox, -iA A ->• B, -iA & A -»- B, A |- B, into his (now institutionalised, despite his protests ) formalisation of intuitionistic statement logic; for it lacks a clear justification in the intuitive model with which he was working. As Heyting effectively admits (56, p.106) it is an arbitrary extra : Axiom X [i.e. np -*• (p -*• q) ] may not seem intuitively clear. As a matter of fact, it adds to the precision of the definition of implication. You remember that p -»• q can be asserted if and only if we possess a construction which, joined to the construction of p, would prove q. Now suppose that j- np, that is, we have deduced a contradiction from the supposition that p were carried out. Then, in a sense, this can be considered as a construction, which, joined to a proof of p (which cannot exist) leads to a proof of q. I shall interpret the implication on this wider sense. A system of intuitionistic logic in which -»• is interpreted in the narrower sense and in which, accordingly, X is rejected as an axiom has been developed by Johansson in his "minimal calculus". The adoption of axiom X does nothing, however, to increase the precision of the enthymematic account of implication Heyting has assumed: precisely the '... no formal system can be proved to represent adequately an intuitionistic theory1, and so on: Heyting 56, p.106.

126

2.9 THE CLAIMS OF RELEVANT

NEGATION

VERSUS CLASSICAL

NEGATION

same account can be used for Johansson's logic. The difference lies in Heyting's smuggling in of the assumption - which remains far from intuitively clear - that a contradiction implies anything, A |- q, from which he gets to: if p f- A then p (- q. This is no sameness of sense, no mere interpretation of implication, but a substantive assumption about what the construction of contradictions leads to - an eminently resectable assumption which minimal logic rightly rejects. Minimal logic also turns out to be seriously irrelevant, most conspicuously at higher degree stages, owing to its uncritical incorporation of positive logic, and thereby of much rubbish, worst of all paradoxes such as A •*•. B -»• A and A & -)A -|B (hardly a big improvement on A & -jA B). But it, in turn, can be rendered relevant in a satisfying way (see 6.2). Also at the higher degree A can come to play a more central role: for it enables the definition of minimal negation as -|A = ^ A ->• A. That is, negation is like denial, where to deny A is (taken) to say that A implies something objectionable. Patently such an account of negation - which works neatly in semantical as well as syntactical form - cannot be exploited at the first degree stage, for even A — A turns under the definition into a third degree formula. (The study of minimal and intuitionistic logics, of relevant and other variations thereof, and, more generally, of relevant and other logics with non-normal negations, is resumed in chapter 6, where higher degree systems are introduced.) §3. Rival negations and rival connectives3 and the determinable theory of connectives such as negation and implication: what negation is3 and derivation of the normal negation rule. Negation is, as we have seen, a crucial notion for understanding the character and variety of logical theories of implication, and the notion which really separates classical from relevant logic. The conventional classical picture is that classical negation is the ordinary normal intuitive negation notion of natural language and logical thought, while alternative negations such as relevant negation are seen as deviant, peculiar, queer and abnormal, as contrived, formal or mathematical abstractions, which correspond to nothing in natural language. Since going relevant essentially involves replacing classical negation by relevant negation, seeing through these pretensions of classical negation is an essential part of seeing through classical implication theory and seeing why a relevant implication theory should replace it. In fact the situation is pretty much the reverse of the conventional classical picture. Relevant negation has a better claim to be the primary negation of natural language - if indeed there is a unique natural negation, which is doubtful - than has classical negation. Classical negation is a depauperate onedimensional concept which distorts natural language and seriously limits the applicability of any logic in which it is the (primary) negation.1 Classical negation may however seem natural for various reasons. First, most of us have been brainwashed by classical logic, with a high success rate, and its concepts are now seen as natural like those of other entrenched theories. Secondly, because classical negation has correctly accounted for one dimension of the natural negation concept (rather as strict implication has in the case of entailment) and given an account of part of what is required for a full account, it is therefore easy to take the naturalness of this part as showing the naturalness of the concept and to overlook the fact that further dimensions have been neglected. One thing classical logic can no longer claim - the claim was never valid is uniqueness of its negation. The appearance of several different negations, relevant negation, intuitionistic negation, connexive negation, orthological 1

This large claim, clarified in what follows, is argued in more detail in NC and UL. 727

2.9 THE USELESSNESS

OF LOGIC TEXTS IN EXPLAINING

NEGATION

negation, and so on has demolished that claim. But this recently found variety raises, at the same time, the serious question, so far postponed, of the general characterisation of connectives, and of negation in particular. What is negation? Or, in the easier formal mode, When is a connective a negation? One cannot expect to look up books on logic and find satisfactory answers to this kind of question, though before the blanketing snowfall of classical logic which obliterated most logical landmarks,1 texts on (traditional) logic used to carry sections or chapters bearing the title 'Negation'. Perusing the older texts, though important in showing that traditional negation is not classical, is only of limited value because the problem we are immediately concerned with only became conspicuous with the advent and proliferation of modern formal systems.2 As remarked, it is none too clear what the negation of traditional logic is like in all its detail - that is, assuming that it is determinate rather than, what is more likely, a determinable with competing determinates. But accumulating evidence suggests that the main negation determinable of traditional logic is not, as has recently been assumed without sufficient evidential basis, a classical one, but rather a connexive one. (The issue is further clouded by the fact that recent researches into the history of logic have been biassed by received classical logical doctrine.) Contemporary texts are of even less use. Mostly, in contemporary logical theory, negation is simply defined by a truth-table, with no discussion, or perhaps, in more traditionally-oriented texts, with a Venn-diagram approach based on inclusion or exclusion of properties. The following passage, which reveals that contemporary theory has scarcely advanced beyond the truth-tables of Wittgenstein's Tractatus,3 is fairly typical: Negations. Negations are among the simplest and most obvious truth-functional compounds. We call one statement the negation of another when the former would be true if and only if the latter, its one component, were false. Often 'not' is used to express negations in English.... We represent the negation of a statement A by prefixing a tilde (read 'it is not the case that', or 'not' for short) to A The whole point of is given by this truth-table for the connective... Any negation ~A is false if its component A is true, and true if A is false (Leblanc and Wisdom 72, pp.10-11). Modern formal systems, however, confront us with many different operators which are called negations and which are said, often correctly, to be interpreted as negations. These operators are different, because, for instance, they 1

The metaphor derives from Findlay 63, p.xi.

2

Many texts are also of limited value because they confuse issues as to the function of negation with ontological issues: see NC, §1.

3

. The Tractarian (complete possible worlds) hang-up, and misdirection, is even more evident in passages like this: J> and Ng. determine the situation of each other in logical space for they totally exclude each other and thereby surround each other. Thus together they map a state of affairs on to logical space, its logical place being determined according to whether it verifies the positive or the negative proposition (Bastable 75, p.292).

m

2.9 THE DETERMINABLE

THEORY OF CONNECTIVES

LIKE

NEGATION

satisfy different logical principles. Yet they also have features in common, e.g. all are one-place operators, taking at least sentences to sentences and often other parts of speech to themselves, and they have considerable similarities. At least they form a cluster or family. But there is more to it than this, and greater exactitude can be had. When in natural language people discuss or make use of negation, often they do not intend to be specific as to the logical properties of the operator, though some participants could of course be quite specific as to what they mean. That does not imply that the latter are talking about something different, or cannot participate in the discussion. For it may well be that the more determinate matters fall under the less specific, determinable, notion under discussion. Relevant negation falls under the determinable, negation, much as the determinant, red (colour) falls under the determinable, colour, to tie the account to Johnson's determinable-determinate distinction through one of his paradigmatic examples, colour-determinate colours (Johnson 21, p.174 ff.). How many different negations, negation determinates, are synthesized under one negation in the one theory of negation, is through the theory of logical determinables, which solves the many-one problem for negation. Negation, like most key logical notions, and like implication in particular, is a logical determinable under which various determinates fall. This theme helps resolve a major problem for both the philosopher and the linguist wishing to recognise a variety of negations - both those present in ordinary language and those wanted for various philosophical and logical purposes - explaining how a language can contain, and its speakers can cope satisfactorily with, so much apparent complication and ambiguity; while also explaining how all these varieties of negation are equally negations. As the logical theory eventually developed relies heavily on the theme that both negation and implication, the main connectives studied, are determinables, more should be said on the general features of logical determinables, and fortunately more can be borrowed from earlier work :How best to describe the situation with regard to key logical notions3 with regard to negation3 implication3 quantification and identity3 itself constitutes a minor problem. To say that implication3 for example3 is ambiguous is misleading; for it is not that there are several unrelated senses of 'implies'; on the contrary it is possible to give a general characterisation of implication as providing a universal guarantee over a class of situations. To say that implication is systematically ambiguous is somewhat better3 only it suggests the technical sense given to systematic ambiguity in the ramified theory of types3 and in any case needs explanation. We shall say that such words as 'everything'3 'some 'same '3 'possible '3 'not' and 'implies ' are logical determinables. In each case there is a common covering sense3 and under this a class of distinct senses3 the determinate senses?1 For example3 the word 'everything' has as a covering sense that of universality of the class of things (taken distributively)3 but specification of the nature of the contraction class leads to distinct senses under the cover. In one such sense3 that demanded by most classical logicians3 "everything exists" is true3 but in another it is false; thus salva veritate may be used to show that the senses under the cover acre distinct. Sometimes the context settles the contraction class3 the extent of the quantifier, and sometimes it does not3 as in the case of generalisations which may support counterfactuals: consider the different ranges of 'every' in 'Every raven is black'. The determinable 'not' has a covering function an exclusion or negating function ... but out of context the specific features of the [function] are not indicated. A generalised salva veritate principle shows that the specific senses are distinct; thus the determinates '"»' and differ in sense since dislikes dancing)' is nonsignificant but '~\(7 dislikes dancing) ' is true. For very many everyday linguistic 1

A synthesis of identity determinates, through the one determinable formula characterising the determinable, is given in EMJB, pp.249-50. 729

2.9 GETTING

INTO FOCUS THE DETERMINABLE

CHARACTER

OF NEGATION

purposes a single logical determinable suffices; the determinate in question in a given context is specified with sufficient precision by the scope of the determinable, or the scope of the determinable in combination with a modifier3 e.g. 'not' followed by 'significantly' as distinct from 'possibly'. Where scoping brackets or particles are unsuitable for the task3 as in spoken English3 variant forms with an appropriately built-in scope may take over; hence such negative particles as 'in- '3 'un- '3 'dis- '3 'non- ' etc. It is not surprising that logicians keen on economy of notation and ideas prefer to reduce all differences between determinates to differences of scope3 and scope to bracketing scope. Given appropriate auxiliaries this can be done; a determinate is replaced by its determinable together with distinguishing auxiliaries and scopes. Thus all negations can be reduced by occurrences of '~'3 which can then be taken to symbolise the determinable3 together with scopes3 e.g. sentence scope3 predicate scope3 and auxiliaries3 e.g. 'T'3 'S'3 '0Such a reformulation does not however eliminate determinate senses3 and it is no concession to the mistaken one word-one determinate sense assumption. For many logical purposes3 moreover3 the clarity obtained by using distinct logical symbols for different determinates is more important than the apparent economy got by relying on scoping devices and modifiers. (GR, pp.239-40). Logical determinability should not be confused with the (nevertheless closely related) notion of systematic or typical ambiguity. A systematically ambiguous notion includes in its prescription a variable, e.g. "of type a", whereas a determinable will have (at least) a particular quantifier in its prescription but no free (ambiguous) variable. For example, in the case of negation the determinable is given by a single recipe which establishes a stable and single covering sense for the core negation function; however the recipe contains a particular quantifier (in this case the quantifier: for some situations) and this can be made determinate with respect to various different determinate instances and scopes. A speaker who has learnt the recipe will be able to operate with and understand a large variety of differently scoped negations. To progress in this difficult area towards further characterisation of the negation determinable, it helps to distinguish purely negative functions from those not purely negative functions, which result from combining a purely negative function with some other functor which is not a negation function. Into this latter class fall a number of negations which involve changing the normal sense of negation, such as intuitionistic or minimal negation, and several of which are odd or queer. Intuitionistic negation, for example, may be analysed as the combination of a classical negation with a possibility functor, thus the intuitionistic negation, often read (better as) 'it is absurd that', may be interpreted as 'it is not possible', i.e. as ~0. Another example of a function which is a negation but not a pure one is provided by the 3-valued connective -i, read 'it is not true that', of GR, which is analysed as ~Tp (i.e. Fp v ~Sp). The further idea of this approach is that purely negative functions can be reduced to some simple or core negation, from which all other negatives can be obtained by compounding. The approach encounters two classes of problems however. Firstly, characterising the core negation (intended to be normal negation) leads to the introduction of logical (or semantical) principles which also enable the characterisation of a larger class of negatives, and perhaps the requisite class. In short, elaborating the approach leads to a bypassing of the basis of the approach. Secondly, it is by no means easy to determine, noncircularly, what compounding is allowable. As the many-valued case suggests every connective that can, with any warrant, be accounted a negation, or more generally negative, which is not ~ itself, can be analysed as a combination of ~ with "positive" functors (though

no

2.9 THE CENTRAL

FEATURES

OF NEGATION

in functionally incomplete logics these functors may not be definable in the logic). This is shown in the 3-valued case in GR (p.280, where there is a much fuller discussion of the conditions under which a one-place many-valued connective is interpretable as a negation; see p.278 ff). The result is fairly trivially generalised to all many-valued logics (given a suitable liberal definition of 'positive'). And, using features of the universal semantics subsequently introduced, an analogous decomposition can be obtained for any sentential negative.1 While a negation may be analysed in this fashion, not every combination of positive functors with negation ~ yields a negative. Necessity, •, which can be analysed (preserving entailments) as is not a negative. Differently, such functors as 'It is disbelieved that' are not negations, though they can be analysed in terms of Given an initial listing of positive functors of at most modal or entailment strength, these unwanted classes of functors can be removed by specifying firstly that ~ occur at most once as (or only an odd number of times) in the analysing phrase, and that the positive functors used in the analysis be at most entailment strength. The approach evidently presupposes quite a lot, including an unfortunate logical atomism. Whether a function which proves negative under analysis counts as a negation or not remains, of course, in many cases a matter for decision. Whether such a functor is a negation turns on whether or not it preserves sufficiently many sufficiently important negation features. And determining which these features are is crucial not just for this reason, but because the whole approach through analysis is open to the damaging objection that if positive functors can simply be listed why not negatives also. But such mere listing, without the principles underlying the listing, offers little illumination. The approach, unless better managed, is a dead-end. To state the central features of negation however, whether syntactically or semantically is to presuppose some logical apparatus. For a syntactical approach, the logic DLL, which is common ground to many rival theories, is taken for granted, though a characterisation which supposes even less is also given. To be a negation a one-place connective must exhibit sufficiently. (1) Involutary behaviour: -i~iE. A

at least one of

-i-iA and -i-il. -nA -* A

must hold, and normally both. The involutary requirement may be restated semantically, e.g. where -. is normal it is a 1-1 function of period 2, for every (dedu^cive) situation a, -i-»A holds at a iff A holds at a, for every A. Since Identity is an involution, involutary behaviour is not enough. (2) Inversion behaviour: at its weakest inversion is exhibited in the Modus Tollens principle MT. A ->• B, -»B -fr-iA. But it is not unreasonable to strengthen this requirement to Rule Contraposition: A B ->B -»- TA - so that a basic framework for negation has already been provided with system BCL (of p.123). Alternatively, it may be suggested that it is enough that any contraposition principle be satisfied (but if any is, it seems that the form given, linked to MT, should be). 1

Again the result is rather trivial in the absence of restrictive accounts of 'positive connective'. And duly strengthening the account of positive may rightly restrict what counts as a negative, i.e. the interlinkage of "positive" and "negative" tends (not unexpectedly) to the analytic.

131

2.9 FURTHER

FUNDAMENTAL

FEATURES

OF NEGATION

(3) Interchange behaviour: some of the following De Morgan principles should be satisfied: n(A & B) -,A v n B, v -fi + -,(A & B), -i(A v B) ->A & -,B, TA & -»B -»• -r(A v B). Given DLL the second and fourth of these principles can of course be proved, but in connexive systems for instance such proofs fail. More generally, inversion and interchange principles are interrelated. A normal negation will satisfy all interchange principles. In sum, whereas a negation need only satisfy sufficiently many of the principles listed, and minimally need exhibit only some inversion behaviour - to do some convenient legislation, Rule Contraposition - a normal negation exhibits all three classes of features. Yet a normal negation is by no means classical, and need not satisfy fundamental traditional principles. (4) Traditional (exclusion and exhaustion) tactical laws of thought should hold: viz: quirements which seemed the hardest of all, to traditional logicians, now appear, since and more recently of paraconsistent theory, the requirements given.

behaviour: at least one of the syn-i(A & -iA), and A v -,A. These reand most central features of negation, the rise of intuitionistic theory, the weakest and most peripheral of

To reformulate these important traditional conditions1 semantically requires use of the factual situation T (or of some distinguished subset P of possible worlds). Then where -i is entirely normal, (1) it is normal and (2) at least where a is situation T, and more generally for every a in P, -jA holds in a iff A does not hold in a, for every A, i.e. as regards a, negation is both exclusive (only if) and exhaustive (if). It will follow that for an entirely normal negation the principles of classical logic hold for all possible worlds, including T.2 The class of situations for which condition (2) holds, i.e. for which negation is exclusive and exhaustive, is called the class classically controlled by that negation. The variation available in the control class provides another dimension along which negations and normal negations may vary, a further determinable dimension in negation. Classical negation is simply the special case where all situations are classically controlled by the negation, whereas even with an entirely normal negation only some situations need be classically controlled. Thus an account of negation as entirely normal does not abandon but merely generalises the classical account. Given the syntactical features of negation, semantical characterisations can be derived; in particular, the minimal or weak rule, and the normal or * rule, can be derived and explained. A semantical framework of deductive situations, like that for DLL (but admitting considerable generalisation upon it, especially in the case of the minimal rule) is taken for granted, as are several facts already pointed out and argued for. It is, in particular, a fact that there are deductive theories, and situations where deductive reasoning goes on, which are inconsistent, i.e. for some A both A and -tA hold. There are similarly such theories and situations which are incomplete, i.e. for some B neither B nor ->B holds. An adequate theory of •deducibility must include both. Put differently, the semantical analysis must allow for situations a and b such that, for some A and B, I(A,a) = 1 = I(-iA,a), 1

Subsequently, in chapter 4, these conditions will be stated more accurately, with negation a further step removed from classical form. Then the condition is simply the normal rule with a* < a (not a* = a), and the negation though satisfying traditional requirements is not entirely normal.

2

The latter condition is explained, in a way, by the fact that T is the predicate which eliminates scope distinctions.

132

2.9 LIBERALISING

THE MISTAKEN

CLASSICAL

RULE FOR

NEGATION

and I(B,b) ^ 1 ^ I(-iB,b). The classical rule for the evaluation of negation quite illegitimately excludes these cases. Therefore, the classical rule for negation is wrong. The classical rule mistakenly restricts consideration to a proper subclass of deductive theories and situations, those that are consistent and complete, i.e. effectively to complete possible worlds.1 Accordingly the classical rule has to be abandoned. The question is what replaces it, how is it liberalised? There are several constraints on this problem which enable a solution to be obtained. The rule for negation, though it need not be, indeed cannot be, "Boolean", i.e. allow the assessment to be entirely done in the same situation or theory, should be "Fregean", i.e. make the evaluation of ->A in terms of that, or those for A, and more generally assess a complex expression analytically through its parts. In short, what is sought is a rule which gives I(-|A,a) in terms of I(A,d) for appropriate d related to a. Subsequently (in chapter 13) it will be shown that the general rule for negation where functors which require evaluation over nondeductive situations are also to be taken into account2 can take the general form I(-iA,a) = 1 iff Sa{d:I(A,d) 4 l}. But for the present the scene is restricted to that where only transmissible functors (those that can be fully evaluated over deductive situations) are involved. And what is sought is more information as to the character of situations d such that I(A,d) ^ 1 when I(-iA,a) = 1. The complementary aspect of inconsistent and incomplete situations is the key to the answer. There is a certain duality to be observed between such situations. For example, where a theory d is inconsistent in that both C and ~C belong to d, then a complementary situation d' to which neither C nor ~C belong is incomplete. Similarly as regards incomplete situations. While we know there are such situations - the complementary situation d c is one such - how do we know they are deductive, i.e. deductively closed? In the case of theories - the case taken as representative there is at least one such theory which is deductively closed, namely a*, where a* is given by the class of statements A such that TA is not in a. Theory a* is deductively closed when a is in virtue of Rule Contraposition. For suppose C is in a* and C H- D; to show D is in a*, i.e. -|D is not in a. Since TC is not in a, and by Rule Contraposition H- -»C, -»D is not in a. Now how to semantically evaluate negation is evident from this duality. Whether I(-iA,c) = 1 or not is determined, not in general by I (A,a) but by the value of I(A,d) for certain other related situations d, such as a*: I(-»A,c) » 1 precisely when for such d I(A,d) f 1. The initial form of the negation rule is accordingly the minimal rule already familiar from intuitionistic semantics: 1

The historical route to the * operator and the normal negation rule, which is what renders inconsistent and incomplete situations semantical objects, also began from these facts, as is explained in IDRL. The point counters the frequent charge that these things were obtained in an ad hoc way. The question, why change the classical negation rule rather than (say) the conjunction rule?, which could arise at this point, though it has been largely answered in §§3-4, is further considered in 13. It is worth noting that of course one can - in the investigation of other notions, not deducibility vary the &-rule, as we shall in later chapters where we are concerned with, for instance, epistemic notions: but implication in its basic forms is the present quarry. And in the proper investigation of deducibility we must, as the argument in the text shows, change the classical negation rule anyway. Thus the principle of minimum variation (or mutilation), like the argument of §3, commends retention of the normal conjunction rule.

2

Indeed the rule is a rule, with but minor variations, for any one-place connective.

133

2.9 DERIVATION

OF THE STAR RULE FOR NORMAL NEGATION

I(nA,A) = 1 iff, for every deductive situation d, if Sad then I(A,d) ^ 1. Relation S, which gives the thing the appearance of formal exactitude, but is really so far a veneer to cover over inexactitude, is the relation of "certain duality". Fortunately, S can be explained, or better eliminated, by appeal to other features of negation evaluation. For Rule Contraposition is not the only condition normal negation, satisfies. It is a fact that Double Negation holds for normal negation, that for every deductive theory or situation, if — A holds in it so does A, and conversely. Semantically, this imposes the following condition: that I(A,a) = 1 iff for every d such that Sad there is some e (which must in sufficiently comprehensive models, such as canonical ones, amount to a itself) for which Sde and I(A,e) = 1. Thus for each a, the following picture results:

It follows, as is immediate from the way * was introduced, that a** = a. And the picture suggests the rule, where all the further d situations coincide with a*, that I(~A,a) = 1 iff I(A,a*) ^ 1, the * operation replacing the S relation. But they do not force it. Further facts about core negation do however force uniqueness, specifically the De Morgan laws. Consider the fact that ~(A & B) -*•. ~A v ~B. Validation of this principle (given DLL) requires d uniqueness, i.e. for every d such that Sad, d = a*. For consider what counterexamples to ~(A & B) ~A v ~B require within the framework of DLL semantics, i.e. with & and v evaluated latticewise. Then for some a, I(~A & B,a) = 1 ^ I(~A v ~B,a). Thus by the S rule, for every d for which Sad I(A & B,d) + 1, i.e. I(A,d) ^ 1 or I(B,d) ^ 1. Also, as I(~A,a) i 1 t I(~B,a), for some d, say d^, Sad^ and I(A,d^) = 1, and for some d, say and I(B,d2) = 1.

All this is perfectly possible unless d^ = d^ : otherwise no

counterexample is forthcoming, since nothing stops I(B,d^) f 1 and I(A,d2) f 1. It follows then that the semantical rule for normal negation is the normal rule, I(~A,a) = 1 iff I(A,a*) ^ 1, where a** = a. And for each deductive situation a, a* is the unique deductive situation with respect to which inversion occurs. It is evident that greater generality may be obtained by lifting the restriction a** = a on the normal rule (the resulting unrestricted star rule is studied in chapter 6), but (the effect of) the minimal rule is not thereby recaptured. More important for present purposes are further impositions on the 1

Often further explanations of the * function beyond that given are demanded though exactly what is sought is rarely clarified and the demands are seldon backed by substantive reasons. However we do endeavour to explain, picture and show off the roles of * function both in NC and 3.1 and 3.2 and later sections. 154

2.9 CLASSICAL

NEGATION

ILLICITLY

SMUGGLES

IN (MAXIMAL) CONSISTENCY

REQUIREMENTS

rule. For an entirely normal negation it is further required that T = T* (so establishing that for some a, a = a*), while for the classical determinable a = a* holds always. So results, substituting a for a*, the familiar classical rule, I(~A,a) = 1 iff I(A,a) ^ 1. This rule is, as we have seen, much too restrictive. It does far more than is required merely to fix the meaning and function of negation; it restricts us to certain possible set-ups, set-ups where tautologies always hold and contradictions always fail, and accordingly makes unavoidable the suppression of tautologies. The classical rule conflates requirements; it conflates a negation function with consistency and completeness requirements, a conflation that can be brought out by transforming the classical rule to the equivalent form I(~A,a) = 1 iff I(A,a') f1 1 and a' = a for some a*. Instead of being stated explicitly the restriction a = a' , to maximally consistent situations, has been smuggled into the negation requirement. It is rather as if by writing a consistency requirement into the formation rules for negation, we defined statements so that only consistent statements were well-formed. We can however abandon both the maximality requirement and the consistency requirement without changing the negation function, and without being stuck with a weak or queer negation. Normal negation is neither weak nor queer because it behaves like classical negation over consistent and complete worlds. Since its behaviour diverges from that of classical negation only over a class of situations not admitted by modal logic, it is not the negation introduced by our normal rule which is different, but the class of situations that the rule entitles us to consider. A major and recurrent theme has been that the classical account of negation, for instance, in terms of truth tables and relativised to worlds, is inadequate. It is inadequate for a variety of reasons, all of which come down in one way or another to the fact that classical negation applied generally rules out a range of inconsistent and incomplete situations which are crucial for important logical purposes, especially for the guaranteeing of leading properties of implication, for the elimination of paradox, and for the treatment of intensionality. The difficulties can be surmounted, as we are beginning to see, by generalising classical negation to a normal negation. It is not, then, that we rule out classical negation as in some way illegitimate; classical negation is a special case, namely the restriction of normal negation to situations a such that a = a*, or equivalently, when taken as universal, a restriction of situations to consistent and complete ones. It is a familiar objection that the negation rule used in the construction of impossible situations - the normal rule - dispenses with the usual sense of negation and replaces it by a "queer" weak sense. This however makes the pervasive, but entirely false and groundless, assumption that the usual or ordinary negation is classical (see further NC). In a sense the complaint is simply a way of restating the already-criticised objection that only possible worlds, and the classical negation rule they satisfy, are usual and ordinary - impossible situations are "queer" and can be dismissed from consideration. But there are other defects as well in this view. The normal negation rule cannot be regarded as simply involving a different sense of negation, in the way for example that intuitionistic negation may be said to involve a different, even "queer", sense of negation. For intuitionistic negation and classical negation are such that there are common situations for which they prescribe different behaviour, e.g. the factual situation T itself. But for the supposedly normal "queer" negation and classical negation there need be no situation for which they prescribe different behaviour; for the normal

735

2.9 HOW CLASSICAL

NEGATION

REMOVES SCOPE AND OTHER

DISTINCTIONS

negation behaves like classical negation over all the situations classical negation recognises, namely complete possible worlds. Hence the difference must be seen rather as a difference in the scope, or extent of negation rather than the simple introduction of a queer negation, and it will not do to dismiss the normal negation rule as simply involving "a different sense of negation". Nor can it be dismissed as "not a negation at all", since it agrees with classical negation where classical negation is defined. The difference between the two negation rules and the different situations they enable us to consider is the semantical analogue of the syntactical distinctions of negation position and scope. In fact the abandonment of the rule ~A e a iff A I a (or a = a* as it is expressed in terms of the * functor) no more involves introducing a queer sense of negation than recognition of the difference between ~$A and $~A (with $ a propositional functor) involves recognising a queer sense of negation. Recognising situations where ~A e a diverges from A 4 a, in addition to the classical ones for which a = a* is much the same as recognising that there are many functors like $ for which $~A diverges from ~$A, as well as functors for which $~A iff ~$A. We must drop the equation a = a* (~A I a = A i a) if we are to duly acknowledge the distinction between and $~B. This distinction is directly reflected in set-up behaviour as the distinction between B 4 a (for ~$B), and ~B e a, where a is the set-up picked out by functor $ i.e. a = {C:$C}. For then a = a* iff (C)($~C = ~$C). Thus the incompleteness of {C:<J>C> reflects the fact that for some C, neither 4>C nor $~C holds; for example, neither D nor not-D is in the class of propositions which are necessary, obligatory, or believed or desired by Arthur. Correspondingly the inconsistency of {C:$C} reflects the fact that for some D, both D and ~D fall within the set. The rule ~B e a = B I a amounts then to elimination - inadmissible elimination - of the scope distinction between ~$B and $~B. By evaluating functors over sets of worlds (or model sets), as well as worlds (or model sets), semantics for modal logic is able to reinstate one of these scope distinctions and drop HE>B > $~B, i.e. to recognise the incompleteness of {C:$C} (which is of course essential to deal with modal functions such as necessity which are incomplete). But it cannot satisfactorily reinstate the other distinction and drop $~B > ~$B, i.e. duly recognise the inconsistency of {C:$C}. Nor can stepping up to sets of worlds (or model sets) eliminate those paradoxes which result from the logically identical behaviour of tautologies, or the consequent propositional identity of all tautologies, for example. For the use of set operations cannot increase the level of discrimination.1 ' Adopting a = a* everywhere involves insisting that there is only one possible scope for negation with respect to functors which pick out situations - the (extensional) scope of classical negation. But recognition of other possible scopings is an essential step to dealing with intensional functors. There are in use in natural language a far greater variety of scope distinctions than can be satisfactorily recognised by classical logic or the classical theory of negation.2 1

Multiplying up set operations can do something to increase classical discriminatory power, as Cresswell's roccoco constructions (in 73) have shown. But they do not do nearly enough, and what they do achieve can be accomplished far more simply and satisfactorily, just by abandoning the classical basis which is inadequate to the task of representing intensional discourse.

2

Of course classical logic can recognise some scope distinctions in functors. For example V, defined VB B B, is extensional, but there is a scope difference between V~C and ~VC. 136

2.9 EXPLAINING

WHAT NEGATION

REALLY IS

The classical theory (which is'a consistency theory) amounts to the assumption that there is only one negation determinate - classical negation - which is characterised by being exclusive and exhaustive over all situations. Given the determinable recipe for negation, however, one is not only able, but obliged, to recognise a much larger range of negations and to deny that classical negation holds any special or privileged position among those. Classical negation is not the "real" or "usual" negation, or even the one which occurs most frequently in ordinary language. This latter privilege seems to go in fact to the determinable itself; for most frequently the scope or range of the negation is simply left open. The privileged position of classical negation in current logical work is due entirely to the prejudice in favour of the extensional, and the belief that extensional logic is the only true or real logic (a prejudice extensively criticised in EMJB). Classical negation does have a special position as the only negation occurring in extensional logic; for it is of course unique when the class of describable situations is restricted, as in extensional logic, to the factual world (see GR, p.280). We are now in a position to try and answer the vexed question: What is negation? Negation is a logical determinable. The determinable may be isolated initially syntactically and more satisfactorily by adding semantical features. Syntactically negation is a one-place functor, which is at least sentence forming on sentences and typically, in natural languages, also operates on other unsaturated sentence parts such as predicates and modifiers, but is only very dubiously subject-forming on subjects (despite the calculus of individuals).1 The sentence connective (component) must satisfy at least some inversion features, minimally (to take the bound imposed) Rule Contraposition, and for normal determinates a good deal more. The recipe for the determinable can now be given, semantically, thus: ~A holds at a iff for certain situations like d suitably related to c A does not hold at c. The determinability is reflected in the phrases certain situation and suitably related,both of which afford dimensions for variation. Several of the leading determinates have already been discerned. Negation can be defined, rather uninformatively, thus: B is a negation of A iff for every deductive situation a, B e a iff ~A € a. Notice that while this account appears satisfactory, a corresponding strict account over classical situations would not be: for then any contradiction at all would count as negation of a tautology, and so on. Then, the negation of A = the B such that B is the negation of A. The definition is satisfactory given that uniqueness up to coentailment only is required.2 It may be objected that the theory of negation advanced is unsatisfactory; that negation should be characterised independently of any such connection as f or or 'not' or 'it is not the case that'. But this is to expect far too much. Negation is an independent primitive notion which cannot be defined outside its own circle of connections. For example, it is simply demonstrable that is not definable in terms of the positive connectives '&', 'v', '-v', '•«•', etc., of sentential logic, or of these together with the quantifiers of predicate logic (see GR, chapter 5); in short, negation cannot be positively characterised. 1

Since this text is restricted to sentential logics, these additional forms are not further considered in what follows. The omission does not upset the account. In any case, such nonsentence operators can be approximated by way of sentence operators, especially given help of auxiliary operators and scoping devices.

2

Beyond that it fails since where ~B is the negation of A so is ~~~B, for instance. Repairing this sort of deficiency calls for the introduction of nondeductive situations, as in later chapters, especially 13 and 1A where the theory of negation is further elaborated. 137

2.9 ON THE GENERAL CHARACTERISATION OF AND TYPES OF IMPLICATION

The general picture of implication can also be clarified through the theory of logical determinables, and puzzlement thereby removed as to how so many different connectives, of enthymematic and nonenthymematic character, of entailmental strength and of material-conditional feebleness, can all be implications. The answer is that implication is a determinable with a classifiable range of determinates. The determinable feature is that already mentioned, of general conditionality, that (in connected contexts) a statement holds when another does over some so far unspecified class of situations including the factual one T, the second statement thus giving some sort of universal guarantee of the first.1 The determinates, as we are beginning to see, are many and varied. Conditionals which support counterfactuals provide one important group of determinates, entailment and logical implications another. Entailment is a sort of implication: logically necessary relevant implication (as argued in chapter 10): this does not imply that there is a single relation of entailment that can be pinned down in one formal system. Indeed the argument will be that there are at least two entailmental notions of philosophical and logical importance, one of them linked to propositional identity and independence notions and another tied to the transmission of content and information. But the differences between these notions emerge only at the higher degree, so they are not a present concern. As well there are of course enthymematic and irrelevant surrogates for entailment such as the various strict implications. The classifications to be taken into account in evaluating entailment must include (as already argued in 1.8) nonmodal worlds, both incomplete situations and, more perplexing for the trad philosopher, inconsistent situations, since both sorts are perfectly admissible deductive situations. One familiar reason for this is that entailment amounts to logically valid argument, and logical validity makes a universal guarantee, over all (deductive) situations. A central feature of valid as opposed to practical argument is its universal applicability, that the truth of the premisses provides a guarantee of the truth of the conclusion which can be relied upon, not just in some particular circumstances, but in all circumstances where such a guarantee could be required. A practical argument, by contrast, is one which makes assumptions applicable only in particular situations and which can therefore only provide a (truth) guarantee limited to these situations. But the circumstances under which we generally require such a guarantee must include nonmodal worlds. For we do, repeatedly, encounter such worlds in arguing from or about those sets of propositions which are picked out by creatures' beliefs, opinions, hypotheses and assertions: an implication which would enable us to perform such operations as investigating or discussing the consequences of someone's beliefs or assumptions would need to provide a (truth) guarantee over nonmodal worlds. For such intentional functors select sets (not necessarily unique) of statements which may be inconsistent and are always incomplete; we may, for example, fear both C and ~C, and believe neither D nor ~D. And the same holds for the deductive closures of these sets. If only modal worlds were available, a creature's beliefs, opinions, fears, etc., would be subjected to complete distortion, e.g. by the addition of necessary truths and the removal of all inconsistencies, before they would be amenable to logical treatment at all. A person's "beliefs", thus processed, may bear very little relation to what he actually believes. Either everybody's actual incomplete set of beliefs remain forever beyond the reach of logical discussion and investigation, or one must 1

Hence the attempts by Russell and many others to explain implication as a sort of universal conditional, e.g. p -»• q through (x) (Px > Qx) for a suitably restricted universal quantifier. (This involves as well an attempt to represent the semantics in the syntax, which is of course alright if the syntax is, unlike Russell's, rich enough.)

J3S

2.9 THE RANGE OF FUNCTORS DEMANDING

ASSESSMENT

OVER NONMODAL

WORLDS

change such beliefs before one can investigate them.1 Yet creature's beliefs, opinions, theories etc., certainly seem to be proper, and important, subjects for rational discussion and logical argument. If deductions from incomplete sets of statements, which accordingly require assessment over incomplete worlds, are central to philosophical reasoning, deductions from inconsistent sets of statements, requiring inconsistent worlds for assessment of the entailments, are important in logical and mathematical reasoning.2 For we encounter inconsistent worlds in the shape of premiss sets of reductio arguments, in the form of, what is not so uncommon, inconsistent theories, and in the description of impossible situations and states. Moreover, if the wellknown limitative theorems are to be believed, the sets of theorems of sufficiently rich deductive systems, such as arithmetic, are always intensional set-ups, since these systems are either inconsistent or incomplete. In other words, provability functors, like most intensional functors, characteristically select nonmodal situations. For similar reasons conviction-carrying and logical obligation functions of entailment demand nonmodal worlds. These features of entailment are a matter of the following rationality principles: that where A entails B then, for any person x, if x accepts A then he ought logically to accept B, if x believes A then he ought thereby to believe B, if x is prepared to assent to A then he ought thereby to be prepared to assent to B, and if x is convinced of (believes) A then he could be convinced of (could be brought to believe) B. Entailment is so intimately bound up with these rationality principles that the principles are often used to characterise entailment; thus for example it is claimed (e.g. by Hare 52) that A entails B iff one could not rationally accept A but reject B. But this is a further case of requiring a guarantee from incomplete situations, in this case from the class of statements which a person accepts. The same point can be made syntactically, through the notion of suppression, which is soon to be explained: Suppression causes such rationality principles to fail because there is no guarantee that the suppressed proposition, which is needed to obtain the consequence, fails under the relevant intensional functor, even if the stated premises do. But there is no obligation, for example, to accept a conclusion obtained from a given premiss if the derivation depends essentially upon something which is not accepted, as a proof involving suppression may do. The obligation to accept the conclusion of a valid argument if one accepts the premisses derives from the fact that the accepted premises are sufficient on their own for the obligated conclusion. The problems raised for rational intensional functors are but a special case of the problems insufficiency and suppression create for the individuation of statements in terms of their logical content. For suppression equates the logical powers of logically distinct statements. Thus, for example, where a statement T is generally suppressible, we are unable to distinguish in logical power between A and A & T, for their consequences (and what they are consequences of) are identical. To separate the statements with distinct logical powers that suppression implications conflate, it is essential to deploy nonmodal worlds and to demand maximum variation among worlds. Two statements have the same logical content iff they belong to precisely the same worlds. Hence statements are logically distinct, i.e. have a different logical content, iff they belong to distinct worlds. Some 1

Even then only a piecemeal investigation could be achieved, because no one could be brought to believe all necessary truths, even granted - what seems false that they could be converted to any particular one. Hence too the inadequacy of attempts like those of Hintikka 62 to treat epistemic logic semantically using only modal worlds, or equivalently, only classically extended state descriptions.

2

The points that immediately follow condense those of 1.8.

139

2.9 THE PLAY ON 'POSSIBLE'

IN 'POSSIBLE

WORLDS'

logically necessary statements have, however, distinct content from others, and some impossible statements have distinct logical contents. To separate these statements we need worlds which distinguish between necessary truths and set-ups which distinguish distinct impossible statements. For this it is necessary to have a variety of situations to which tautologies do not belong and likewise of situations where contradictions do hold. For adequate individuation of statements then - individuation that is needed for such associated interpretations1 as inclusion of logical content and container models of entailment and for the vindication of important topic preservation and non-deducibility theses - nonmodal worlds are essential. Strict implication, which permits consideration only of certain alternative worlds to T, modal worlds, cannot even claim to consider all consistent situations, let alone all those which can consistently be considered. Strict implication is sometimes erroneously regarded as ultimate because its implication is assessed over "all possible worlds". But, as we have seen, only complete possible worlds (maximally consistent theories) are admitted; and were all consistent situations considered, and negation symmetry duly conceded (through Contraposition), the (inconsistent) worlds of DML would result. Moreover 'possible' here is ambiguous between 'internally consistent', and, what is different, 'consistently specifiable or treatable' in the sense of conforming to rules which guarantee coherent logical behaviour. A situation would be inconsistent in the first sense if it ratified some statement and also its negation; in the second if some statement were both ratified and not. But it is only if 'possible' is used in the latter sense, that of conforming to suitable situational rules, that we can claim that when we have looked at all possible situations we have considered all we can reasonably be expected to consider. And the only way in which situational rules can be justified as suitable for a given connective is by considering its intended interpretation. For this reason (among others), the definition of strict implication, of A B, as ~0(A & ~B), is not self-justifying as a definition of entailment. It has seemed to be because it has been confused with the general requirement on implication, that A implies B iff for no (consistently specifiable relative to suitable world evaluation rules) world c does A belong to e without B's also belonging to c. Thus the defender of strict implication would need to argue that, for the intended interpretation as entailment, we need only look at complete possible worlds. There are compelling reasons, however, for looking beyond these worlds, to inconsistent or incomplete worlds. The ambiguity in 'possible' has however made it easy for bandwaggoners to jump onto convenient parts of the relevant enterprise, and acclaim their discernment and vindication of 'impossible possible worlds'. The latter expression delivers no contradiction because '[im]possible' is being used in the two different'senses, and the second occurrence, of 'possible' in such a way that possible worlds do not differ from worlds. By such a strategem the appearance of working within the older conceptual framework of modal logics is retained, whereas in fact a major shift beyond that unduly restrictive framework has been made. §10. Suppression and its connections with insufficiency and irrelevance; positive and negative suppression characterised, the disastrous effects of suppression, and Lewy's arguments that the negative paradoxes and Disjunctive Syllogism do not involve suppression refuted. The choice between strict first degree "entailment" and relevant first degree entailment turns on the issue of the treatment of non-contingent statements, since both treat contingent statements in essentially the same way. To elaborate the contrast:- strict continues to 1

Such associated interpretations are given in 3.3.

2

All this section, bar the critique of Lewy, is based on V. Routley 67 and subsequent modifications thereof.

140

2.10 THE TRADITIONAL

PROHIBITION

OF SUPPRESSION

OF (NONCONTINGENT)

STATEMENTS

treat non-contingent statements in the way material implication treats all statements, that is, as suppressible where true, while (deeper) relevant systems treat all statements, contingent and noncontingent alike, in the way strict treats contingent statements, that is, as non-suppressible. A crucial issue for the first-degree choice then is the defensibility of strict treatment of noncontingency, and especially the strict assumption that necessary truths, unlike contingent truths, can be suppressed. Suppression occurs where an assumption needed for the correctness of an argument is omitted, not stated or exhibited, so that the argument is incomplete and the conclusion is not necessitated. A suppression principle is one that suppresses information required to proceed generally from antecedents to consequents. With suppression principles antecedents are not (everywhere) sufficient on their own for consequents. Analytically then, suppression destroys sufficiency. For if a needed assumption (or alternative) is omitted the premisses of the argument cannot be sufficient on their own for the conclusions. The basic implication relation, as one of sufficiency, must accordingly be suppression-free. Similarly for entailment and logical sufficiency: avoidance of the suppression of logical statements is essential. In fact a good initial way to test for the presence of suppression in logical principles is to substitute the words "(is) logically sufficient for" for ->•. It will then be seen that many principles which were not implausible when stated with 'if-then' or even 'logically implies', lose their plausibility once the requirement of a full statement of assumptions is made explicit in this way. Suppression has been traditionally prohibited in valid argument at least since Aristotle, who says of valid syllogism that a syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. I mean by the last phrase that they produce the consequence, and by this, that no further term is required from without in order to make the consequence necessary (Prior Analytics, 24 a , 18-21, emphasis added). The passage seems clearly to be making the point that the premisses must be sufficient, on their own, for the conclusion. The premiss of a valid argument must be sufficient for the conclusion just as the cause must be sufficient for the effect it produces; and just as it is incorrect to say that some cause produces some effect if something further is needed to produce it, so it is incorrect to say that some conclusion is validly deduced if some further assumption is needed, but left unstated. There is nothing in the passage to suggest that Aristotle intended the statement to apply only to contingent statements. In fact in traditional syllogistic, there is no difference in treatment between contingent and non-contingent statements, and from a valid syllogism such as All elephants are animals; No stones are animals; Therefore, no stones are elephants one could not derive "No stones are animals; therefore no stones are elephants", as a valid argument. Specified provision was made for enthymemes, as arguments whose premisses needed supplementation in order to be valid, but there was no exception where the premiss omitted was a necessary one. In traditional logic then suppression has been regarded as illegitimate, and not just the suppression of contingent truths. Moreover, it is still implicitly assumed to be a valid objection to the validity of an argument that the stated premisses are not sufficient for the 141

2.10 CHARACTERISATION

OF POSITIVE

ANV NEGATIVE

SUPPRESSION

claimed conclusion, even where the suppressed statements are logically necessary. For example, it is a common criticism of the Euclidean deductive system that axioms, like those of continuity, which are needed for the proof of some theorems, are not included in the axiom set, the assumption being that the theorems do not follow properly from the insufficient axiom set. Similarly xt is a common criticism of logicism that mathematics is not deducible from logic because logic on its own is not sufficient. Many similar examples could be adduced to illustrate that people remain reluctant, in their natural reasoning, to practice the general suppression of necessary truths strict implication preaches. Suppression is informally characterised as the making of implicit or unshown but needed assumptions in a proof or argument. Once a more precise characterisation is attempted, however, it becomes evident that there are a number of different ways in which something can be assumed but remain unshown. For example, the implicit assumption might consist in reliance on either the acceptance of some true statement or the rejection of some false one without explicit indication. For a statement can be (universally) assumed not only by setting things up so that it is impossible to think without it, so that it is tacked on as an implicit further premiss in any and every argument, but also by taking its rejection to be unthinkable. This is just as much a way of assuming it, of taking it as unchallengeable or unrejectable, or of assuming further information not shown in the argument, as in the other case. In the one case, then, a statement A is treated as suppressible by treating it as if we could not think anything without it, and in the other by treating it as if ~A were unthinkable in any situation. These two modes of suppression are closely related but their difference is important. The distinction is reflected formally by the difference between positive and negative suppression. Positive and negative suppression are normally correlated and can be transformed into one another by contraposition principles. Positive suppression occurs when a non-redundant true or proven statement is deleted from an antecedent position in a deducibility or implication statement. Negative suppression occurs when a non-redundant false statement, or one whose negation is proven, is deleted from a consequent position in a (multiple consequent) deducibility statement or implication statement. Although these characterisations apply to higher degree forms of suppression as well as first degree forms, treatment of explicitly higher degree forms of suppression is postponed until the higher degree is introduced. It should be noted however that most of our general remarks about the damaging nature of suppression and the reasons for avoiding it apply equally to the higher degree forms. In fact neither the traditional nor the modern (Cambridge) treatment of suppression goes essentially beyond the first degree theory. Syntactical characterisations of first degree suppression are as follows:A is positively suppressed in Af- T when A, A}-T (.is true, provable, etc.) but A|- F. There is a matching formulation in terms of implication: A is positively suppressed in B -»• C when B & A -*• C but not B -»• C. Thus a statement is suppressed in an implication when, although not stated as part of the antecedent and not a consequence of the antecedent, it is presupposed in obtaining the consequent from the antecedent.1 Positive suppression, which corresponds to the traditional notion of an enthymeme, can easily be rectified by conjoining the suppressed statement to the insufficient antecedent, rendering it sufficient. A is negatively suppressed in A|— T when A|-A, T (is true, provable, etc.) but Aj- T. Similarly with implication, where B -»• A v C but not B -»• C. Negative suppression has not attained the traditional recognition that (positive) suppression achieved, in part no doubt because the traditional picture of argument was always one of multiple antecedents but a single consequent. In neither scholastic nor traditional logic were multiple consequents considered; so the technical means for 1

This is the account relied upon in Routley2 70. 142

2.10 SUPPRESSION

AV01VANCE

BV WORLV VARIATION

ANV THE MAXIMUM VARIATION

PRINCIPLE

representing negative suppression were simply not available.1 This helps in explaining too why the suppression features of Disjunctive Syllogism were only occasionally glimpsed (though the main reason why its suppression features were mostly not observed was, of course, the consistency hang-up). There are illuminating semantical parallels to both the formal and informal concepts of suppression. In semantic terms, A is positively suppressed in B ->• C when C holds in every situation where B holds and A holds, but C does not hold in every situation where B alone holds. Similarly A is negatively suppressed in B C when either A holds or C holds in every situation where B holds, but C does not hold in every situation where B does. It is a corollary of the definitions that we can avoid positive suppression of a statement only by being prepared to consider situations where it does not hold, and negative suppression by situations where it does hold. If there are statements such that we are never prepared to consider situations in which they do not hold, or in the negative case do hold, their wholesale suppression is permitted. Subsequently (in chapter 3) we prove that DML, and its first degree extension FD, is free from wholesale suppression by showing that for every statement, necessary and contingent alike, there is some situation to which it does not belong, and some situation to which its negation does belong, i.e. every statement fails to hold somewhere. In contrast, an analogous result in strict implication holds only for contingent statements. First degree strict implication allows the wholesale positive suppression of tautologies, because it takes them to hold in every world, and the wholesale negative suppression of the negations of tautologies, because it lets them hold in no world. To avoid wholesale suppression it suffices, however, to assess implications over complete possible worlds and only two additional worlds - one, say, to which all contradictions belong and another from which all tautologies are absent. Although the two worlds would prevent the suppression of tautologies relative to contingent propositions, they do nothing to prevent the suppression of tautologies relative to other tautologies. To ensure that an implicational connective permits no suppression, it is essential to have an adequate range of variation of necessary statements, for instance, in incomplete worlds, just as complete worlds have as regards contingent propositions. For adequate variation we need to consider not merely some incomplete or inconsistent worlds, in the manner say of modal systems S2 and S3, but an adequate range of such worlds, and in particular worlds which distinguish every logically distinct pair of tautologies. Variations among intentional set-ups, as well as worlds, should be the maximum consistent with normal evaluation conditions. It should be the case that for every statement B which is not a consequent of A there is some situation where A holds, but B does not. Any violation of this maximum variation principle2 will allow suppression somewhere. 1

Medieval and traditional logicians were not alone in their limited perception. In 70 Routley2 quite erroneously asserted that negative suppression could not easily be rectified syntactically. We are obliged to Kielkopf, who pointed out in 74a how the syntactical account of negative suppression could be framed, thereby making even clearer the symmetry between positive and negative suppression. Negative suppression is repaired, of course, by disjoining the suppressed statement to the consequent.

2

Although a definition of consequence based upon this principle, like one based upon the definition of suppression itself, would of course be circular, it is nevertheless a useful guide as to whether a suppression-permitting implication has been confused with entailment.

143

2.10 THE EQUIVALENCE

OF SUFFICIENCY

AND SUPPRESSION-FREEDOM BUT NOT OF

RELEVANCE

Indeed, maximum variation in the worlds over which an implication is assessed, sufficiency of the antecedent of the implication for the consequent, and suppression-freedom of the implication, are logically equivalent notions. First, one statement is logically sufficient for another iff there is no suppression. For a condition is sufficient for an outcome.iff it is not in need of completion for guaranteeing the result, i.e. the condition is able to guarantee its outcome without depending on or presupposing anything else, i.e. the implicational relation is suppression-free. Secondly, sufficiency is logically equivalent to maximum variation. For by the very meaning of sufficiency, a condition A is sufficient for B iff A, operating quite on its own, i.e. independently of everything else, is enough to guarantee B on its own. Hence if, and only if, we have a case where the antecedent is sufficient for the consequent, we should be able to consider situations where everything distinct from (not logically included in) A varies independently, i.e. so that everything else is falsified in some of the situations considered where A holds. In order to obtain sufficiency of antecedent for consequent we should, just as in the case of testing for causal sufficiency, assess a purported implication over a class of worlds in which the antecedent is held constant (as a member of the set-ups) but everything independent of it is appropriately varied. In both cases if there were some condition (for example, in the causal case, air pressure) which although independent of the antecedent of a purported implication, could not be varied, it would be impossible to determine whether or not it played an essential role, and therefore impossible to determine whether or not it could be dropped from a statement of conditions sufficient for the consequent. But it is just this position, of having independent conditions which we are unable to vary, which we are forced to adopt in the case of necessary statements if we assess implications only over complete possible worlds. Why is suppression to be avoided? Most obviously, it is responsible for relevance-violating paradoxes, positive suppression for positive paradoxes and negative suppression for negative paradoxes. Indeed moves used in deiriving paradoxes of strict implication provide the most striking examples of (first degree) suppression. Since A, B v ~B |- B v ~B but A f B v ~B, B v ~B is positively suppressed in A (- B v ~B. Similarly, since C & ~C |- C & ~C, A but C & ~C /f- A, C & ~C is negatively suppressed in C & ~C |- A. Thus unrestricted suppression directly yields irrelevance. But the converse does not obtain. Though elimination of suppression eliminates the paradoxes, elimination of the paradoxes and relevance violations does not guarantee absence of suppression, because certain limited kinds of suppression do not lead to relevance violations. Therefore the satisfaction of relevance requirements is not itself sufficient to guarantee suitability of an implication for interpretations which require suppressionfreedom. Equally important, suppression has an extremely damaging effect on both primary and associated interpretations of entailment. In terms of the primary interpretation, that of deducibility or valid argument, suppression runs counter to the whole tendency and main function of logical argument, which is the making explicit of assumptions, used in argument, so that they can be examined, questioned, thought about, discussed and accepted or rejected separately. A logic which undermines this assessability and explicitness function through allowing the hiding of assumptions fails crucially in one of the central functions of reasoning and deduction. An argument with a suppressed assumption is much less readily assessable than one in which all the assumptions are explicitly stated. Because a large number of statements could fill the gap there is also serious loss of clarity and precision.

144

2.10 CATALOGUING

THE MANY DAMAGING

EFFECTS OF

SUPPRESSION

A related defect of much importance concerns the role of deductive argument in carrying conviction - the "compulsion" feature. It is only if all the assumptions in the arguments are stated and accepted that the conclusion is compulsory. If some assumption is used but not stated, then in accepting the incompletely stated assumptions we do not accept a set completely sufficient for the conclusion. And should we demur at the further needed assumption - and the possibility that we may do cannot be dismissed and should certainly be allowed for, even if the statement is very widely accepted - we are not compelled to accept the conclusion. The point holds good irrespective of whether the suppressed statements are contingent or not; for one can - not at all irrationally - doubt necessary truths, or accept inconsistent statements (see Routley2 75 and EMJB, 8.12). But the matter goes beyond the conviction feature, for even if the suppressed statement were one which were rarely doubted, we might still want to know whether it was used to obtain a particular conclusion. We may, and often do, want to know not merely whether the conclusion is true, but whether it can be obtained from this set of premisses. We expect a correct assessment of deductive responsibility. Deductive logic has never been concerned with the willy-nilly churning out of true statements implied by some true set of premisses, we care not which. A major function has always been the correct assigning of responsibility for these conclusions, and with the converse relation, assessing exactly what is involved in asserting some set of statements.1 Hence its important traditional uses in criticising what someone says as insufficient for the conclusions he draws, and criticising what he says by looking at its consequences. It is not peripheral, but essential, to deduction that it should be able to fill these roles, and both presuppose correct assigning of responsibility for the conclusions drawn. In terms of this notion of valid argument, the modern systemic notion which claims to have replaced it - that of the churning out of theorems from some of the axioms, we care not which - is of much more restricted applicability. But suppression has a disastrous effect on all these functions. By omitting some premiss without which the deduction of some conclusion is not valid, it misrepresents the premiss from which this conclusion is obtained, and hence responsibility for the conclusion. To agree to accept partial responsibility as good enough here is like agreeing to say that somebody was responsible for the dinner when he peeled the potatoes and the cook did the rest. The first statement cannot be accepted as an elliptical, but allowable, way of making the second statement. And similarly suppression enables us to obtain as causally responsible a partially sufficient rather than a fully sufficient causal condition. Likewise we cannot assess exactly what is involved in asserting that A If we are unable, for the purpose of drawing some conclusion C, to separate A from some other statement B, and to separate A's consequences from B's. But if B is suppressible when it occurs conjoined with premiss A for some conclusion C, this is just what we are unable to do. We have assessed, not what is involved in A, but what is involved in A and B, which may be a very different matter indeed. There is good reason then for the traditional prohibition on suppression; for suppression hampers the performance of a number of functions deducibility has traditionally been required to carry out, and success in which can be taken as a 1

Through responsibility sufficiency and suppression may be linked with notions that Ackermann and Anderson rightly emphasized in analysing entailment, that the conclusion follows from the premisses, and that the premisses are used in deriving the conclusion. But of course the linkages with these notions (which are brought in later) may be obtained in other ways also.

145

2.10 HOW SUPPRESSION

DESTROYS

MEANING

CONNECTIONS

DEDUCIBILITY

REQUIRES

necessary condition for correctly characterising an implication as deducibility. These functions require of deducibility a (truth) guarantee over all deductive situations. Whether one argues, then, from the meaning of 'deducibility' as determined by ability to perform a number of traditional functions, or from the semantic analysis of implications as yielding hypothetical truth-guarantees over a class of worlds, the result is the same; an adequate account cannot be obtained if assessment is made only over modal worlds. Perhaps the most important objection to suppression derives from deducibility as a meaning relation between statements. B should be deducible from T only if there is a connection of meaning between T and B. But this connection may be destroyed if suppression is allowed; for the suppressed statement, which although used no longer appears in the premiss set F, may be just what originally made the meaning connection between F and B. Once this used statement has been dropped off, F and B may no longer have the right connection of meaning (e.g. inclusion), or worse still, may have no connection at all. We find the latter situation where the suppressed statement provided the only link between premisses and conclusion, as, for example, A may do in instances of A & B •*• A. By just such a use of suppression paradoxes are produced, as was argued. If the meaning connection is taken specifically to be one of inclusion of content, any suppression will lead to misrepresentation of that relation. For, as a matter of definition, suppression is incompatible with the sufficiency of premiss for conclusion. But sufficiency of premiss A for conclusion B is a necessary condition for A's containing or including B. For something X contains something Y only if X alone contains Y; if we have to add something further to X to obtain Y then X simply does not contain Y. A contains B only if we do not need to go beyond A to obtain B, i.e. if A is sufficient for B, B's consequences are a subset of A's consequences. But if the logical content of a statement is given by its logical consequences, the logical content of B is contained in that of A.1 In this way suppression leads to the misrepresentation not merely of the logical consequences of a statement, but also of its meaning. For we are then unable to differentiate it, in terms of its logical consequences, from other different statements. When we suppress a statement we discount its meaning, by treating it as adding nothing. No distinct statement can always be treated in this way. For suppose that some statement A is generally omissible. Then, using Conjunctive Simplification, A & B A, since A is generally omissible (by hypothesis), B implies A i.e. any statement is sufficient for A. Therefore A is included in the content of every statement. But it is reasonable to suppose that some statements are entirely disjoint in content; therefore this generally omissible statement can have no content. Hence it is not a distinct statement, whence, contraposing, no statement should be generally omissible. All these features of deducibility, then, provide reasons for saying that every statement sometimes occurs essentially and has its own bit to add. This leads to the Anti-Suppression Principle: for every statement p there is some statement q such that the consequences of q are a proper subset of the joint consequences of p and q. There is no privileged class of statements which are generally suppressible. 1

Similarly a watershed law such as A & B A satisfies both the inclusion and the sufficiency models for the same reason - that if the premiss A is sufficient for the conclusion A (as it is), it must remain so upon the addition of further premisses, just as it must continue to include the conclusion upon the addition of further premisses.

2

Or perhaps we should say underprivileged. For while the generally suppressible statement is privileged in being protected from rejection, it is underprivileged in that its meaning is discounted. 146

2.10 L0G1CO-TECHNICAL

REASONS OFFERED

FOR SUPPRESSION

VO NOT JUSTIFY IT

It should be evident by now that the technical shortcuts in argument suppression allows are indeed bought at a heavy price.1 Suppression does not give any more genuine reasoning power, because, as the characterisation indicates, any genuine results obtained by using suppression can also be obtained by making explicit the assumptions suppression permits to be hidden. This may be illustrated by a comparison of the real increase in reasoning power of the allegedly "stronger" first degree strict implication gives as compared with first degree entailment. Because, given Contraposition, first degree entailment differs from strict first degree at bottom only in disallowing the suppression of necessary truths, essentially the same results can be obtained by making these assumed necessary truths explicit in the argument. Thus deductive power for applied areas, such as mathematics, should be substantially unaffected. It is true of course that an implication which allows suppression of a class of statements will produce more theorems and thus more "results" than an otherwise similar one that does not (especially higher degree theorems), but in a good sense such an implication does not yield any further independent or applied results, only apparently varied results which are parasitic on the suppression principles themselves.2 A further reason for dropping suppression is the increased economy and generality to be obtained by doing so. The same class of genuine results can be reached with less in the way of assumption inputs. It is important to notice that this sort of notion of economy is different from that of the classical logician, who is inclined to praise the economy, simplicity and strength of classical systems. But his notion of economy is characteristically a trivial and syntactical one, economy of expression or minimisation of axioms (in number or length of symbols), which has nothing to do with true economy of assumption or content, or minimising on what is assumed. (To illustrate this, consider the assumption that everything is true, which from the point of view of the syntactical notion of economy and of minimisation of axioms is a highly economical assumption, but which, from the point of view of real economy, of minimising what is assumed, is a highly uneconomical one.) But it is the semantical concept of economy, that of economy of assumption in argument, that is wanted, and which has been traditionally seen as a virtue in most reasoning contexts.3 Suppression destroys this feature also, since in a suppression implication there are, at the first degree level, always some statements (the suppressed class) which are assumed to hold in every situation, and others which are assumed to hold in no situation. These statements thus become tacked on (either to the antecedent, or in the negative case to the consequent) as further assumptions in every argument, whether needed or not. For example, in the possible situations of strict implication, every necessary truth is assumed to hold in every situation, and thus is constantly, even if invisibly and usually unnecessarily, present in the assumption set for every argument. Thus every argument carries not only its 1

Considered as a piece of logical technology, suppression implications are logical equivalents of the SST, nuclear power, and other typical technological fixes in that they allow cleverness in technique to obscure and override the fact that very heavy costs are being incurred for an unnecessary item, and that there are sound, if less glamorously high-powered, alternatives.

2

This complex matter is discussed in more detail in the next section.

3

To say that minimisation of assumption is a virtue is not of course to imply that an argument which fails to so minimise is invalid, only that there is more to assessing the merits of an argument than validity.

147

2.10 THE

CLAIM

THAT SUPPRESSION

OF LOGICAL

LAWS IS INEVITABLE

REFUTEV

load of hidden necessary assumptions but also its load of unnecessary assumptions. This loss of discriminatory power means that the virtue of true economy, the use of minimal assumptions for a conclusion, can never be attained. An associated advantage obtained by dropping suppression is increased generality. A suppression-free logic is one which enables the making of a minimum of assumptions (logical and non-logical) for obtaining the conclusion, and minimal assumptions about the actual world and the situation or world in which the argument is applied. The class of situations in which an argument holds is larger, because the class is not restricted by further assumptions unnecessary to the argument, in contrast to the suppression case where situations contain further assumptions beyond those needed for the argument. So increased economy of assumption is accompanied by increased generality, and Increased range of application. Thus, the classical-style logician's prized "economy" and "strength" are in fact undesirable properties, incompatible with true economy of assumption and generality of application, that the supposed strength of classical logic, which derives from suppression, is in fact a major weakness. By way of summary, then, suppression creates many damaging effects, both for the primary interpretation of deducibility and for associated interpretations such as propositional identity and inclusion of content. With suppression such solid virtues of implication as economy of assumption, generality of application, assessability and explicitness are being traded in for superficial technical facility, which gives no genuine increase in reasoning power. Despite the evident disadvantages of an analysis of entailment harbouring suppression, it is sometimes suggested1 that the suppression of logical laws anyway is an inevitable and even harmless part of the deductive process, for it is accepted practice to use (and therefore to presuppose) logical principles in deduction without needing to state them as part of the premisses. It is true that it is a characteristic deductive procedure to use logical laws as principles of inference, as for example when we argue that this object's being red entails its being coloured, on the grounds of the truth of the principle that for any object, being red entails being coloured, but such use of a logical principle is not a case of suppression. The conclusion of the argument, that this object is red entails that It is coloured, is obtained by instantiation from the universal premiss P.

For every object, being red entails being coloured,

and not, as the argument that it is a case of suppression must suppose, by suppression of P in the argument (P and this object is red) •*• this object.is coloured.2 Entailments like the particularised "this object's being red entails its being coloured" are valid because they are instances of valid general laws, not because the suppression of necessary truths is permissible in valid argument. Application of logical laws as principles of inference is by instantiation, and an instantiated proposition cannot reasonably be said to be presupposed by the proposition which instantiates it.3 But since suppression is explained syntactically in terms of 1

See e.g. Bennett 69, p.230, and references therein.

2

Insistence on re-presentation in this form is a Lewis-Carroll fallacy, a matter already discussed in 1 (p.30).

3

For reasons elaborated in Routley2 70. First, the instantiated proposition implies its instance and not vice versa, whereas a presupposed statement is implied by the statement which presupposes it. Also if 'presupposes' were to cover instantiation, a true statement would be able to presuppose a false one.

148

2.10 TWO IMPORTANTLY

DIFFERENT WAYS OF USING A LOGICAL LAW

DISTINGUISHED

presupposition, instantiation does not count as suppression. Nor does it count as suppression under the semantical characterisation of suppression. The potential of a law for use as a principle of inference, its ability to support instancing, is modelled semantically not as the requirement that the law holds in every world, but as the requirement that every set-up conforms to the principle, i.e. that ho world can provide a counterexample to the law.1 But as the suppression of a statement is defined there must not only be no counterexample where the suppressed premiss holds, there must be a counterexample in some situation where the suppressed premiss does not hold. Yet no counterexample to a law, and therefore to its instances results even in those situations where the law does not hold. The difference between a situation's conforming to a law and the law's holding in that situation is the semantic reflection of the difference between the use of a law as a principle of Inference and its use as a premiss in an argument. It is not merely that such a distinction is formally viable, it is also clear in terms of the functions of entailment why it should be drawn in this way. Genuine suppression must lead to the stated premisses being insufficient for the conclusion on its own; in contrast an instantiation of a law where the antecedent is sufficient for the consequent does preserve this property. But the harmful effects of suppression on the functions of entailment discussed above derive from the insufficiency of the antecedent for the consequent. The confusion between these two ways of using a logical law has led to the idea that the suppression of necessary truths is justified where that of contingent truths is not. But the distinction, as well as being theoretically clear, is easily seen in actual examples. We may be entitled co argue that p Q => p Q , on the (unstated) grounds that A => A is valid, but we are not entitled to conclude that "the world is round ->- p => p" on the grounds that p => p is necessary and that "the world is round and p => p -»•. p => p". The latter case reveals that the suppression of necessary truths has as little to recommend it as the suppression of contingent truths. The main argument to the inevitability of suppression is not however from the alleged positive suppression of logical laws in deduction, but from a failure to discern negative suppression, from the failure to observe anything the matter with Disjunctive Syllogism and its mates, from the unavoidability then of the hard Lewis paradoxes, and from the fact that positive suppression principles can be demonstrated using contraposition principles given in the inevitability, harmlessness, or whatever, of negative suppression. And traditionally suppression was just positive suppression. The distinction between these kinds of suppression is important as is taking account of them both; for if one adopts the traditional account of suppression as a kind of incompleteness of the premiss (an enthymeme), which can be remedied by augmenting the premiss, negative suppression will be completely overlooked and only positive suppression recognised for what it is. And it is perhaps because Disjunctive Syllogism involves negative and not positive suppression that its suppression features have been overlooked even by those who, like Lewy and Moore (see Lewy 76 and Moore 62), have been worried by positive suppression. But negative as well as positive suppression produces paradoxes, and it is only possible to lay responsibility for the paradoxes where it belongs - to suppression - if both kinds of suppression are recognised. Both kinds of suppression entail a lack of sufficiency, a dependence on something other than the content of the antecedent, and suppression of both kinds is obnoxious to the cluster of associated interpretations which do require sufficiency of antecedent for consequent. 1

The distinction between holding in and conforming to is crucial as regards applications of the theory, e.g. in the philosophy of science.

149

2.10 POSITIVE

AND NEGATIVE

SUPPRESSION

ARE TWO SIVES OF THE ONE BAP COIN

The traditional concept of premiss suppression, that of omission of further

needed information which should be added, or conjoined, to the premiss set corresponds, as we have seen, to positive suppression. However, because, as remarked, traditional logic failed to allow for multiple conclusions, there was no easy way for traditional logic to formulate or recognise negative suppression as a variety of suppression or as the dual of positive suppression, and its omission from traditional consideration demonstrates again limitations of that approach. For the inability to take account of negative suppression in addition to positive suppression has given rise to many problems, especially concerning the origins and reasons for the paradoxes, not only for traditional logicians but for others, like the Cambridge philosophers, who are tempted to recognise positive but not negative suppression. The basic reasons for objecting to positive suppression are equally reasons for objecting to negative suppression, which is the dual of the positive notion, and every bit as bad. Given the resources of modern logic and formal semantics it is easy to see that they are two sides of the one bad coin. Not only are they normally connected through Contraposition, so that each is convertible into the other, but they have similar effects. For example, each is responsible for its own class of paradoxes, positive suppression for the positive paradoxes and negative suppression for the negative paradoxes. The process by which suppression produces paradox can be traced through in the way already illustrated. The failure to recognise or understand negative suppression meant that no complete or systematic account of the origin of the paradoxes could be obtained. Even if positive suppression was recognised as causally linked to the positive paradoxes, the fact that a class of paradoxes (negative paradoxes) was produced without use of positive suppression seemed to show that there was no real connection between suppression and paradoxes. Ignorance of negative suppression also led to the persistent failure to recognise the defects of negative suppression principles such as Disjunctive Syllogism, and their responsibility for the negative paradoxes. Thus the real cause of the paradoxes could not be understood, and the connection is still denied by those who fail to recognise negative suppression (cf. Lewy below), while the so-called "intuitiveness" of negative suppression principles such as Disjunctive Syllogism remain the main prop of paradoxical implications. In case it should be thought that negative suppression is a doubtful matter compared to the traditional notion of positive suppression, it is worth pointing out that the duality of positive and negative suppression is implicitly recognised in the strict position itself. Strict implication was arrived at by modifying material implication so as to disallow the suppression of contingently true propositions, considered by founders of strict implication to be illegitimate. But the strict implication position thus implicitly recognises negative as well as positive suppression, and the connection between them. For if material implication is modified so that only positive suppression of contingent truths is disallowed, the system arrived at is not strict implication but one of the intuitionistic systems (of 6.1). In order to obtain strict implication it is necessary to prohibit not only the positive suppression but also negative suppression of contingent falsehoods. This is reflected semantically in the structure of strict implication's possible worlds, which include not only worlds where contingent truths are not included (which eliminates positive suppression of these statements) but also worlds in which contingently false statements are included (which eliminates negative suppression of contingent statements). The counterparts of such anti-suppression mechanisms in the case of necessary truth and falsehood are, of course, the incomplete and impossible worlds of the semantics of (first degree) entailment.

ISO

2.10 THE ARGUMENT

FROM SUPPRESSION

TO STRICT

IMPLICATION

NOT FAR-REACHING

ENOUGH

A fundamental reason why we should consider incomplete and inconsistent worlds is that doing so eliminates the suppression of necessary truths and falsehoods, which is inevitable if we can consider only complete possible worlds. Suppression of necessary truths is eliminated by the consideration of impossible and incomplete set-ups in precisely the same way as assessment of implication over worlds other than T avoids the suppression of contingent truths allowed by material implication. For such worlds differ from the factual world T in that they may omit true contingent propositions and admit instead their false, contingent negations. Within the framework of world requirements, worlds exhibit maximum variation of their contingent members. It is this variation which guarantees the freedom from suppression of contingent truths achieved by strict implication, and the resulting individuation of contingent propositions. Assessment of implications in worlds (for instance intentional worlds) which vary in a similar fashion not only their contingent but also their non-contingent members, enables the privileges strict implication accords to contingent propositions to be extended to non-contingent propositions. The fact that strict implication thus implicitly recognises (the defects of) negative as well as positive suppression in the case of contingent statements raises difficulties for anyone who would argue for the suppression of necessary truths and falsehoods permitted by strict implication on the basis (i) of rejecting the notion of suppression altogether, since, as we have noted, it is built into the foundational theory motivating strict-implication, the' strict theory being implicitly based on such a notion in the contingent case, or on the basis (ii) of accepting prohibition of positive suppression of necessary truths but rejecting the notion of negative suppression as valid in this case. This latter position is sometimes assumed in the argument for strict implication based on the alleged unrejectability of Disjunctive Syllogism, since the latter is a negative suppression principle. It is also implicit in the position of Lewis, who rejected higher degree principles allowing the positive suppression of necessary truths but thought negative suppression, as exemplified in Disjunctive Syllogism, quite unexceptionable. Recently it has been given an explicit statement in Lewy 76, whose contention that Disjunctive Syllogism exhibits no suppression is either simply mistaken if a full account of suppression analogous to that implicit in strict implication is used, or else relies upon an account of suppression at the necessary level which is more restricted than that already employed by strict at the contingent level, i.e. as an account of positive suppression only. The position of those who would recognise the validity of the notion of negative suppression in the contingent case when motivating their choice of strict implication rather than material implication, but deny its validity in the necessary case when it looks as if it might reveal the inadequacy of strict implication, is, then, motivationally inconsistent. On the basis of the usual reasons for rejecting material implication in favour of strict implication, negative suppression must be seen as the dual of positive suppression, and as equally damaging. Lewy correctly arrives at the conclusion (76, p.113) that a possible, and prima facie plausible, solution to Lewis's positive paradox argument is to reject Expansion, A -*•. (A & B) v (A & ~B), on the ground that it illegitimately suppresses a necessary premiss, namely B v ~B.1 But Lewy thinks that this 'solution will not do in the end', because it does not work in the case of other paradoxes, in particular Lewis's negative paradox. 1

Anyone inclined to doubt the suppression story, should look at Levy's energetic account of the mechanism at work through Expansion.

15/

2.10 LEWY'S CASE AGAINST A SUPPRESSION

ANALYSIS

IS BASED ON MISTAKEN

DUALISATION

...if the trouble with Lewis's second [positive] proof is due to illegitimate suppression, then the trouble with his first [negative] proof must also somehow be due to illegitimate suppression since from the intuitive point of view it seems absurd to deal in one way with one of the proofs and in a radically different way with the other proof (76, p.113). Those who question forms of Contraposition, e.g. minimalists and certain dialecticians, would hardly agree, but let this pass. Lewy next claims that in the negative argument it is only Disjunctive Syllogism that can involve suppression: 'it is entirely clear that no illegitimate suppression, of any kind, is involved in any of the other premisses' (76, p.114). Connexivists might well dissent; but let that pass too. For the remarkable, and quite unjustified, move Lewy makes that we do want to contest is this:Now if Lewis's [positive] proof suppresses "B v ~B", i.e. the conclusion which he professes to deduce from his premiss, then his [negative] proof presumably suppresses the negation of his premiss, that is, [the] proof presumably suppresses "~(A & ~A)" (76, p.114). Lewy has gone well off course. In the positive paradox suppression is removed by conjoining the consequent, B v ~B, to the antecedent. Thus dualising, suppression in the negative paradox is removed by disjoining the antecedent, A & ~A, to the consequent, giving A & ~A B v (A & ~A). Lewy has presumably latched onto the negation of the premiss, A & ~A, rather than the premiss itself, because he did not see how impossible statements could be suppressed. Moreover the correct dualisation leads to precisely the expected duality between Lewis's negative and positive paradox arguments, not to Lewy's disanalogies. And so correcting suppression in the first Lewis proof, which Lewy rightly locates in Disjunctive Syllogism, leads not to Lewy's unsatisfactory result, but to rejection of Disjunctive Syllogism itself. Ltet us consider Lewy's several observations in turn, wirh a view to rectifying them. The main observation concerns paradox avoidance through the incorporation of suppressed statements: ...if the two proofs are entirely parallel, we ought to be able to avoid paradox in the [negative] proof by adding the allegedly suppressed [formula] (76, p.114). Lewy (wrongly) conjoins ~(A & ~A) to the antecedent of ~A & (A v B) -*• B, and shows that the result of doing so, i.e. ~(A & ~A) & ~A & (A v B) B, still leads to paradox. But proper rectification of the fault disjoins A & ~A to the consequent of Disjunctive Syllogism, yielding ~A & (A v B) B v (A & ~A). And this repair destroys Lewis's negative argument. That no paradox can be derived upon replacing Disjunctive Syllogism by the corrected form in the negative paradox follows at once from the facts that the corrected form is a theorem of DML and that DML is free of paradoxes which violate weak relevance. Thus the short answer to Lewy's question (p.115) as to why the replacement of Expansion by its corrected expansion in the positive paradox leads to no paradox, while his replacement in the negative paradox still leads to paradox, is that he has made the wrong replacement, and more generally, that he has failed to grasp the phenomenon of negative suppression, of suppression of impossible propositions in conclusion sets. For similar reasons, Lewy's attempted (if somewhat obscure) 3-valued explanation of the difference misses the point.1 For given the correct repair to Disjunctive Syllogism, the positive and negative cases fare in exactly the same way with respect to the 1

The reason why the 3-valued Lukasiewicz matrices do make the requisite discrimination is given in 5.3. At bottom it is that the third value of can represent inconsistency as well as incompleteness. 752

2.10 THE IRRELEVANCE

OF THE PRINCIPLE

OF BIVALENCE

TO DISJUNCTIVE

SYLLOGISM

3-valued Lukasiewicz matrices Lewy adopts. Neither Expansion nor Disjunctive Syllogism are 3-valued tautologies. But both their suppression-free repairs are, i.e. ~A & (A v B) -»-. B v (A & is a 3-valued tautology whereas Lewy's ~(A & ~A) & ~A & (A v B) ->• B is not. The further upshot is that Lewy's location of the significance of his findings is seriously astray. Contrary to Lewy, formula suppression does occur in the negative paradox; A & ~A, which is in fact the "premiss" in Lewis's negative paradox, is suppressed in the paradox argument, only negatively. Contrary to Lewy, the negative paradox does not "suppress" the metalogical principle of bivalence ('that every proposition has one and only one of the two truth-values, truth and falsity', p.116). It is not difficult to guess the (quite invalid) route by which Lewy reached his conclusion regarding metalogical principle suppression: he leapt from the breakdown of Disjunctive Syllogism in one 3-valued case, to failure of bivalence as a necessary condition! It is not: Disjunctive Syllogism is rejected in relevant logics which retain bivalence. (Lewy appears to totally neglect bivalent intensional logics, though strict implicational logics are among them.) Accordingly bivalence is not sufficient for Disjunctive Syllogism. Lewy's claim that Given the principle of bivalence, every step that Lewis uses in [the negative] proof is an intuitively acceptable entailment step (p.116), is accordingly not just false; it is unfounded. Had Lewy got suppression right he would have found that Disjunctive Syllogism was a negative suppression principle, since A & ~A is suppressed in A & (~A v B) -*• B (because A & (~A v B) -*•. B v (A & ~A) distributing, but not A & (~A v B) B); and so, for this reason among others, the principle is rather far from generally intuitively acceptable. And he would not, presumably, have gone on to make the exaggerated claims he makes on behalf of Disjunctive Syllogism. He contends, firstly: The only reason one could give [as to why Disjunctive Syllogism is not a true entailment proposition] is that true entailment propositions must not assume the principle of bivalence (p.116). The only reason? This should be enough to make one suspicious of Lewy's repeat of the Lewis sufficiency-of-bivalence thesis. And in fact we have already indicated major reason why Disjunctive Syllogism is not valid, namely (bivalent) semantical arguments, and a case based on illegitimate suppression. These points also refute Lewy's second contention, that The most one can say against [Disjunctive Syllogism] is that we may wish so to define "A entails B" that B should follow from A with the help of the smallest possible number of extra premisses or principles (pp.115-116). There is much to be said against Disjunctive Syllogism (see below), but this is not one of the telling points. For Lewy's suggestion would write off most of the alternatives to strict implication - including relevant logics, which accept such inferences as A, A -*A - along with Disjunctive Syllogism. %11. Relevant logics and normal relevant logics characterised; the case against Disjunctive Syllogism and Rule Antilogism elaborated; logical tradition and intuition further consideredj and the coherence of refined intuition; and relevant resolution of the Lewy paradoxes. In virtue of the theory of suppression, we can finalise the conditions we choose to impose upon relevant sentential logics, in terms of systems DLL and DML: A relevant logic is a weakly relevant logic which conservatively extends DLL. Thus DLL and DML are relevant logics, and so are such systems as E, R and T (of ABE) and P (of Arruda-da Costa). But analytic

753

2.11 CONNECTIONAL,

RELEVANT,

ANV NORMAL RELEVANT

LOGICS

CHARACTERISE!?

implicational systems and weakly relevant connexive systems are not relevant logics in the narrower sense because, though weakly relevant and so relevant in a wide sense, they do not conservatively extend DLL.1 And the I systems (of chapter 7) and Mingle systems such as EM and RM+ (of ABE) are not relevant logics because, although they conservatively extend DLL, they are not weakly relevant. That is, conservative extensions of DLL and weakly relevant systems only properly overlap. Systems that are not relevant but are closely related to relevant logics, such as I and RM, have of course provided much valuable information regarding relevant logics. But that does not render them relevant, or even "semi-relevant". We adopt the narrower characterisation because what we are especially interested in are weakly relevant implicational logics which are suppression-free at least at the first-degree,2 i.e. logics which explicate implication as sufficiency and accordingly have a transitive reflexive implication which eschews suppression (see p.80). Even so the characterisation of relevant logics proposed has a stipulative element, converting certain necessary conditions for a relevant logic into necessary and sufficient conditions. Alternative conditions could have been chosen, in particular the lower bound DLL could have been selected differently, e.g. as the system DML or as the first-degree system FD, and most logics so far accounted relevant logics would have been included. Most, but not all: for logics, sometimes accounted relevant, with non-normal negations, such as relevant versions of intuitionist and constructive logics and of Latin-American paraconsistent logics (studied in chapter 6) would be excluded. One of the advantages of the wide characterisation proposed is, then, that constructive logics, which are weakly relevant, are relevant logics, and that relevant rivals to Heyting*s intuitionistic logic are relevant logics, relevant intuitionistic logics. Moreover there is nothing to stop, and much to recommend, sub-classification of relevant logics, so the sorts of relevant logics can be distinguished. The relevant logics on which we concentrate, and which we are particularly concerned to defend, are normal relevant logics, i.e. relevant logics which conservatively extend (and so include) the system FD of first-degree implication (studied in 3.1 ff.). FD is a minimal system as regards negation-normal relevant systems which retain truth-functional logic, and it is a watershed system; for it keeps in, as well as all classical tautologies, the right principles, especially such system distinguishing principles as Simplification, Addition, Distribution and Rule Syllogism, and it keeps out the bad guys, most notably among variable-sharing principles, Expansion and Disjunctive Syllogism. The substantial case against these bad principles is not that each leads, in the logical context of DLL, to violations of weak relevance, though this is true. They are not thereby rendered "fallacies of relevance", without firstly the (correct) assumption that DLL is correct, since in different frameworks they do not lead to irrelevance (e.g. Disjunctive Syllogism in connexive settings), and secondly the (controversial) assumption that violations of relevance are damaging, so damaging as to constitute fallacies (see 5.2).3 The substantial case does not 1

Nor are extensions of systems such as P-W-I to include &-v-parts (cf.11.3); for P-W-I lacks Identity. Systems relevant in the wide (weak) sense are sometimes referred to as connectional or connectedness logics.

2

In fact suppression-freedom guarantees weak relevance, which can accordingly drop out of the characterisation.

3

The case against Disjunctive Syllogism, like the case for the normal negation rule, spreads over several sections. Nor are the cases so far elaborated confined to this text. The argument concerning the *-rule spills over into NC, and the argument against Disjunctive Syllogism extends into V. Routley 67, (footnote 3 continued on next page)

154

2.11 INTERVERIVABILITY

OF OTHER BAP GUYS WITH DISJUNCTIVE

SYLLOGISM

concern relevance, preservation of which is a derivative matter. The main case is, on the syntactical level, that these principles are suppression principles and destroy sufficiency, a fact reflected on the semantical level by restriction of the situations that ought to be considered in the evaluation of entailments, to possible worlds. The restriction occurs because, when only possible worlds are considered, necessary truths (let D be a representative example) hold everywhere, and impossibilities (e.g. E) hold nowhere, so the semantical analysis is unable to discriminate between A - * B , A & D - + - B , A ->• B v E, A & D BvE. In short, the analysis is then unable to detect positive suppression of necessary truths such as D and negative suppression of impossibilities such as E. A further explanation of positive and negative suppression, and of what is wrong with suppression, will occupy us shortly; but first we want to present the semantical case against the bad variable-sharing principles. Within the framework of DML, which is accepted as sound if incomplete by the orthodox opposition and indeed by most of the opposition, Expansion, Rule Antilogism, and Disjunctive Syllogism are derivationally equivalent, so detailed discussion of what has seemed the hardest of these bad principles, Disjunctive Syllogism, should suffice. And the semantical points made against Disjunctive Syllogism are readily applied against the other principles. Derivational equivalence of the principles is established by the following derivations, which maintain relevance line by line:a d A & B - » - C - » A & ~ C - > - ~B, using Disjunctive Syllogism. By Rule Contraposition, A & B - > - C - « ~ C ^ ~(A & B). Hence ~(A & B) v ~B, by T5. But A & B - * C - » C - * . ~ A v ~B, using Rule Syllogism; so A & B -»• C -»A & ~C A & (~A v -fi), by Rule Factor. (Rule Factor is derivable using Identity, Simplification and &-Composition.) But A & (~A v ~B) ->• ~B, since Disjunctive Syllogism is given. The result then follows using Rule Syllogism, ad A & (~A v B) -»• B, using Rule Antilogism. By Distribution, A & (~A v B) (A & ~A) v (A & B). Since by Simplification, A & ~B •*• A, by Rule Antilogism, A & ~A — B . But — B -*• B, so by Rule Syllogism, A & ~A B. Also A & B -* B, so by v-Composition, (A & ~A) v (A & B) B, whence the result follows by Syllogism again, ad A & (~A v B) •* B, using Expansion. By Expansion, ~B -»-. (~B & A) v (~B & ~A). But ~B & ~A ~A, so by v-Composition, Identity and Rule Syllogism, ~B •*. (~B & A) v ~A. Hence, by Rule Contraposition, Identity and Rule Syllogism, ~B -»•. (~B & A) v ~A. Hence, by Rule Contraposition and de Morgan and Commutation principles, A & (~A v B) B. ad A ->-. (A 6. B) v (A & ~B), using Disjunctive Syllogism. ~A & (~A v B) -> ~A by Simplification, and ~B & (~A v B) ~A by Disjunctive Syllogism; so by v-Composition (~A & (~A v B)) v (~B & (~A v B)) -»• ~A. Hence by Distribution, (~A v ~B) & (~A v B) ~A. Finally by Rule Contraposition, Double Negation, de Morgan principles, and Rule Syllogism, Expansion results. (An, irrelevant derivation, which first derives A B v ~B from Simplification by Rule Antilogism, etc., is undoubtedly shorter.) The semantical falsification of Disjunctive Syllogism reveals at once how it fails in inconsistent situations, and why it is quite correctly - contrary to classical perceptions - rejected as invalid. Let d be some non-trivial inconsistent situation where both A and ~A hold, but, since non-trivial, some B does not. Situation d at once counters Disjunctive Syllogism: since both A and (~A v B) hold at d but B does not, by the entailment rule, B is not deducible from (footnote 3 continued from previous page) which contains a neat argument of the following form:- By the arguments of chapter 1, ^-methods are not adequate for deducibility. But what DS tells us is that a "proof" of B from A by these unsatisfactory methods together with A itself is sufficient for a proper deduction of B. So DS is incorrect. 155

2.11 SEMANTICAL

AND INTUITIVE

FALSIFICATION

OF DISJUNCTIVE

SYLLOGISM

A & (~A v B). Thus it suffices for the falsification of Disjunctive Syllogism that there are non-trivial but inconsistent deductive situations and theories. But such theories and situations abound. Many of the major philosophical theories of all time are of this kind, as are leading mathematical and scientific theories (on both claims see PL and UL). And lesser theories and systems of the same kind are even more plentiful, and readily accessible. Consider, to illustrate, the system DML° obtained from DML by adding as further axioms the theses, concerning constant object A^, AQ and ~AQ. Then DLM° is certainly inconsistent; but it is not trivial, because an arbitrary B is not derivable, as the semantics reveal. Indeed A^, ~A j-B is not derivable; and hence DML° serves to falsify AQ & (~AQ

v

®>

an

instance of Disjunctive Syllogism.

It is too often and too quickly claimed that there are, and can be, no intuitive counterexamples to Disjunctive Syllogism.1 Similarly it has been claimed that Disjunctive Syllogism, and its mates such as A 4 ~A + B, cannot be invalid in the sense of permitting inferences from what is true to what is false (cf.1.6). All this is mistaken - as semantics for relevant dialectical logics such as systems DL and DK (of 6.5) quickly reveal. Deductive situations like d are easy to envisage; indeed theories and rationally-held belief sets of this type are of not uncommon occurrence. Consider the following case which provides a pattern for a whole range of intuitive counterexamples:- Let R be the Russell set, the class of all those classes which are not members of themselves. Let A be the statement that R e R, i.e. that the Russell set belongs to itself. Then ~A is that statement that R i R, i.e. that the Russell set does not belong to itself. By the intuitive paradox argument - convincing to very many of the untutored and to a growing number of the tutored - both A and ~A are true. Let B be any false statement; e.g. that there is no ice at the South Pole. Then, by the semantical rules for & and v, A & (~A v B) is true while B is false. So Disjunctive Syllogism is false. Even if one disputes these truth value assignments, as one reasonably may, it is hard to deny that such assignments may be made in (perhaps misguided or mistaken) deductive theories - and that is enough for intuitively convincing counterexamples. To be specific, let d be a situation generated by a dialectical set theory, containing both A and ~A as theses, which is non-trivial because of the non-provability of some statement B' (e.g. A e A); that is, d is a situation where both A and ~A hold but B' does not. Nor is it just that situations like d where naive dialectical set theories hold are readily envisaged and reasoned within and about; it can actually be proved that there are such non-trivial dialectical set theories (see Brady and Routley 78, Brady 79). In a similar way, intuitively appealing counterexamples to Disjunctive Syllogism are a by-product of the theoretical admissibility of semantically closed languages (of the kind English to all appearances exemplifies). For such 1

Thus for instance in Bennett 69, e.g. pp.213-4 where it is said that 'the search for arguments' against the negative paradox A & ~A -*• B 'has been underway for decades, and has so far most miserably failed'. It no longer fails, now that paraconsistent logics have been clearly discerned. In any case, it depends what one is prepared to count as argument. Similarly Geach implies that there are no 'actual or even possible counterexamples' to Disjunctive Syllogism (77, p.495). What little argument he offers for this erroneous proposition Inadmissibly assumes, however, that negation is classically exclusive, so that where ~A holds A cannot hold.

156

2JJ

FURTHER INTUITIVE

COUNTEREXAMPLES

TO DISJUNCTIVE

SYLLOGISM

languages admit the semantical paradoxes yet are not trivial; and again the non-triviality of formal languages of this semantically closed sort can be demonstratively established. Naturally there is no need to go as far afield as the logical and semantical paradoxes in seeking examples of situations which are inconsistent but non-trivial: though the paradoxes provide appealing and striking examples of inconsistent situations, equally appealing examples can be designed which deploy inconsistent, but non-paradoxical, theories or inconsistent belief systems (a range of such examples may be found in Routley2 75). Several such examples have already been indicated (in 1.7) - physical theories such as dialectical quantum mechanics, mathematical theories such as those of infinitesimals, philosophical theories such as those of Hegel and of Engels - but more philosophically mundane examples are easily recalled; for example, real-life situations involving inconsistent obligations (see Lemmon 65, and also 8.3 and 8.4), and situations where inconsistent laws or directives hold, or again the bizarre world of absolutely naive realism where all deliverances of perception are taken as veridical, e.g. the stick in water is both straight (by tactual perception) and also not straight (by visual perception), though of course not everything holds.1 These examples reveal some of the sorts of situations which anyone, rightly uneasy about the negative paradox, A & ~A -*• B, should be led to expect, once she has grasped the semantic rule for entailment, namely that A & ~A •*• B fails to hold in virtue of deductive situations, like d, where A & rA holds but B does not. But the situations which so result - there are commonly many such where B is irrelevant to A - serve, given only normal composition and disjunction rules, to falsify Disjunctive Syllogism. The lynch-pin of all the arguments that strict implication is entailment or, less extravagantly given the flawed character of strict implication, the best rational reconstruction of entailment - is that rejection of Disjunctive Syllogism, even if now contemplatible, is highly counterintuitive. The semantical argument presented is designed to show, however, that the rejection is not at all counterintuitive, but is perfectly natural once it is observed that the assessment of entailment is not restricted to negation-consistent possible worlds but involves inconsistent situations. It is then obvious in fact that the principle is false. Let us record, too, that we do not find Disjunctive Syllogism intuitively correct or convincing, and that we do not find - for that matter, some of us never have found - its rejection in any way counterintuitive. So let us hear no more such unsupported armchair assertions as that 'we all feel a strong inclination to accept Disjunctive Syllogism'; for, as an empirical claim, this is false even of unrefined intuition. It remains to explain however why some find Disjunctive Syllogism intuitively compulsory and its rejection so counterintuitive. The real answer appears to lie in a narrowness of vision, in the restriction of the situations envisaged, over which entailment is taken to be assessed, at least to the consistent, and commonly, under referential pressures, to the factual world T only. But, as we have argued, there is no good reason for such restrictions, and a solid case against them. These points are explained in more detail in EMJB, where it is argued that what is deductively assessible, like what is conceivable and what is the object of intuition, is by no means restricted to what is consistent. 1

How material objects really are, i.e. are in T as distinct from the worlds of immediate perception, is determined by a none-too-clearly articulated theory, which, among other things, consistencizes perception in a way characteristically based upon stability (or qualified invariance) of objects under a class of admitted transformations (e.g. physical transfer, daytime change of illumination, temperature, humidity and wind, etc.).

157

2.11 DEMOLISHING

THE ARGUMENT

FROM TRADITION

FOR DISJUNCTIVE

SYLLOGISM:

THE

REASON

Not unexpectedly however, there is an extreme reluctance among classicallyoriented logicians to give up Disjunctive Syllogism and Rule Antilogism, or to even contemplate their invalidity. Various specious or defective arguments are dragged in in their defence. Thus for example, the argument from tradition, and from the historical record, for Disjunctive Syllogism and Antilogism is all too frequently encountered. The rebuttal of the argument is three-pronged. Firstly, Disjunctive Syllogism does not, contrary to common assumption, enjoy an undisputed or unblemished historical record. Tradition is far from as uniform as it has often been represented. Nor is tradition static. Material implication, now so fashionable, was a decidedly minority position during the heydays of Megarian and Stoic logics, and what amounts to something like strict implication appeared to be only one of several competing analyses of validity of argument. Though in medieval logic strict implication appears to have become the dominant position it remained only one of several positions and by no means went unchallenged. The paradoxes of implication 'generated as much controversy among medieval logicians as they have done among modern logicians' (Moody, 75, p.374). Most important, Disjunctive Syllogism was rejected as invalid1 by several logicians. The extent of this historical rejection is only beginning to emerge, largely because historical research has concentrated on seeing past logicians as anticipating modern orthodoxies (but partly because, especially as regards classical times, nonstandard work has not survived), and research into the history of non-standard logic (and the important obligationes-literature) is only in its infancy.2 Perhaps most striking because it is for precisely the right sort of reason and anticipates recent relevant thinking by almost 5 centuries - is the rejection of Disjunctive Syllogism by the Cologne commentators, who in 1493 said that even though every step in the derivation of B from 'A & ~A' [i.e. in the "hard" Lewis argument] is formal, one cannot accept the last step [which uses A v B, ~A (- B]. This is because 'A & ~A' can be taken in two ways, absolutely, as a virtual contradiction, or for the sake of the argument. In this proof, it is accepted for the sake of the argument, and since both A and ~A have been conceded, one cannot use one part of the formal contradiction to deny the other part. That is, 'A v B, ~A (- B' has to be rejected! (Ashworth 74, p.135)3 On its own, in the factual world, A & ~A cannot hold, but for the purposes of the proof situations other than the factual have to be considered; in particular, for the hard paradox argument, situations where both A and ~A are conceded. But in such situations ~A does not eliminate A, in the classical fashion. Hence what can be concluded from either A's holding or B's holding is no more than what is already conceded, that A holds; certainly not that B holds. Apparently both Javellus and De Soto took similar positions; De Soto 'also appealed to common usage to support his doubts about the paradoxes: who would say that if you are a stone, it follows both that you are and that you are not' (Ashworth, p.135). Both the Cologne commentators and Javellus also rejected (according to Regius) the view that anything is deducible from the impossible, 1

More exactly, the consequence relation, modus tollendo ponens, represented ~A, A v B B, was denied formal validity.

2

For a more detailed history of connectional and relevant logics, which also expands what follows In the text, see tne newer papers of DRL.

3

The revealing exclamation mark indicates Ashworth's view of the matter. are fortunate that she recorded the Cologne position.

158

We

2.11 WHY THE ARGUMENT

FROM TRADITION

IS OF COURSE FALLACIOUS

on the grounds that in a case such as "Man is an ass, therefore the stick is in the corner", there is no relation of dependence between the propositions, and the negation of the consequence is not incompatible with the antecedent (Ashworth, p.136; italics added). The grounds given are absolutely right and reflect precisely the themes of the third position on the nature of conditionals (from the famous debate in antiquity about conditionals) presented almost 2000 years earlier, by those who introduce the notion of connexion [and] say that a conditional is sound when the contradictory of its consequent is incompatible with its antecedent1 and who also rejected disconnected conditionals with (temporally) necessarily false antecedents or necessarily true consequents. It would be surprising if there were not a more continuous tradition representing these views. Secondly, tradition is as poor and inconclusive - or as adequate - an argument for the retention of a logical principle as it is in any other area. Logic is not a closed shop, and the business of logicians, their elegant and miserable codifications and logical principles, are just as much open to criticism and discussion and just as little settleable by the weight of numbers or democratic vote (whether among present or past philosophers), or by historical track record, as philosophical ones and scientific ones. Logicians should no more allow themselves to be browbeaten by tradition than philosophers or scientists. In this case, moreover, there is good reason to believe that mainline traditional adherence to Antilogism and Disjunctive Syllogism may be a collective blind spot, based on a confused account of negation. There are plenty of examples of arguments which are traditionally sanctioned or which have widespread intuitive appeal, but which are invalid (Aristotle, ~(A •*• ~A), is just one example). It is not that tradition shows nothing - standing the test of time is something - but it is by no means a guide to correctness, as long standing but mistaken scientific theories show. The "classical" accounts of validity and deducibility, in terms essentially of (first degree) strict implication are of this sort: of long but not uncontested standing, and mistaken. Thirdly, there were competing traditional themes. In this area tradition is not even a consistent guide, since the traditional adherence to principles such as Antilogism and Disjunctive Syllogism commonly conflicts with other more basic traditions, such as the prohibition of suppression, the containment picture of deducibility, and the requirements for soundness or validity of connection or relevance. Each of these traditional requirements2 conflicts with the classical accounts, which Disjunctive Syllogism is commonly taken to secure (and does secure given DLL). Thus arguments from tradition to the classical accounts, or 1

Sextus Empiricus, quoted by Kneale2 62 p.129. Kneale2 suggest that the third position may be that of the Stoic Chrysippus on the ground that Chrysippus is said to have adopted the account of soundness in terms of incompatibility. But that account is open to a range of positions, including paradoxical positions such as strict theories, which eliminate (or trivialise) connexion. There is however better evidence that Chrysippus adopted the third position, as well as some competing evidence that Chrysippus*s theory of implication collapsed to a strict one. These matters are elaborated and straightened out in the newer papers of DRL, where too the popular view that the third position is one of strict implication is criticised and rejected.

2

The traditions are of varying age, the containment picture being apparently of much more recent origin than the other requirements which go back to classical antiquity.

159

2.11 MAJOR WEAKNESSES

JN THE ARGUMENT TO DISJUNCTIVE

SYLLOGISM

FROM

INTUITION

to the acceptability of deducibility paradoxes, are broken backed. Consider, for example, the traditional containment models for deducibility, according to which the conclusion of a valid argument is contained in the premisses. With a few plausible and historically widely accepted assumptions, such as that necessary propositions are not contained in every proposition, such models for deducibility are inconsistent with maintaining Disjunctive Syllogism (see below or FD). Much the same applies in the case of the traditional prohibition on the suppression of premisses, including necessary premisses, clearly stated by Aristotle, and an important part of traditional theory. Insofar as much traditional theory also accepted Simplification and, with Aristotle, Antilogism, that theory is patently inconsistent.1 This suggests that traditional acceptance of Disjunctive Syllogism and Antilogism is partly based on a failure to appreciate what they really do. Only the argument from intuition, from the intuitiveness of Disjunctive Syllogism, rivals the argument from tradition in popularity. But intuitiveness is as'weak a reed as tradition. For, firstly, there are so many examples of intuitively appealing arguments, even traditional arguments, which further reflection, refined intuition, or investigation, has revealed to be specious, even pernicious, that it is astonishing that such a reason as intuitiveness should be advanced as justifying the retention of principles, such as Antilogism and Disjunctive Syllogism, which can so clearly be shown to be implicated in paradoxical consequences. One has only to think of the readiness with which people embrace as obvious not only philosophical falsities, but such fallacious though appealing principles as the Monte Carlo fallacy, not to mention such arguments as Zeno's paradoxes and Kant's antinomies, to-realise how fallible a guide "intuition" may be. Nor can the paradoxes of implication - any more than these aforementioned paradoxes seriously be accepted as intuitive but just surprising (pace Lewy 76, p.132). There is a second, and really .serious, weakness in the argument from intuition. The alleged intuitiveness of Disjunctive Syllogism depends on a consistency assumption regarding worlds, a theoretical assumption, which remains unexamined, but when exposed turns out to be eminently rejectable. Once this is seen, and it is seen that the underlying assumption is decidedly theoretical and at the very least in doubt, the intuitive appeal of the principle tends to evaporate. The certainty often felt about the principle depends upon taking for granted questionable assumptions, upon living the logically unexamined life. Since Disjunctive Syllogism does stand or fall with a high level theoretical assumption, it cannot, with any methodological soundness, be guaranteed by appeal to largely inaccessible private certainties such as intuitions, but should be assessed on the basis of the sorts of general theoretical and public considerations already introduced which lead to its rejection. To realise, as anyone who has proceeded beyond the elementary stage of philosophy must, that intuition is no infallible guide, is not of course to say, with the Quine and Co., that it is bankrupt.2 That is too like saying that 1

The inconsistency does not damage Aristotle's position, given - what independent evidence seems to show - that his position was a connexive one (which qualified Simplification). Indeed, since Aristotle took the avoidance of contradiction to be logically fundamental, this contributes further evidence for the hypothesis that Aristotle's working logic can be given a connexive interpretation. An alternative story floated in NC, for which there Is some evidence, is that Aristotle's position vacillated between a connexive and a classical one. Such vacillation helps to account for competing traditional (Aristotelian) logical theories and the ongoing confusion of classical and connexive logical theories.

2

The bankruptcy charge is pragmatically self-refuting when advanced by those who defend Disjunctive Syllogism, or say classical logic, by appeal to intuition.

160

2.11 REFINED

INTUITION

A RELIABLE

GUIDE; DISJUNCTIVE

SYLLOGISM

A MODAL

ILLUSION

weather forecasting is no different from astrology on the grounds that the former is by no means infallible. In such areas as the logic of implication, intuition tends to be specially fallible, and the feeling that a particular principle is innocent must always be treated with caution and tempered by examination of its logical and semantical behaviour. Even so intuition remains a guide, if a quite fallible one, a guide moreover that can be much refined in the way attempted in what follows (especially chapter 3). The familiar comparison of intuition with perception (made, e.g., in Nowell Smith 54) helps rectify the perspective. We don't accept all the deliverances of perception, which sometimes provides apparently inconsistent data: because of illusions, etc., unqualified perception is not an absolutely reliable guide. But the fact that perception can sometimes lead us astray, doesn't at all imply that it should be e;iven up as a method of discovery or confirmation, or is bankrupt. Perception can be crosschecked, errors can be filtered out, and it can be improved and extended by instrumentation and other techniques. So in analogous ways, can intuition be adjusted and refined. It takes but little if any adjustment to see that Disjunctive Syllogism is a modal illusion, in contrast to other principles of DML which are intuitively correct. Lewy has, however, gone so far as to claim that 'our intuitive, that is, preformal, concept of entailment is inconsistent' (76, p.132). It is indeed worse than that; for on Lewy's view our intuitive concept is trivial, any proposition entails any other. Two of Lewy's array of arguments for his inconsistency thesis have already been faulted. One argument is simply that in the case of the negative Lewis paradox we are all led intuitively to accept the premisses but reject the conclusion. As we have seen, this is false: enough of us are not so led, and, more revealing, some were and are inclined to question or reject premisses of the Lewis argument without being aware of their paradoxical fallout. Another argument, no more satisfactory, is based on the alleged failure of the intuitive notion of suppression to resolve the paradoxical situation in a consistent fashion. In his later attempt (p.153-4) to elaborate this argument by showing that there can be no 'satisfactory reconstruction of the intuitive concept of entailment' by 'putting a "sieve" on strict implication', in effect to exclude all cases of suppression, Lewy goes seriously astray. Not only is he in error in claiming that one case required for such a sieve is of a radically different sort from the other cases, which forbid (positive) suppression, and 'can only be excluded by forbidding suppression of a metalogical principle (viz. the principle of bivalence)' (p.153, first italics added) given that the required exclusion can be effected by prohibiting (partial) negative suppression, which is the dual of (partial) positive suppression; but also he is mistaken about the character of Anderson and Belnap's theory of entailment, which he supposes involves 'rejection of the principle of bivalence' (p.135, also p.154). Moreover, Lewy's own work has been seminal in the design of a sieve for entailment, which strains out from provable material implication, in a natural way, exactly the theory of first degree entailment, i.e. exactly Anderson and Belnap's tautological entailments. Lewy defined a relation -**• of "strict entailment" as follows: A-»-B iff A => B is a substitution instance of some tautology A' => B' with ~A' and B' purely contingent, i.e. essentially by replacing 'contingent' in the exactly analogous definition of Smiley entailment by 'purely contingent' (see pp.150-1); and he conjectured that strict entailment is transitive and does characterise tautological entailment. There are however no purely contingent wff in the intended sense (as Clark 80, Dunn 80 and Urquhart have each shown), with the result that Lewy's first conjecture is vacuously true and the second immediately false. Clark and Dunn have independently repaired Lewy's definition in such a way that Lewy's conjectures

161

2.7 7 ARGUMENTS

THAT THE INTUITIVE NOTION OF ENTAILMENT

IS INCONSISTENT

REFUTED

prove correct, that is, strict entailment as reconstructed does provide a sieve for first degree entailment.1 Lewy's "additional" case for the inconsistency of the intuitive concept of entailment fares no better. The argument is this:... consider the following three propositions (1) S-necessitation [i.e. Smiley-entailment] is a sufficient condition for "A entails B" ; (2) Entailment is unrestrictedly transitive; (3) It is not the case that (A & ~A) entails B. This triad is patently inconsistent; yet we feel a strong inclination to accept all the three propositions. ... each proposition brings out an intuition we have in the area, but these intuitions are inconsistent (76, p.133). We feel no inclination to accept (1), especially given the new sieve which rectifies Smiley-entailment (an early attempt at a sieve) along with Lewy's strict entailment. The new sieve does not upset premisses (2) and (3) at all, and provides an appropriate modification of (1). Nor could Lewy's argument for (1) persuade us otherwise, unless we granted him - what is fallacious - the conversion of an A-proposition. Furthermore (1) has considerably less claim than (2) and (3) to be intuitively based; indeed we suspect that for very many philosophers (1) is not intuitively assessible at all, in view of the character of Smiley-entailment (discussed on pp.76-8) and the way all substitution-instances of complex entailments have to be grasped in assessing (1). Thus (1) does not bring out an intuition most of us have concerning entailment. Were, however, 'sufficient' in (1) replaced by 'necessary' it would yield a proposition that many of us would be led, by a proof (such as Lewy indicates) to accept. The proof simply does not convert to yield a sufficient condition. The remaining part of Lewy's case for his inconsistency thesis is based on paradoxes he himself has devised - what we shall call Lewy Paradoxes. His argument, if filled out, would no doubt be that, as with the negative Lewis paradox, we are intuitively inclined to accept all the premisses and argument steps in these paradoxes but to reject the conclusions. What we will show, by examination of key steps in the Lewy paradox arguments, is that essentially the same reasons that led us to reject Disjunctive Syllogism and its mates lead us to reject Lewy's steps. In short, the general, intuitive, theory already reached (and reflected in the formalism of DML at the first degree entailment stage) is enough to undermine Lewy's argument, to show not only that we are not all inclined to accept the argument steps, but that we ought not to be inclined to grant the steps. Consider, to begin, Lewy's paradox 1 (p.117 ff.). In barest outline the argument is that (A) and (B) jointly entail (C), but (B) is necessary, so, what is certainly paradoxical, (A) entails (C), where (A), (B) and (C) are the following statements:

Clark's repair (80, p.10), which resembles the Lewy original in being purely syntactical, has apparently been endorsed by Lewy. Dunn's reconstruction (80, p.49), in terms of truth and falsity trees, though equivalent to Clark's, is somewhat more elegant and suggestive of extensions beyond first degree sentential applications (for truth and falsity trees generalise to semantic tableaux, as developed subsequently). 762

2.11 DISMANTLING

ARGUMENTS

FOR THE INCONSISTENCY

THESIS FROM LEWY

PARADOXES

(A) The proposition expressed by the English sentence "There is no brother who is not male" is the proposition that there is no brother who is not male; (B) The proposition that there is no brother who is not male is necessary; (C) The proposition expressed by the English sentence "There is no brother who is not male" is necessary. The argument as Lewy presents is, does not simply rely on the by now obviously defective principle of Necessary Premiss Suppression, but on the more complex principle A & B

C, DB, 7A, 7C -*A -* C

(Lewy).

where V reads 'it is contingent that A' (or else on the analogous statement with 'if then* supplanting -t> ). This restricted torm of premiss suppression Lewy claims to find intuitively acceptable (cf. p.119). But it is not, and is open to similar counterexamples and countermodels to those that refute Necessary Premiss Suppression.1 Moreover, Lewy offers however no case for the principle beyond its alleged intuitive acceptability and the fact that it follows using one half of the definition of entailment Strawson has proposed, namely strict implication restricted to contingent antecedent and consequent. The latter is not much of a recommendation, particularly given the discrepancy of the Strawsonian account from what it aims to explicate, which admits (indeed, draws its paradigmatic cases from) entailments between non-contingent propositions. The next two paradoxes Lewy offers are open to the same objections. propositions he considers are respectively: Paradox 2

The

Paradox 3

(A') Caesar is dead E Russell is a brother;

(A") Caesar is dead E this has shape;

(B') Russell is a brother E Russell is a male sibling;

(B") This has shape E this has size;

(C') Caesar is dead E Russell is a male sibling.

(C") Caesar is dead E this has size.

and the paradoxical conclusions are, respectively: (A') entails (B')

and

(A") entails (B").

As the arguments are the same in both cases, it suffices to consider one, say the first, which goes as follows:by transitivity of E. & (C') entails (B') by transitivity of E. & (B') entails (C') is necessary (and (A') and (C') are contingent). from 2 and 3 by Suppression principles, e.g. entails (C') Lewy. from 4 by first degree entailment principles, 5. (A') entails (A') & (C') by 1, 5 and transitivity of entailment. 6. (A') entails (B*)

1. 2. 3. 4.

(A') (A') (B') (A')

DML supplies countermodels like those it supplies to Disjunctive Syllogism. Counterexamples are like those elaborated in 3.8. The following choices for A, B and C illustrate the style: A : a knows that (p) (p -»• p) and q, with q: There is ice in Norway. B : a knows that (p) (p •*• p) (p) (p -»• p) and (p)(p p) + all theorems of modal logic C : all theorems of modal logic (or any long enough conjunction) and q.

163

2.11 FAULTING ARGUMENTS

TO STRICT-IMPLICATION

AS A RATIONAL

RECONSTRUCTION

The conclusion is paradoxical, but the conclusion that results if the illegitimate suppression in step 4 is removed, namely (A') & (B') entails (B'), is not paradoxical. Thus the paradoxes illustrate nothing more than the known troubles with suppression; they make no new difficulties for entailment.1 Lewy suggests a variant upon paradox 2, obtained by making (B') a propositional identity. But then, although (A') may guarantee (C'), (A') & (C') certainly do not entail (B') (except by paradoxical theses), i.e. step 1 fails. In paradox 4 the conclusion Lewy finds mildly paradoxical, that there are exactly 10 brothers entails that there are exactly as many brothers as brothers, is no more paradoxical than a specual case of one of the premisses involved, premiss (C), namely that "There are exactly n s and there exactly n \|fs" entails "There are exactly as many 4>s as ^s", where \(/s are s. Paradox 4 gives (per impossibile) even less support than the other paradoxes to Lewy's inconsistency thesis: it generates none of the difficulties Lewy imagines (p.131) it makes for relevant logic. Since intuitions which, like relevant ones, require of entailment some connection, are commonly not inconsistent, displacement of intuition, for example in favour of some "rational reconstruction" of the "defective" notion - is not called for, only its refinement to discriminate better among higher degree entailment principles. Lewy, however, lumbered with his inconsistency thesis, insists that the most rational reconstruction of the inconsistent intuitive concept of entailment is Lewis's theory (76, pp.133-4). But what case Lewy does present for this claim is insubstantial. He contends firstly that 'clearly, Lewis's theory is the simplest reconstruction'. This is far from clear. For on Lewis's theory, strict implicational system S2 explicates entailment at the sentential level. But in certain important respects systems S4 and S5 are simpler, in another respect SI is simpler, and in most respects classical logic S is simpler still. This merely emphasizes however that simplicity is not much of a guide to correctness. Nor is it clear that S2 is simpler than various relevant logics. Lewy's second contention - which is unsupported, but which presumably relies on evidence we have already disputed - is that Lewis's theory is much the best because, for sufficiently rich languages, there is in any case no theory of entailment, however complex, which would enable us to accept all those entailment propositions which on an intuitive level we wish to accept and reject all those which we wish to reject. And, so far as I can see, there is no hope of such a theory (pp.133-4). Irrespective of whether what is stated in the reason (i.e. following the 'because') is correct or not, the reason does nothing at all to show that Lewis's theory is much the best reconstruction: all it does is to delimit theories of entailment, as bound to diverge at least at some points from intuitive data. Since every

Suppression is not all that is wrong with the arguments. The principle relied upon in steps 1 and 2, namely (A = B) & (B = C) -»• (A = C), of transitivity of material equivalence, is not a first degree entailment, and is not generally correct. Lewy is aware (see fn.l, p.124) that the principle is rejected as a genuine entailment, and even advances this as a reason for not presenting certain further paradoxes. Subsequently repaired versions of Lewy's paradoxes, with material equivalence replaced by a genuine conditional, will be put to real work, e.g. against exported transitivity principles in chapter 3.

164

2.11 SOME REASONABLE

GROUNDS

FOR CONCERN ABOUT THE LOSS OF DISJUNCTIVE

SYLLOGISM

theory is going to conflict with some intuitive data, since people's intuitions clash (e.g. on some - mistaken - intuitions Disjunctive Syllogism is correct, on others it is not), the requirement imposes no real limitation, no constraint on theory choice. Thus Lewy has given no basis to support his second claim. Lewy's third contention is that the consequences of accepting Lewis's theory are less counter-intuitive than the consequences of accepting any of the alternative theories. In particular ... it is less counter-intuitive to accept that (A & ~A) entails B than to reject the principle [~A & (A v B] entails B (p.134). This contention likewise does nothing to support Lewy's best rational reconstruction claim unless two rather large assumptions are made, firstly that less counterintuitive theories are better reconstructions and, secondly, that Lewis's theory is less counterintuitive. As to the first, if a theory is bound to be counterintuitive anyway, it is by no means evident that a somewhat less intuitive theory may not afford a better reconstruction, e.g. classical logic than strict S2. The second assumption we have already disputed at length: we have argued that Disjunctive Syllogism is counterintuitive and tried to explain why; and it seems to us that a creature's "logical intuitions" have become seriously blunted if not atrophied if it ceases to find the Lewis paradoxes counterintuitive. In sum, Lewy provides but little satisfactory guidance on the central issues of the character of better theories of entailment and how to choose among such theories. Moreover what technical guidance he does offer, in the shape of a proposed sieve, leads, when duly repaired, to first degree entailment and ipso facto, to the rejection of Disjunctive Syllogism. Rarely are arguments for Disjunctive Syllogism advanced which are based, in the end, on other than intuitiveness and tradition. There are however more solid, if less often heard, grounds for concern about the rejection of Disjunctive Syllogism. A major ground is the loss of deductive and theoretical power incurred, especially as regards more mathematical theories. How do things stand? It is certain that many uses of Disjunctive Syllogism can be avoided or else replaced, e.g. all those that reduce to applications of y in a consistent theory, and so are admissible. It is also certain that not all uses can be avoided, for instance those that trivialise an inconsistent theory. But such uses are, in any case, illegitimate. The theme to be defended (it will take more than this section, or this book, to establish the theme) is that no real results, no legitimate ones, are sacrificed by ditching Disjunctive Syllogism. But for the present only some methods for replacing uses of Disjunctive Syllogism are outlined, and the situation in the case of total failure of such methods considered. The disjunction concerned is either extensional or it is not. If it is intensional then Disjunctive Syllogism is, as Anderson and Belnap argue, but a disguised version of a principle such as A & (A -*• B) -»• B, which can replace it (at least in many settings). If however the disjunction is extensional then commonly its introduction is by Addition: A -»-. A v B or B -*• A v B. Consider then ~A & (A v B) -*• B. If A v B comes from B then Disjunctive Syllogism can be replaced by Simplification, ~A & B ->• B. If A v B comes from A, Disjunctive Syllogism is nothing but the spread paradox ~A & A -»• B, and its use should be rejected. Differently, the argument may take the form: A v B, A -*• C but ~C, so ~A. Hence by Disjunctive Syllogism B but B -»• C. So C. An argument of this sort occurs in one of the versions of Russell's paradox. The argument can be 165

2.12 EXTENDING VML CONSERVATIVELY:

PROVABILITY

replaced by application of v-Composition (A

AS VERIVABILITY

FROM THE TRUE

C) & (B -*• C) -»-. A v B

C.

Often, again, applications of Disjunctive Syllogism require only the corresponding rule form y; and the admissibility of the rule may be established with respect to the framework concerned. In subsequent chapters (especially 5) the admissibility of y is established for a large class of relevant logics, and work (particularly of Meyer, some of it reported in DRL) has established the admissibility of y for much stronger logical theories (much more remains however to be done). It may be however that these and other methods fail. This is either because the background system is inconsistent or else for other reasons. If the system is inconsistent, then the effect of Disjunctive Syllogism should not be alternatively engineerable: its use is illegitimate unless the system is independently trivial. Otherwise, the system is consistent, but replacement methods so far available fail. (There is no reason to expect effective replacement methods, and reasons not to expect foolproof methods). In this event, there is nothing for it but to say: for the present any results not relevantly derivable without appeal also to Disjunctive Syllogism are only legitimately achieved with the additional super-assumption of consistency (see further Routley DS). To say this is not onerous or an admission of failure: for relevant derivability is the test of correctness. §12. Intensional lattice logic, and one way of eliminating Modus Ponens. There remain some minor - but easily compensated for - losses in trading Antilogism in, as we have (in 2.6), for Contraposition. One of those is that obvious ways of introducing theorems of the form (- B in BL are lost in DML. Just as one can no longer define a logical truth }f- B as an ob true in every model, so correspondingly one can no longer define an unconditional theorem (in the defective way that depends on paradoxes of implication) as an ob B such that r h B whatever F. Since it is not the case in DML that A V ~A (- B V ~B other classical methods such as defining t as A v ~A and an unconditional theorem |- B as an ob such that t^-B, cannot be adopted (since t would not be well defined): but they can be simulated. For t can be introduced as a constant ob. And though t can be conservatively added to DML, as the semantical methods show - that is to say, its addition adds to DML no new theorems not involving t - its addition does give the appearance of much increasing the power and scope of DML, largely as a result of the way theoremhood can be defined in DML with t added. DMLt results then from adding to DML the constant ob t, subject thus far to no axiomatic constraints. Define |- B = t (- B. DR4.

A, Af- B

(- B

DR5.

A, f-B -fr ( - A & B

Modus Ponens. Adjunction.

Modus Ponens follows using the definition from Transitivity, Adjunction from &-Composition. The simple derivation of Adjunction and particularly Modus Ponens in DMLt represents, it seems to us, a considerable conceptual gain: Modus Ponens loses its entrenched position and becomes but a special case of Transitivity - and all theses of logic become at bottom entailments! There are some theses concerning t, e.g. T9.

(- t

T10.

f- (t & t) v t

;

but some of the enrichment one might perhaps have expected from adding t - for example that A v ~A is a thesis (as it is in BL, under an appropriate definition of t) - is not forthcoming without the addition of further postulates governing t, as the semantics can again be applied to show.

166

2.12 STRETCHED

HOVELS: THE ANTICIPATION

OF SUBSEQUENT

SEMANTICAL

DEVELOPMENTS

To model t we reintroduce 0. A DMLt model M is a structure <0,K,*,I> where 0 is a subset of K. The semantical rule for t is this: I(t,a) = 1 iff a e 0. Adequacy Theorem for DMLt. (i) A A is theorem of DMLt iff A (-A is DMLt valid. Hence (ii) A is a thesis of DMLt iff for every DMLt model M and every a e 0, 1 (A,a) - T, i.e. iff jf. A. The only new detail in the proof lies in completeness; and there define 0 as C {a e K :t e a}. c Corollaries 1. DMLt is a conservative extension of DML. For suppose (- A is a statement of DML and a theorem of DMLt. Then I A is DMLt valid. But as it does not contain t establishing its validity concerns only inductive clauses for operations of DML; so A f- A must be DML valid. Hence 11- A is a theorem of DML. 2.

f- A V ~A is not a thesis of DMLt.

For a countermodel with a e 0 but a ^ a* can be produced. To obtain (- A v ~A as a thesis it helps to embark on our next series of extensions of the semantical analysis. Not that such embarkation is essential just for this cause; for, as it will turn out from our voyaging, |-A v ~A can be modelled within DMLt semantically just by the requirement: if a e 0 then a = a*. But there are other causes to consider, in particular the question of the semantical analysis of the first degree of entailment, that is the logic obtained by extending the operations &, V, ~ to apply to statements such as A B - though permitting no nestings of |- (thus A (-B is not at the first degree stage simply considered a further ob).1 First, however, let us cope with [•A V ~A. A stretched DMLt model M is a structure M = <0,K,5,*,I> where 5 is a reflexive and transitive ordering on K which imposes the following requirements: *i)

if a 5 b then b* 5 a*;

Oi)

if a < b and

Oa then Ob, i.e. 0 is a filter;

Ii) if a 5 b and I(A,a) = 1 then I(A,b) = 1, for each initial ob and for every a,b e K. Relation 5 can be taken as a partial order on situations: for it is modelled as set inclusion _£. *i) and Oi) are needed for the following lemma which extends Ii) for initial obs to all obs. Hereditariness Lemma. For every ob B and every a, b e K, if a 5 b and I(B,a) = 1 then I(B,b) = 1. Proof is by induction over the construction of obs. Semantics using stretched models are of course adequate for DMLt. More important, the law of excluded middle, f-A v ~A, can now be readily modelled (in a way that anticipates later developments) by the requirement: EM.

if a e 0 then a* 5 a, for a e K.

Adequacy Theorem for DMLt 4- [- A v ~A. The statement is the expected extension of that for DMLt. For soundness of f- A v ~A suppose for arbitrary a e K, I(t,a) = 1. i But note that given the move to the full first degree, the t representation of theses already provides for iterated occurrences of f- . For example, the excluded middle thesis A B v ~(A |- B) is represented by t . A (- B v ~(A f- B), which nests |- (in a quite intelligible way).

W

2.12 INTENSIONAL

LATTICES, AND INTENSIONAL

LATTICE LOGIC ILL

Then a € 0, so by EM, a* < a. Since I(A,a) = 1 or I(A,a) ^ 1, by stipulation of I, I(A,a) = 1 or I(~A,a*) = 1. But by hereditariness if I(~A,a*) = 1 then I(~A,a) = 1. So I(A v ~A,a) = 1, as required. For completeness, extend the previous canonical model to the structure M = <0 , K , *, I>. Then *i) follows using Contraposition, and Oi) and I?) arecimmediate from inclusion features. As to EM, suppose a e 0 and A e a*; to show A e a. As a e 0, t e a ; so A v ~A e a, i.e. A e a o r ~ A e a , i.e. A e a or A ^ a*. But A c a*, so A e a. Stretched models are a lot of trouble to go to in order to ensure Excluded Middle. Fortunately they are good for a great deal more than this. In fact they put us well on the way to providing semantics for a large class of relevant logics, and are already rich enough (given minor adjustments) for such irrelevant systems as RM, Fitch-Nelson logics and stronger I systems. (All that is required to model RM is, as Dunn has shown, an appropriate inductive clause for I(Af- B,a), where iteration of (- is permitted). Stretched models can play an important role in the semantical theory of first degree deducibility (especially where material detachment is not given in the logic), but there is much else they can do before we reach that stage. To illustrate, let us examine two somewhat bizarre applications of the apparatus we have. The first of these applications supplies a formalisation of intensional lattice logic, ILL. An intensional lattice is a structure where is a De Morgan lattice and o is a truth filter of K, i.e. a filter of K that is consistent, i.e. for a e K, not both a e 0 and a* e 0, and complete, i.e. for a e K, either a e 0 or a* e 0. But though the notion of an intensional lattice has featured large in the development of algebraic semantics for first . degree entailment logic, it looks jejune when reformulated as a formal logic. For ILL adds to DMLt the following (paradoxical) postulates: t j—tvB,

~t(-tv C

(t Filtration).

An ILL model is a stretched DMLt model such that 0 is a truth filter on K, i.e. (since 0 is already a filter) a e 0 iff a* I 0 for a e K. Adequacy Theorem for ILL. The statement follows the established pattern. Proof adds the following details to that for DMLt wit^h respect to stretched models. ad Soundness, t (- ~t v B is true in a model M iff, for every a e K, I(t,a) = 1 materially implies I(~t,a) = 1 or I(B,a) « 1, i.e. I(t,a) = 1 = I(t,a*) materially implies I(B,a) = 1, i.e. if Oa and Oa* then materially I(B,a) = 1. But for no a do we have Oa and Oa*, by filter consistency. Hence the material implication is vacuously satisfied, and the axiom scheme is validated. The argument for ~t (- t v B is similar but uses filter completeness: the antecedent of its material implication would require ~0a* and ~0a, which is again impossible. ad Completeness. A canonical model with non-degenerate (n.d.) theories is used. It suffices to add to previous details those for filter consistency and completeness. ad filter consistency. Suppose a e 0, i.e. t e a; to show a* I 0, i.e. t i a*, i.e. ~t e a. Since a is n.d. some ob B is not in a. But by closure ~T v B e a, i.e. by primeness ~t e a or B e a. Hence ~t e a. ad filter completeness. The argument is similar but uses the axiom ~t f- t v C. It is evident from the proof that there are intermediate logics, one which adds to DMLt the axiom t |—t V B and is modelled by filter consistency, and another which adds just ~t (- t v C and is modelled using filter completeness. Even more bizarre than ILL is the logic which does the opposite, which

168

2.12 THE EVEN MORE BIZARRE ANTI-INTENSIONAL

LATTICE

LOGIC

formalises a De Morgan lattice where the filter 0 meets the requirement a e 0 iff a* e 0, for a e K: The corresponding logic adds to DMLt the paradoxical axiom D (- t v ~t So much for curiosities: now to the more serious matter of the first degree of entailment.

169

3.0 DEFINING

DEGREE, AMD INTRODUCING

THE FIRST

DEGREE

CHAPTER 3 THE SHAPE OF THE FIRST DEGREE LOGICAL AND SEMANTICAL THEORY, AND COMPETING PROFILES FOR HIGHER DEGREE LOGICS Thus far most of what has been accomplished formally, in chapter 2, has been confined to statemental relations of zero degree obs (wff). But these deducibility statements admit of truth-functional compounding; they also admit, in natural languages, of iteration or nesting in combination with truthfunctional compounding. The first of the extensions of the logic thereby suggested takes us from simple deducibility to the first degree theory, the second to the higher degree theory. More explicitly, the degree of a wff is a syntactical measure of its intensionality, and indicates the level of nesting of intensional connectives in a formula. In the case where |- (or ->-) is the only intensional symbol and &, Y and ~ the extensional connectives, degree may be defined recursively as follows: sentential parameters are of degree 0; where A is of degree n then ~A is of degree n; and where A and B are wff one of degree n and the other of no more than degree n then A & B and A V B are of degree n and A f B (or A -*• B) is of degree n + 1, for every non-negative integer n. The definition is readily enlarged to accommodate other connectives: further extensional connectives do not increase degree, but further intensional connectives behave like •* in increasing degree by 1. The initial sections of this chapter extend relevant logic, DML, to the first degree. This can be done uniquely, since the first degree theory FD is presently without significant-relevant rivals. To be sure there are irrelevant rivals, such as the better known first degree of strict implication and the deservedly little known first degree of Curry's system LD (Meyer 77); and it is presumably only a matter of time before competing relevant theories, such as first degree connexive logic, are formulated. The objections already made to these competitors carry over to the first degree. Beyond the first degree matters lose this splendid simplicity. Remaining sections of the chapter make a substantial beginning on the examination of high degree complexity, and on the choices between an embarrassment of relevant options. %1. The first degree of entailment and implication, alternative semantics for the first degree theory, and how to dispose of yet other objections to the semantics. It makes perfect sense to combine statements of deducibility truthfunctionally. Thus it is true that either B is deducible from A or that B is not deducible from A, i.e. A If B v ~(A B), and it is false in general that either B is deducible from A or A is deducible from B, so for some A and B ~(A H- B v B |f A). Both statements are significant, because truth-valued. In fact, of course (as already argued in chapter 1, and in ABE), it makes good sense to iterate deducibility statements, as in the statement That A entails B and B entails C entails that A entails C - or with more that's for those who demand them. But nothing so mind-shattering is so far demanded, only elementary truth-functional combinations of implicational statements by &, v and something even levels theorists are prepared to admit in their metalanguage. An easy and painless way to move up to the first degree is to add the following statement forming rules to those of DML (the fullest statement of these rules is given in §3 for system DLL): 170

3.1 MORPHOLOGY,

AND DEDUCIBILITY

AXIOMATISATION,

OF SYSTEM

FD

If a and S are statements (wff) then so are ~a, (a & 3) and (a v 6). The usual first degree considers only initial statements of the form A (- B (i.e. A B), but there is no reason why we should not extend the notation to admit sets of antecedents (or sets of consequents). For the present constant t is omitted: it is picked up again later. Though the tiered account of well formation picks out the first degree neatly without further statement, it facilitates work, and especially later comparisons, to adopt a uniform notation (we use A, B, C, etc.) for statements and obs, and to smudge the tiering. With this approach we have to presuppose that each formula scheme admitted is of at most first degree, i.e. contains no nested occurrences of (- . A glaring deficiency of almost all the logics so far formulated using deducibility as the central notion (all those up to §10) is that they include only theses with (- as the main connective, in particular that they include no truth functional analyticities, such as the law of non-contradiction ~(A & ~A) , and no theory linking truth functions and deducibility, such as the rule of counterexample: if A & ~B then ~(A (- B). The first degree system FD rectifies these deficiencies, though in the course of doing so its own limitations will become apparent. Perhaps the simplest, and most controversial, axiomatisation of FD builds on the logic of truth-functional connectives &, V, ~ by just taking over some axiomatisation of classical two-valued logic S and adding to it the axioms of DML and the rules of DML reformulated as material implicational thesis. Such rules as Transitivity have after all an excessively weak formulation in DML; indeed these rules of DML should hold not merely in first degree material form, as e.g. (T (- B)&(B, A f- C) 3. T,A|-C, but in entailment form, as e.g. (r (- B)&(B, A f- C) )- (T, A C). Such a formulation, however, takes us beyond the first degree, which is the immediate issue, and into the murkier regions of the second degree.

Deducibility formulation of FD: Let us build on the axiomatisation of S that has become known as PM, a formulation that takes ~ and v as primitive and defines material implication: A B ~A v B, and conjunction in the usual way: A & B

=

~(A V ~B).

Rule y: A, A => B

Then FD is as follows:

B

Material Detachment

Axiom schemes. (1) Schemes for S : A =>. A v B

A v A => A A => B o. C V A =>.

A V B =>. B V A (2) Deducibility schemes: A

b

A, B A, A B A

B

A

(r

b b

A

r

B)&(B,

b (-

B &

r

A H B

C)

c^. r

3. r, 4 (• c |- B & C

B

b b b

A V B

-A

b

(r,

A |-

c) & (r,

B |-

A V B -A

A |- B

~B

|- ~A

A

171

c)

r,

A

CvB

3.1 ENTAILMENT*/. AXIOMATISATION OF FV, M W THE ROLE OF RULE y

This axiomatisation of FD is certainly not as economical as it might be since A |- A is redundant, deducibility schema for & are derivable, and deducibility schemes can be used to eliminate truth functional schemes such as A V A => A and A v B => B v A through the derived rule; A |- B A B. But the axiomatisation reveals nicely the interaction of S and DML at the first degree stage. Entailmental formulation of FD; The axiomatisation may be rewritten in the more familiar guise we elsewhere adopt simply by expressing D, A f- C as D & A C, and making minor adjustments. Then transitivity, for example becomes A - > - B & B - » - C 3 . A ->- C. Let us rewrite the deducibility schemes (2) in entailmental form using just familiar connectives &, ~ as primitives. Then FD consists of the classical postulates - of S - together with the following entailmental schemes:A -»- Aj, (A -*• B) & (B (A -»• B) & (A usual, A

C)

A + C, A & B + A, A & B

B,

C) 3. A -»• B & C, A & (B v C) -»• (A & B) V C, with V defined as A,

A

A, A -»- B =>. ~B

~A.

It is time also to formalise the linkage, already presupposed, between and •*•. Connective may supplant relation where the obs are themselves statements or wff. Then where A and B are zero-degree wff, i.e. contain no occurrences of (- or •*•, A (- B iff A B. And more generally, f- may replace where is the main connective of a wff, e.g. A -»• B -»•. A & C B & C may be rewritten A B f- A & C B & C. The differences between |- and -*• are usually taken to be (i)

f- is a metalinguistic relation, where •*• is a systemc connective;

(ii)

-*• is iterable, where (- is not significantly iterable;

(iii)

j- can relate sets of items as in T

A, where •*• can connect only wff.

None of these differences are essential or have much basis except convention; all can be (profitably) broken down.1 For certainly deducibility and derivability can be iterated; entailment, ->-, can be extended, unproblematically, to connect sets of antecedents and consequents; and, as we have begun to see, the object language for metalanguage barrier can be dissolved (see further chapter 15). What justifies FD is, as its formulation makes plain, what justifies DML as the theory of deducibility together with what justifies S (with y duly systemically restricted) as the theory of truth functions, for example that both are analytic on the senses of operators involved. A leading criticism of the axiomatisation of FD given is bound to be, not its lack of economy, but the central role material implication plays in the formulation, especially in the detachment rule. Does this matter? After all, rules in this formulation are not Intended to correspond to deducibility relations, but are essentially detachment devices for getting theorems from theorems. We could if we wished reformulate FD without Rule y - the complex two page formulation of FD in ABE, pp. 207-8, establishes this point - but then Y would reappear as a derived rule (as semantical or metavaluation methods will show). In fact Y plays a central role in easier formulation of 1

A serious weakness of the combinatory logic approach to foundations is that it does not permit iteration of [- .

] 7 2

3.7 THE SEMANTICAL

ANALYSIS

IN TERMS OF WORLVS OF FV

FD because FD is a half-way staging point between DML and fuller entailment systems. In the fuller systems the formulation problem will vanish again since the rules of DML can be systematised with deducibility as the main connective, and y replaced by deducibility detachment, i.e. Modus Ponens. But at the first degree one is obliged to stick with the limited resources of that stage - and that means either use of the desperately poor approximations to good rules that the truth functional resources can provide, e.g. y or an equivalent rule, or else the use of rule formulations, which avoid the use of y in letter but not in spirit. For the, usually unformalised, theory of rules is thoroughly classical in character, permitting, for instance, the ad lib introduction of theorems as extra premisses and the derivation of theorems ex nihilo. In short then, we do not apologise for using y in formulating FD: its use is admissible at the first degree, and so long as it is only applied to appropriately consistent extensions of FD one won't go too far wrong using it. Nor is its use ideologically damaging to us: not all admissible, or primitive, rules need correspond to deducibility theses, and it is often convenient to use rules which do not, such as rules of necessitation and generalisation. So much in the way of excuses for the ugliness of an easy axiomatisation: now to some of the attractions of the first degree. Semantical formulation of FD: An appealing feature of FD is its semantical analysis which is engagingly straightforward. An FD model M is a structure M = (or simply, as for DML, a structure < K , * , I > wTth T = f £x(xeK)T, where K is a set, T e K, * is an involution on K (i.e. a** = a) such that T* = T, and valuation I is a function on initial obs and elements of K with values in {1,0}. Valuation I is extended to all wff of FD, for every a e K as follows:I(A & B,a) = 1

iff I(A,a) = 1 = I(B,a)

I(A v B,a) = 1

iff I(A,a) = 1 or I(B,a) = 1

(v-rule);

I(~A,a)

iff I(A,a*) * 1

(—rule);

= 1

(&-rule);

and

I ( r (- B, T) - I iff, for every b e K, if I(A,b) = 1 for every A e T, then I(B,b) = 1. For a in K distinct from T, I ( r (- B,a) is not defined; knowledge of it is never required in assessment of FD wff. In a slogan, first degree deducibility is a matter of truth preservation, more exactly of value preservation. An FD model differs from a DML model, then, primarily in the role a distinguished element T of K, for which T = T*, plays in the evaluation rule for f- - an evaluation rule moreover which simply replaces the definition of truth in a DML model. And now truth is redefined in terms of holding at T. B is true in M iff I(B, T) = 1 i.e. B holds at the factual world which determines truth; and B is FD valid iff it is true in all FD models. Semantical adequacy, like representation results in algebra, enables easy determination of many important features of the logic. Adequacy theorem for FD. A is a theorem of FD iff A is FD valid. Proof of soundness is by the usual induction of proofs. Suppose, for completeness, that A is not a theorem of FD. We build, in a completely classical way, a Lindenbaum set T to which A does not belong. An FD proof of A from hypotheses A, written A I- A, is a finite sequence of FD formulae ending in A each of which is either an FD theorem or an element of A or is obtained from predecessors in the sequence by rule y. The familiar material deduction theorem holds: namely if A, A i- B then A i-A => B. To construct T, enumerate the wff of FD, An,A ,...,A ,... Define a sequence of sets of wff A_,A_,...,A ,... (with U 1 n u 1 n union T) inductively as follows:-

7 73

3.1 ESTABLISHING

0 A

n

Ai

n

THE ADEQUACY

OF THE WORLDS

SEMANTICS

= FD =

A

iff A , , A , i- A, and n-1 n-1

n-

== A A

T ==

.. u {A ,} otherwise (i.e. if A is not derivable), nn-1 n-1

u

n
A

n

.

Then (i) A i T (ii) T is closed under material detachment and so under Adjunction. The second follows from the first using the theorem A o. B o A & B. For the first suppose D, D ^ B e T but B I T. Since B = A. for some i then A , B I A; hence T, B i A by Augmentation. (i).

But then T, D, D o B i A, i.e. T 4 A, contradicting

(iii) T is negation consistent, i.e. for no D is both D and ~D in T. (iv) T is closed under (- . Suppose T (- D is an FD theorem and T £ T. Let C be the conjunction of elements of T. Then C 3 D is an FD theorem, and so C ^ D e T since C => D e A^. The result then follows by (ii) . The missing proof detail is the derived rule C D-» C => D: But this follows using ~C V C and v-principles. (v) T is prime, and so negation complete. The second follows from the first since the theorem ~A v A e T. For primeness suppose otherwise that for some B, C, B V C e T, B i T, C i T. Then, as in (ii) , T, B 4- A and T, C i-A; and so, by the deduction theorem, T i- B => A and T i- C => A. But, as B3A=. B V C A, so T i- B V C = A. Thus by (ii) and since B v C e T, T 1-A, contradicting (i). A canonical FD model based on T is defined as follows:T = T; K is the set of T-determined prime FD theories, i.e. for a e K , a c c c is prime, closed under Adjunction and closed under the deducibility relation fof T, that is, if T £ a and r f- C e T then C e a; for a e K , a* = {A: ~A i a}; and, for initial wff A and a e K , l(A,a) = 1 iff A e a. c 1. M = is an FD model. In virtue of the properties of T c c c established, T^ e K . Because of the negation consistency and completeness of T, T * = T . c c 2.

By double negation principles, a** = a.

For every FD formula D, and every a e K c for which I is defined, I(D,a) = T

iff D e a.

All but the induction step for j- are as for DML. As to (- , suppose first F C e T. Hence if for every A e T, A € a, for arbitrary a € K c » C e a. But then, applying the induction hypothesis, I(T [- C, T ) = 1.

Suppose conversely

r f- C i T. Define a' - (A:T |- A e T}. Then T £ a', C i a*, and a' is an FD theory. Now by the extension lemma proved for DML, since DML £ FD, there is a prime FD theory a extending a* which excludes C. Then, using the induction hypothesis, for each a e T, I(A,a) = 1, and I(C,a) ^ 1, i.e. I ( r 1- C, T ) 4 l.1 c l A stronger adequacy theorem - which the present construction suffices however to prove - is established using algebraic means in SI, theorem 15.

174

3.7 A FIRST COROLLARY

OF ADEQUACY

: THE FINITE MODEL

PROPERTY

The finite model property and decidability of FD. Among the first of many corollaries of the adequacy theorem is the decidability of FD. Corollary 3.1. Where A is a first degree wff with k occurrences of symbol (or, on the alternative formulation, of +), A is a theorem of FD iff (i) A is true in every FD model on world set K = {T,a ,a *,...,a ,a *} for every c c 1 1 c c such that 0 < c < k (in case c = 0, K = {T}); (ii) A is true in every FD model on world set (a^ and a_. being distinct when i t j).

K^ = {T,a^,a^*, . . .

Proof. Soundness is a special case of soundness in the adequacy theorem; for if A is true in all models it is true in the FD models of (i) and (ii). For the converse, suppose A is a wff with k occurrences of (- but A is not an FD theorem. Then by the adequacy theorem there is a (denumerable) FD model M = < T, K, *, I > such that I(A, T) * I. Now A is a truth function of k deducibility statements, i.e. it is of the form 6[(T1 1- C ),..., (T, (- C ), 1 1 k k where . . . . . a r e also, or consist of, truth functional, i.e. zero degree, wff. In the falsifying model some number c of the k deducibility statements will be assigned value 0 at T. Each such assignment, I(I\ C^,T) t 1, forces the introduction of some pair on which T. and C. are evaluated. In case K contains several such pairs select one. i i K^ be the set consisting of all such falsifying pairs together with T, i.e.

Let

K

= {T, a., a * a ,a *} with c determined as above, and with the first c 1 1 c c of the falsified deducibility statements falsified at a^,..., and the last falsified at a . c

Form the restriction M

the restriction of I to K^.

c

— ^T, K . c

I ^ of M, where I is c c

Then M^ is an FD model, since T and every *-image

of falsifying elements selected from K have been carried over to M . By induction, (a) for every such formula D of A and every a in K c , Ic (D,a) = I(D,a), The induction basis is given by specification of I and the induction steps for truth functional connectives &, v and ~ are virtually immediate, in the case of ~ since a* e K iff a e K . c c ad f- . If I(r (- C , T) = 0 then for each B ± in r I(I\,a) = 1 t I(C ,a) for some a in K by design of K , so I (B.,a) = 1 * 1 (C,a) by induction hypothesis, whence I (ri |- C.,f) = 0.° Conversely if°I(r. |- C.,T) = 1 then c i J J for every a in K, and so for every a in K , if I(B.,a) = 1 for B. in T. then c J J J l(C.,a) = 1, whence I c (r f- C..,T) = 1. Accordingly by (a), I (A,T) t 1, establishing (i). c To establish the remaining half of (ii) either we make a more generous selection from K or we add some dummy elements to K^. Let us take the latter course.

Let J = ^c+i' ac+l*'" * *, a k , a k*^

distinct situations not

included in K; let K^ = K^ u J and define a valuation coincides with I

on K^, and for every initial obj D,

775

as follows:

3.1 THE DECIDABILITY

OF SYSTEM

I (D,a.) = 1 t I(D»aj*) for each a^ and a^* in J. ^

FD

Then

= is an FD model, and further

(3) for every truth functional wff D, I(D, a^) - I t I(D, a^*) for each a. and a.* in J. J J (3) is proved by induction: note the crossover from a^ to a^* and from a.* to a.** = a. in the step for 2

J

J

(y) For each deducibility statement r

i

. . in A, Ik(r±f- C , T) - I (r±f- C^T).

C

If I (T l-C.,T) t 1 then the element a in K which falsifies T C. also c c l c c i falsifies I k (r ± (- C^T). If I (- C^,T) - 1 then for every a in K c > when = 1 f r every B = S every a in K when for ° i i n ^i* ° c» B. in T. I, (B.,a) = 1, 1, (C.,a) - 1. But by (3) for each a in K -K when for fil l k i k i k c i in T. I^CB^^.a) = 1, a) = 1 by properties of material-implication. Hence

V r i i- C i - T ) = 1 (5) I, (A,T) = I (A,T). K.

C

An induction basis for each element I\ \- C^ and D^ in A, at T, has been established, and the induction steps for &, v and ~ are virtually immediate. By (6), I k (A,T) = 1, proving (ii). A simple variation on the argument for (i) is to write the result into the construction of the canonical model used In the completeness argument of Makinson 66. Corollary 1, which shows that FD has the finite model property, provides a simple decision procedure for FD. For to decide whether a wff A is a theorem of FD, or FD valid, it is enough to count the number of occurrences of (- in A, k say, and to test A in all FD models on K^ (see further 5.9). Hence Corollary 3.2. FD is decidable. Decidability of FD may be alternatively established by more algebraic methods (see below, and cf. ABE, p.206 ff.) or by Gentzen methods. Matrices for FD, including the Belnap 8-valued matrices. As the semantics for DML (in 2.6) yielded matrices and algebraic structures for the logic, so also does the semantics given for FD. Moreover matrices for FD may be obtained from classes of models in given model structures in precisely the way they were obtained in the case of DML (p.114). But in contrast with DML, a single finite-valued characteristic matrix will not result; instead a wff with k occurrences of will be connected with matrices, and lattices, whose size is dependent on k. A model M = is, as before, resolved into a pair consisting of a model structure (m.s.) S = and a valuation I on S. Matrices are associated with model structures, each matrix being determined by the class of valuations on the corresponding m.s. A matrix value on S Is represented as a function from K to II, i.e. {l,0}, and accordingly corresponds exactly to a subset of K. Where p and a are matrix values, matrix operations "&", "v" and corresponding to each connective, are defined thus, for every a and b in K: (~p)(a)

= 1 iff p(a*)

(p v a) (a)

;

(p & a)(a)

= 1 iff p(a) = 1 or 0(a) = 1

176

;

= (p

1 iff p(a) = 1 = o(a); a) (T)

= 1 iff for every a

3.7 HOW FV HOVELS DETERMINE MATRICES FOR FV

p(a) ^ 1 or o(a) = 1, and otherwise (i.e. for a ^ T) -*• is not defined or is arbitrarily determined. An example of the latter is the Belnap-specification (which though optional is not chosen without reason): for a / T , (p a) (a) = 1 iff p(d) # 1 always (i.e. p = A) or a(d) = 1 always (i.e. a - K) or else p(a*) ^ 1 and a(a) = 1. The point of such "implicational extensions" should become evident from the full -*- matrix of M^ (it is explained in the next subsection). In view of the definitions, operations & and v are effectively set as lattice operations u and n respectively, and may be evaluated accordingly; and ~p is given by {a:p (a*) ^ l}. Finally, p is a designated matrix value iff p(T) = 1. This completes definition of the matrix S + = <M,D,~,&,v,+> on S, where M is the class of matrix values and D the designated subclass. A matrix satisfies a wff A iff for every assignment to the initial components (variables) of A, the resulting value of A is designated. It has already been proved (p.115), for every fd wff A, that S + satisfies A iff for every valuation I on S, I(A,T) = 1. Hence a further corollary of the semantics for FD is that a wff A is a theorem of FD iff every matrix S + on every f.d. model structure S satisfies A. While this result substantially prunes the class of matrices Lindenbaum representation of the system requires, corollary 1 enables a more spectacular reduction in matrices. Corollary 3.3. Where A is a fd wff with k occurrences of •+, A is a theorem of FD iff A is satisfied by the matrix S + on S = <1,1^,*>. Corollary 3.3 provides a type of matrix decision procedure for FD. considering the initial cases where k is 0 and k is 1.

It is worth

Where k = 0, the matrix S + on is simply two-valued truth tables, with values 1 = {T} and 0 = A, and &, v and -»• evaluated as in truth tables. It follows that a zero degree wff of FD is a theorem iff it is a (classical 1 tautology. The first degree of entailment is, as the axiomatisation adopted also shows, an amalgam of DML, which is 4-valued, with 2-valued classical logic. This suggests, what proves to be the case, that the 8-valued matrices, generated by k = 1, will play a crucial role in the analysis of FD, and that these matrices in turn will be the product of the 4-valued (Smiley) matrices with the 2-valued truth-tables. Where k = 1, there are 8 matrix values, which are conventionally labelled as follows1: +3 +2 +1 +0

= = = =

{T, a, a*} {T, a} {T, a*} {T}

-3 -2 -1 -0

= = = =

A {a} {a*} {a, a*}.

Positive values are designated. Since the matrix values are partially ordered, by set inclusion, the values can be represented by the following Hasse diagram (called dM n , for diagram of M_).

1

Subscripts are omitted from elements of K, i.e. a^^ is written just a and a as a*

177

3.7 T H E PARTIAL M^ MATRIX FOR FV A W ITS DIAGRAM

Since the diagram represents a distributive lattice, matrix tables for operations & and v can be simply read off the diagram, in terms of lattice operations n and u:&

-3 -2 -1 - 0 + 0 +1 +2 +3

~

V

-3 -2 -1 - 0 + 0 +1 +2 +3

-3

-3 -3 -3 -3 -3 -3 -3 -3 +3

-3 -3 -2 -1 - 0 + 0 +1 +2 + 0

-2

-3 -2 -3 -2 -3 -3 -2 -2 +2

-2

-1

-3 -3 -1 -1 -3 -1 -3 -1 +1

-1 -1 - 0 -1 - 0 +1 +1 +3 +3

-0

-3 -2 -1 - 0 -3 -1 -2 - 0 + 0

-0 -0 -0 -0 -0

-2 - 0 - 0 +2 +3 +2 +3

+3 +3 +3 +3

+0

-3 -3 -3 -3 + 0 + 0 + 0 + 0 - 0

+0 +0

+1

-3 -3 -1 -2 + 0 +1 + 0 +1 -1

+1 +1 +3 +1 +3 +1 +1 +3 +3

+2

-3 -2 -3 -1 + 0 + 0 +2 +2 -2

+2 +2 +2 +3 +3 +2 +3 +2 +3

+3

-3 -2 -1 - 0 + 0 +1 +2 +3 -3

+3 +3 +3 +3 +3 +3 +3 +3 +3

+2 +1 +3 + 0 +1 +2 +3

For example, &(+l, +2) is given by +1 n +2, i.e. +0; v(-0, +0) by -0 u +0, i.e. {T} U {a,a*} = {T, a, a*}, i.e. +3. The table for negation, already presented, can be simply computed from the equivalence d e ~p iff d* i p. For example, where p = +3, i.e. p = {T, a, a*}, d* I p is always false for every d. So ~p = -3. Again, where p = +2, i.e. p = {T, a}, T e ~p and a* e ~p are both false, but a e ~p is true, so ~p = {a}, i.e. ~p = +2. The matrices presented have the two-valued form

F

F

F

T

F

T

T

T

where T and F stand in for designated and undesignated values respectively. For truth functions of first degree entailments it is only this blade and white picture that matters (full colour vision is important for the higher degree). This explains why can be undefined, or arbitrarily defined, except at T. All that matters for satisfiability as far as -> values are concerned at the first degree is a partial matrix giving T and F values; for example, where k = 1 the following partial matrix (shown on the left)which is determined for the T part of + :

178

3.7 DETERMINING

THE H

MATRIX BV SUPERPOSITION

-3 -2 -1 -0 +0 +1 +2 +3

->-

-3

T

T

T

T

T

-2

F

T

F

T

-1

F

F

T

T

OF PARTIAL

MATRICES

at a -3 -2 -1 -0 +0 +1 +2 +3

T

T

T

-3

F

F

T

T

-2

F

T

F

T

-1

a a

-0

F

F

F

T

F

F

F

T

+0

F

F

F

F

T

T

T

T

-0

+1

F

F

F

F

F

T

F

T

+1

+2

F

F

F

F

F

F

T

T

+2

+3

F

F

F

F

F

F

F

T

+3

*

a

a

a a

a

a

a a

a a a

-0

a a

a

a

a

a

a

a a a a

The partial matrix on the right tabulates p -• a at a, evaluated according to the Belnap-specification. Value a indicates that p -*• a maps a to 1, value blank to 0. Similarly in the partial matrix for -*• at a* under the Belnapspecification set out below on the left, value a* indicates that p -»- a maps a* to 1, a blank that it maps a* to 0. And similarly also in the matrix for the T part of , T or designation shows that T is mapped to 1, i.e. that the value, whatever it is, is designated. Different notation is used so that the partial matrices can, like two-tone maps, be overlayed to give a full picture, in the (Belnap) 8-valued matrix for ->• exhibited on the right:-

- -3

-> at a* -3 -2 -1 -0 +Q +1 +2 +3 -3

a* a* a* a* a* a* a* a*

-2

- 2

-1 -0'+0 +1 +-2 + 3

-3 +3 +3 +3 +3 +3 +3 +3 +3

a*

-2 -3 +2 -3 +2 -3 -3 +2 +3

a*

-1 -3 -3 +1 +1 -3 +1 -3 +3

a* a*

a*

a*

-0 -3 -3 -3 +0 -3 -3 -3 +3

+0

a* a*

a*

a*

+0 -3 -2 -1 -0 +0 +1 +2 +3

+1

a*

a*

-1 -0

a*

+1 -3 -3 -1 -1 -3 +1 -3 +3

+2

a*

+2 -3 -2 -3 -2 -3 -3 +2 +3

+3

a*

+3 -3 -3 -3 -3 -3 -3 -3 -3

The Belnap -* matrix is the superposition of the three preceding partial matrices (ideally with F now erased in the first), the values being represented as before, with +3 through Taa*, +2 through Ta, etc. Thus the full matrix can be read off the preceding partial matrices, which however are computed largely case by case. The most tedious, the matrix for -»• at T, results from the equivalence: T e P a iff T i p or T e a , and a i p or a e 0, and a* I p or a* e a . Suppose, for example, p = - 3 . Then as no element of K belongs to p , T e -3 a, for every assignment to a . Or suppose, p = +0 and a = + 1 . Then as T e a , a * e a and a ^ p , T e p + O, i.e. +0 ^ +1 is designated. A more intelligible assignment function for -> at a (and analogously -»- at a*) is the following (with and dominating or): (p -»- a ) (a) = 1 iff p(a) ^ 1 or a(a) = 1 and p(a*) 4 1 or a ( a * ) « 1 and p(a*) 4 1 or o ( a ) = 1 and p(T) 4 1 or a(a) = 1 and p(a*) 4 1 or a(T) = 1. (The analogous function for a* simply substitutes a* for a throughout). Several of these clauses are what might be expected; only the last two conjuncts are a little surprising. The adequacy of the function for the matrices given is straightforwardly verified. Of course, other more "obvious" functions work, e.g. the S5-ish (p a)(a) = 1 iff for every c in K, p(c) 4 1 or o(c) = 1, but they supply different -»• matrices.

779

3.1 MATRIX PROOF OF THE WEAK RELEVANCE OF SYSTEM FV

The ~ and v matrices together with the partial matrix for at T constitute the partial MQ matrices, to which, and analogous matrices where k > 1, algebraic analysis of FD will correspond. As indicated, the full matrices for ~ & V and are called, after their discoverer, the Belnap (8-valued) matrices. Although it has subsequently been found that many of the tasks the matrices are good for can be accomplished using simpler matrices (e.g. the 6 element crystal lattice introduced in §6), still they have played an important role in the development of the theory of relevant logics, not least in establishing weak relevance for these logics, as well as various refinements thereof (see ABE, p.253). Weak relevance of FD. Relevance emerges as a byproduct of the semantical theory, in several ways. One way, now pursued, is by a matrix argument. Corollary 3. 4. A -*• B is a theorem of FD only if A and B share a sentential parameter. Proof. Suppose A and B share no parameters. The following assignments in the MQ matrices are accordingly possible: all parameters in A are assigned value +2 and all parameters in B are assigned +1. connectives that

It then follows by induction over

(i) every subformula of A is assigned value +2 or -2; and every subformula of B is assigned value +1 or -1. Verification is a matter of inspecting the tables for &, v and ~ (and in analogous arguments for higher degree systems that for as well). But (ii) ±2 -»• ±1 = F always (similarly ±1 -> ±2 = F); that is, whenever C has one of values +2 or -2 and D one of values +1 or -1, C -»• D is assigned a nondesignated value (-3 in the Belnap matrix). Accordingly (since -3 is not a designated value), A B is falsified in MQ, and is not a theorem of FD. Suppression freedom and paradox freedom of FD. The main results are built up in stages. It can be assumed, without serious loss of generality that the initial wff are sentential variables. (1) No truth-functional wff holds in every world. For consider any arbitrary truth-functional (i.e. zero degree) wff B containing just initial wff p ls ...p . n Choose a situation a such that none of pi...p hold in a but pi,...p e a*. Then n n a ^ a*, and ~Pi,..., i a by the ~ -rule. Here, and subsequently, we sometimes abbreviate 'D holds in b' by ' D e b ' , 1 We show first that all negations in B can be driven down to operate just on initial wff. More precisely, for every wff C there is a wff C' obtained from C by repeated applications of these transformations (applied in turn to negations from outer to inner) to subwff replace

~(Ci & C 2 )

by

c2)

by

(~CX & ~C 2 )

by

Cj

~(Ci v —Ci

(~Ci v ~C 2 )

such that no iterated negations occur in C' and all negations apply to initial wff alone, and such that C e a i f f C ' e a for every situation a. For first

1

~(Ci & C 2 ) e a

iff

Ci & C 2 I a*

iff

Ci I a* or C 2 i a*

iff

~Ci e a or ~C2 e a

In terms of this elementary translation, between I(D,b) = 1 and D e b , the valuation approach followed in the text can be transformed into the model set approach to FD of Routley2 70, and vice versa. The basis and details of the interconnections between these styles of logically equivalent worlds semantics are elaborated in EMJB, 1.17. ISO

3.1 DERIVATION

Similarly

(Ci v

OF VARIATION

c2)

e a

~C i e a

AND SUPPRESSION

FREEDOM

PRINCIPLES

iff

(~Ci v ~c 2 )

iff

(~Ci & ~C 2 ) e a, and

iff

Ci e a.

e

a.

Now if C is not already in reduced form C', apply &- and v-rules repeatedly, listing the order and maintaining bracketing arrangement, to C. Each time a wff which is not an initial wff preceded by a negation appears, replace the negated wff according to the specification (which is uniquely determined by the main connective of the unnegated wff). When the procedure is completed, and negations occur only preceding initial wff, reassemble the wff by applying the &- and v-rules in the reverse order to that listed. Then the resulting reduced wff, C', holds in a iff C holds in a. Moreover C* contains the same initial wff as C. We now show by induction that no reduced wff C' containing only initial wff pj,... ,pn holds in a. There are these cases, since induction bases are provided both for negated and unnegated initial wff:C' is of the form Ci & C 2 , where Ci does not hold in H and C 2 does not hold in H. Then by the &-rule C' does not hold in H. C' is of the form Ci v C 2 , where Ci does not hold in H and C 2 does not hold in H. Then by the v-rule C' does not hold in H. Therefore, assembling connections established, B does not hold in a. (2) Every truth-functional wff holds in some world. For consider any truthfunctional wff D containing just initial wff Choose a situation a such that qj.a.q^ e a but none of

a*.

Then ~q1,...~qm e a also.

Let

D' be the reduced form of D; then D f a iff D' e a. But, by induction, on the basis of the fact that qJ...,qm e a, D' e a. There are two cases, one given by the &-rule, the other by the v-rule. Important special cases of (1) and (2) are

Hence D e a.

(3) Every tautology fails in some situation, and (4) Every contradiction holds in some situation . (1) and (2) establish Corollary 3.5(a). There is no wholesale suppression of truth-functional wff in FD. For every truth-functional wff is such that it fails in some situation and its negation holds in some situation. A better result can however be obtained. (5) For every pair of truth-functional wff A and B which do not share a sentential variable, there is a world where the one holds and the other does not. For let A contain just initial wff q^.-.q^ and B just initial wff pj ...p^. Consider a situation a, with mate a*, such that p 1 ...p

4 a, q1 ...q e a, p1x ...p e a*, and 1q ...q I a* . n m n m Then the conditions are precisely as in (1) and (2), and by the same arguments, A holds in a and B does not hold in a. Thus the worlds of FD do provide adequate variation. Corollary 3.5(b). FD satisfies the maximum variation principle (of 2.11) and accordingly is suppression free. Given the situational characterisation of entailment, summed up in the evaluation rule for the following entailmental principles (al)-(a5) follow respectively from (l)-(5). (al) No truth-functional statement is entailed by every statement. (a2) No truth-functional statement entails every statement. It follows from (al)

181

3.1 INTENSIONAL

LATTICES A W THE ALGEBRAIC

and (a2) that a first degree entailment A the falsity of A or the truth of B.

ANALYSIS

OF FV

B is not true simply in virtue of

(a3) No tautology is entailed by every statement. Accordingly such positive paradoxes as A B is not true simply in virtue of the falsity of A or the truth of B. (a3) No tautology is entailed by every statement. Accordingly such positive paradoxes as A T, where T is a tautology, are rejected. (a4) No contradiction entails every statement. Accordingly such negative paradoxes as ~T -»• B are rejected. Accepting (3) and (4) is not merely sufficient to eliminate all standard paradoxes which negate (a3) and (a4); it is also the case that, given that worlds satisfy conjunction, negation and implication rules, accepting (3) and (4) is necessary for eliminating these paradoxes. Thus, given normal situation conditions, (3) is equivalent to (a3), and (4) to (a4). Accordingly the best way to eliminate the paradoxes for a situational implication, the way that retains normal set-up conditions, is to stop suppression. (a5) Where A and B are truth-functions, A entails B only if A and B share a sentential variable, i.e. FD satisfies Belnap's weak relevance requirement, WR. Thus the argument for (a4) provides a simple alternative proof of corollary 4 above. It also follows using (a4) that such false classical laws a s A + B V B + A are rejected, and in general, connexivity paradoxes of material implication are eliminated. Algebraic analysis of FD. Intimately connected with the matrix analysis of FD, given by corollary 3, is an algebraic analysis which ties matrices with algebraic structures, in the case of FD with a specialisation of De Morgan lattices now called intensional lattices.1 These lattices have already been introduced, and formalised, In 2.12. But, to recall, and to adjust the notation (which in 2.12 was semantical) to avoid clashes:- an intensional lattice I is a structure 1 = <M, D, n, U, N, O sometimes written, for short, <M, D, N, < > where M is a De Morgan lattice under < with operations u and N, i.e. M is a distributive lattice and N is an involution, i.e., for a and 3 in M, NNa * a and where a < 3 then N3 < Not; and further (T)D is a truth filter of M, i.e. B is a filter which is consistent and complete.2 The lattice M , already introduced in matrix investigations, furnishes a very appropriate example of an intensional lattice; for it has played a fundamental role in algebraic studies of relevant logics. Where M + = {+0, +1, +2, +3} and -M

= M

= {-0, -1, -2, -3} (i.e. -M + is the set-theoretical complement

of M relative to the elements M° of M ), where N(±p) = +p for every element p + o (in M°), and where < is the lattice ordering already exhibited in the Hasse diagram for M (which can be written down independently as a set of orderings on 1

These were called, in Routley 72 on which this subsection is directly based, Belnap lattices or icdlwtfs (a familiar Welsh term, unpronouncible even by Poles) and in Belnap 67, more longwindedly, intentionally complemented distributive lattices with truth-filters.

2

Some of the important properties of intensional lattices, especially the conditions under which a De Morgan lattice has a truth filter, are summed up (or referenced) in ABE, p.l93ff., and need not be repeated, since not used, in these predominantly semantical investigations. In particular, it is shown that a necessary and sufficient condition for a De Morgan lattice to have a truth filter is that N have no fixed point, i.e. that (N3) a ^ Na always. Remarking on what he takes to be 'the high intuitive plausibility of (N3) and (T) as conditions on propositional negation', Dunn (who wrote the section (footnote continued on next page) 1SZ

3.1 HOW MODEL STRUCTURES

DETERMINE INTENSIONAL

elements of M°), M q = <M + U M_, M + , N, < > . tediously) verified that M

AND DE MORGAN

LATTICES

It is straightforwardly (if a little

so defined is an intensional lattice. o

M q may be seen as made up from more basic components, in two interrelated ways.

Firstly, as already shown in essential details, M^ can be synthesised from

a three world set {T, d, d*}, its elements M° = M + U M being those of the power set P({T, d, d*}). Then D° consists of those elements to which T belongs and lattice ordering < reduces to set inclusion. Secondly, M q can be obtained as the direct product of the two-valued lattice 2 De Morgan lattice D^.

with diagram

J^with four-valued

On this picture, M q consists of two superimposed-four

element lattices, the upper lattice being picked out as designated by value 1, and the lower lattice being signed as undesignated by 0. Products of M , or o of D q times 2 provide all the intensional lattices required for algebraic analysis of FD. But on a deeper analysis these basic intensional lattices are generated from world semantics for FD. FD model structures determine intensional lattices. An intensional lattice I(S) = <M, D, <~i4 u, N, < > on m.s. S = - and likewise I(K) = I(S) on K is defined as follows: M - PK, i.e. the set of subsets of K; D = {p e M: T e p}; for p, a e M, p H 0 = { a e K ; a e o & a £ a}, p U a = { a e K: a e ' p v a e c r } , and Np = {a e K: a* I p}; and p < o iff p c o. (Thus the definition is an elaboration of the usual definition of a distributive lattice through a field of sets.) Lattice I(S) differs from De Morgan lattice L(S) = <M, n, u, n, < > on S - only in the addition (and definition) of D, i.e. all elements of L(S) * L(K) are defined precisely as for I(S). L({d, d*}) is of course D q (see 2.6, p.110). Since I(S) determines D, but not operation •+, I(S) is intermediate between L(S) and the full matrix S + on S. Intensional lattices are in effect obtained from matrices generated for relevant logics by replacing colour photographs of -+,s behaviour by black and white ones which show only designated places (see the derivation of M q above). Both the De Morgan lattice L(S) on S, and the intensional lattice I(S) on S may thus be read off the matrix S + on S, by replacing mappings by their extensions, i.e. the subclasses of K they map to 1, operations &, v on mappings by operations N, n, u on their extensions, and defining p S a as (p •*• o) (T) = 1, i.e. as p -*• a € D. Conversely, matrix operations &, v may be read off (or identified with) corresponding De Morgan lattice operations, and designated values of p + o read off p S a when D is determined as in intensional lattices (cf, p.117). (footnote 2 continued from previous page in question in ABE, p.194) continues: Indeed it is hard to think of anything more absurd than supposing that a proposition could be its own negation, and it is hardly less absurd to suppose that it should be impossible exactly to divide the propositions into the False and the True, a proposition being true iff its negation is false... Is this so? According to the venerable principle of identity of opposites, adopted by many philosophers from Heracleitus through Hegel, some propositions are identical to their negations. And according to genuinely phenomenological investigations of logico-semantical paradoxes, various paradoxical propositions are logically tantamount to their own negations. These features can be built into semantical theories which assign some propositions both of value True and value False and some propositions neither of these values (see §2) - semantics of a sort Dunn himself has advanced. 183

3.1 ALGEBRAIC

SEMANTICS

FOR TV INTERPRET

WFF IN INTENSIONAL

LATTICES

Lemma 3.1(1) Where S is an FD model structure (or a more comprehensive m.s. for a higher degree system such as E or I or R, or a DML m.s.), L(S) is a De Morgan lattice. (2) I(S) on FD m.s. S (or on more comprehensive m.s.) is an intensional lattice, i.e. L(S) is a De Morgan lattice and D £ M is a truth filter. Proof. (1) L(S) without operation N is the subset lattice of K (as defined e.g. in Szasz 63, p.34), and accordingly is a distributive lattice. Further, N(Np) = p since a** = a, and Np < Na iff a < p; hence L(S) is De Morgan. (2) Given (1) it is enough to show that D = {p:T e p} is a truth filter. Suppose, for consistency, that, on the contrary, for some p, p e D and Np e D. Then T e p and also T e Np, i.e. T* I p, so T k p which is impossible. Suppose, for completeness, that, again on the contrary, for some p, p I D and Np £ D. Then T i p and also T ^ Np, i.e. T* e p, so T e p which is impossible. Algebraic semantics for FD, which are rather artificial, involve interpretations of FD wff in intensional lattices. Such an interpretation in an intensional lattice I consists really of two parts (the artifice starts here); an evaluation in terms of I of each first degree implication (fde), i.e. of each FD wff of the form C -»- D, i.e. of each DML wff; and truth functional rules to account for first degree compounds of fdes. Given an intensional lattice I, an evaluation v on I (for an fde A) is an assignment of lattice elements [matrix values] of M to initial wff, i.e. sentential variables (of A). It is extended as follows for zero degree wff B and C: v(~B) = Nv(B);

v(B v C) = v(B) U v(C);

v(B & C) = v(B) n v(C).

I verifies FD wff Ci ->• C2 for evaluation v iff v(Ci) § v(C2). I verifies fde A iff I verifies A for every evaluation; otherwise I falsifies A. Lemma 3.2. I(S) on S = verifies fde A iff v(A, T) - 1 for every valuation v on I(S). Proof. Given v(p, a) define v(p) = {a: v(p, a) = l}. Conversely given v(p) = p define v(p, a) = 1 iff a e p. Then since p = {a: a e p} = {a: v(p, a) = l}, v(p) = {a: v(p, a) = l}. Thus, in any case, v(p) = {a: v(p, a) = l}, from which it follows by induction, as in a previous lemma, v(B) = {a: v(B, a) = 1} for any zero degree wff B. Then v verifies Ci C 2 iff v(Ci) < v(C 2 ) iff {a: v(Ci, a) = 1} c {a: v(C2,a) = 1} iff v(Ci + C 2 , T) = 1. Theorem 3.1. Where A is an fde, A is a provable entailment (of FD, and its conservative extensions) iff M verifies A, i.e. M , like D (of 2.6), is character^ o o' o ' istic for fde. Proof. A is provable iff A is true for every valuation on m.s. S = , i.e. iff I(S) verifies A. But I(S) on S is M Q . Actually M q , unlike D q , is characteristic for any truth-functional compound formed from a single fde. Given an intensional lattice I, an evaluation for fd wff (for wff A) is a pair where v is, as before, an assignment of lattice elements to sentential variables (of A), and w is a valuation assigning truth-values 1 and 0 to wff as follows, given the extension of v: Where A is a zero degree wff w(A) = 1 iff v(A) e D; where A has the form Cj C 2 , w(A) = 1 iff v(Ci) < v(C 2 ); and otherwise w(~B) = 1 iff w(B) 4 1, w(B & C) = 1 iff w(B) = 1 = w(C), and w(B v c) = 1 iff w(B) = 1 or w(C) = l.1 I verifies A for evaluation 1

The evaluations are like those given in Belnap 67 and repeated in ABE, p.207, where however valuations in intensional lattices are tendentiously linked with propositional modellings. Such a propositional modelling proceeds to identify what are often accounted distinct propositions. 1S4

3.1 PROOF OF ADEQUACY

OF THE ALGEBRAIC

SEMANTICS

FOR FD

(for A) iff w(A) = 1; and I verifies A iff I verifies A for every I-evaluation. Finally, A is lattice valid iff every intensional lattice verifies A. * Lemma 3.3 I(S) on S = (or an extension thereof) verifies FD wff A iff v(A, T) .« 1 for every valuation v on m.s. S. Proof extends the preceding lemma for fdes. Given v, define w(A) = v(A, T) and given define v(A, T) = w(A). The adequacy of these definitions follows from the following points. (i) Where A is a zero degree wff w(A) = 1 iff v(A) e D iff {a: v(A, a) = 1} e D iff T e {a, v(A, a) = 1} iff v(A, T) = 1. (ii) (iii)

v(Ci + C 2 , T) = 1 iff v(Ci) § v(C 2 ). v(~B, T) = 1 iff v(B, T) t 1, etc.

Theorem 3.2. (Adequacy of the algebraic semantics). Where A is a FD wff, A is provable in FD (or any conservative extension thereof) iff (1)

for every FD m.s.S,I(S) verifies A; and

(2)

A is lattice valid.

Proof. A is provable iff A is true for every valuation on every m.s. S « , by the adequacy theorem for FD, i.e. iff I(S) on S verifies A for every S, by the previous lemma, so establishing (1). It follows at once that if A is lattice valid then A is provable, so disposing of the completeness half of (2). The soundness half of (2) is proved by the usual inductive argument and all cases can be directly verified. But the work can be short cut. The schemata of S and the sole rule y are lattice valid simply in virtue of truth functional requirements w functions satisfy; and the deducibility schemata of FD just mirror logically lattice principles and so are inevitably verified. With algebraic semantics for FD - much as with world semantics - tighter results than (2) are to be had which much more severely restrict the class of models that have to be considered. As (1) suggests, only certain nicely behaved lattices need to be involved. The lattices generated by the m.s. K • {T, a ,...,a *} for each integer k are among these. Where K = {T, a ,...a *}, k k define N

• I(K).

N

plays the role with respect to FD wff that M q plays for

fde, and of course M q = N 1 .

But N^ is not alone in this roler

products of M q ,

and also of D q , can occupy similar roles (as can multiples of other lattices, such as the crystal lattice introduced later in the chapter). The sought result k using N can be derived immediately, whereas analogous results using product lattices of M q and D q call for some algebraic preliminaries. Theorem 3.3. Where A is an FD wff containing k occurrences of symbol -»-, A is k provable in FD (or any conservative extension thereof) iff N verifies A. Proof. A is provable iff A is true for every valuation on m.s. S • , by a corollary of semantical adequacy, i.e. k iff I(S) verifies A, by the preceding lemma, i.e. iff N verifies A. Although the main work (by Belnap and co-authors) on algebraic analysis of FD has been in terms of the products M^, for each k, of M^, in fact there is a disconcerting twist in the definition of M^ which is not based on the direct product of M taken k times, and what is going on is rendered rather more trans° k parent by considering the direct product 2 x D , i.e. 2 x D x...xD with D tatcen k times. ~ o o o

US

3.1 PRELIMINARIES

IN SHOWING THE FIRST VEGREE ADEQUACY

OF 2 x V k and M k

The point of the second product can be immediately glimpsed. For consider a wff of FD with k implications. Each implication can be taken account of by one occurrence of D (generated by a different pair {a., a.*}) which characterises ° J J that implication, and the overall truth function of the k implicational subformulae is taken care of by 2 Moreover ,2 x D does, in a good sense, exactly k k _k the work of the product M . Proving all this and the adequacy of 2 x D and M requires several preliminaries and readily draws in some wider ranging results. Lemma 3.4. (1) The direct product L } x...xL de Morgan lattice. (2)

of De Morgan lattices L , — , L n is a

Where L ^ ) , — ,L(lO are De Morgan lattices on mutually disjoint

*-complemented sets K. (1 < i < n), L(l)K.) is a De Morgan lattice, i i i (3)

Where

< ^ <

n

is

a

set of mutually disjoint *-complemented sets

and L(K^) is a De Morgan lattice on K^, L(UK^) is isomorphic to L(Kt)x...xL(Kn). (In fact n need not be finite.) Proof. (1) is a consequence of the componentwise definition of operations in a direct product (see e.g. Szasz 63, p.197). In particular since, by definition N

= for p e P(K.), and since , p > < I 2 n i 2 n x i i n iff for every 1 < i < n p . < a . , NN

=

and K i n l x * n i* "n N < BWjj,... ,an> iff < ^ . . . . , p n > . (2) is like the proof that L(S) on FD m.s. S is a De Morgan lattice; for the union of disjoint ^-complemented sets is a *-complemented set. (3) Where K. has n. elements then the elements of L(UK ), given by 1 1 1 . . I n 1 PCUK^) are 2 in cardinal number, and the elements of L(Kj)x...xL(K ), given by P(Kj)x...xP(Kn), are 2 n r K '

in

number.

Hence there is a 1-1 correspondence

between the sets concerned given by a 1-1 function. 1

Now order P(Ki)x..,xP(Kn)

1

1 and represent the i "* member a. x = <sub.K l i ,...,sub.K i n > with sub.K. x j the i*"* subset of P(K.); and order P(UK.) so that the i member = sub.KiU.. .Usub.K . Let 3 i i i x x n

h be the 1-1 function which maps a^ to B^;

then h is the required isomorphism.

For, first, h(a.r\x.) = h(<sub.K nsub.K ,...,sub.K nsub.K >) x 3 = (sub.Kxnsub.K, i j ')U...U(sub.Kx Hsub.K n j n) x i j i x n 3 n = (sub.K U...Usub.K )n(sub.K U...Usub.Kn ), xi x n j i 3 since sub.K <~)sub.K„ = A for 8 ^ a. x a j 6 = h(ai)n?i(aj). fc(a..Ua.) = Ma^ufctoj) directly. Also h( Na ± )

= Hffi(a±)

Uh(a.) x

= N(sub.K U...Usub.K ) x i x n

7zN(a.) x

= Nsub.K U.. .UNsub K . i l in

For and

1U

3 . 1 T-HOMOMORPHISMS

ANV PRODUCT

LATTICES

Let a. € K., then a. e 7z(Na.) iff a. e 1 J i 3 J a^ e N(sub a e

U.. .Usub^fO iff a^ e Nsub^K^.

UK^ a e h(Ha^)

iff a e Hh(a^),

(OF ARBITRARY

Nsub.K.; J ]

CARDINALITY)

VEFINEV

and also a. e N?z(a.) iff 3 1

Hence, generalising, for every

i.e.fc(Ntt^)= Nfcfa^) as required.

Thus h is

a homomorphism. The direct product of De Morgan lattices may however lack truth filters, even when some of the lattices have truth filters, unless the lattices satisfy further conditions. Furthermore, De Morgan lattice homomorphisms may not preserve the property of belonging to the truth filter. To evade these difficulties, the homomorphism relation is duly strengthened and certain suitably defined product lattices are focussed upon.1 A T-homomorphism (or truth-preserving homomorphism) of an intensional lattice I with truth-filter D into an intensional lattice I* with truth-filter D' is a homomorphism h of I into I' such that whenever p e D then h(p) e D'. A T-isomorphism is a 1-1 T-homomorphism.

To define the product

in terms of M

taken n times, divide the elements M° of M q into its parts M + » {+0, +1, +2, +3} and M (M )

n

= {-0, -1, -2, -3}.

Let (M + ) n be the direct product of M + n times and

the similar direct product of M .

Then, by definition,

M° - <(M + ) n U (M_)n, (M+)r , N, where N and < are defined componentwise on ( M j n U (M ) n (i.e. set M11 for short); that is, for and + n n in M°, N - , and < < o l f . . . , a n > iff p ± < o ± for every i such that 1 < i < n.

The definition of

may be generalised to products of arbitrary cardinality (i.e. n need not be finite) using indexed sets. Firstly, products of intensional lattices are defined generally. Where s ort an { M ^ D^, N^, ^ i ^ i g i N ' is the structure ^ » i s <3f,D,N,<> indexed set of intensional their product defined thus: D - lattices j-j^* D is the direct product of the indexed sets D ^ -D « X i € iN^ M i~ D i^* M = D U -D, and for { c ^ } ^ and { B ^ ^ i n iff

d

for

every

1 in

M, N ^ } . ^ ) 711611 w h e r e

^ilelN i is the cardinality of set IN, M C = ^gjjj 1 ^'

= { N ^ } . ^ and i a j ^ I

is M

<

for each

ieIN, and c i Q Finally, for present definitions,

where T e K^ define a (T-) truth filter T p for the direct product P = L(Kj)x...1(K.)x...L(K ) of De Morgan lattices as the class of ordered sets of th ^ P whose j elements belong to the truth-filter T^ of I(K^). In such a case where a filter is distinguished (as on the working paradigm designated truth values are), we write of a filtered product. Q

Lemma 3J5. (1)M

is an intensional lattice.

(2)If for some a e K, a = a*, De Morgan lattice L(K) on K has a truth filter D = {p:a e p}. (3)The filtered product P is T-isomophic to I(UKi). (4)Where 2 = I({G}), the two element Boolean algebra with elements 1 = {G} and 0 = A and truth-filter 1, M is T-isomorphic to the filtered product 2 x D o > 1

This concentration proves to involve little serious loss in generality in view of embedding theorems, such as that established below or that presented in ABE, p.201. m

3.1 ALGEBRAIC

(5)

REPRESENTATIONS

Where D

= D

Of FIRST DEGREE WFF IN PRODUCT

x...xD

product 2 x D n .

11 times,

LATTICES

is T-isomorphic to the filtered

(Note: each D q can be taken to be generated by a

different set {aj, aj*}). Proof. (1) Proof is like part (1) of the previous lemma, and relies heavily upon the componentwise definitions. (3) Let h be the isomorphism given in the previous lemma part (3). It suffices to show that if a. e T then h(a.) e T, where T - {p e I(UK.):G € p}. l p i 1 i Now supposing o^ e T , then s u b ^ j

e T

j•

Hence G e sub^K^.

Hence G e

t

and h(a^) e T. (4) is a special case of (5). (Note that every Boolean algebra furnishes an intensional lattice, since a maximal filter is a truth filter.) (5)

The mapping for a =

^

h{a) = <sg(a), ^ / W ^ ) }^ , where homomorphisms f and sg are defined from M q onto D as follows: o = 1;

f(-3) = / ( + 0 ) = 4

f(-1) = f(+l) - 2;

/(-2) = /(+2) = 3

/(-0)

= /(+3)

sg(a) = 1 iff a € D ( M Q ) ; sg(a) = 0 iff a 4 D . 1 Then L

m" + 2 x D n is a T-isomorphism.

Theorem 3.4.

Where A is an FD wff with k occurrences of symbol

a is provable in

system K iff the filtered product 2 x D^ verifies A, and iff M^ verifies A.

--

Proof. The result follows from the previous theorem using parts (3) and (5) of the preceding lemma. For N^ is T-isomorphic to 2 x D^ and thereby to M^. Thus, for example, if M^ falsifies A the evaluation obtained by composing the T-isomorphism function with k the falsifying evaluation falsifies N . Corollary 3.6.

Every finite intensional lattice is isomorphic to a sub-lattice of N

for some finite k, and to a sublattice of M

Q

for some finite c.

2

Proof is the same as the stronger corollary in Belnap 67 (except that given the inverse diagram ID(A) of A, D(A) the disjunction of all members of ID(A) is defined). A better embedding theorem than the corollary affords can however be obtained by algebraic means combined with semantical. Theorem 3.5. Where I = <M, D, n, u, N, §> is an intensional lattice, there is an m.s. S = such that I is T-isomorphic to a sublattice of I(S). Proof. Let K be the set F of all prime filters of I. For a e K, define a* = {m e M : Nm i a} and T = D. Then T e K, since D is a prime filter, T = T* since D is consistent; and a* is a prime filter. For, m, n e M, 1

The mapping was taken from Belnap's miscellaneous notes, unpublished.

2

Had the results been carried through - as they could have: see subsequent theorems on sequences of FD wff - for sequences of fd wff, Belnap's corollary would go through intact.

IBS

3.1 PROOF OF A MAIN EMBEWING

m n

n

e a*

m U n e a*

iff

N(m n n ) i a

iff

Nm U Nn

4a

THEOREM

FOR INTENSIONAL

a iff m

e a*

LATTICES

4a and Nn

4

iff

Nm

iff

m e a* or n e a*,

and n e a*

by a similar argument. Now for m e M, define h(m) = {a e F:m e a}, i.e. as in Stone's representation theorem, the mapping h associates with each element m the set of prime filters to which m belongs. For a T-homomorphism it is required (i)

for m, n e M, h(m n n) - h(m) n h(n),

(ii)

for m, n e M, h(m U n) = h(m) U h(n),

(iii) (iv)

for m e M, ?i(Nm) - Nfc(m), if m € D then h{m) e D' = {p £ M':T e p).

Proofs of (i) and (ii) are standard. a e ft(Nm) iff

As to (iii), for any a e K;

Nm e a

iff

~(m e a*)

iff

~(a* e h(m))

iff

a e THh(m) , by definition of Np, for p e M'.

For (iv), given m e D, m e G. Thus, since T c K, T e {a e F:m e a), i.e. T e h{m). Hence h(m) e {p e M':G e p}, i.e. h(m) e D'. _ Now define T(S) = <M,D,n,u,N,i> as follows; M = {fc(m):m e M} and D =_Oz(m):m e D}; and the operations of I(S) are the operations of I restricted to M. Since M', the power set of the set of all prime filters of I, includes M, and since I(S) is closed under operations, I(S) is a sublattice of I(S). Accordingly h is a T-homomorphism of I into I(S). Further, since h is 1-1, it is a T-isomorphism. For if m ^ n then m S n say. By Stone's prime filter theorem there is a prime filter ai say such that m e aj and n I aj, and hence fc(M) *Hti). Corollaries 3.7. 1. Every De Morgan lattice is isomorphic to an N-complemented ring of sets. 2. Every intensional lattice of cardinality a is T-isomorphic to a c ?a sublattice of M for some c < 2 Corollary 1 is proved in the course of proving the theorem, while corollary 2 depends upon combining the theorem, and the bounds it furnishes, with previously established T-isomorphisms. Corollaries like 2 provide the basis for more algebraic proofs of the adequacy theorems established above. The algebraic analysis for FD is at the same time an algebraic analysis of the first degree of all higher degree systems with the same degree as FD. Since all the main candidates for logics of entailment and of relevant implication share this common first degree, the analysis affords an analysis of the first degree of entailment and of relevant implication, as well as of the first degree of many irrelevant logics (see 7.7, where main results concerning the common first degree of a diverse class of logic are established). To signal the greater generality of the results, that they are established not merely for FD, the results are sometimes stated for all conservative extensions of FD, i.e. for extensions of FD whose stock of theorems in the syntax of FD does not exceed FD itself. It is trivial, of course, given the characterisation of conservative extension, that results which hold for FD hold for the first degree part of con-

189

3.2 SOMETIMES

SPECIOUS

REASONS

TOR ALTERNATIVE

FIRST VEGREE

SEMANTICS

servative extensions of FD. What is less trivial is that such results hold for the first degree of such systems as E, R and I, which are in fact conservative extensions of FD, and it is such less trivial results we want to anticipate here, and not repeat in full subsequently when (as in 7.7) we have shown that E and I are conservative extensions of FD.1 Thus it is taken for granted that results for FD hold for the first degree part of systems such as E and R. §2. Alternative semantics for the first degree, and disposing of various objections to the first degree theory. The semantics furnished for FD, simple and recent though they are, have already encountered vigorous opposition, on two main scores; firstly, the use of worlds in the semantical theory, and especially of incomplete and even worse inconsistent worlds, and secondly, and connected with the latter, inclusion of the *-operation. Neither kind of objection should carry much weight. The objection to talk of worlds is based on the false assumption that worlds must be interpreted either ontically or epistemically. The epistemic option raises serious difficulties as to how interpretation is to be achieved at all, particularly in the case of impossible worlds which are, it is claimed, epistemically inaccessible, and even if it should somehow be done (e.g. along lines indicated in Belnap's 77) what it has to do with semantics proper. The ontic option raises all the familiar issues as to the ontic status of worlds beyond the actual that are directed against modal logic semantics, and, given the types of worlds involved, more. The conventional choice is however a false one: the interpretation of FD intended is neither ontic nor epistemic, but neutral (in the sense explained in EMJB). Objects such as worlds can be spoken about and deployed semantically though they in no way exist and often could not exist, and though they do not reduce to epistemic or other like construction. The objection to the •-operation seems to -he based on the seriously mistaken assumption that the operation was cooked up out of nowhere just for the semantical theory and that no further or independent account can be given of it - an objection already disposed of (see especially 2.9). Because of the objections however, however misguided, there has been some interest in semantics for systems like FD which remove either worlds or the •-operation or both. Semantics of all three further types for FD can readily be obtained from that already given. The methods used in deriving these further semantical analyses apply quite widely to systems other than FD. 1. The elimination of worlds. In simple cases, such as FD and modal logics like S5 and higher relevant logics like _P none of which require relations between worlds, worlds may be absorbed into valuation functions, so yielding what will be called a pure valuational (or V) semantics2, which comprise purely valuational functions. Worlds can drop out, because (at least at the sentential level) they serve only as carriers of different sets of value assignments, and these can be carried alternatively by different valuations, i.e. the pair consisting of I(A,a) can be fully supplanted by the valuation function for each world a. From this perspective, worlds are but a perhaps convenient way of organising sets of value assignments to initial wff; and since worlds - allegedly - cause metaphysical troubles it is convenience better dispensed with. 1

By considering implicative extensions of intensional lattices many of these results could be established, building on present algebraic apparatus: cf. Belnap 67. Implicative extensions are considered in chapter 7.

2

In Latin America where many semantics of this sort were first worked out, they are called semivaluational semantics.

190

3.2 PURE VALUAT10NAL

SEMANTICS

WITHOUT

WORLVS

The absorption of worlds may be effected by the following V-translation which transforms world semantics into pure valuation semantics:- with respect to a given FD model, for each a in K-{T} I(A,a) translates a(A) (with Greek letter chosen to correspond to Roman letters for worlds), and I(A,T) translates to T(A). a, which is a function from wff to II = {1,0} is called a local valuation, and x which is in effect a distinct member of the class of such valuations, a valuation. Under translation then a model M = is replaced by a pure valuational model M^ = <x,fi> where ft is the set of functions such as a yielded by application of the interpretation function I to worlds, i.e. the set of a such that for a e K, I(A,a) translates, for arbitrary A, to a(A). Such pure valuational models will serve for systems such as DLL and (as Routley and Loparic 78 show, pp.316-7. for systems such as Pi. For FD it remains to cater for the *-operation. But it is plain that operations on worlds transform to operations on functions, i.e. what is perfectly in order, certain functions on functions. That is, l(A,a*) translates to A*(A), where the operation * on functions has properties analogous to those on worlds, i.e. a** = a, and T* = x, i.e. * can be removed as far as valuations proper are concerned. A pure valuation model M^ for FD is a structure M^ = <x, ft, *> where ft is a set of local valuation functions from [initial] wff to II = {1,0}, x e ft, and * is an operation on elements of ft such that for a e ft, a* e ft and a** = a, and T* = T. Furthermore, it is either required that the following conditions be satisfied by every a in ft, or a is extended to satisfy the conditions: a(A & B) - 1

iff

a(A) = 1 = a(B),

a(A v B) - 1

iff

a(A) • 1 or a(B) - 1,

- 1

iff

a*(A) = 1;

a(~A)

T(A

B) • 1

and

iff for every local valuation a, if a(A) » 1

then a(B) • 1. A wff A of FD is V-valid, i.e. valid in the pure valuational semantics, iff T(A) " 1 for every pure valuation model, or alternatively (suppressing models) iff T(A) • 1 for every valuation T. Adequacy Theorem 3.6,for FD under the V semantics. V-valid iff A is a theorem of FD.

Where A is a wff of FD, A is

Proof uses the previous adequacy theorem and the V-translation, and perhaps also its inverse. Suppose A is a non-theorem of FD. Then there is an FD countermodel M = such that I(A,T) t 1. So by the V-translation there is a pure valuation model M^ = <x, ft, *> such that x(A) ^ 1. Hence A is not V-valid. Conversely, where A is not V-valid, there is a pure valuation model <x, ft, *> under which x(A) if 1. By the inverse of the V-translation (since it is a translation, the mapping is a 1-1 function and so has an inverse), there is a model such that I(A, T) ^ 1. For simply replace a(A) by I(A,a), a*(A) by I(A,a*), x(A) by I(A, T), etc., throughout the counterexample argument. Accordingly, A is not a theory of FD. Alternatively, soundness under the V-semantics may be established directly by the usual style of inductive argument over length of proofs. Corollary 3.3.

For each wff A of FD, A is FD valid iff A is V-valid.

It follows that in the case o f F D - and similarly for many other systems, including all so far considered - worlds are expendible in the semantical analysis. Of course the elimination of worlds is more a matter of appearances than substance, since local valuation functions depict worlds through what holds and what does not, but here as elsewhere in human enterprises appearances are taken to be what count. 191

3.2 THE 4-VALUED AMERICAN PLAN CONTRASTED

WITH ITS AUSTRALIAN

COUNTERPART

2. The Australian plan and the American plan; the elimination of *, and fourvalued semantics for FD. There are often good reasons to take seriously theories that are either inconsistent or incomplete with respect to negation. If we simply adopt such a theory as our guide for an assignment a truth-values or holding values, whatever we are going to make epistemically or ontically or neutrally of such values, two possibilities stick out for how the assignment of truth-values is to be made. One way, which we call the Australian plan, is simply to assign 1 to a wff that is in the theory and 0 to a wff which fails to be in the theory. Evidently, on the Australian plan, we assign exactly one of 1, 0 to each wff of a theory at that theory (or whatever we take to realize a theory, ontically or neutrally or epistemically). But, where our theory is inconsistent or incomplete, there clearly will be formulas A such that both A, ~A get assigned 0, or else both are assigned 1. The American plan is different.1 On this plan, we adhere resolutely to the view that ~A holds iff A does not hold, everywhere. But we allow that, in addition to being assigned a simple 1 or 0, a formula may be assigned both 1 and 0, or neither. Specifically, still being guided simply by our theory, we may assign 1 to A if it is in the theory and its negation is not; 0, if A is not in but its negation is. These are the classical circumstances. But if the theory is nonclassical as regards A, the American plan bids us assign both of 1, 0 to A if both of A, ~A are in our theory; neither, if neither A nor ~A is in the theory. Evidently, on the American plan, every formula is assigned at a theory (or whatever we take to realize that theory) one of four values: 1, 0, both, or neither. But, though the usual negation rules apply (in the sense that ~A is assigned at least 1 iff A is assigned at least 0, and dually) the Australian "strange situation" in which both A, ~A are assigned 1 gets an American counterpart in which A is assigned both 1, 0; similarly, the Australian assignment under which neither A nor -^A.hold has an American counterpart in which A is assigned no value at all. Thus -each -assignment on each plan has a counterpart under the other plan. Neither the Australian plan nor the American plan constitutes in itself a semantics for anything; rather, these are merely distinct plans which may be employed in conjunction with other intuitive and formal insights. Each plan has its advantages: the Australian, for its firm 2-valuedness; the American, for its more classical generalization of the classical treatment of negation, and removal of the * operator. Nor, indeed, should the differences between the Australian and the American plans be taken as all that significant. So far as relevant logics are concerned, anyway, a hard part of their semantic analysis lies 1

Despite the nationalistic terminology, already in use and taken from Meyer BV, neither plan has simple national affiliations (as Meyer subsequently admits pp.93-4), and adherents of each plan, such as they are, have been aware through much of their researches of the other plan. Of the three Meyer associates with the Australian plan, R. and V. Routley and Meyer, only one is strictly an Australian, Meyer himself being an American; and R. Routley had earlier used the American plan in the investigation of negation inconsistency and incompleteness (see, e.g.,70 pp.297-98) before the * operator, which enables the Australian plan to function, was adopted. Of the three Americans Meyer couples with the American plan, Anderson appears to have little to do with such a semantical (as distinct from algebraic) approach, while Belnap (and perhaps also Dunn) appears committed to an epistemic modification of the American plan on which there can be real truth-values, just true and false, as well as (four) told values. For according to Belnap 77, while we allow for situations in which, epistemically, we have been both told true that and told false that A, we have to agree deep down that A cannot really be true and false (and, mutatis mutandis, for incompleteness). Do we? The American plan was first developed formally, largely but not solely by Americans, for the study of incompleteness (see for instance R. Thomason 66 and 69) but the duality of the method was soon evident. On the much longer history of such 3 and 4 value approaches, see PL. 192

3.2 BRINGING THE AMERICAN PLAN FOR FV TO FRUITION

in the story that is told about a relevant •», reducing to a less problematic issue the question of whether we shall be Australians or Americans in the analysis of This is all the clearer in that, as the motivating explanation above makes plain, we can always obtain an American valuation from an Australian one and vice versa. The American plan does not extend very naturally however to higher degree logics: when nested occurrences of -*• are taken into consideration, simple correlation between the plans breaks down. At the first degree the Australian plan, as implemented through the semantics for FD already given, is readily transformed to yield a semantics for FD under the American plan, what we shall call the Dunn semantics1 for FD. Under the "American plan" the value set II = {1,0} is enlarged to include two further values, both 1 and 0, represented {1,0} (i.e. II itself), and neither 1 not 0, represented A or { }. The resulting value set is accordingly 1111= {1,0, {1,0} { }}, or, what comes to the same, and facilitates briefer presentation {{l}, {0}, {1,0}, { }}, or n for short. Initial wff are assigned any one of these four values; that is an initial wff A may either hold (i.e. has value 1) or not hold (0) or both hold and not ({1,0}) or neither hold nor not ({ }). By having the four values made up from two values, 1 and 0, several of the advantages of a two-valued semantics can be carried over to the four-valued scene. Obviously there is room for intermediate three element value sets {1, 0, {1,0}} and {l, 0, { }} with the third value representing, respectively, overcompleteness (or inconsistency) and incompleteness. Such valuation sets have been exploited in much logical work. The American plan may be combined with a world model structure - then each initial wff is assigned at each world one of the four values from n - or alternatively worlds may be absorbed into the valuation functions, as previously explained. Since the latter strategy, of eliminating worlds, is that favoured by those heavily involved in the elaboration of the American plan, such as Dunn and Priest, the Dunn semantics will be taken to be a. world-free one, i.e. purely functional. It is easy to recover four-valued worlds semantics from the Dunn semantics, as will be shown. A Dunn model M^ for FD is a structure M*5 -
1

1 e a(A & B)

iff

1 e a (A)

and

1

0 e a(A & B)

iff

0 e a(A)

or

0 e a(B).

€

a(B), and

Not because the semantics that follow are due to Dunn, but because of Dunn's enthusiasm for semantics of this type, and because they represent an easy extension of semantics under the American plan for first degree entailment theory which Dunn was the first to work out (see Dunn 71). Dunn also applied the American plan to the irrelevant logic RM (see his 76). While the plan is fairly easily applied to other irrelevant logics in the vicinity of E and R, until very recently it resisted extension to higher degree relevant systems (see 4.7). The semantics that follow are due, in so far as they are due to any individuals, to Priest, and Routley and Loparic. They are given in Priest 78. and indicated in Routley and Loparic 78, p.318. In both the presentation of the Dunn semantics and the proof of their adequacy we make some use of Priest 78. Calling such semantics the 'Dunn semantics' has good historical precedent.

793

3.2

VARIOUS REPRESENTATIONS

OF THE 4-VALUEV

(CONJUNCTION)

RULES

Were the local valuations two-valued functions to {{l}, {0}} these clauses would furnish exactly the classical truth-table conditions. Since then 1 £ a(C) iff a(C) = {1} and 0 £ a(C) iff a(C) = {0}, the clauses just amount (upon absorbing curly brackets into underlining) to the classical rules a(A & B) = 1

iff

a(A) = 1

and

a(B) = 1, and

a(A & B) = 0

iff

a(A) = 0

or

a(B) = 0.

Similarly, in the case of other connectives v and, most important, classical conditions are simply aped - only relative now to four values generated from the two classical values.1 But, looked at differently, the clauses amount to a short-hand way of setting down much messier four-valued clauses, which run as follows: a(A & B) = {1} iff a(A) = {1} and a(B) - {1} ; a(A & B) = {1,0} iff a(A) = {1} and a(B) - {1,0}, or a(A) = {1,0} and a(B) = {l}, or a(A) = {1,0} = a(B) ; a(A & B) = { } iff a(A) = {1} and a(B) = { }, or a(A) = { } and a(B) = {l} or a(A) = { } = a(B);

and

a(A & B) = {0} iff a(A) = {0} or a(B) = {0}, or a(A) = { } and a(B) * {1,0}, or a(A) = {1,0} and a(B) = { }. As is soon evident however, the four-valued clauses may be represented more elegantly in other, interwoven, ways. First, they may be summed up in a matrix that has already been presented (p.115 above) and explained and derived (p.117). Secondly, the clauses, like the matrix, may be encapsulated in and read off a fourvalued lattice, specifically the fundamental lattice D q (of p.117). &

{I}T,I

{l,0}B,2

{ }N,3

{0}F,4

{1}T,1

{1}T,1

{l,0}B,2

{ }N,3

{0}F,4

{0}F,4

{1,0}B,2

{l,0}B,2

{1,0}B,2

{0}F,4

{0}F,4

{1,0}B,2

{ }N,3

{ }N,3

{0}F,4

{ }N,3

{0}F,4

{ }N,3

{0}F,4

{0}F,4

{0}F,4

{0}F,4

{0}F,4

{I}T,I

{1}T,1 {1,0}B,2\^ N ^>{ }N,3

~

Three formulations of the matrices and the lattice are here combined, the different matrix values reflecting different origins of the values and roles for the matrices:-

1. On the left are the values of n, from the Dunn semantics. These originate under the American plan, where every wff either holds only, {1}, fails only, {0}, both holds and does not, {1,0}, or neither holds or does not; { }. But the values can be seen as derived in other ways, as 2 and 3 make plain. {0}F,4

2. On the right are the values 1, 2, 3 and 4 of the fundamental De Morgan lattice D presented in 2.7 (p.117). These were defined in terms of a world a

So it is also with respect to three values in the case of tukasiwicz's system -fc^ a n d as regards RM^' a three valued extension of EM.

194

3.2 RULES FOR v, ~ AW

->, AW

DUNN

MODELS

and its * image a* as follows: 1 = {a, a*}, 2 = {a}, 3 = {a*}, 4 = A. This shows how the values arise directly under the Australian plan; and it also suggests how the two plans may be interrelated, by associating function a with the pair . 3. In the centre are the values T, B(oth) N(either) and F of the Boolean algebra 4, which can be viewed as composed of classical truth-values from II = {1,0}. For 4 is but the Cartesian product II x II of II with itself, the values being T <1.7 1>, B = <0, 1>, N = <1, 0> and F = <0, 0>.1 Then the & matrix is simply the lattice or Boolean operation of intersection defined pointwise (i.e. in terms of II2). But this gives no significant advantage to a Boolean construal since on other approaches the & matrix can equally be read off lattice structures, e.g. from D^. Nor is the Boolean approach the only one that generates the four values from two values. For of course lattice products can be formed in the same fashion. Moreover, n can be usefully viewed not in product terms but through selections. Its four values represent all selections from II, . i.e. N = P H . for v:

1 € a(A v B) iff 1 e a(A) or 1 e a(B),

and

0 £ a(A v B) iff 0 £ a(A) and 0 e a(B). Disjunction may be defined in the usual way, so the v matrix may be calculated from those for & and ~ given. More simply, the v matrix may be read off underlying lattice D q , in terms of lattice join. for

1 e a(~A)

iff

0 £ a(A),

0 e a(~A)

iff

1 £ a(A).

and

With two-valued local functions these reduce precisely to the classical travesty of negation: a(~A) = 1 iff g(A) = 0 and a(~A) = 0 iff a(A) = 1. In explicitly four-valued form they take the form a(~A) = 1

iff

a(A) = 0;

a(~A) - {1,0}

iff

a(A) = {1,0}

;

a(~A) = { }

iff

a(A) = { }

;

a(~A) = 0

iff

a(A) = 1.

and

Thus, in lattice terms, 1 and 0 invert and {1,0} and { } are fixed points under negation. for ->: t(A -> B) = {1} iff for every a

(in M b ) if

1 e a(A) then 1 £ a(B) and if 0 £ a(B) then 0 £ a(A);

and

x(A -*- B) = {0} otherwise. This rule too is a straightforward and natural extension of a classical rule. A wff is true in a Dunn model M^ = : a four-valued function yields a pair of two-valued functions, and con1

This is how Meyer BV see p.94 ff.

prefers to construe the logical lattice L4, of Belnap 77: j

3.2 INTERCONNECTING

FV ANV VUNN MODELLINGS

THROUGH

MATCHING

versely two two-valued functions multiply up to a four-valued one. While the alternative route is shorter and (even) more obvious, since worlds have already been removed, it is instructive to make the interconnection with the original FD semantics. However a feature of the alternative route, the starring of functions, is wanted. For every a, a* is defined thus, with respect to every initial wff A of FD: 1 e a* (A)

iff

0 I a (A)

0 e a* (A)

iff

1 I a (A)

(* map)

Since a is a function, a* is also a function, likewise to n. Just as a was extended from initial wff to all wff by an inductive definition, so a* is extended also, by the same definition. Fact 1. The * map extends to all zero degree wff of FD. Proof is by induction over number of connective occurrences of are cases for each connective: ad

1 e a*(~A) iff 0 e a*(A)

& and v.

There

by the extension of a* to all wff

iff 1 i a(A)

by induction hypothesis

iff 0 I a(~A)

by the extention of a to all wff.

0 e a*(~A) iff 1 € a*(A), i.e. iff 0 i a(A), i.e. 1 I a(~A) ad &.

1 e a*(A & B) iff 1 e a*(A) and 1 e a*(B) iff 0 i a(A)

and 0 I a(B)

iff 0 I a(A & B) 0 e a*(A & B) iff 0 e a*(A) or 0 e a*(B) iff 1 I a(A)

or 1 I a(B)

iff 1 I a(A & B) ad v.

Similar

Observe that * is, as * map reveals, an operation on a, and it can in principle be repeated, i.e. an operation * can be defined, using the * map, on fi. Then a** = a and T* = T. The main interconnection between modellings is captured in the following definition: The pair matches a iff, for every initial wff A of FD I(A,a) = 1

iff 1 e a(A),

I(A,a*) = 0

iff 0 e a(A)

and

] i

vML;

j

where a = {A: I(A, a) = 1} and a* = {A: I(A, a*) = l}.1 Fact 2. The matching connection (MC) extends to all zero degree wff of FD. Proof is by induction over the number of connectives involved. There are three pairs of cases:ad

I(~A, a) = 1

ad &. 1

iff

I(A, a*) = 0,

iff

0 e ot(A)

I(~A,a*) = 0

iff l

iff I(A, a) = 1

iff

1 e a(~A)

iff 0 e a(~A)

e

I(A & B, a) = 1 iff I(A, a) = 1 = I(B, a)

Thus worlds are generated as set-ups in the fashion of Routley2 70.

196

a(A)

3.2 PROOF OF THE THEOREM

INTERCONNECTING

iff 1 e A ( A )

and

VUNN ANV FV MOVELS

1 € a(B)

iff 1 e A(A & B) I(A & B, a*) = 0 iff I(A,a*) = 0 iff 0 e A ( A )

or

I(B, a*) = 0

or

0 e a(B)

iff 0 e A(A & B) ad v : similar. Fact 3.

Where matches a, matches a*.

Proof. Suppose matches a*. Theorem 3.7(i) Where M = is any FD model there is a Dunn model M b =
such that I(A, T) - 1 iff x(A) - {l}. (ii)Where M b = <x,

M

= T(A)

is a Dunn model there is an FD model

such that

= {1} iff I ( A , T) = 1.

Proof (i). Let M be given. Define M b as follows: ft is the class of a such that for some a in K, a matches and x matches . M is a Duiin model, because T e fl since T = T* e K, and x is a map to {{0}, {1}} since T = T*. For then by (MC) and the rule, 1 e x(A) iff 0 I x(A), so excluding values {1,0} and { }. ~ It suffices to show, by induction, that for every wff A, (I) I ( A , T ) - 1 iff 1 e T ( A ) , i.e. iff X ( A ) = { L } . The induction steps for connectives extending interconnection (MC).

& and v are as in the proof of fact 2

ad Suppose, first ^(A two cases

By the evaluation rule for

B) 4 {l}.

there are

Case 1. For some a in M b , 1 e a(A) but 1 I a(B). Then, for some a in K (that for which a is ), I(A, a) « 1 and I(B, a) t 1, by the extension of (MC) to all zero degree wff as in fact 2. Hence I (A -»• B, T) 4 1, as required. Case 2.

For some a in M b 0 e a(B) but 0 I a(A) Now a matches for some

element of K. Hence, by fact 3, a* matches ; so a* € M b . Thus by the * map, fact 1, 1 e o*(A) and 1 I a*(B). Therefore by fact 2, I(A, a*) = 1 and I(B, a*) ^ 1, hence I(A B, T) t 1. For the converse suppose I (A B,T) ^ 1. Then for some a e K, I (A, a) « 1 and I(B,a) ^ 1. Then where a matches , 1 e a(A) and 1 k a(B), by fact 2. 4 Hence by the -*• rule, x_(A B) i {l} completing the subproof that I (A +

B, T)

• 1

iff

x(A -»• B) - {1}, i-e. iff 1 e x (A -»• B) ,

and so completing the proof of (I), and of (i), (ii) Let M b be given, and define M a = {C: 1 e a(C)T and a* = {C: 1 e a*(C)7 Through the connection a = {C: 1 e a(C)}, T* « {C: 1 e x*(C)}. I(C, a) = 1 iff C e determines a, a* determines a*. Since a* matches a , and also matches a * . 197

as follows: K is the class of elements for a in fl, with C an initial wff. a determines a. T = {C: 1 e x(C)} and a. Finally, as regards *, where a = {C: 0 I a(C)}, by * map, It suffices to show

3.2 VARIATIONS

(II) (III)

ON THE VUNN SEMANTICS

M is an FD model;

: 4-VALUEV,

WORLVS

SEMANTICS

and

x(A) = {1} iff I(A,T) = 1 for every FD wff, i.e. 1 e x(A)

iff I(A,T) = 1, since _x is two-valued.

ad (II). T e K since l e f t . T = T* since T* = {C: 1 e 2*(C)} and X* = T_. a**= a, since a** = a. * is an operation on K, since a and a* are in K and application of * never leads beyond a and a*; and * is an operation since appropriately 1-1. ad (III). The argument is by induction, and the cases for proof of the matching connection, i.e. fact 2. ad

Suppose X^(A -*• B) ^ {l}.

& and v are as in

There are two cases:-

Case 1. For some a in ft, 1 e a(A) and 1 I ot(B). Thus for some a e K, I(A, a) = 1 and I(B, a) = 1, by fact 2, both A and B being zero degree wff. Hence, I (A -*• B, T) t 1. Case 2. For some a in ft, 0 e a(B) and 0 i a (A) Hence by fact 1, 1 e a*(A) and 1 i a*(B). Now a* determines a class a* in K, and by fact 2 again, I(A, a*) = 1 and I(B, a*) i 1. Hence again I(A B,T) t 1. Suppose conversely, I (A -*• B, T) ^ 1, so that for some a e K, I (A, a) = 1 and I(B,a) ^ 1. Since a e K, for some a in ft either a determines a or a* determines a. Since A and B are zero degree, by the matching connection (MC) and fact 2, either 1 e a(A) and 1 I a(B) or 1 e a*(A) and 1 I a*(B). But in the latter event, by * map 0 e a(B) and 0 I a (A). Thus in either event, 1 d x^(A -*• B), by the -*• rule. Hence ;r(A -+ B) ^ T l ) Corollary 3.8.

Where A is an FD wff, A is a theorem of FD iff A is Dunn valid.

The Dunn semantics for FD admit of obvious, but interesting, variations. First, worlds can be reinstated and the batch of functions in a Dunn model reduced to one, i say. A Dunn worlds model for FD is a structure where i is a function from initial wff and worlds to n. The translation from Dunn models to Dunn worlds models is like that from pure valuation models to worlds models; specifically, A ( A ) = I (A,

x (A)

a), for a e K - { T }

and

= i (A, T).

Secondly, the semantics can be made more explicitly four-valued. There is competing notation for the values, and there are competing interpretations for the values. To connect with other work, we settle on the values T, read 'holding only' or 'on only' (sometimes read 'true only'), JJ for both, read 'on and off', etc., N, for neither, read 'neither on nor off', and F, read 'off only'. B represents overcompleteness, overdetermination or inconsistency, and N underdetermination or incompleteness. A four-valued model M*5 for FD is a structure M^ = <x, ft> where ft is a set of functions from initial wff to {T,B,N,F} and x is a function in ft which takes only values T^ and f\ These functions are extended from initial wff to all wff of FD by, firstly, in the case of connectives &, v and ~ the four-valued matrices already given (in setting out the Dunn semantics), and secondly, in the case of -»• by the evaluation rule: x(A B) = T? iff, for every valuation a, if a(A) = I or B (i.e. a(A) = T or a(A) = B) then a(B) = T or B, and if a(B) = F o r N then a(A) = _F or N. Otherwise, of course, x(A B) = Where T^ and 1$ are taken as designated values and 1? and IJ are non-designated - one of three natural construals the rule simply requires proper preservation of designated and nondesignated values. Since the explicitly four-valued semantics for FD is but a recasting of the Dunn semantics, its adequacy is evident.

198

3.2 THE

(UNOBJECTIONABLE)

0NT1C COMMITMENT OF THE FIRST VEGREE

SEMANTICS

In the four valued and Dunn semantics all reference to worlds has been removed. Of the Q-ontic commitment (i.e. ontic commitment in the sense of such reference theorists as Quine) of the FD semantics to both worlds and functions, only Q-commitment to functions remains. The Q-ontic commitment is indeed only of the order of that of the semantics for classical sentential logic. Moreover, in the Dunn semantics only the two classical truth values, 1 and 0, are required: it is merely that their exhaustiveness and exclusiveness is no longer assured. The functional translations and alternative semantics thus reveal that worlds occur inessentially in first degree semantical theory. They have then parallel consequences to those that the elimination of descriptions (in the limited contexts where it can succeed) has for what there is, for what is a subject of discourse, and what exists - namely, none at all as to what exists, but a substantial bearing on what subjects can be encompassed in a given vocabulary. It means, in particular, that we have provided semantical analyses of first degree systems that should be acceptable to those who are prepared to talk quantificationally of functions but not prepared to admit quantificational talk of worlds. In this respect first degree systems compete directly with classical logic. Yet further semantics for FD. The alternative semantics for FD, and other first degree systems, outlined in this section, designed primarily to remove worlds or the * function or both from view, by no means exhaust the supply of semantical analyses of FD. Other alternatives, in the shape of matrix and algebraic semantics, were studied in §1; yet other alternatives, such as range and content semantics will emerge in §3; and different semantics again will appear as special cases of subsequent investigations, e.g. metavaluational semantics.1 On top of l The naive metavaluational interpretation of FD, though sound, is not complete: both its form and its failure are instructive. Such a metavaluational v is a function from wff to {1,0}, i.e. to two holding-values (or more generally to several values) such that for every FD wff A and B, v(A & B) = 1 iff v(A) = 1 = v(B); v(~A) = 1 iff v(A) 4 1; v(A v B) = 1 iff v(A) = 1 or v(B) = 1; v(A

B) = 1 iff U ^ A B and ' FD if v(A) = 1 then materially v(B) = 1. The first three classes are familiar truth-tabular evaluations. What distinguishes metavaluations is the rule for -»(and like rules for other, typically degree-increasing, connectives) which introduces syntactical provability considerations into the "semantics". The ->• rule says that even when truth preservation is required (it is characteristically dispensable), it is not sufficient for an implication to hold. Provability of the implication however closes the gap. A wff A of FD is (tentatively) metavalid iff v(A) = 1 for every metavaluation v of FD. Soundness theorem. A wff A of FD is a theorem of FD iff A is metavalid. Proof. It is immediate that the rule and the classical schemata are metavalid in virtue of the truth-tabular rules for &, ~ and v, and requirement on -»•. To conclude the inductive argument for soundness it remains to establish the specifically entailmental postulates adapted from DML. One example will serve to illustrate the very distinctive metavaluational method of verification, ad (A B) & (A C) =>. A (B & C). Suppose then v(A -* B A ->• C) = 1; to show v(A B & C) = 1, i.e. I r- DA -»• (B & C) and if v(A) = 1 then materially v(B) = 1 = v(C).

By supposition however, (- ™ rU

A -*• B and I-

r D

A

C, whence by

the very postulate in question (A -»• (B & C); and if v(A) = 1 then materially rD v(B) = 1 and v(C) = 1, as required. Completeness fails however. For such a wff as ~(p •*• q) is not a theorem of FD but is, because (p q) is also not a theorem, metavalid. FD calls for a more sophisticated metavaluational modelling.

199

3.2 A VETAJLEV EXPLANATION OF TABLEAUX METHOVS (FOR FV)

this there are semantics for FD which will not be encountered (except for passing mention) in this text, but details of which are available or can be devised from what is available, such as three-valued and significance semantics for FD (developed in SL), many-valued semantics, both finite and infinite valued, and, closely connected therewith, Boolean-valued semantics (piven in effect in Meyer BV), fuzzy (logic) semantics and probabilistic semantics. 1 Just as semantics for FD are far from uniquely determined, so it is with higher degree extensions of FD. There are various interconnected semantics, some of which will be developed in what follows, others of which remain to be worked out (even where their analogues for FD are known), and presumably others which remain to be discerned (e.g. perhaps superior semantics for systems R and NR). Alternative tableaux and proof-theoretic methods for FD. Several of the semantics given for FD admit of reorganisation as semantical tableaux, and these can in turn be transformed into methods such as natural deduction and ^entzen-sequents procedures for FD. Fairly general methods for converting semantics for relevant logics into more proof-theoretic analyses of these types are assembled in chapter 11: but the general methods, like the semantics, simplify substantially in the first degree environment. Here one tableaux method for FD is presented in fair detail; other methods, which are essentially variants upon it, are sketched; and certain other connected more proof-theoretic methods for FD are merely indicated. Consider, firstly, tableaux methods for FD, which are based on the two-valued worlds semantics. The methods - which will be explained in a self-contained way differ from the tableaux methods for modal logics (considered in detail by Kripke 63 and 65 in particular) primarily in the following features: firstly, tableaux are introduced in pairs, i.e. whenever a new tableau is called for by the occurrence of A B on the right of a tableau t^ (in fact, the main or initial tableau) a pair of (auxiliary) tableaux, t^ and t^*, are introduced; secondly, rules for wff of the form A -*• B are provided only when the wff occurs in the initial tableau tg t^*), since such wff will not occur elsewhere in virtue of FD well-formation; and, thirdly, the rules for negation are those for relevant negation, not classical negation. A tableau consists of a left and a right column, set on a sheet. A sheet, e.g. a sheet or rectangle of paper, has two sides. Where t is a tableau its opposite t* consists of a left and a right column set on the opposite side of the sheet t is set on. Thus where t is a tableau so is its opposite t*. (The opposite t* can of course often be more conveniently represented diagrammatically by a separate rectangle on the same sheet of paper as t's rectangle.) With tableaux constructions are made, by stages. Each stage of a construction consists of a system of alternative sets of tableaux. A construction for a wff A (of FD) is started by putting A on the right of the main tableau t^, which consists of two columns, a left (interpreted as the truth column, or as showing what holds) and a right (interpreted as the false column). (The motivating idea is to find a countermodel to A, if there is one; otherwise A will be valid.) This is the initial stage, stage 1, of the construction. The stages of the construction continue by the application of rules for analysing wff in tableaux into their subformulae. Application of one of these rules, applied to wff of the form B C appearing on the right of t^, introduces further tableaux, auxiliary tableaux. So result in general a set of tableaux, consisting of the main tableau and auxiliary tableaux. In addition, by application of rules &r (for a conjunctive wff on the right of a tableau) or v£ l

Naturally, probabilistic semantics for FD, and for relevant logics more generally, will be based not on classical probability theory (which reflects the faults of classical negation) but on relevant probability theory, such as that introduced in UL. 200

3.2 RULES FOR CONTINUING TABLEAUX CONSTRUCTIONS (FOR FV)

(for a disjunctive wff on the left of a tableau)or a system of such sets results, each set of the system being called an alternative set. Subsequent stages of a construction after the initial stage result then from continuing the construction by application of the following rules (which, except for -*• rules, apply to any tableau t, main or auxiliary^ in an alternative set) in any order of application:(i.e. the rule for a wff with main connective ~ in a left column): where ~A appears on the left of tableau t, put A on the right of its opposite t*. ~r: where ~A appears on the right of t, put A on the left of t*. Note that t^* = t^ and (what is automatically ensured) t** = t. hi (i.e. the rule for a wff with main connective & in the left column): where A & B appears on the left of tableau t put both A and B on the left of t. &r: where A & B appears on the right of tableau t, then t (and also t*) is extended in alternative pairs t and _t (and correspondingly .t* and » ^ is put on the right of t ana B on the right of ^ T h e alternative tableau formulation provides a more precise way of saying what is more simply expressed thus: where A & B occurs on the right of t either put A on the right of t or put B on the right of t. Kripke offers the following informal explanation of how the next stage is obtained in application of &r : 'if the original ordered set [of tableaux] is diagrammed structurally on a sheet of paper, we copy over the entire diagram twice, in one case, putting in addition A in the right column of the tableau t and in the other case putting B' (63, p.73). He then gives a more formal statement of the matter* in effect as above. vl: where A v B appears on the left of t, begin alternatives ^t and ~t and put A on the left of ^t and B on the left of „t. When a tableau splits into alternatives, all entries already made are included in both alternatives, vr: where A v B appears on the right of t, put both A and B on the right of t. where A B appears on the left of initial tableau tg - the only tableau for which the case occurs - for each tableau t in the construction (whether so far introduced or yet to be introduced) there are alternatives ^t and _t, and A is put on the right of ^t and B on the left of ^t, i.e. (less exactly; either put A on the right of t or put B on the left of t. -»r: where A B appears on the left of initial tableau t^, begin a new tableau t and put A o n the left of t and B on the right of t. Only this rule generates new tableaux (as distinct from alternatives). An FD construction continues until either it closes or else, as it turns out, none but redundant rules can be applied, since otherwise only initial wff remain. A rule is redundant if either it applies to a wff in a closed alternative set or its application is already provided for (i.e. applying it would merely duplicate the situation). A tableau is closed iff some wff A appears on both sides (i.e. in both columns) of a tableau, a set of tableaux is closed iff some tableau in it is closed, a system of tableaux is closed iff each of its alternative sets is closed. Finally, a construction is closed iff at some stage in the construction a closed system of alternative sets results. As an example consider how the construction for A -*• B =>. ~B -»• ~A, i.e. ~(A -*• B) v (~B + ~A), closes:

201

3.2 PROOF THAT TABLEAUX CLOSURE COIHCWES

WITH VAL1V1TV

Main tableau t Q (=t0*) ~(A-»B) V (~B-+~A) ~(A-B) ~B->~A

}•initial stage (say) 1 stage 2, by 1 vr } stage 3, by ~r

A -»• B set of tableaux

Auxiliary tableaux tj* (Strictly t * is the) opposite side of t^.) \ stage 4, J by -*r A

j stage 5, J by ~r B

i stage 6, { by ~i

In stage 7, by application of the set splits into alternatives, each sheet of which contains all previous wff. Alternative 1

Alternative 2 tp and t^ are represented in miniature

C

V

1

n n

N - 1

A

A

B

B

A

B tableau closes

Since each alternative set is closed the construction for A (unremarkably)

B

~B

~A is

closed.

Not unexpectedly, closure coincides w i t h validity; that is Theorem 3.8. A is FD valid iff the FD construction for A is closed. Proof of the theorem reduces to the two lemmas which follow. To reduce the number of cases in these lemmas, w e will assume again that V is defined. The ryles for V are then easily derived from those for & and The proofs are essentially adaptations of those given by Kripke (in 63, pp.76-80). Lemma 3.6. Where the FD construction for A is closed, A is FD valid. Proof. Suppose, on the contrary, that A is not valid. Then there is a n FD model M = < T , K , * , I > such that I(A,T) ^ 1. This in fact provides an induction basis for the proof, by induction on n, that at the nth stage of the construction there is an alternative set S and a function a mapping tableaux of S into elements of K, w h i c h maps t^ to T and maps t* to a* whenever it maps t to a, such that

202

3.2 MAPPINGS INDUCTIVELY ESTABLISHED BETWEEN TABLEAU CONSTRUCTIONS ANV MODELS

©. If t is a tableau of S, a = ot(t), and B is any wff on the left of t a n d C any wff on the right of t, then I(B,a) = 1 and I(C,a) 4 1. Induction basis: n = 1. There is one tableau t_ w i t h A o n the right, and I(A,T) = 1. Moreover a(t ) = T. Induction step; assume © nolds for n (so there is an appropriate S and a); to show it holds for (n + 1). Stage (n + 1) must result by the application of a rule to some tableau t of S; and there are just these cases ad &£. B & C appears on the left of t, and by induction hypothesis, for a = a(t), I(B & C,a) = 1. Hence I(B,a) = 1 - I(C,a), so application of rule preserves the requisite properties of function a, thus "validating" &£. ad &r. B & C appears on the right of t, and I(B & C,a) 4 1. Thus either I(B,a) ^ 1 or I(C,a) 4 1. Rule &r correctly allows for these alternatives. Tableau t is replaced either by ^t w h i c h is like t except for B on the right or by „t w h i c h is like t but with C on the right, and alternative sets S

I

result.

1

and S <•/

If I(B,a) 4 1 then S^, in place of S, will satisfy requirements of © ;

otherwise where I(C,a) 4 1,

does the job.

ad~£. ~B appears on the left of t, and so for a = cx(t) I(~B,a) = 1, whence I(B,a*) 4 1, since by ~Z B appears on the right of t* and a* = a(t*), requisite properties of a are preserved. ad ~r. Similar. ad -+1. B C appears on the left of t Q , and so for T = a ( t Q ) I(B C,T) = 1. Thus for every a in K, either I(B,a) ^ 1 or I(C,a) = 1. Correspondingly each tableau t is replaced by alternatives, either by ^t with B on the right or ^t w i t h C on the left, so yielding alternative sets S^ and S 2 I(B,a) 4 1 then S w i l l ad

B

(for each t).

If

satisfy © ; otherwise where I(C,a) = 1, S 2 will.

C appears on the right of t^, where I(B

C,T) 4 1, and so for some

a' in K, I(B,a') = 1 4 I(C,a'). By ->-r a n e w tableau t' is begun w i t h B o n the left and C on the right. Accordingly, at the (n + l)th stage extend a by a(t') = a'. Then the extended function a satisfies requirements. Since the FD construction for A is closed, for each alternative set S in the construction there is a tableau t w i t h some wff on both right and left of t. There is however a function a mapping tableaux of S into elements of K w h i c h satisfies ©. Let a = a(t). Then by © , I(D,a) = 1 and I(D,a) 4 1, which is impossible. Hence A must be valid. Lemma 3.7. Where the FD construction for A is not closed, A is not FD valid. Proof. Suppose the FD construction for A is not closed. The construction nonetheless terminates after a finite number of stages (with redundant rules best not applied). The reason is that A has only finitely m a n y subformulae, into w h i c h it breaks down in a finite number of stages at t^, apart from implicational subformulae»which each permit, however, only a single transition to tableaux other than t n (so infinite cycling cannot occur).1 1

A detailed proof that the FD construction for A terminates can proceed by defining immediate descendant in the fashion of Kripke 63, pp.77-8, and then showing that, in v i r t u e of the FD rules, the tree structure reflecting this relation is finite.

2 03

3.2 ALTERNATIVE PROOF PROCEVURES : WRINGING CHANGES ON TABLEAU METHOVS

Since the construction is not closed, the terminal stage of the construction contains an alternative set, S say, which is not closed. In terms of S define an FD model M = as follows: T is the main tableau T Q of S, K is the set of tableaux of S, * is the opposite operation on tableaux of S, and for initial wff A and a in K, I(A,a) = 1 iff A appears in the left column of a. Then n. For any wff D, if D appears on the left of a tableau a then I(D,a) = 1, and if I) appears on the right of a then I(D,a) 4 1. Proof is by induction on the number of occurrences of connectives in B. The induction basis is immediate from specification of I and the fact that S is not closed, so in no tableau does an (initial) wff appear on the right as well as on the left. In the induction step there are these cases: B & C appears on the left of a. Then, since the construction has terminated, by B and C both appear on the left of a whence by induction hypothesis I(B,c) = 1 and I(C,a) = 1, and so I(B & C,a) = 1, as required. B & C appears on the right of a. Then by &r either B or C appears on the right of a in alternative S. If B, for example, does then by induction hypothesis I(B,a) t 1, whence I (B & C,a) ^ 1. ~B appears on the left of a. Then by ~£, B appears on the right of a*. By induction hypothesis, I(B,a*) ^ 1, whence I(~B,a) = 1. The case where ~B appears on the right is similar. B -*• C appears on the left of T. Then by -*•& for everv a in K either B is on the right of a or C is on the left of a. Thus, applying the induction hypothesis, for every a in K, either I(B,a) ^ 1 or I(C,a) = 1; whence I(B C,T) = 1. B C appears on the right of T. By ->r a new tableau a is begun with B on the left of a and C on the right. Thus I(B,a) = 1 I(C,a), whence I(B -»• C,a) j 1. Since A appears on the right of t Q (=t), by IT, I(A,T) ^ 1. But M is an FD model (the fact that S is not closed is important because otherwise I would not be well defined); so M provides a counter-model to A, and A is not FD valid. Corollary 3.9. A is a theorem of FD iff the FD construction for A is closed. Proof. Combine the theorem with the adequacy result for FD. Alternatively, the theorem offers the way to an alternative proof of the adequacy of the semantics of FD. For completeness can be alternatively established by showing that where the FD construction of A is closed then A is a theorem of FD. Semantic tableaux methods provide an easy decision procedure for probability and validity with respect to FD. For, in virtue of the elementary finite model property of FD, they provide finite search procedures: after a finite number of stages the construction for an FD wff A terminates, and A is provable or not according as the construction is closed or not. Tableaux methods may be converted into other proof methods, such as Gentzen and natural deduction methods, by tableau reorganisation and transformations, and they may also have changes wrought upon them in the way that semantics for FD were replaced by alternative semantics. Detailed examples of changes of the first sort are given in chapter 11 for full relevant logics, from which analogous results for FD will follow as special cases. Much- as the two-valued semantics for FD were converted to four-valued semantics without a star operation, so two-column tableaux methods for FD can be converted to four-column semantic tableaux methods without starred tableaux. In such a system a tableau has in general, then, four columns representing T, F, B and N; but the main tableau tg has as before only two columns. The new tableau rules simply mirror those of the four-valued semantics for FD. Four column tableaux may - as before, in the semantical analyses - be given an isomorphic representation, which reduces the impression of four values to that of two-looked-at-differently. The new columns are T, t (for not T), F and f. A wff that previously went into B now appears in both T and F, one that went into N

204

3.2 FROM TABLEAUX TO COUPLEV TREE METHOVS

now appears in t and f.

The translation is summarised in the following table: T T t t F f F f B T F N

Like a double helix of molecular biology, the resulting truth and falsity tables may be unthreaded to give separately truth (T and t) and falsity (F and f) tableaux. Sometimes, as with first degree entailment, these tableaux may be given a one-sided representation, i.e. only the T and F columns need be recorded, much as with analytic tableaux for classical logic where only one of the truth and falsity columns need be displayed. If now the truth tableaux, in a one-sided representation, are organised in connected trees, then we arrive essentially at Dunn's neat relevant adaptation of Jeffrey's "coupled trees" procedure for classical sentential logic. What Dunn noticed (in 76) is this: that in order to arrive at precisely the classically valid arguments with his coupled trees modification of analytic tableaux, Jeffrey had (in 67) to introduce two ad hoc technical complications (corresponding to the two Lewis paradoxes). If these are removed, what results is not classical argument but first degree entailment. The basic idea is that in order to test the validity of an argument, A (-B, one constructs two truth trees, one for A (coming down) and one for B below (going up). Since each path in the tree for A represents an alternative set of truth conditions for A, and similarly for B, it is natural to require that every path in the upper tree 'cover' some path in the lower tree in the sense that every [sentential variable or negation thereof, i.e. every] ... atom ... that appears in the covered path also appears in the, covering path. Thus every way in which A is true is also a way in which B is true (Dunn 76, pp.151). %

The rules for breaking down these truth-functional wff A and B to atoms, in ways that reveal what make them true, are the following familiar rules (namely, those already used in §1 for arriving at reduced form): A & B A B

A V B A

B

~~A A

~(A & B) ~A ~B

~(A V B) ~A ~B

The second and fourth rules have branching conclusions, which represent (as in tableaux) alternatives; these are the source of the tree structure. The following examples, with the arrow in the first indicating covering or closure, illustrate the method, and how it lets through good arguments and keeps out bad ones: p & q P

p & ~q P ~P q

q V r

~q q V ~q

Thus p & q q v r i s closed under the method, p & ~p |- q v ~q is not. More generally, A (- B is closed (under the coupled tree method) iff every path in the

205

3.3 REASONS FOR ASSEMBLING APPLICATIONS AT THE FIRST DEGREE STAGE

tree for A covers the same path in the tree for B. Theorem 3.9. A f- B is closed iff A |- B is a theorem of DML, i.e. iff A -*• B is a (correct) first degree entailment. Proof is not simply a special case of the corollary concerning semantic tableaux, since the bottom tree is a truth tree, not a falsity tree as the (exemplified) linearising of a semantic q v r tableau for A f- B, with A on the left and B on q the right, would yield. (Such a coupling of trees r would supply yet another proof procedure for DML). However, an argument rather similar to that for semantic tableaux, which shows that A (- B is DML valid iff the coupled tree construction for A B is closed, works. The argument is in fact like that presented by Dunn (76, p.159). Alternatively, coupled tree constructions can easily be connected directly with proofs in DML (since in the absence of Modus Ponens the difficulties in proofs of Cut are bypassed). While tableaux methods, of a variety of sorts, extend readily from first degree entailments to the full first degree and indeed, though less readily, to the higher degree theory, coupled tree methods may appear confined to the initial stage, to treating arguments between truth-functional wff. That they are not so ' limited, to truth-functional wff, McRobbie's work (in 79) makes especially plain. 1 But obtaining pleasant methods of this sort for full relevant sentential systems, or even in a way without artifice at the first degree, remains an open problem. §3. Harmonious consequences of} and applications of, the first degree theory; additional connectives and further constants under the theory} and alternative formulations of the theory. There are said to be good reasons for assembling certain applications of the theory at the first degree stage, rather than at the stage of first degree entailment or at higher degree stages. On the one hand, there are but few natural applications at the first degree entailment stage, i.e. as regards DML, that do not recur, or cannot be repeated in a more general setting at the first degree. The main reason is that whenever such assertions as that 'A entails B' or 'B is deducible from A' are significant so, it is generally conceded, are compounds of these by (extensional) connectives such as &, v and The same applies to other notions that can be analysed using the same sort of semantical apparatus, e.g. causal sufficiency. On the other hand, it is commonly alleged that analogous extrapolations break down beyond the first degree. There is something in this. There are, firstly, several notions important in the philosophy of science whose higher degree structure is either not well-defined, e.g. iterated statements of causality or causal sufficiency, or else is so far peripheral, e.g. higher degree inclusion of content or traditional suppression. Failure of iteration or other typical higher degree features does not however exclude higher degree behaviour, which may result from, for instance, the interaction of an intensional functor with entailment, as in the statement of the transitivity of causation, that A causes B and B causes C entails that A causes C. There are, secondly, several notions, again of importance, whose higher degree is acclaimedly not well-defined, and, whether it is or not, where the main problems of analysis already appear to arise at the first degree; for instance, such l

Given that |-A iff t f- A the restriction to arguments is no serious restriction; and t can always be eliminated through finite conjunctions of wff.

206

3.3 FIRST DEGREE APPLICATIONS VERSUS HIGHER DEGREE APPLICATIONS

notions as probability, confirmation, verification, evidence, etc. Thirdly, the analysis of notions that do extend, easily or naturally, to the higher degree, fragment into different analyses corresponding to competing higher degree entailment systems. By contrast, FD is the stable first degree part of all these systems, so the applications also are comparatively stable. The first degree applications have the advantage also, like the semantical theory on which they depend, of simplicity: by contrast again, the higher the degree theory is rather more complicated (as chapter 4 will make evident). Despite the build-up, the main applications made of relevant logics will not be of the first degree theory but of full sentential systems which make allowance for a higher degree (in chapters 8 and 9 especially). This is partly for organisational reasons, but mainly because the objections to use of higher degree theories do not hold much water, as should have begun to emerge and will be brought out further in what follows. For example, many of the connectives or functors alleged not to iterate combine and iterate perfectly well in natural languages like English, without use-mention or other confusion. Probability is just one example among many. Even where there is doubt about iteration, combination of entailment with other operators, as for instance of cause with entailment, may force an adequate logical treatment to be of more than first degree. Many of the connectives and constants (zero-place connectives) subsequently added to higher degree implicational logics (especially in chapter 5) can also be added to FD. For example, FD can be conservatively extended by the addition of classical negation -», axiomatised in terms of paradoxical principles, A & -»A -*• B and B •*•. A v -»A, which serve to rule out inconsistent and -incomplete situations, and so ensure the evaluation rule I(->A, a) = 1 iff I(A, a) 4 1. But the treatment is like that in the higher degree case, and can be transferred intact from the larger setting. By no means all the connectives subsequently considered can of course be treated in FD in a way that simply reflects higher degree analysis. For any connective, such as fusion, which is degree increasing cannot be introduced at the first degree without displacing the central connective In short, a mix of intensional degree-increasing connectives cannot be considered in a n interesting way in the first degree setting. This illustrates, from another angle, the serious limitations of the first degree as a medium for logical analysis; and shows again how unnatural a stopping-point the first degree (and its more fashionable metalinguistic analogue) is. Even so much can be accomplished in FD, for these reasons:- First, the isolated behaviour of some intensional connectives, simply in combination with truth functional connectives and not in degree multiplying combinations with -»-, is of independent interest. The point holds both for iterable connectives, such as inconsistency, and for dubiously iterable connectives such as cause (which can however lead, once again, to degree multiplication in combination with other connectives such as •*). Secondly, several connectives, though they admit syntactically of higher degree compounding, exhibit essentially first degree logical behaviour, and can be modelled in what amounts to a first degree setting. The most striking example of this phenomenon is provided by S5 modalities where iterations of modalities reduce to a single (the final) modality, because M[]A GA and M'O •**• OA where M and M' are any modalities. While such modal reduction principles cannot be presented in a first degree logic, the effects of the higher degree, since logically the same as the first degree, can be obtained. Another example, of a different sort, is afforded by modal systems in the vicinity of SO.5, where there is no distinctive higher degree logic (systems of this sort are considered in detail in chapter 14). Simply to introduce the familiar S5 modelling condition for necessity, • - namely, I([]A,T) = 1 iff, for every a in K, I (A,a) = 1 - in a first degree

207

3.3 FIRST VEGREE ANALYSIS OF FUNCTOR • ANV CONSTANT t

setting is unprofitable, except in so far as it exposes the usual background assumptions of modal logic semantics. For DA then behaves like a further initial wff, satisfying no distinctive logical conditions. Because of first degree situational variation (see §1), whatever A there will be situations where A fails to hold. Hence ~ | - D A (a feature that will be exploited in chapter 9). There are, however, various equivalent approaches in terms of which • can be introduced in the first degree theory, all of which involve an additional piece of semantical apparatus, which has already been used. Syntactically, • is a one-place connective on wff (i.e. where A is a wff of FD so is GA), which is degree-increasing. Thus [TA, like B -»• C, gets evaluated only at T in FD models. The rule for I([]A,T), to be pointful, will have to be in terms of A holding not at all worlds in K but at certain select worlds in K, namely the possible ones^. If we fudge the distinction between regular worlds (those where theses hold) and consistent worlds in the fashion of modal semantics, then the rule for • can be given in terms of the set 0 already introduced in the evaluation of constant t, thus: I([]A,T) = 1 iff I(A,a) = 1 for every a in 0. The result is the same as that obtained by use of the familiar rule for • from semantics for normal modal logics, including often S5, namely I([]A,T) = 1 iff for every a for which RTa, I(A, a) = I, upon equating Oa with RTa. The rule may be arrived at in yet another way, upon observing that the result is the same as that for evaluating t A, that is defining DA as t -*• A - a move adopted by Ackermann and Anderson at the beginning of the contemporary relevant logic enterprise. For I(t A,T) = 1 iff for every a for which I(t,a) = 1, I (A, a) = 1, i.e. iff for every a such that Oa, I(A,a) = 1, i.e. iff I([]A,T) = 1, upon evaluating constant t as before (in 2.12), i.e. I(t,a) = 1 iff Oa. Thus, upon extending FD by t, to give system FDt, • is obtained for free. Its axiomatisation does not need to be separately worked out (though it easily can be); and ways of eliminating t (or rather presenting it through classes of tautologies) also eliminate •. An FDt model M is a structure M = where T e 0 c K and otherwise all is as for FD models, except that (as in 2.12) t is now evaluated using 0. To axiomatise FDt it is enough to add to FD the single axiom: t. Adequacy Theorem for FDt 3.10. A is a theorem of FDt iff A is FDt valid. But minor details are added to the corresponding result for FD. t is valid since T e 0. For completeness, define 0 as {a e K : t e a}. Then immediately, t e a iff a e 0, and 0 c K. Further T e 0, since,°as (- t, t e T. FDt gives rise to several theorems concerning t, e.g. |-t =>. p v ~ p and j-p v ~p t, and also contains an attenuated logic of modality with such principles as DA ^ a (whereas DA -*• A exceeds first degree resources), •A & (A -*• B)=>. DB, etc. But in order to obtain a richer logic of modality, and to investigate logically such central questions as to whether entailment is logical implication, including thus properties of D(A -»• B), the resources of first degree logic (and its metalogical analogue) have to be much exceeded. 1 The richer higher l Although the systems so far modelled using FD and FDt model structures are, as contemporary logics go, very weak, the model structures themselves suffice for the modelling of much stronger systems. The system S5~, Lewis S5 with relevant negation added, studied in Meyer BV is just one of many examples: S0.5~ and SO.5 are others. For modal and relevant logics that are essentially first degree are tractable in such a framework.

20S

3.3

FIRST PEGREE ANALYSIS OF CONSISTENCY OR RELATIVE POSSIBILITY

degree logic of t and the theories of modality thereby supplied are studied in 5.1. If it is now beginning to look as if the first degree theory is not good for very much, that too is wrong. For within the framework of the first degree theory we can redo, mostly much more satisfactorily, almost all of Carnap's rich semantical work. The reason is that very much of this work was set at, or at the equivalent of, a first degree level, and even what extended beyond the first degree was done in terms of S5 and so resembles a first degree modal treatment. Thus redoing the work is largely a matter of replacing first degree modal theory systematically by first degree relevant theory, state descriptions by worlds (possible and not). Reworkable in this way are, for example, not merely the accounts of range and content (considered below) but the theories of sense and denotation, of extension and intension (of Meaning and Necessity and elsewhere); not only the theory of semantic information and the basis of the semantical theories of probability and confirmation (both tasks undertaken in UL), but the wealth of notions dealt with in Carnap's theory of logical probability (most fully presented in Logical Foundations of Probability), e.g. qualitative, comparative, and quantitative notions of probability, probabilistic relevance, etc., etc. The task of showing how resetting Carnap's semantical theory relevantly pays off, in terms of the attractive nonparadoxical results turned out, can be profitably combined with the business of indicating how notions conflated in modal theory live their separate lives in a relevant setting, for instance how consistency and relative possibility relations, wrongly reduced to one-place connectives in modal theory, really behave. In fact the first degree theory can be alternatively formulated in terms of consistency principles, and points made earlier (in chapter 1) concerning the logic of consistency thereby vindicated. The entailmental principles (al)-(a5) established in §1 (in the course of investigating weak relevance) yield in turn analogous relative possibility or consistency principles. To make the connections, the following identification, argued for at length by Lewis and long and widely accepted, is adopted: A 0 B

~(A+ ~B)

;

in words, A is consistent with B, or B is compatible

with A for all that A says, iff A does not entail the negation of B. 1 The relation is symmetrical, and, provided that A is logically possible, reflexive, i.e. A 0 A. Given the definition the modelling condition for relative possibility is immediate: Restricted O-rule. Where A and B are truth-functional, A 0 B holds at T iff, for some world a, A holds in a and B holds in a* (~B does not hold in a). The following b-principles follow either from the a-principles or from the principles (l)-(5) of §1 that yield the a-principles: (bl-2) The relative (im)possibility of two truth-functional wff is never a function of the absolute (im)possiblllty of one of the wff alone. In particular, no truthfunctional wff is inconsistent with every proposition. (b3) The relative possibility of truth-functional wff A with a tautology is not a matter of the relative possibility of A alone. (b4) No contradiction is inconsistent w i t h every proposition, i.e. every contradiction is consistent with some wff. i

Note that 0, consistency, differs from fusion, intensional conjunction except under special conditions such as those supplied by system R which at the same time collapses modalities, as in the R thesis, NA •**• A. The issue is taken up again in chapter 5, where fusion is studied. 209

3.3 STRICT AND ABSOLUTE POSSIBILITY CONTRASTED WITH RELATIVE POSSIBILITY

(b5) Where A and B are truth-functional, A is inconsistent with B only If A and B share a sentential variable. The claim that 0 is a genuine relative possibility notion rests in part on these results, in part on the further fact that because entailment is suppression-free, the possibility of the negation of the consequent is assessed relative just to the antecedent without taking into account the truth of other statements, such as tautologies. If B is relatively possible with respect to A then B is possible for all that A says; hence A and B may be relatively possible even though one of them is impossible. In contrast, the strict possibility relation defined in terms of strict implication,assesses the impossibility of the negation of the consequent relative to the antecedent plus the body of tautologies; and the concept of absolute possibility of a statement, interdefinable with strict possibility, assesses the relative possibility of that statement relative to the body of tautologies. For it has already been shown that the total body of tautologies holds in a iff a = a*; so strict possibility, symbolised '•', and absolute possibility,symbolised '0', can be adequately modelled as follows: Restricted •-rule: For truth-functional A and B, A • B holds at T iff, for some c such that c = c*, A and B hold in c; Restricted O-rule: For truth-functional A, OA holds at T iff for some c such that c = c*, A holds in c. Interconnections of the possibility notions follow: it also follows that each of (bl)-(b5) breaks down for strict possibility. It follows too, where A and B are truth-functional, that the entailment A B never holds simply in virtue of the impossibility of A or the necessity of B; it is always a relational matter requiring a connection between A and B. In addition, the modelling explains some of the otherwise puzzling features of strict possibility, such as (Lewis's consistency postulate) that an impossible proposition is strictly impossible relative to anything. Relative possibility is a sharper and more useful notion than the Lewis notions, for in terms of it one can define and take account of these other possibility notions, whereas they are insufficient for a definition of relative possibility. Though the Lewis notions can be taken account of semantically by way of the body of tautologies, a syntactical definition requires more resources. The standard move of replacing the body of tautologies by an arbitrary tautology naturally fails, since it depends on the identification of tautologies; as we discriminate among tautologies, a single tautology cannot represent the body. Since, however, a = a* iff, for every p, if p holds in a p holds in a*, a syntactical definition can be had by enriching FD by propositional quantifiers. Using the particular (P) quantifier rule: (Pp)B holds in a iff B holds in a, for some p, it follows from the O-rule that H.

OA is true iff ~(Pp).A -»• p & ~p is true.

Conversely, promoting this equivalence to a definition (as in Hallden 49) the O-rule follows. It follows from H that A - j B is true iff (Pp).p v ~p

A => B is true.

But a more revealing enthymematic connection, accenting suppression features of strict implication, can be proved; namely that for truth-functional A and B, A B is valid (provable) iff (Pp). taut (p) &. P & A -*• B is valid (provable), where 'taut (p)' abbreviates 'p is a tautology'. I.

If (Pp). taut (p) &. p & A •*• B, then for some tautology t^, for every FD

model, for every world a, if t^ holds in a and A holds in a then B holds in a. By restriction the same is true for every world a such that a = a*. But in case a = a*, t 1 holds in a. Thus for every world a such that a = a* if A holds in a

210

3.3 FROM RELATIVE POSSIBILITY TO THE SEMANTICAL DEFINITION OF CONTENT

then B holds in a, whence A

B.

2. If A H B then, for every FD model for every a for which a = a* if A holds in a then B holds in a. Let C.,...C include all subformulae in A and B; form t„ = I n l (C, v ~C,) &....& (C v ~C ). Then for every a, if t„ holds in a, then if A holds 1 1 n n 2 in a, B holds in a. We sketch an argument for this: a full argument can be obtained by considering tableaux constructions above. Any model construction showing that when a = a* if A is in a B is in a, need involve only subformulae of A and B. But as far as subformulae of A and B are concerned a = a*, since, for such a subformula C, C holds in a iff C holds in a*, as C v ~C holds in a and in a*. Hence the condition that a = a* can be replaced in the model construction by the condition that t^ holds in a. The desired result then follows by particularisation on t^. It also follows for truth-functional A and B. A => b is valid (provable) iff (Pp). p p & A -»• 3 is valid

(provable).

(Here '=>' may also symbolise positive or intuitionistic implication.) These results may display the comparative suppression features of material and strict implication given entailment as a base. The success of relative possibility so defined in explicating the intuitive notion of the possibility of one statement for all that another statement says, depends semantically upon the discriminatory power of worlds to provide variations which individuate statements, and to distinguish the content of p from the content of p & T, for t an arbitrary tautology. The same features enable the definition, in terms of entailment, of logical content and inclusion of meaning notions which improve on the accounts given by suppression implications. A requirement of adequacy on sought definition is that they lead to the results that (for truth-functional A and B) A entails B iff (the meaning of) B is included in the meaning of A, and, (the meaning of) B is included in the meaning of A iff the content of A includes the content of B, where the inclusion relation is the familiar (lattice ordering) relation. Symbolising 'the content of A' by 'c(A)', we can secure the second of these points by defining: (the meaning of) B is included in the meaning of A = . c(A) => c(B), provided we can define 'c(A)' in turn so that contents can be appropriately ordered by an inclusion relation. Now we know that (in a typical model) A B iff for every situation a if A holds in a then (materially) B holds in a, i.e. iff for every a if B does not hold in a then A does not hold in a, i.e. iff {a:A does not hold in a} ^ {a:B does not hold in a}, using the usual set-theoretic (or property theoretic) definition of inclusion. Comparison of this equivalence w i t h A -»• B iff c(A) £ c (B) reveals that all the desired features follow if w e connect c(A) w i t h {a:A does not hold in a}, that is, the content of A with the class 1 of situations where A does not hold. I We have adopted the familiar class terminology though a treatment of content as a property of statements would be more apposite. In fact everything we do is compatible with such a treatment, which can be simply obtained by reconstruing {a:A does not hold in a} as Xa(A does not hold in a) where X provides property abstraction, or, for that matter, as a{a:A does not hold in a} where a is a 1-1 function.

211

3.3 INVESTIGATION OF LOGICAL CONTENT PRINCIPLES

The intuitive point of this connection (essentially that proposed by Popper 59) is clarified by making the familiar connection between not holding and falsification, so that situation a falsifies A iff & does not hold i n a. Then the logical content of £ is given by the class of set-ups which falsify A. (We have incidentally widened the definition to take account of the content of all FD formulae, including ill-formed formulae.) The account of falsification leads at once to such classical results as that if A entails B and a situation a falsifies B then a falsifies A. As usual we say that A has no logical content iff c(A) = A, i.e. iff no world excludes it, and otherwise that A has some logical content; and that A has total logical content if c(A)=V, i.e. the class of all worlds in the canonical model. Then of course a statement has no content if no situation falsifies it, and total content if every situation falsifies it. Thus ill-formed formulae have no logical content. Since the class of situations which falsify A & B includes the class which falsify A, c(A & B) £ c(B). It is this result in particular, as Popper noticed, which warrants regarding c as a content notion: for there is a good sense in which A & B generally says more than A.. But if A entails B then B does not add anything to A, i.e. c(A) £> c(A & B). Indeed the fact that (in a typical model) A -»• B iff c(A & B) = c(A) - a fact commonly used in characterising entailment 1 - provides a further warrant for regarding c as a content notion. Several content principles follow at once from (l)-(5); comparable inclusion principles are adduced below. (cl) Every truth-functional wff has some content. (c2) No truth-functional wff has total content. (c3) Every tautology has some content. (c4) No contradiction has total content. (c5) If A and B are disjoint truth-functional wff, then they have distinct (noninclusive) contents. In particular, then, any two tautologies with distinct variables have distinct contents, and similarly distinct contradictions. By contrast, the same definition of logical content for strict implication leads to results such as that some statements, namely necessary truths, have no content, whereas others, the negations of necessary truths, have total content (see, in effect, Carnap 42). Thus, strict implication, far from capturing a natural and inevitable account of logical content and necessary truth, embodies questionable positivistic views of the nature of necessary truth. This view of necessary truths as without content can also be explained as following from the view that they can always be suppressed. For using Hallden's principle, a proposition which does not add anything to any proposition either has no content or its content is included in that of every proposition. But some statements are quite disjoint in meaning from others. Therefore necessary truths, like illformed formulae, have no logical content. Conversely, the view that necessary truths have no content has been used to prop up the claim that they can be suppressed. Compounding rules for logical content follow from the definition of content and the conditions on normal situations, namely: c(A & B) = c(A) U c(B), and !

Thus Hallden, for one, defines entailment in this way (in 49): Hallden's account of entailment also drops DS (see 48). Wittgenstein, for another, is committed to such a characterisation. As will be explained in chapter 14, these characterisations, good for the first degree where their intuitive base lies, break down at the higher degree.

212

3.3 THE CONTENT SEMANTICS, RANGE PRINCIPLES, AND RANGE-CONTENT INTERCHANGE

c(A v B) = c(A) H c(B) 1 , where fl and IJ are normal set or lattice operations. Also c(~A) = {a:A e a*} = N{a:A I a} = Nc(A), where operation N is De Morgan lattice complementation. Accordingly a recursive definition of logical content of truth-functional wff can be provided given assignments of content to atomic wff, and in this way a "content" semantics for first degree entailment can be obtained. It may be objected that as an interpretation this semantics is unsatisfactory because it uses the allegedly unfamiliar and ad hoc operation N. In fact N, which is (by now) neither of these things, can be eliminated by giving a joint recursive account of content and range, where the logical range of A, symbolised 'r(A)', is essentially that notion which has been elucidated at length by Carnap (42, 62). For defining, situation a verifies A as A holds in a, and r(A) = ^ {a: a verifies A}, it follows trivially that the range of A is the class of worlds where A holds. Range principles, which contrast favourably with Wittgenstein's and Carnap's range principles (see Carnap 62), follow at once from (l)-(5) above. Thus, for example, (dl) No truth-functional wff has total range. D-principles (d2), (d3), (d4) and (d5) are obtained from (cl), (c4), (c3) and (c5), respectively, by replacing 'content' by 'range' and interchanging 'some' and 'total 1 . Since it has become fashionable to identify propositions with ranges (or better, to suppose that they are isomorphic), the contrast between modal and entailmental accounts is all the more striking and important. Now given both the range and content of each initial wff we can determine the range and content of each truth-functional compound without use of star or N. For, firstly, c(~A) = {a : ~A I a} = {a* : A e a*} = r(A), and similarly r(~A) = c(A). That is, negation interchanges range and content. Secondly, we already have compounding rules for content as regards other truth-functional connectives; and there are dual principles for range, namely r(A & B) = r(A) H r(B) and r(A v B) = r(A) U r(B). Plainly then, by associating with each initial wff both a range and a content, use of N can be eliminated. Thus with the range and content apparatus we can provide a further starfree modelling for DML, and thereby at the same time furnish an interpretation for Dunn's modelling in terms of topics for first degree entailment. An rc model l

These compounding principles do not hold for rival notions which also have some claim to be counted as content notions; e.g. for the signification of A, s(A), so defined that s(A)={C:A C}, the analogous principle s(A & B) = s(A) U s(B) fails. It is almost immediate (in terms of FD) that signification provides a content nonexpansion account of valid argument, that there is a valid argument from A to B just in case the content or information conveyed by B is included in that conveyed by A. For A B is true iff s(B) £ s(A), i.e. iff for every wff C if B -»• C then (materially) A -> C. One half uses material Conjunctive Syllogism, the other half Identity and rule y. Hanson's "positive account" of content (in 80, pp. 663-5) adds to signification as defined little but complexities concerning wff-proposition relations.

273

3.3 RANGE-CONTENT ("NEGATION FREE") SEMANTICS, AND INFORMATIONAL SEMANTICS

(range-content model) associates, with every initial wff A a pair « rc(A). rc(A) may be extended recursively to all truth-functional wff by the following connections, which are of course derivable: rc(~B) = = ; rc(B & C) = ; rc(B v C) = . Finally, for the given model, B C is true iff c(B) £ c(C) & r(C) £ r(B), i.e. iff c(B) £ c(C), i.e. iff r(C) £ r(B). A B is rc valid iff A B is true for every rc model. It follows that first degree wff A B is rc valid iff A -» B is valid, i.e. iff A J- B is DME valid. This refutes the claim (made e.g. by Hanson 80, p.662) that with first degree relevant logic truth-preservation and content-nonexpansion accounts of validity diverge. For the account of validity in DML, and indeed for all relevant logics studied in chapter 4, _is a (generalised) truth-preservation account, requiring just that wherever, in whatever situation, the premisses of a valid argument hold so does the conclusion. 1 And, as shown, this coincides at the first degree (and indeed at higher degrees, as UL explains) with the content nonexpansion account, according to which 'an argument is valid just in case the content of the conclusions does not exceed the combined contents of the premisses', i.e. the contend of the conjunction of the premisses, with moreover 'content' defined in the usual, negative, way, in terms of 'the states of affairs ruled out' (p.662). Dunn's informational semantics for first degree entailment (in 71) does not analyse elements of the ordered pair in the modelling in terms of range and content. It simply interpret[s] every truth functional wff A as a pair of sets where X^ and X^ are subsets of a given set X called a universe of discourse. X are called topics; and the pair

The elements of is called

a proposition surrogate, whose components are to be thought of, respectively, as the topics that A gives information about and the topics that ~A gives information about (Dunn 71, p.363). Without further interpretation such as range and content afford, this is not particularly informative. However an abstract, modelling open to various further interpretations does result. Let I be a function assigning to each sentential variable a proposition surrogate. Then where 1(B) = <X, ,X_> and l

The widespread assumption that 'relevance logicians reject the truthpreservation account as by itself insufficient for validity' (Hanson 80, p.662, and many others) is mistaken. The most that is true is that evidence for this assumption may be found in the early work of some relevant logicians, notably Anderson and Belnap. The addition that it is widely supposed relevant logicians make (and even must make) to the truth-preservation account, 'appropriate relevance between ... premisses and conclusion', follows from the truth-preservation account (as shown in §1). Naturally such an addition does not follow from what they do reject, modal accounts of validity, i.e. truth-preservation restricted to (certain) complete possible worlds.

214

3.3 TOPIC SEMANTICS FOR FIRST DEGREE ENTAILMENTS AND THE FIRST DEGREE

1(C) = , I can be extended inductively to all truth-functional wff, much as before, thus: I(~B) = < X 2 , X ; L > ; I(B & C) = ^ u Y p X 2 PI Y 2 > ; I(B V C) = <X, n Y

X2 u Y2>.

Then

a first degree relevant implication A -*• B [is] valid in a universe of discourse iff for every interpretation I in the universe of discourse, if 1(A) = <X X 2 > and 1(B) = , then Y £ X ± and X 2 £ Y 2 ; and [is topic] valid iff it is valid in every universe of discourse (p.363). Theorem 3.11 (after Dunn). First degree entailment A -*• B is provable (or valid) (1) iff A -*• B is topic valid, and (2) iff A -*• B is valid in a universe of discourse consisting of a single topic. Proof of (1). It is a matter of direct verification and the usual induction that every theorem of DML, every provable first degree entailment, is topic valid. Most cases simply apply lattice principles. Consider then Contraposition. Suppose A -»• B is topic valid, i.e. always Y^ £ X and X 2 £ Y H e n c e , as I(~B) = < Y 2 , Y > and I(~A) = < X 2 ,

and

c Y 2 and Y

£ X^

~B -*• ~A is topic valid.

For the converse of (1), if A ->• B is topic valid, then, as a special case, it is rc valid, and so valid. As to (2), if A -> B is provable then of course, by (1), it is valid in a single topic domain. Conversely, if A B is valid in a universe of discourse consisting of a single topic, then A B is true in every rc model comprising just one world pair, i.e. where K = {a,a*} for some a. Hence A B is true in a model on K, and so (by a result of 2.6) valid. The topic semantics can be extended to the full first degree in somewhat the fashion that the algebraic semantics was extended in §1 from first degree entailments to the first degree, namely by introducing, since the apparatus for evaluating first degree entailments does not permit of iteration, a further valuation function which evaluates truth-functional compounds. A topic evaluation for an fd wff A is a pair <w,I> where I is as before in the topic semantics, and w assigns to each sentential variable one of the truth values 1 and 0. Alternatively, to knit the valuation functions more tightly together, 1 the proposition surrogates can be taken as correct or not, according as their propositions are true or not, and w determined for initial wff A so that w(A) = 1 or 0 according as the propositional surrogate 1(A) of A is correct or not. Function w is extended to all wff of FD as follows: where A has the form B -»• C, w(A) = 1 according as B C is valued under I; and otherwise, w(~B) = 1 iff w(B) ^ 1, w(B & C) = 1 iff w(B) = 1 = w(C), and w(B v C) = 1 iff w(B) = 1 or w(C) = 1. A wff A is true in an FD topic model with valuations < w , I > iff w(A) = 1; and A is FD topic valid iff A is true in all such models. Theorem 3.12. Where A is an FD wff, A is provable in FD iff A is FD topic valid. Proof. Soundness may be proved as usual by direct verification of postulates. Truth-functional postulates are authomatically ensured since w is truth-functional: and also in virtue of this remaining postulates flow from conditions on I. Conversely, if A is not provable in FD than A is not FD valid, so there is a (canonical) FD countermodel M = for which V(A,T) t 1. Form the rangecontent model on M and define 1(B) = rc(A) for truth-functional B; and set w(B) = V(B,T). Then <w,I> provides a topic evaluation such that w(A) ^ 1. Hence A is not FD topic valid. 'The looseness of fit that can be got away with is perhaps a little surpising. It is because there are no postulates that inextricably mix first degree entailment and truth functions.

215

3.3 VAN FRAASSEN'S "FACTS" MODELLING EXTENDED; AND MEANING CONNECTION PRINCIPLES

What amounts at bottom to a variation on the range-content modelling is given by van Fraassen (in 69; also ABE §20.3), where r(A) is characterised as the class of facts which make A true 1 and c(A) as the class of facts which make A false. Not surprisingly, situation a verifies A can just be mapped to: class of facts a makes A true. Just as the rc modelling can be extended to a full first degree modelling, so van Fraassen's "facts" modelling can, by introducing a further partly truth-functional valuation function, be extended to a modelling for FD. Content and range-content modellings at once supply inclusion of content relations, and so inclusion of meaning relations (in the sense of that elusive notion). And in a relevant setting, inclusion of content provides one account of meaning connection which meets Bennett's demands (69, p.214, already discussed in chapter 1) that where there is an entailment there is a meaning connection and that for some 0 there is no meaning connection between P & ~P and Q. 2 Indeed, the following desirable principles follow from the account of inclusion of meaning given: (el) No truth-functional wff is part of the meaning of every statement. (e2) No truth-functional wff includes in its meaning every statement. (e3) No tautology is part of the meaning of every statement. (e4) No contradiction includes in its meaning every statement. (e5) For truth functional A and B, B is included in the meaning of A only if B shares a sentential parameter with A. Where propositional identity, equated with identity of logical content, is defined as coinclusion of logical content it follows further that tautologies are propositionally distinct from other tautologies when they contain distinct sentential parameters. Similarly for contradictions. Using similar definitions for strict implication would yield such disastrous results as that there is just one necessary proposition, which is included in the meaning of every statement, and just one contradiction, which includes in its meaning every statement. The inclusion features established provide only a part of the traditional containment account of entailment. According to this view the conclusion of a valid argument was an unpacking of the premisses, and it was impossible to get out of a valid argument something which was not already implicitly assumed or contained in the premisses. What the container model adds to the inclusion model then is the assumption of a class of properties which, because of their close connection with logical content, are assumed to be preserved, in one direction or the other, over entailments. It is on this view of valid argument that non-deducibility theses, such as the thesis that an evaluative conclusion cannot be deduced from non-evaluative premisses, are based. But l

2

In fact van Fraassen's reading for his T*(A) coincides with Carnap's informal rendering of L-range: 'We take as the L-range of A the class of those L-states which make A true' (42, p.104, our change of variable). Thus T*(A) comes down to r(A), in Carnap's terms, the class of states-of-affairs where A is true. These demands can even be satisfied by an inclusion of meaning notion defined in terms of an implication as inadequate at higher degrees as that of system I: see UL and chapter 7.

3

Whether propositional identity coincides with identity of logical content (even) at the first degree will be further considered in later chapters. That A has the same content as (B v A) & A, for example, has cast some doubt on the coincidence. But in any event, identity of content is necessary for propositional identity, and this suffices for the main point in the text.

216

3.3 FEATURES OF PROPERTIES FOR WHICH THE CONTAINER THESIS HOLVS

suppression implications destroy this important feature of valid argument, because they allow one to get out subject matter which has not been put into the premisses, but which has been assumed in the course of the argument. How can we better characterize the class of properties which according to the container thesis, is preserved over entailments, since clearly the thesis is not asserted for all properties? They are properties which turn on the subject matter of a statement. The extent and importance of the class is best indicated by examples: they include statement-properties such as existentially loaded, tending to corrupt morals, controversial (in the sense of containing controversial material), evaluative, racially prejudiced, ill-advised, politically biassed, conflicting with quantum mechanics, about geography, morally unacceptable, capable of being falsified by experience, displaying an ignorance of psychology. What all these properties have in common is that a statement has them if it contains a statement having the property or commits one to a statement having the property. But whatever this sense of 'commit' is, it must at least be stronger than strict implication. For suppose, what seems clear enough, that some contradiction, e.g. that a cube is two-dimensional, is not evaluative, racially prejudiced, or politically biassed. It follows, using strict implication as valid argument and the container principle that properties of this kind are preserved over valid arguments, that no statement is evaluative, politically biassed or racially prejudiced. But no collapses of this scandalous sort are obtainable with the implication of FD, because of the satisfaction of the relevance requirement . For simplicity, consider the problem of characterising the notion of commitment or containment concerned in the case of purely truth-functional wff, the form in which the problem naturally emerges at the first degree. Any such truth functional wff can be re-expressed, without change of logical content, in a form where the negations are driven down to apply at most to sentential variables (see the proof of (1) above in §1). The initial problem then reduces to this: given that a conjunction A & B, or a disjunction A v B, has a given containment property E of the subject matter sort, do its components have the property, and alternatively given one or both its components do, does it? Given that A has E, A is about geography, then clearly A & B is about geography; likewise, given that A V B is morally unacceptable then A is. If A v B were morally unacceptable and A were not then we could do something which was not morally unacceptable, viz. A, and thereby bring on what was unacceptable - which would collapse the notion. Thus, generalising, if EA V EB then E(A & B) and if E(A v B) then EA & EB. The converses of these principles also appear to be true for containment features. If A & B contains feature E, or commits us to E, then one or other of A and B must do the same; and if both A and B commit us to E then A v B surely does. If, for example, A is morally unacceptable and B is also, then A V B is morally unacceptable; for if it were not we would expect to get away with one or other of A and B with impunity. Though most of the sample properties cited above appear 1 to satisfy these conjunction and disjunction conditions (and likewise l

For relevant senses and applications of their predicates. The qualification is important. For instance, one of Kielkopf's alleged counterexamples (in 74a and 75) assumed the transformation of 'That ... is ill-advised' to - what is quite different - 'It is ill-advised to say of'. Kielkopf's other counterexample (in 75) is correct. The statements A and ~A may each be capable of empirical falsification though A v ~A is not, countering the condition EA & EB only if E(A v B) for such a functor. More damagingly, it may be claimed on similar grounds that evaluativeness is not a content-loading property.

277

3.3 TRANSMISSION PRINCIPLES FOR CONTENT-LOAVING

ANV RANGE-LOAVING PROPERTIES

analogous quantificational principles), they do not exhibit uniform behaviour with respect to negation. This turns out not to matter: we may satisfactorily characterise the properties on the basis of features they do satisfy generally. Thus we define a propositional property E as content-loading iff for every truthfunctional A and B, both E(A & B) iff EA V EB and E(A V B) iff EA & EB. Further, a proposition is content-loaded with property f (f-content-loaded) iff it has content-loading property f; and f is put into premisses of an argument iff these premisses are f-content-loaded. Thus, for example, we establish whether evaluative notions are put into premisses by checking whether individual premisses are evaluative. It can now be shown that the container thesis is correct for entailment though not for suppression implications. For in a valid argument of FD with truth-functional components no content-loading property can be got out which has not been put into the premisses. Our result is a consequence, using appropriate definitions, of the following Theorem 3.13. Where E is a content-loading property, if A + B and E(B) then E|A), for truth-functional A and B. Proof takes for granted some terminology and results from Anderson and Belnap 62 (repeated in ABE). We suppose that A -»• B, which is given, is in normal form, 1.*;. has the fp"n A v, . v A &...& B where each A. is a r primitive conx n 1 m i junction and each B . is a primitive disjunction. Now suppose further that E(B), i.e. E(Bj &...& B ).

It follows, by iterating the coudition, if E(A & C) then

E(A) v E(C), that E(B.) v E(B„) V...V E(B ), i.e. E(B.) for some 1 < j S n. l l m j primitive disjunction B^ is of the form jp^j V...V |p k |, where each atom, i.e. a variable or the negation of a variable. tent-loading, E j pj | &...& E [ p k |.

Since A

Now

is an

Since E(B^), and E is con-

B is true, A^

® j » • • •»

A

B n

j•

Since A^ -+ B. is explicitly tautological one of the conjuncts of A^ is among |p J , ... , ] p k l ; A. with a 5 i 5 n. l

thus by the condition, if E(C) then E(A & C), E(A^) for each Thus E(A,) & ... & E(A ), whence E(A. v ... v A ) i.e. E(A). 1 n I n

Dually, a statement property D is range-loading iff it satisfies the dual conditions, f^r every truth-functional A and E that D(A & B) iff DA & DB and D(A v B) iff U)A v DB. Thus, for example, descriptive and logically acceptable appear to be range-loading properties. Every content-loading property is matched by a range-loading property, e.g. E is matched ^ y (E~). Dually^ too, the following is provable: ° Theorem 3.14. Where D is a range-loading property, if A -*• B and D(A) then D(B). It also follows that if A + B then for every content-loading property f, f(B) ^ f(A), i.e. if A entails B then f(B) is contained in f(A) as far as contentloading properties go. Dually, if A entails B then f(B) contains f(A) for rangeloading properties. These principles give the cash value of the container model. Analogous proofs break down when entailment is replaced by suppression implications; for there is then no guarantee that properties are transmitted across explicitly tautological entailments, since where suppression occurs explicitly tautological entailments may have no atoms in common (see Anderson and Belnap, 62, pp.13-14). Moreover it is pretty obvious that analogous theorems do not hold for material and strict implication; for we can find statements p and q such that E(q), e.g. q is evaluative, but ~E(p) and ~E(~p), i.e. such that ~E(p & ~p). Indeed, by applying the FD semantics, we can give countermodels to analogous results for strict and material implication. Consider the following restricted E rule: Where A is truth-functional, E(A) holds at T iff for some situation a,

27$

3.3 THE l/IWI CATION OF M O N - V E V U C m L l T V THESES

WITH RELEVANT

LOGIC

such that (a t a* and) TR £ a, A is not in a, where it is required of the relation R e that ifTR^aj and TR £ a 2 then aj = a.^.

Thus if A has feature E then there is

a unique situation from which A is excluded. It follows for property E so defined that E(A v B) iff E(A) & E(B) and E(A & B) iff E(A) v E(B) for truthfunctional A and B, i.e. E is indeed a content property. Now consider a model on set K = {T,a,a,*} so defined that TR^a, p holds in a but q does no^, and p holds in a*. Then p & ~p -3 q and E(q) and not E(p & ~p) are true in this-m^del. Hence, where E is a content-loading property, A -J B and E(B) do not guaranty that E(A). Thus strict implication wrecks the container model; with strict implication one can get out what "one has not put in. If, for example, evaluativeness is a content-loading property, f.d. entailment, in contrast to strict implication, vindicates the historical thesis of the non-deducibility of evaluative statements from non-evaluative statements. Non-deducibility theses - such as that of the non-deducibility of value from fact, of contingency from necessity, of certain types of necessity from contingency, etc. - are of immense importance in philosophy and its applications in the sciences. That many of them break down with strict implication reveals often not the inadequacy of the theses but the failure of strict implication, a failure that can sometimes be rectified by relevant implication. The approach to non-deducibility theses so far considered, through contentloading and range-loading properties, is only one of the approaches that relevant logics afford. It is furthermore an approach that has not been worked out beyond the first degree setting. Both other, less limited, approaches arfeT removal of the inhibiting first degree restriction call for investigation. Important and appealing though the first degree theory FD is, there are substantial reasons for moving beyond it. Most obviously, entailment and implication admit - like many intensional connectives that philosophers have tried to restrict to the first degree - of iteration; and there are many nontrivial arguments, in philosophy, mathematics and elsewhere, involving them which do not admit of satisfactory first degree representation. There are also other more technical reasons for moving on. One is in order to obtain a pleasing syntactical formulating of the first degree theory which does not depend on qualified use of principles such as y that are peripheral at best and hostile at worst to the relevant enterprise. Another is that objections to the first degree on the grounds that it requires not just consistency but completeness of the base world T, that the law of excluded middle is out of place in logics that allow for the failure of y, are easily met at the higher degree (see the features of basic system B, chapter 4). Development of the first degree theory had, in any case, only limited objectives; now that many of these objectives have been sufficiently accomplished, we shall move beyond it. §4. The need for, and shape of, higher degree investigations. An initial classification of types of systems and classes of questionable principles. Beyond the first degree lies the second degree, and beyond that the higher degrees. And legitimate compounding principles lead ineluctably to the second degree and to progressively higher degrees. The logics of entailment, of implication, of probability, and of intensional notions generally, do not stop at the first degree. Iterated statements of deducibility make, as we have argued, perfectly good sense. Moreover they have a logic - such second degree principles as (A B) & (A C) ->-. A -> (B & C) are rather evidently correct - and it is past time we had a good theory of them. The first degree is no final stopping point in the quest for adequate applied theories of entailment and implication. A logical theory which stops at the first degree stage, as metalinguistic theories do, is quite inadequate to the linguistic and evidential data. Positions which admit a higher degree, but which collapse the whole higher degree to

279$

3.4 RIGHT ANV WRONG DIRECTIONS IN THE SELECTION

OF HIGHER VEGREE

SYSTEMS

the first degree, vary in their degrees of inadequacy. The extreme classical position (which few hold any longer) that identified deducibility with the converse of material implication and thereby reduces the higher degree of deducibility to Boolean logic is patently inadequate. The vintage modal position, according to which S5 strict implication captures deducibility, and which reduces strict implication (along with its modal reduction) to the first degree, is only a shade less inadequate (cf. chapter 1 again). Despite the temptation of logicians to manufacture strong higher degree logics by collapsing evident distinctions between various sorts of iterations or by eliminating iterations - Portation which equates A -*• (B -+• C) with A & B -»• C is one of the more blatant and damaging examples - uncorrupted logical sense lends weight to very few such reductions. Very roughly,1 the adequacy of an entailmental system varies inversely with the extent of the entailmental reduction principles it includes. Such reductions are linked to strength of systems, and strength has hitherto been considered a virtue. It is not (as will be argued); nor does a principle of strength set the onus of proof. The correct procedure is not that dictated by strength - to admit a (questionable) principle until it is discredited - but quite the reverse - not to accept a questionable principle until it has been duly accredited. As will quickly emerge, some of the commonly accepted principles of stronger higher degree relevant logics do not have particularly good credentials. Accordingly they should not be accepted. A pernicious influence on the sorting out of correct and incorrect higher degree principles has been logicians' penchant for studying pure entailment systems. It is simply assumed that a study of the correct principles should be an interesting one (such as the study of a "pure calculus of entailment" may be, even though the calculus is decidable) and not just (say) a study of substitution instances of A + A. Simply expecting a strong and interesting pure system to emerge is a good way of going wrong. For the fact is that the interesting correct principles of entailment are the result not of multiplying up arrows but of interaction of entailment with other logical operations. Multiplying up entailments on their own is simply a (and a simply) splendid source of incorrect entailment principles. (This is the first of our complaints as to the procedure of Anderson and Belnap, in ABE and elsewhere, in arriving at system E). More generally, an assemblage of parts approach to the selection of logical principles can be decidedly misleading: usually it is important to take some account also of the whole into which the parts are to be fitted. Analysis needs to be combined, as so often, with synthesis. Even enlightened intuition is not very firm in the region of iterated implication, an"d it is too commonly swamped by prejudice or dogma, or swayed by methodological principles (such as those of strength, or parts-analysis) that do not bear very much examination. Part of the explanation of the dimness of logical intuition in the higher degree region is undoubtedly that the iterated entailment statements that do occur are rarely of more than second degree, and even in theoretical areas where the deductive methods get most intensive use most implication or deducibility statements are of the first degree.2 1

The inverse correlation is very rough, firstly because some reduction principles are more damaging than others, and secondly because some principles are correct, e.g. there is a lower second degree bound on correct systems of entailment. 2 TO admit this is not to retreat on our claims as to the inadequacy of entailment theories which cut out at the first degree; the formalisation of the second degree theory of entailment remains an important and necessary part of a complete theory of entailment and of one which heeds evidential data.

220$

3.4 CONDITIONS

OF ADEQUACY

ON SYSTEMS

FOR

ENTAILMENT

Because of the uncertain standing of many higher degree entailment and implication principles we shall approach the higher degree in a rather different and more cautious way than we approached the first degree. We do not entertain any doubts as to the exact shape of the first degree logic. Some of the outlines of the exact shape of the higher degree remain obscured in darkness however; that is, we remain uncertain or undecided about various higher degree principles. Rather than dogmatically, and somewhat arbitrarily, selecting at the outset a higher degree system (or systems) as our candidate(s) for entailment, we shall examine a considerable range - we should like to suppose the whole range, but even that is almost certainly unduly ambitious, especially given the way in which new principles that merit examination keep emerging - of likely candidates for the formulation of sentential entailment systems which contain at least the extensional connectives. Nor do we exclude the possibility - though we think substantial fragmentation unlikely - that entailment splits into various different, (equally) correct notions at the higher degree - e.g. strong or propositional entailment, replacement entailment, affixing entailment, and reduced entailment - corresponding to the main styles of semantical analysis we shall encounter. There are nonetheless important constraints limiting the systems which are considered candidates. The constraints appear to rule out all systems studied before Ackermann 56 and, on one impoitant count, Ackermann's systems as well; so the published study of entailment candidates proper is less than 25 years old. To begin with, a satisfactory entailment system should include connectives &, v, This is not to exclude the occurrence of other connectives in combination with these; the set is simply a minimal set (the case is like that argued in GR, chapter 5). I. A satisfactory {->, &, v, entailment system should conservatively extend the first degree theory FD; that is, though it extends FD a first degree theorem of such a system must be a theorem of FD. The extension should be a proper one; for a satisfactory system should include some second degree principles which are not simply substitution instances of first degree theorems. In particular, a satisfactory system should, it seems to us, include theorem versions of the fundamental rules of first degree entailment, i.e. of the systems DML. Normalising these rules, by reflecting them in entailment theses, and supplying the resulting system with a conventional and uncontroversial rule structure (consisting of just the rules of Modus Ponens and Adjunction already defended, and indeed derivable using t), leads to the basic full {->-, &, v, system, the following system AQIAl. A2. •A4. A5. A7'. A9. D4'. R2.

A A B2. A & B -»• A A3. (A - B) & (A C) A -*• (B & C) A A V B A6. (D & A C) & (D & B -»• C) -»-. D & (A V B) -»• A A9\ A B -*. ~B ~A R1. A, B -6 A & B

(A ->• B) & (B -> C) -»-. A -»• C A & B B B A V B -»• C A -> — A A B, A

B

All but the second degree principles have been defended as correct in arguing for DML. So in arguing for the correctness of A, it suffices to argue for B2, A4, A7 1 and D4'. Correctness will then follow by induction on the lengths of proof in system A. The arguments for the second degree principles (and their substitution instances) are all of the same kind, arguments from interpretations of entailment, in terms of valid argument, as inclusion of logical content, etc. 4$

3.4 THE ARGUMENT

TO THE COMPLEX STRUCTURE

OF SUFFICIENCY SYSTEMS

Consider, to illustrate, what is perhaps the most controversial of the higher degree principles, B2, under the valid argument interpretation. If there is a valid argument from A to B, which may be represented A, ..., B, and a valid argument from B to C, represented B, — , C, then there is a valid argument from A to C, represented by putting the two arguments together, thus A, ..., B, B, C, abbreviated A, ..., C. Not only does this validate Rule Syll, but, since the argument from antecedent (if) to consequent (then) is itself a valid argument, that is the argument from A, ..., B and B, ..., C to A, ...[,B,B,]...C is itself valid, it also validates Conjunctive Syllogism, i.e. B2. The argument from entailment interpreted as inclusion of logical content is a little different berause inclusion, unlike entailment and sufficency, does not naturally iterate (where the content c(A) of A is represented as a class or property). In this case the argument for B2 is that there is a valid argument from the antecedents c(A) £ c(B) and c(B) £ c(C) to the consequent c(A) £ c(C), for the excellent reason that c(A) c(B) £ c(C); put differently, the conjoined antecedent is logically sufficient for the consequent. II.

A satisfactory entailment system should properly include system AQ.

Conditions I and II already lead to systems larger than AQ. For A V ~A is a first degree theorem, but it is not a thesis of AQ (as the semantical models of chapter 4 show). Thus AQ has to be expanded at least to a (minimum) system A which allows the recovery of FD. A direct syntactical argument shows the cheapest such system to be AQ + {A v ~A}: this is system A. Conditions I and II place a lower bound on the semi-lattice of sufficiency systems, and relevance will place an upper limit. The resulting class of relevant implication systems is an enormous and structurally complex class of systems, very many details of which remain to be discovered. III. No satisfactory entailment system violates weak relevance. Hence any principle which added to A leads to irrelevance is to be rejected. Ill does not follow from I and II. There are systems (such as system I of chapter 6, and the "classical relevant" systems) which conservatively extend FD and include A but which violate weak relevance. Condition III is not fully effective, but that does not diminish its power or importance. It suffices to rule out all the ordinary, noxious, systems as unsatisfactory. The standard paradox arguments see to that. It leaves open however (as the unwarranted procedure in ABE of arguing from the entrenched system E does not) the question of whether such principles as

and

A -»• B -»•. C & A A A A

+ C& B

Factor Mingle

can be added to the minimal system A preserving relevance. For many systems the answer is of course in the negative. Factor produces irrelevance in even weak affixing systems (see chapter 7). And Mingle leads to irrelevance, not just in system E, but in sub-logics of E which cater for logical necessity in E's fashion - for example in the following way: ~(D -»• F) D -> F •*. D F, by Mingle and Contraposition. D -»• F -*. ~(D F) D -*• F, by Restricted Commutation. Now with E style necessity D , where °A (A -*• A) ->A, A •*• B **. (A -* B) and D A & nB ° (A & B). Thus since every necessity claim is an entailment, D (A & B) -»•. ~ D (A & B) -+ °(A & B). Hence ° (C C) & D (D D) -»•. (C C) & ° (D -* D)) ° (C -»• C) & ° (D D) whence ~((C C) & (D D)) -+. (C C) & (D D) , and so ~(C + C) D D and irrelevance. Such derivations of irrelevance depend however upon other principles available in the system with respect to which irrelevance is

222$

3.4 WHAT IS WRONG WITH RELEVANCUEV

PARADOXES

SUCH AS MINGLE

established. And it often needs further argument to show that irrelevance is not attributable to other systemic principles, rather than the added principle. For example, it is Factor in combination with Affixing and other principles that leads to irrelevance (see 7); by removing Affixing relevance can, it seems, be restored in certain systems containing Factor. The argument to the correctness of other principles of the base system adopted is accordingly crucial in faulting principles as irrelevant or as inducing irrelevance. Mingle, by contrast with Factor, is a good example of a relevancised paradox. It is a false principle, closely related to a paradox which has however a relevant look about it. B does not entail A A for arbitrary B, as happens with stronger strict implication, but B A A wherever B A. The incorrectness of Mingle is, it seems to us, quite evident: A is not sufficient for the law of identity in A, nor is this part of the logical content of A. But intuitive counterexamples to Mingle are not so readily designed, since the consequent A A is a logical law and so never actually false; and though A -»• A commonly fails to hold it is a moot point whether the many (deductive) situations where it fails are readily accessible to unrefined intuition (they are inaccessible to classically warped intuitions). For a direct counterexample to Mingle (details of which will be made good in chapter 4) consider a situation d where A holds, but A -»• A fails to hold, e.g. because d is non-normal (as in the case of some S2 worlds) and no implications hold in d, or because d is a world where implications never hold between identical statements (for reasons developed in 11.2), or most simply because d is not suitably theorem-complete, i.e. regular. Even given such counterexamples, a problem is left slightly open (cf. FM) as to what is really wrong, not just with Mingle, but with a variety of sometimes proposed implicational principles of the form A -*•. B C (where A may be a sentential variable). For some of these principles, e.g. A (A B) -»• B, do no damage to relevance in quite strong systems. The explanation of what is wrong with all such principles as entailment principles is, however, very simple given S5 modal principles (which we shall later defend: the simple explanation is not open to Anderson and Belnap who go out of their way to reject S5 principles). What is wrong with entailmental principles of the form p =>. B =» C is shown by contraposing and exposing the form of the entailment B =» C. Then (B -*• C) =» ~p. But p, and so ~p, may well be replaced by a contingent statement, whereas (B -»• C) is necessary (for, by S5 principles, => Thus p =». B =» C leads to a violation of the content principle DN, that a necessary statement never entails a contingent one. (If a necessary statement were to entail a contingent one then the contingent one would have to be part of the content of a necessary one, which is impossible since a necessary statement conjoined with a contingent one may not itself be necessary.) One argument for the correctness of the S5 principle used (other arguments are set out in detail either in EI or subsequently in chapter 10) is that the antecedent ^ C is a logical statement (i.e. one of non-necessity) and so if true necessarily so and if false impossible. An important upshot which should be recorded is: IVS. A satisfactory entailment system should conform to Ackermann's requirement, that there are no correct principles of the form A B -»• C with A a sentential parameter. The Ackermann requirement is not so simply defended where the arrow connective represents an implication as distinct from an entailment. And indeed too many relevant theorists who have accepted the Ackermann requirement for entailment have also taken A (A -* B) -*• B, for example, to be a correct implicational principle. Such principles, sometimes supposed to distinguish entailment from lawlike or natural non-necessary implication, we approach in a more piecemeal fashion later in the chapter.

223$

3.4 VOUBTFUL

CLASSES OF PRINCIPLES THE ADEQUACY

CONDITIONS

LEAVE

OPEN

IVS is a special case of a more general precept of fallacy avoidance, a precept which also includes III, namely, IV. A satisfactory (entailment) system should avoid the fallacies enumerated in chapter 1. Other conditions also transfer from chapter 1, notably the non-classical constraint : V. A satisfactory system should be paraconsistent; it should enable the nontrivial logical treatment of various inconsistent theories. Ideally, it will be universally paraconsistent and admit self-referential reasoning without breakdown. For a universal deductive logic, which is one of the things we seek, will do as much. A corollary of V is that no satisfactory system will contain y as a primitive rule, whence in particular Ackermann systems are not satisfactory (cf. 1.7). However dual classical requirements are also important. VI. A system which is not explicitly dialectical, e.g. a pure sentential system, should accommodate the rule y of material detachment as an admissible rule, and more generally. VII. A satisfactory system should allow for the enthymematic recovery of classical, strict and intuitionistic logics. But few principles beyond the first degree are required to satisfy VII, and especially VI. A main difficulty with VI is that additional principles can easily lead to its breakdown; however, (as will emerge in 5) such additional principles are mostly undesirable principles. The conditions of adequacy so far marshalled leave a great many things open; for example, they fail entirely - as probably they should - in resolving the choice of correct entailment system, only restricting the choice to an extensive class of relevant systems. Nor, accordingly, do the conditions settle the status of many interesting but often dubious principles separating these candidates for entailment (i.e. systems competing to explicate the logic of entailment at the given sentential level). These perhaps dubious principles cluster in certain groups, which will be called without undue prejudice doubtful classes . Associated with the doubtful classes is a division of normal entailment logics, i.e. of entailment logics with normal conjunction, disjunction and negation; in other words, the doubtful classes induce a typology of entailment logics, some types of which serve to structure subsequent investigations. The initial typology adopted resembles that explained for modal logics (in 1.4). It is a semantically based classification of normal entailmental (and implicational) logics. Type I Logics: Nonreplacement oc Propositional Identity Systems. Syntactically these logics are distinguished by the failure of the principle of substitutivity of coimplicants, SE, A •»• B -fr D(A) -»• D(B) (i.e. of substitutivity or replacement of equivalents in one sense). Correspondingly the logics are distinguished semantically by the fact that values for implicational formulae may be assigned arbitrarily in outer worlds; i.e. effectively, for an outer situation a, I(A B,a) is assigned 1 or 0 by stipulation of the valuation function I, exactly as I(C,a) is assigned 1 or 0 arbitrarily by I for initial wff C. For the inner worlds, where I(A B,a) is determined inductively in terms of values

22 4

3.4 TYPE I ANV TYPE II LOGICS, AND ASSOC!ATEV

VOUBTFUL

CLASSES

for I(A,c) and I(B,d) for appropriate classes of c and d, the evaluation rule for may be either that used for type II systems or that used for type III systems below. Relevant Type I systems are investigated in depth in chapter 14, where the point of such systems is explained. Type II Logics: Replacement Systems. Syntactically these logics are characterised by the satisfaction of replacement, SE, but the failure of (over?) strengthened Replacement principles in the shape of the Affixing rules: A -*• B C A C+B Prefixing, i.e. where A -*• B is a theorem of the system so is C -*• A •*. C -*• B; A + B - 0 B -* C A C Suffixing

and

Correspondingly replacement systems are marked out semantically firstly by the rejection of outer worlds, i.e. there are no outer (or implicationally arbitrary) worlds in the modellings, and secondly by the determination of the value of I(B •*• C,a) by a so-called neighbourhood, or second-order, semantical rule. The following semantical rule for -*• is subsequently derived (and equivalent neighbourhood-type rules are also adduced in chapter 13, where replacement systems are studied): I(B

C,a) = 1 iff Ra[B][C], where

[B] = {c e K: I(B,c) = 1}, i.e. [B] is the range of B, the class of worlds where B holds, and R is a 3-place relation on KxP(K)xP(K), with P(K) the power set of K, i.e. R is a relation whose first place is taken by worlds and whose remaining places are taken by sets of worlds, in fact by those that are ranges. One of the attractions of this rule - the range rule — is that a cogent argument can be advanced for it (unlike some semantical rules which are arrived at rather more in the fashion of Popperian conjectures). The range rule can be reformulated dually as a content rule I(B

C,a) = 1

iff c(C) c c(B), —a where c(B) = {d e K: I(B,d) = 0} gives the logical content (in Carnap's sense) of B. That is to say, according to the content rule, B -*• C holds at situation a iff, as seen from a, the logical content of C is included in that of B. Thus the rule provided a situationally relativised version of the traditional view of entailment as inclusion of logical content. Type I and type II logics are separated by the first class of problematic or doubtful principles, namely Doubtful Class 1: Replacement principles:- These include not only the replacement rule SE that distinguishes type I from type II systems, but various systemic strengthenings of the rule, such as:A • B & D(A) D(B), A • B & D(A) -»• D(B), A * B + . D(A) D(B), where in each case D(A) is some well-formed frame containing A. An instance of the second principle is the scheme (A •**• B) & ~A •* ~B. Other principles of this class, e.g. (A •»• B) & ~B ~A, result by admissible elementary transformations; but strengthenings such as (A B) & ~B -»• ~A, take us to further classes of questionable principles (e.g. Contraction principles). Obvious strengthenings of Replacement principles which enable the proof of SE - and so are at least as problematic - are those which fail in pure replacement systems, most notably Affixing principles and their variants. Doubtful Class 2: Affixing principles and their equivalents. The basic prin-

225$

3.4 TYPE III AbJV TYPE 11/ LOGICS; AFFIXING AND EXP0RTATII/E PRINCIPLES

ciples are Prefixing and Suffixing rules.1 The equivalents include schemes such as Augmentation, B -»• C A & B -+ C (a conjunctive strengthening principle) and also rule versions thereof, schemes which have encountered much criticism recently in the study of conditional logics (see 1.5). Whereas Replacement principles are in doubt primarily because they give questionable results when coentailment is construed as propositional identity, Affixing principles are in doubt in part because they appear to yield counterintuitive results when implication is taken to represent conditionality (generally). Affixing principles of course separate type II and type III logics. Type III Logics: Affixing Systems. Syntactically these logics are characterised by the satisfaction of the Affixing rules, whether as primitive rules or derived rules. For example, Affixing rules can be derived (see 4.2) in logics which extend DML by SE using just the conjunction axioms: A (B & C) A -> C B -v C -+. (A & B) C. Affixing systems are distinguished semantically by the reduction of the evaluation rule for to a "first-order" rule, that is to say, in place of a relation R on KxP(K)xP(K), a relation R on KxKxK, i.e. on worlds in every place, can be adopted. Affixing systems are the main logics studied in subsequent chapters. Affixing systems may, and in pure cases do, reject systemic normalisations of their characteristic principles, that is to say they may reject Exported Syllogism. Hence the class which distinguishes type III and type IV logics :Doubtful Class 3: Exported Syllogistic principles, namely the following arrow strengthenings of Affixing principles: A B C-+A-*. C B and A B -»•. B C A C.2 In A and its extensions these principles are interderivable; so we can speak simply of Exported Syllogism. Exported Syllogism is the key principle separating the Exportative or Strong relevant logics, which are virtually the only relevant logics studied in ABE, from the Nonexportative or Deep logics which are the main focus of subsequent investigations in this text. Why Exported Syllogism is in serious doubt is explained later in the chapter. Type IV Logics: Exportative (i.e. Exported Syllogism) Systems. Syntactically these systems are distinguished by the inclusion of Exported Syllogism. Semantically it turns out that all the usual Exportative systems can be modelled using reduced models, that is (as chapter 4 explains) models where the ordering relation < can be defined in terms of the base world T, as a < b = n f RTab.

1

Although Prefixing and Suffixing stand or fall together in systems, such as extensions of A, which contain full Contraposition principles, they go their separate*ways in some weaker systems which lack full Contraposition, e.g. in certain systems proposed in recent endeavours to formalise the logic of conditionals: see 1.5. Such systems are considered again in chapter 11.

2

The authors differ as to the dubiousness of these principles. For example, while the first author thinks that there is not much doubt but that as entailment principles these principles are incorrect, some other authors think that without much doubt they are correct. The second view at least has the virtue of resolving all the doubtful cases so far assembled. The authors also differ, though perhaps less markedly, as to the status of certain other doubtful classes.

226$

3.4 MODAL AHV RELEVANT

TYPOLOGIES

COMPARED,

AND TYPE 1/ LOGICS

Strictly Replacement systems include Affixing systems which include Exportative systems, but the typology can be made exhaustive by an obvious application of the method of difference, as already indicated in the case of type II systems. Let us distinguish where necessary, then, Pure Replacement Systems and Pure or Nonexportative Affixing systems. As remarked, the typology resembles the rather similar classification already made (in chapter 1) of strict modal logics; but it does not quite coincide with it. There type I strict logics are systems in the vicinity of SO.5; type II strict systems are those resembling SO.7 -SI (see WSL) which permit strict replacement; type III systems are systems in the vicinity of S2 which incorporate affixing rules or their modal analogues, i.e. Becker rules such as A B -t> D A -»• D B , or some thesis equivalent such as D (A & B) -*D A ; and type IV systems correspond to modal systems extending S3 which include strong D transitivity principles such as A + B A °B. But the comparison, though good, is not exact semantically, because nonnormal worlds do not feature in the entailment case in quite the way they do in the modal case. This classification of strict modal logics leaves out (and in 1.4 relegates to a footnote, fn.2, p.39) what is usually taken to be the most important class modal logics, namely normal modal logics. Indeed much work on modal logic proceeds as if normal modal logics are the only modal systems, that is within a delusional framework. Normal modal logics are distinguished syntactically through the rule of necessitation, A -«• °A, and semantially by the requirement that all worlds are normal or regular, i.e. there are no worlds where theorems do not hold. The question arises as to whether the typology, designed for relevant systems, can be usefully extended to include systems analogous to normal modal systems? And the answer is: In a rough way, Yes. Type V Logics: Necessitation Systems. These systems are characterised syntactically by a slight (but needed) generalisation of the necessitation rule which in systems like E takes the form, A (A -+ A) -* A, namely the rule A

(A -*• B) -* B

(Theorem Suppression).

The rule is so named because it permits the suppression of a theorem in an internal position in a wff, namely of A in (A -* B) ->. A B. It has as a special case th& scheme ((A -*• A) B) -* B, Suppression, which reflects suppresion of A + A in an internally nested position. The suppression features do however have the technical advantage that they enable the development of a modal theory in terms of "entailment" without recourse to new primitives. For necessity, N, can be defined, as in E, thus: NA (A A) ->• A. It is immediate by the rule that Necessitation holds, i.e. A -6 NA, and hence a weak form of N distribution over -*•, A B, NA -6 NB, holds. Also by Suppression, NA -»• A. But to establish convincingly that Necessitation Systems have modal features (though in a relevant frame), it would have to be shown that N distributes appropriately with respect to •*• and &; and such a demonstration requires principles that weaker Necessitation Systems may lack. What is really required is a proof that (A •*• A) -*• A

t -> A

(A-B, for Anderson-Belnap),

since (A A) -*• A was introduced to sutstitute t -* A while avoiding use of t, and since given A-B it is easy to prove, by Replacement, both NA & NB -»• N(A & B) from axiom A4, and A -»• B - NA -*• NB, by Rule Prefixing. However A-B is (at least) a third degree principle, proof of which requires considerable logical resources, primarily those of Exportative Systems together with those characterising t. Constant t conforms to the schemes t A -*• A and (t -»•' A) -*• A (see 5.1).

in

3.4 IMPORTANT

DOUBTFUL

CLASSES:

COMMUTATION

ANV RESTRICTED

COMMUTATION

PRINCIPLES

From the first, by Affixing, (A -*• A) •*• A •*•. t •*• A. And by Exported Syllogism and the second, t -»• A -*•. A •*• A •*•. t -»• A, whence, as the final clause t -»• A implies A, t •*• A -»-. (A •*• A) •*• A. Such systems, if they incorporate Exportative principles symmetrically (i.e. both prefixing and suffixing forms), while they admit derivation of stronger distribution principles such as A -*• B NA •*• NB (since A -»• B -»-. t -*• A -»•. t -»• B) - thereby forcing •*• into an entailment interpretation - though they are quite strong may still be much weaker than system E where such derivation of modality from entailment was perfected. For example, a Necessitative System with a full modal structure may lack the Contraction and Reductio principles (introduced shortly) that E includes. Unlike normal modal systems a really neat (indeed simplistic) semantical condition does not appear to correspond for Necessitation Systems. What does correspond semantically is a however a condition on accessibility from regular worlds (for every world a there is a regular world x such that Raxa, i.e. seen from a, regular world x precedes a). Type V logics are of course separated from logics of the other types so far defined by a further class of dubious principles namely Doubtful Class 4: Restricted Commutation principles. These include not only Implication-restricted Commutation, [A (B -»• C) D] B C -»-. A D, and Necessity-restricted Commutation, but such connected principles as Implicationrestricted Assertion, (A -»• B) -*. (A B) C C, and the pair ((A A) & (B -* B) C) -»• C and ((A -»• A) B) -»• B and their specialisation, (A A) •*• A-•A, as well as the rule A (A B) B and its specialisation A (A •*• A) A. In stronger systems many of these principles are interderivable, in weaker systems they diverge. Restricted Commutation principles immediately suggest another class of principles, of considerable importance in the choice of stronger relevant logics, and also doubtful, namely Doubtful Class 5: Commutation principles, e.g. Commutation, A -*• (B -»• C) B -»•. A •* C, and Exported Assertion, A A -* B •» B, and its specialisation, A A A -»• A. It is rarely suggested of course that principles of this class 1 hold for entailment, but it has been seriously contended that they hold for natural or lawlike implication or for the conditional (see Meyer 68a, Barker 69, and Bacon 71). As entailment principles and the Commutation principles of class 5 were quite properly criticised by Ackermann 56 and later by Anderson and Belnap (see ABE), but both went on - erroneously it is shortly argued - to adopt the restricted versions of class 4 as correct entailment principles. Commutation principles deliver further classes of systems; specifically Exported Assertion - which is modelled in Affixing systems by the condition that if Rabc then Rbac, or revealingly in operational semantics by the condition that a © b = b © a - yields Type VI Logics: Commutative Systems. attention in what follows.

But such systems attract little explicit

Exported Syllogism does not suffice on its own for reduced modellings, but the combination of it with Contraction principles is sufficient. The core Contraction principles are Ass, A & (A -»• B) ->• B, and its derivative in system A, J

A notable exception is provided by Lewis's early proposals (12 and 14) for entailment, where the associativity principle for intensional disjunction AV(BVC) BV(AVC) led directly - since AVB ~A -*> B - to Commutation. While some of these systems collapsed into two-valued logic (see Parry 68), others relevant logics - did not (see chapter 5). 11$

5.4

TVTE V I M C G I C S ; A,VP

• CONTRACT!CM,

REVUCTJO

ANV OTHER VOUBTFUL

TYPES

Rule Contraction, A v (A B) -t> A B. Rule Contraction is in fact a common denominator of several familiar contraction principles which cause a headache for more sensitive entailment theorists, and it leads to a further significant class of principles concerning which there is much room for doubt. Doubtful Class 6: Contraction Principles:- These include as well as A & (A B) B and the Rule, the following principles, all of which readily vield the Rule: Absorption (or Contraction) Importation Commuted Self-Distribution Self-Distribution (or Frege)

A - (A - B) ->. A ->• B A - (B -> C) A & B -C A > B A - (B C) -<-. A ->• C A (B - C) A - B A ••» C

Corresponding to these principles, and distinguished by A & (A * B.) B, are Type VII Logics: Trivialisable Systems, so named because the addition of unrestricted abstraction (or comprehension) principles to these systems leads, by variants of the logico-semantica1 paradoxes, to triviality of the systems (one of the arguments is presented in 4.1, others in ?9 below). Given the correctness of system A, there is only one further class of formulae of better known relevant systems R and E (studied in ABE) that remains in doubt, namely Doubtful Class 7: Reductio and Counterexample principles. Among these principles are Reductio (A -* -A) -»• ~A, and its equivalent, A * B »-. A ->• —B -»•. —A, and Counterexample, A & —B ^(A B) . Note that though system A and also the first-degree logic FD both guarantee rule forms of some of these principles, e.g. A -»• "^A ~A, and A ->• B -C> & --B), they do not underwrite the higher degree implicational forms. Since the rule forms are tantamount in both A and FD to LEM, A v —A, legitimate doubts about Reductio principles are not, cannot be, of an intuitionistic cast. Well known positions like intuitionism that would challenge Reductio would also question negation principles already argued for and accordingly taken for granted in A. Certainly once the constraints of & v normality are abandoned the class of principles that might be accounted doubtful expands sharply. In particular, the negation principles of Double Negation and Contraposition are often challenged, though we have not found any good reason to doubt them in the case of sufficiency conditionals - as distinct from enthymematic conditionals, where forms of Contraposition do fail. Although Contraposition and Double Negation principles are (taken to be) correct for the main intended interpretations so the two forms of Double Negation, and the four forms of Contraposition (before iterated negations are considered) are not reckoned doubtful - nonetheless alternative negation determinables cannot be satisfactorily avoided and are worth investigating in a relevant setting. Accordingly, in a later chapter, chapter 6, we consider systems which abandon each form of Double Negation and various forms of Contraposition. For even though Double Negation and Contraposition principles are analytic on the central sense of negation, there are other negation-like notions, such as (intuitionistic) impossibility, for which not all these principles hold. In a similar way alternative conjunction and disjunction determinables are also considered (though in less detail). The weakening of negation principles allows for variations, retaining relevance, of the positive part of the logic, in particular for minglish principles, of which Mingle, A -*•. A -+ A, is just one important example (other examples are the EM axiom scheme (A ->• B) (A -> B) -> (A B) and the evidently false principle A ->- C -*-. B -»• C -»•. A ->• (B -+ C)),and also for strengthening of the disjunction principles. Such options as the latter lead to other branchings of

229$

3.5 DOUBTFUL

PRINCIPLES, ANV NICETIES

IN THE CHOICE OF SYSTEMS

the tree of systems, e.g. to Strong Disjunction Systems, which contain such principles as (B -*• C) & (A

B V D) •*•. A •*•. C V D

Urquhart;

principles, resembling factorisation principles, which emerge fairly naturally under the operational semantics (again dealt with in later chapters, especially 11). Factorisation principles themselves appear to lead to a branching of the tree of normal entailment systems, before Affixing principles are included. These principles lead to a further class of logics, Factor systems, and to a corresponding further Doubtful Class: Factorisation and Summation principles:- These include as well as Factor the following principles: A ^ B C V A C V B A-^B-w. A & B - » - A 1 A B -»-. A - » - A v B j

Summation Inclusion

The investigation of these principles and the case for and against them is taken up in chapter 7. The doubtful classes so far distinguished, more and more casually, by no means exhaust the doubtful class of principles even within the normal framework to which we have, for the most part, limited the classification. A great many more examples will soon appear (two sections hence), when principles of strength are considered and stronger relevant logics investigated, e.g. Strong Distribution principles. It is important, in a full investigation of options, to carry all dubious principles that are serious candidates for inclusion under main intended interpretations1 along for further logical and semantical investigation. The point applies especially to doubtful principles of such systems as E and R which have been put up as correct (which is why principles of these classes and the types of system they induce have been duly labelled.) One question is whether or which, if any, of these doubtful principles, especially those of E and R, are genuine sufficiency principles? As we have hinted, we suspect that few are. (Reductio principles are the main exception.) However for the purpose of much of the formal development we leave the question open. §5. The matter of choice of logic; and why E is not entailmentnor R the system of relevant implication. It is already evident (from preceding sections) that there are very many relevant logics; indeed there is greater variety than is to be found with modal systems, even when the first degree is fixed, since relevant logics make many important distinctions which modal logics obliterate. This fact has not been sufficiently appreciated, or stressed, in previous work on entailment and implication, and the question of how to choose between the systems has not received the attention it warrants. Choice must be a matter of careful consideration of the properties of the different systems available and the sorts of characteristics required for different interpretations, for example for entailment. Choice should not be a matter of plumping for the first known relevant system on the scene historically, i.e. Ackermann's main system, or an equivalent, such as E. Given the variety of systems, and the assortment of questionable principles, it is evidently premature, not to say methodologically unsound, at this stage of investigations, to light upon one system out of the range. It is hardly surprising that reasons subsequently devised for adopting 1

Naturally many other principles which are not serious candidates, which even induce irrelevance, are of much technical interest, and will be studied for that reason.

230$

3.5 HOW STRONG RELEVANT

LOGICS CAN INCLUDE SUPPRESSION

PRINCIPLES

such a haphazardly selected system are unimpressive and look contrived. With respect to choice it seems that progress is best made, initially at least, by trying to determine in what sort of areas a system appropriate for a particular interpretation such as entailment lies, and by trying to impose some bound on and delimit that area, rather than in trying to arrive without further such ado at a unique system as logical representation of entailment. It may also be that there is some indeterminacy as to where entailment lies and that a class of systems represents the end result of investigations; for conditions of adequacy may not uniquely determine a system at the sentential level. It has been contended however that choice is a fairly straightforward matter. We should select the strongest relevant system satisfying necessity requirements, or perhaps the strongest system satisfying these requirements and certain others (cf. Belnap 59, BT). This line of thought is especially likely to appeal to those who see the trouble with strict implication as being just the paradoxes and who would be satisfied with strict implication were it not for its paradoxical features. By going relevant the removal of the paradoxes is achieved, and once this is achieved we should strive, it is thought, to be as much like stricters as possible. Or to put it differently, once the paradoxes have been removed we should attempt to maximise power in the classical mode. The idea is appealing because it reflects a common idea among people dissatisfied with strict implication, that to obtain entailment the paradoxes are "simply" peeled off. (For example, this idea appears in a queer form in Smiley's criterion, already criticised in 2.1. The idea is that A entails B iff A strictly implies B and B has a $ (e.g. meaning) connection with A.) The idea is nevertheless mistaken1 because it is based on an incorrect diagnosis of what is wrong with strict implication: it treats the paradoxes as the whole of the disease when they are simply among the more conspicuous symptoms. As we argued earlier, at bottom the paradoxes are the product of suppression, and this is undesirable not just because it produces the paradoxes: it is an independently undesirable feature which destroys many of the properties an adequate account of entailment should have, such as logical sufficiency. Suppression must be got rid of in order to obtain an adequate account. In terms of this one can see that the strength criterion must give the wrong results. For the strongest relevant systems allow a considerable amount of suppression of relevant noncontingent truths, since it is plain that a wide degree of suppression is compatible with relevance. Thus it is possible to add certain sorts of suppression permitting principles and rules which do not interfere with systemic relevance. Examples are provided by the following suppression rules: A B & (D -»• D) + C - & A B - » C , and A -»•. B & t + C - f r A -»•. B -»• C, where t represents some conjunction of theorems of E or of R. Theorem 3.15: E or R are (1) enlarged by these suppressive rules which however (2) preserve weak relevance. Proof. (l)If the second rule, for example were admissible in E would be provable, since B & t C -»•. B & t + C (in any system K. includes A -*• A), that B & t C B C. But the latter is even in 3-valued logic RM3 (on which see, e.g. ABE, p.470) nor subsystems of RM3 as E and R. 2 Proof of (2) is by the Belnap 1

and R, then it tnat, like E and not provable therefore in such matrices (of §1)

Even if it can eventually (and in hindsight be made to work for full entailment systems.

2

This argument was supplied by J. Slaney.

231$

3.5 PUTTING RELEVANCE I hi ITS PLACE

under which the rule is universally valid. Likewise there are many other principles objectionable for entailment basically stemming from suppression, but which can be argued against independently on the basis of counterexamples - which strong relevant systems include, as we shall argue iij the cases of E and R. If these considerations are correct, we shall have to look for an account of entailment not among the strong relevant systems but among the weaker systems. The upshot of this is that (weak) relevance should be seen as no more than a necessary condition for implication (or entailment), and as neither a sufficient condition nor as leading together with other minor qualifications to a sufficient condition. That the antecedent should be appropriately connected in meaning with the consequent, something for which the variable sharing condition may be taken to give a formal necessary condition at the sentential level, is a consequential feature of good entailment and implication relations - a feature that will emerge from accounts of them, but it is not the crucial or central feature of the relation of entailment, which aims after all to explicate the relation, not of connection of meaning, but of logical sufficiency, valid argument and so cn. How it ought be is not how it has been. A preoccupation3 vith relevance has underlain the selection of systems E and R, which for some time were thought to be the strongest relevant systems, or in the case of E that also satisfying necessity requirements. Relevance has been taken to be central to entailment, in direct contrast to what we have contended, namely that the disease of strict and material implication and other paradoxical systems is suppression. Irrelevance is merely a symptom of that disease and not of the essence. It is not what is reaiiy wrong with those systems, and they are not cured merely by removing irrelevance. (We have already seen in 2.9 how the disease of suppression produces the symptom of irrelevance.) The relegation of relevance and promotion instead of sufficiency and suppression-freedom as central to entailment, helps to explain, what may have puzzled readers, why the text concentrates particularly on more economical relevant systems, especially in view of the fact that almost all of the supporters of relevant logic - as against strict implication and other options - have assumed, without much argument, that it is the overpowered exportative relevant systems that are of philosopical and logical importance. Indeed in the work of some of the foremost proponents of relevant logic, Anderson and Belnap, and in much of the accompanying literature, it is assumed that a particular strong system E can be located as the analysis of entailment at the sentential level. Elsewhere (especially in the papers of Meyer, but also, independently, in Barker 69 and others), and often as a connected thesis, it has been proposed that implication - as a relevant, lawlike (but not logically necessary) connection - is captured by the system R: indeed R is too often called 'the logic of relevant implication' and its implication known as 'relevant implication'. We intend to challenge these claims - in particular, the claim that E is the analysis of entailment, and the claim that the philosophically and logically important relevant systems are the strong relevant systems, especially E, R and T. 2 We want to sketch further reasons for our belief that, on the J

0r rather, an acclaimed preoccupation: relevance.

2

TO recall details of these systems for readers not so well versed in Entailment :(continued on next page)

for little study has been made of

232$

3.5 SETTING THE SCENE FOR CRITICISMS OF S/STEMS E AND R

contrary, it is among the nonexportative relevant systems that the correct analyses of entailment, deducibility, and implication will be found. Unlike most of those who criticise E and R as analyses of entailment and implication we do not object to E and R on the grounds that they do not contain such "correct" principles as Disjunctive Syllogism. As we have already argued (in FD and above) employing a semantical argument now apparently substantially accepted by champions of E and R, Disjunctive Syllogism is not a generally valid argument, since it fails for inconsistent situations. Disjunctive Syllogism undoubtedly has a major role in producing implicational paradoxes (for reasons already traced) yet the main case for it rests on nothing stronger than its supposed intuitiveness and its role in traditional argument. But the intuition derives from a vision restricted to the possible: it depends on ignoring inconsistent situations (where both A and ~A can hold though B does not, and Disjunctive Syllogism is accordingly falsified) and hence is unreliable over the range of situations implicational assessment must consider to be adequate for the analysis of entailment. The feeling for the intuitive correctness of Disjunctive Syllogism, is, in short, a result of the operation of the prejudice in favour of the possible aided and abetted by the prejudice in favour of the historical (cf. 2.9). Even for someone who rejects paradoxical analyses of entailment such as strict implication and agrees that an acceptable analysis should be weakly relevant, the case put up for lighting on a strong relevant system such as E is far from compelling. For (as we have already seen and will continue to see) there are important classes of implicational systems which do not suffer from the paradoxical disease and which satisfy the same conditions of relevance and 1

(continuation from

previous page)

In Anderson and Belnap's 'favourite formulation', E has primitive connective set {-»-, &, v, ~ } with N defined NA A -»• A -*• A, and has these (moderately well separated) postulates: For [El. [E3.

A (A

&

A & B -*• A (A -*• B) & (A

V

[E4. |_E6.

A -»• B B A •*• B) •*• A

C) ->-.

[E8. A A V B [_E10. (A -*• C) & (B -»• C)

A V B

B & C

C

[E12. E14. Rules:

E2.

A

B

B -> C

E5. E7.

A & B -»• B NA & NB ->• N(A & B)

E9. B A V B Ell. A & (B V C) E13. A + ~B

E.

A

C

B

A -»• B, A -o B

&I.

A,B

(A & B) V C

&N

&V

B -»• ~A

A & B

System R results by the addition of EO.

A -»•. A

B

B

(EAssertion),

and this enables the deletion of El and E7 and of either E3 or E12. Alternatively, EO maybe replaced by Commutation, A (B C) -•. B A •*• C. For system T, delete the "necessity" axioms El and E7 and in part compensation add the permuted mate of E2, namely B -»• C A -*• B -»•. A -*• C, and identity, A A. Specifically then, T ha3 as axioms E2-E6, E9-E14, and both E2(P). B C A -»• B A -»• C and E7(I). A -»• A. 233$

3.5 THE ACTUAL WAV THE AXIOMS OF E WERE REACHEP

necessity as E. Thus neither entailment nor E can be represented as 'the logic of relevance and necessity' in the way that the subtitling of ABE suggests. We have already rejected analyses of entailment which, though relevant and in some cases meeting necessity requirements, differ from E at the first degree. Nonetheless a considerable variety of the rival systems have the same stable first degree as E, just as rival strict implicational systems share the first degree, differing only with respect to less readily assessible higher degree principles. Since it is primarily the first degree that is important for practical argument and for deductive systems and scientific and mathematical reasoning, the choice between implicational systems having the same first degree cannot be made just with simple intuition as a: guide. As with strict implication, then, there is a substantial and difficult issue as to which relevant systems are adequate for the analysis of which concepts. In the scramble to obtain new formal results on strong relevant systems, there has been too little and too superficial consideration of this issue. At the same time there has been a premature selection by the proponents of relevant implication of E as the system of entailment . The case for E as an analysis of entailment is inextricably bound up with the case for selecting E from the infinitely many systems there are to choose from, even when relevance and necessity are required. The actual choice of E is however a function of other factors, such as authors' motives in including or excluding this axiom or that rule. And in fact the actual choice of E was in large measure a matter of historical accident, the system being got largely (as reviews and papers of Anderson and shortly thereafter of Anderson and Belnap indicate) by modification of Ackermann's system IT' - modification to replace rules of II' regarded as undesirable (at least as primitive rules), namely the rule (6) B, A B + C - f r A + C of commutation and, more importantly, the rule y of material detachment. Only a decade later was it established that E coincided (theoremwise) with the system IT' from which it was derived by (nonad-hoc) variations. The situation with respect to the choice of the axioms of E has, in the case of some axioms, scarcely progressed at all in the interval. Consider the negation axioms of E. Nowhere in Anderson and Belnap's work is much of a case made out for the choice of these axioms. In ABE, for example, we are simply invited to look at the negation axioms of Ackermann's system II', which after transformation, are adopted as the negation axioms of E; yet every one of the negation axioms adopted has been seriously questioned. More generally, the choice of axioms for both negation and disjunction is not, to put it mildly, convincingly motivated. One of the main devices employed in guiding the choice of axioms, use of subscripted natural deduction methods, becomes decidedly ad hoc when negation and disjunction are reached, the rules simply reflecting axioms already happened upon in II'. Moreover,

x

Not just bv Anderson and Belnap (in ABE) and their followers, but by outsiders such as Keilkopf 77. An unfortunate outcome of the premature selection of E as entailment (at the sentential level) has been an increased scepticism concerning the virtues of relevant logics over the implicational frameworks by those who perceive the weakness of the case for E as the analysis of entailment. By attempting to focus exclusively on a system with, we believe, major defects as an analysis of the relevant concepts, one considerably weakens the case for relevant logics as a whole, a case which we believe is a very strong one.

234$

3.5 THE STRUCTURE

OF THE ARGUMENT

TO E IN ENTAILMENT

[I. a. ABE)

alternative systems of axioms are readily mirrored in natural deduction formulations (see chapters 7 and 11). In Ackermann, from whose work the axioms of E are drawn, the same certitude is not to be found: ...concerning the system Stre [i.e. II"], the concept of a "rigorous" implication is not so well established that one must use it or an equivalent [e.g. E], One could say thatit is the most inclusive of this type1 (Ackermann 58). In fact Ackermann did not suppose that any of the negation or disjunction axioms of E could be seriously questioned, but he did foresee (see the end of 58) that central pure entailmental axioms of E might well be rejected, specifically Exported Syllogism (in both forms) and Absorption. One would expect, or hope, that a substantial work such as ABE would assemble a coherent and appealing route to the main system advanced, the socalled 'cdlculus E of entailment* (title of chapter 4 of ABE, where system E is introduced) - not, one might add, that this is ever done, or even often attempted, in the case of the orthodox rivals of E, or that one any longer even expects such a case there. In the case of E one is going to be disappointed (again): for the complex strategy for arriving at E in ABE fails to meet natural expectations. The procedure in ABE is, in outline, as follows (the roman numerals correspond to the chapters of ABE):(I) The pure calculus of entailment is arrived at by chopping back intuitionistic and strict implication so as to obtain, respectively, relevance and necessity. (II) Negation axioms are added to E^ giving the system E of entailment with negation. (II) builds on (I). ^ (III) A new beginning is made, and the logics of tautological entailment and of the first degree of entailment are investigated. (IV) The resulting systems E^, of (II) and FD of (III) are amalgamated, and enlarged, to yield system E. An important subplot in (I) and (II) from the point of view of the final choice of E is the development of the theory of necessity, with necessity internally defined (even in E ) thus: NA (A •*• A) -*• A. Even if the strategies of (I)-(III) succeeded, system E would not result, since E goes beyond simple amalgamation of Ev, and FD (or E,,, , as it is called in ABE), as the details of rde (IV) reveal. We have already seen that the procedure in (II) is open to serious criticism, namely that no justification of the negation axioms, taken (after minor adjustment) from Ackermann, as principles of entailment is given. The procedures at the other stages of the argument also leave themselves open to severe criticism. We consider the stages in turn. Anderson and Belnap concede to themselves on the very first page of their text (ABE, p.3), as 'obvious and satisfactory principles which we recognise intuitively as valid' both Frege and Exported Syllogism, principles already 1

This last claim is, however, seriously astray\ for there are proper extensions of Ackermann's systems which are of the type in question: see §6.

235$

3.5 BREAK-VOUN

OF THE ARGUMENT

TO E+ FROM NECESSITY

AND

RELEVANCE

registered as doubtful, namely A -*• (B -*• C) -»•. A -»• B A C and A -*• B B C A -»• C - two principles rejected in Lewis's final theory of entailment (cf. ABE itself, p.9) and questioned by several other workers on entailment. This initial piece of bravado sets the scene for Anderson and Belnap's subsequent program (p.5) for homing on the pure calculus of entailment: entailment is to be located essentially by peeling off defective principles from strong systems such as classical and intuitionistic implication; It is not (to be) located by the proper procedure of assembling none but principles warranted as correct. As usual the way up - accept none but recognisably correct principles - and the way down - reject only recognisably incorrect principles from a previously given framework - are not one and the same: the results tend to be very different because the way down includes many principles - many of those that are neutral or doubtful or even unreliable - that the way up does not.1 In the way down of ABE we are presented first of all with the strong and paradoxical system H of intuitionistic implication in the guise of "natural deduction". This system is modified in two ways, first to take account of necessity, second to take account of relevance; and these two modifications are combined to give pure entailment, i.e. 'combining necessity and relevance leads naturally and plausibly to the pure calculus E^ of entailment' (p.6. our italics). The assumptions thereby implicitly made are large; in particular, it is suggested that i) all that is wrong with intuitionistic implication as a logic of entailment consists in isolable violations of necessity and relevance (and likewise all that is wrong with S4 strict implication, which meets necessity requirements, and R implication, which meets relevance requirements, are isolable violations of, respectively, necessity and relevance). It is also in fact assumed that ia) violations able. It should and that more or operations, that adequacy.

of necessity and of relevance of be obvious that this surgery may less may be amputated, and also, the result may well fail to meet

systems are separably removnot be uniquely performable, as with all subtraction hoped for standards of

ii) the root problems of classical-intuitionistic implicational theory consist in failure to observe necessity and relevance requirements, and that there is nothing deeper or more demanding from which these requirements issue. iii) all correct pure implicational principles are already captured by intuitionistic implication (albeit in bad company). Hookers, strong stricters and others, will object to assumption iii), hookers and S5 stricters on the grounds that Peirce's law ((A ->• B) -»• A) •* A or some of its amazing variants have been omitted from the initially given. An S5 stricter attached to strict Peirce's law in the form ((a -* B) -*• a) -»• a where a is of the form C -*• D might argue, for instance, as follows: the system ESP obtained by adding strict Peirce to contains no obvious fallacies of necessity. After all ESP conforms to the Ackermann matrix (p.40) which shows that the system contains no theses of the form p -»-. B -*• C. And if ESP does violate requirements of relevance the fault must lie elsewhere in To dispose of such a stricter, requirements beyond relevance and necessity are called for (e.g. an adaptation of those applied against Peirce's law in 1.4 above). More generally, to revert also to ia), a system may be pruned down to relevance and necessity in many ways, with different results, depending *A three valued (entailment) logical picture with values true, false, and neutral could accommodate both the way up, with only value true designated, and the way down, with both true and neutral designated values.

236$

3.5 FAULTS IN THE ASSUMPTIONS ANDERSON ANP BELNAP RELV ON

on what one starts with and what one chooses to keep in and to keep out. Our main complaints concern, however, assumptions i) and ii). Anderson and Belnap themselves perhaps do not really believe i); for they supply (on p.94) a formula. (A

B

B - A)

A - A

(Albert)

which is a theorem of both R and S3 and conforms to requirements of both relevance (in the variable sharing sense) and necessity (in the full MaksimovaCoffa sense, ABE, p.237 ff.) but which is not a theorem of E . Albert is only one of many theses of intuitionistic logic which, though conforming to relevance and necessity requirements, are not included in E. In sum, relevance and necessity are not jointly sufficient for the (non-unique) selection of from as given. How on earth did Anderson and Belnap arrive at E^ then? They do throw out in passing further selection criteria - naturalness and plausibility (p.6) and strength (p.26) about all of which we shall have caustic things to say shortly but the answer is, of course, that they knew where they were going (always a help in reaching a destination), having previously isolated what they correctly conjectured was the pure entailment part of Ackermann's system II'. They knew, from Ackermann, that satisfied the requirement of necessity, and they had discovered (Belnap BT) that E^ met the requirement of relevance. These, undoubtedly very important, necessary conditions on a correct system do not however select - still less E - from a spectrum of other systems that also meet the conditions. Nor do the other auxiliary principles serve to complete the task as we shall argue in the case of E: E is far from as strong as it might be given only relevance and necessity requirements, yet it contains several implausible principles, and naturalness is, almost, all things to all men. The real reason that Anderson and Belnap have to resort to such weakkneed supplementary criteria as naturalness, strength, plausibility and the like is that they have not got to the heart of the matter: relevance and necessity are not core features of entailment, not really of the essence so to say, but by-products that flow from these other features. In short, we are rejecting assumption ii), something we have done in the case of relevance from the Introduction on. Deducibility of B from A requires the sufficiency of A on its own for B, which in turn requires the avoidance of suppression of logical assumptions. E^ does not avoid suppression, and we can locate the stages in the natural deduction route to E^ of (I) where suppression enters. It enters with the introduction (pp.8-9) of subproofs in so-called "natural deduction" formats and unrestricted rules of repetition in subproofs and reiteration into subproofs. Repetition in subproofs allows the suppression of logical laws as the following two cases which permit the suppression of Identity reveal (the notation is that of ABE):1

B

hyp

C

b)

1

(A -*• A) -+ B

hyp

2

A

hyp

2

A

hyp

3

A

repetition

3

A

repetition

2-3 -+I

4

A -> A

2-3 -+I

1-4 -»I

5

B

4-1

4

A

A

5

B

C

A

A

6 237$

(A -»• A)

B

B

1-5 -+I

3.5 HOW REPETITION

AND REITERATION

FACILITATE

SUPPRESSION

Anderson and Belnap have, of course, to restrict use of repetition in order to remove such irrelevance as occurs in the first example a), but there are ways and ways of legislating to retain relevance, and they choose to impose restrictions which in fact remove suppression of logical principles of the sort illustrated in a) but not that which leads to suppression of the parallel type that figures in b). Reiteration into subproofs likewise facilitates suppression, though of a different kind, as the following examples help reveal:B ->• C A

(B

hyp

B

C)

D

hyp

hyp

B -> C

hyp

A

hyp

A

hyp

A -> B

Reit

A

B

->E

(B •* C) •* D

-E

B ->• C

Reit

D

-+E

A -»•. B B

C

C A

C

(Double) Reit -+E

D

(Double) Reit

-+E C

-*l

A -*• B -*. A -»• C A

(B -> C)

(B

C) -*•. A

+1 B

A

A •* (B

C) •+ D

B

C A - D

C

The right hand example also yields, by a further repetitive step, the principle (A (B B) -*• D) •*. A -> D, a conspicious example of higher degree suppression, of an (irrelevant) instance of identity from an interior position. But by imposing further restrictions, most simply and drastically by prohibition of reiteration altogether, higher degree suppression can be eliminated.1 As far as varying restrictions is concerned, however, the issue here, there is a great deal of room for variation (as we shall see in chapters 7 and ll): there is nothing absolute about the particular restrictions Anderson and Belnap impose. Move (II), the addition of negation axioms, we have already dealt with: there our criticism is not so much the end result - though the reductio principle does remain in question - as the method of arriving there. Much the same goes for move (III). Subject to some interpretational restrictions on the class of sentences admitted, e.g. to exclude uncovered non-significant sentences, the system FD arrived at is correct: but the methods of arriving there leave something to be desired, especially the repudiation (ABE, pp.161-177) of Disjunctive Syllogism, a matter which is considered in detail in 5.2. By contrast, move (IV) both leaves something to be desired and issues, we claim, in an incorrect end result, namely E. The main trouble with the amalagamation (IV) lies, from our standpoint, not with the new conjunction postulates which we have already defended (in ways 1

What is important and will concern us later as a separate issue is whether there is an easy route to suppression-free systems through natural deduction methods.

2 38

3.5 UNIPERSUASII/E EXCUSES FOR THE INCLUSION

OF ALFONSO

JN E

Anderson and Belnap do not consider), but with the new mixed postulates and in particular with the axiom E7, which in unabbreviated form is as follows:((A -*• B) - A ) & ((B

B) -»• B)

(A & B

A & B) -»• A & B

(Alfonso).

Anderson and Belnap give us no reason to believe that this principle is correct; rather they offer two excuses for including it in E. One excuse is that they require it to prove inductively the following necessitation rule: A

A

A -»> A;

and the other is that when propositional quantifiers are added E7 can be proved. The excuses are hardly persuasive, the first because it is not apparent that the rule sought, which would have every theorem entailed by the law of identity in itself, is at all desirable; and the second because the extension to propositional quantifiers, which is not without its own conceptual problems, does nothing to help E itself. There are, moreover, reasons for thinking that E7, which violates conditions of obviousness on axioms, is false. Suppose that A and B are arbitrarily selected statements which are entailed by the law of identity in themselves, e.g. they are defined as such. And suppose as seems true, that most statements are not entailed by the law of identity in themselves. Then it does not follow that the conjunction A & B of A and B is entailed by the law of identity in itself; and certainly nothing seems to guarantee it. The trouble caused by the embarrassing appearance of Alfonso at the marriage of E^ and FD could have been avoided, if sneakily, by reformulation of so as to replace restricted commutation axioms of E^ by Ackermann's rule (6) of commutation1 (or by an approach using constant t). For then Alfonso would simply emerge as a theorem; and objections would have to be (in part) redirected, e.g. to apply against ( 6 ) or use of t. *

*

*

*

*

*

*

x

The formulations of E without (6) primitive and E with (6) primitive, system E* let us say, though theoremwise the same, are not equivalent even under sentential extensions. Compare, for instance, system EA, i.e. E + ~(A •* ~A), with system E'A, i.e. E' plus Aristotle. Then Boethius's principle, A •+ B ~(A •+ ~B), is a theorem of E'A, but not of EA. Firstly, Boethius may be proved in E'A as follows using (6). By ESyll, A B B •* ~A A •+ ~A, whence using Contraposition also A B A -* ~B A ~A, and A B ~(A -*• ~A) -> ~(A ~B). Then Boethius follows by (6). Secondly, by a metavaluation argument (given in this section), EA has no implicational theorems, i.e. theorems in the form C -»• D, not already theorems of E. Boethius is not a theorem of E, as e.g. the 8-valued matrices of 2.11 show, so Boethius is not a theorem of EA. E'A is in fact the same system theoremwise as EB, i.e. E plus Boethius. For, as above EB £ E'A. Conversely the inductive argument showing that necessitation, A -t> NA, is an admissible rule of E extends directly to EB since Boethius is an implication and C •* D •+ N(C -»• D). Then apply the theorem scheme NB -»• (A •*. B -»• C) A C of E (ABE, p.29) in deriving (6) thus: A B f C, B A B C, NB A C. EB, and hence E'A, is an inconsistent but non-trivial connexive logic.

239$

3.5 PRAGMATIC

CRITERIA

FOR SySTEM CHOICE ARE WO GU1VE TO CORRECTNESS

More generally it may be that even if the motivation of E in ABE is unsatisfactory, there are alternative ways of going about the business which reveal that E is well motivated. The most promising approach here is the direct determination of E by applying criteria for choice of system, that is in case, conditions of adequacy on a correct account of entailment. Anderson and Belnap and coworkers, in trying to narrow down the range of choice before the leap to E is made, have advanced and applied various criteria of selection. These criteria (or conditions of adequacy on a correct system), which may be glimpsed throughout ABE in the case of E, are set out more systematically in explaining why R is an interesting system (ABE, p.349 ff.),1 and have been presented elsewhere in the case of E, in most detail in Belnap BT. For E the primary, fundamental, criteria are again those of relevance and necessity. The assumption appears to be that these criteria determine a choice of systems among which selection can then be made by a second, supplementary, set of criteria, namely such pragmatic considerations as systemic strength and power and syntactic organisation. There is a familiar and sweeping objection to all such pragmatic criteria: namely that such criteria are no guide to correctness. And there is a familiar, if less than adequate, reply: that correctness is nothing but acceptability, or warranted assessibility, or some such, which is gauged by just such pragmatic criteria. To avoid immediate stalemate we shall have to proceed a little differently, and this we shall try to do in two ways, firstly by revealing the inadequacy of the secondary criteria, both separately and in combination, and secondly by pointing directly to defects in the sorts of system to which application of these secondary criteria appear to lead. Recall that, unlike most of those who criticise E as an analysis of entailment, we do not dispute that relevance is an important necessary condition which any system that is to have a claim to capturing entailment must satisfy; but we think that relevance is an outcome of more fundamental features of entailment and implication - as we have already proved in the case of first degree theories. Nor do we dispute more central features of the complex criterion of necessity (as embodied in the Maksimova-Coffa account in ABE), namely that necessity must distribute over entailment and that Ackermann's requirement should be met by a satisfactory account of entailment (the remainder of the account, now encapsulated in the slogan 'No non-necessitative entails a necessitative' we shall however severely criticise in chapter' 10). There is, in short, a substantial area of common ground as regards Anderson and Belnap's primary criteria for selection. The main dispute then - until we reach the issue as to whether the modal superstructure of the system or systems selected should be of an S5 type or should diverge in some other direction from an S4 formulation - is as to the secondary criteria suggested which we shall consider one by one, and also cumulatively. §6. The criterion of strength, and relevant (and otherwise appropriate) extensions of E and of R. The appeal of strength and power in modern civilization needs but little present explanation; it is a corollary of the prevailing worldview (often referred to as the dominant social paradigm, and contrasted with the alternative environmental paradigm). That appeal applies not merely in the choice of life styles and as regards what is taken as valuable and as newsworthy, but also in the choice of logics and as what is taken as valuable and newsworthy in logical results. In both cases strength and power x

We are of course not disputing the claim that R is interesting (the text shows that), though opponents of relevant logic might well do so. More to the point, being interesting is neither necessary nor sufficient for being correct.

240$

3.6 THE WIDESPREAD

APPEAL OF THE DEFECTIVE

CRITERION OF STRENGTH

can be damaging: in neither case should their desirability be taken for granted. Yet strength is simply assumed to be a desirable feature of a theory of entailment in ABE (e.g. the 'strong and natural' lists of valid entailments, p.26, p.27). 1 In the choice of a logic, strength (howsoever defined, a problem yet to occupy us) can be undesirable, because it results in incorrectness, or because it seriously restricts the interpretation or application. 2 With E and R both these things happen so we shall argue: principles of commutation fail for implication and their arrow-restricted versions are incorrect for entailment, and the inclusion of principles which reduce nestings of arrows has seriously restricted an important application of these logics, namely the application to logical reasoning about the inconsistent. There is little reason for supposing that certain of the secondary criteria proposed for choice of entailment theory give any guide to correctness (as already remarked). Worse, in some cases, such as strength, the criteria militate against correctness; for a principle of strength will encourage the throwing into a system of extra and perhaps quite unreliable principles which strengthen it, whether or not they preserve correctness. Just this sort of phenomena was exhibited rather clearly in the early temptation to strengthen system R by the addition of Mingle, A A •» A. The principle on its own (or even added to R^) involves no irrelevance, and it is very congenial to the thinking that underlies R [and E], that the suppression of [logical] truths, especially the 'archetypal principle' of Identity, is perfectly in order provided no loss of relevance ensues (A A -»• A comes from A & (A •* A) •*. A •*• A by suppression of A A.) Only the horror of irrelevance in the context of the full system R led to its withdrawal. A basic motivation for the concentration on the strong relevant systems appears, then, to be the assumption that entailment will be found among the strongest relevant strict systems because what is wrong with the strong strict systems which meet necessity requirements is essentially that they lack relevance. 3 Thus according to Belnap a satisfactory theory [of entailment] will isolate the disease of raw irrelevance, keeping everything of the standard theories that can possibly be kept (60, p.15). Fix the irrelevance, it is assumed, and we have our solution to the problem of entailment or at least the major part of it. The position once again takes relevance to be not merely a necessary condition for a correct entailment connection, but the fundamental part of a sufficient condition for adequacy. If irrelevance were the real reason for the unsatisfactoriness of strict systems, the combination of the criteria of relevance and strength should, it would certainly seem, determine the theory of entailment (modulo, x

It is much the same in Belnap's thesis BT, which is far more systematic about conditions of adequacy on a formal theory of entailment than ABE: see Belnap's 60, chapter 1.

2

For a criticism on such grounds of modal reduction principles - (bi-) Conditionals reducing the degree of iteration of modal functors and thereby increasing systemic strength - see Jennings and Schotch 80. On the basis of such criticism they correctly reject the modern picture of modal logics as the study of modal reduction principles.

3

Similarly non-logical implication is to be located, if not as the strongest relevent system, at least among the strongest relevant systems.

241$

3.6 RELEVANCE

ANV STRENGTH

[AND NECESSITY)

LEAD FAR BEYOND R {AND E)

perhaps, principles which strict systems exclude on pain of collapse which it now becomes feasible to add, e.g. A / B /•. C D). Conversely, in assuming that entailment will be found among the strong strict systems it has been assumed that relevance is a crucial part of the story and that obtaining relevance is the crucial thing that needs to be done to modify strict systems. In showing that the combination of strength with relevance does not lead to R, but far beyond it, we are, then, not merely criticising the criterion of strength, and refuting Belnap's .thesis that E is the system that satisfies his requirements in a formal theory of entailment (in 60), but at the same time continuing the campaign to dislodge relevance from its central position. What is wrong with aiming for the strongest relevant system (insofar as it is uniquely determined, e.g. within some given framework) is that this does presuppose that irrelevance is what is wrong with strict and classical systems. But if» as we have argued, this is not the major fault but a byproduct, it is plain that the criterion of strength may well select an incorrect system. For a system may allow damaging suppression but still be relevant through allowing the suppression of statements whose omission does not interfere with variablesharing. Rather evidently, the strongest relevant subsystems of classical logic S (that is systems with S which are relevant but which maximize on principles of S) will be just such systems, and will allow limited suppression, and will accordingly be incorrect. We shall take it, for the time being, that proper inclusion is at least a sufficient condition for strength,1 i.e. if L^ c l^ then L^ is stronger than Lj, where L^ c l^ iff every theorem of logic Lj is a theorem of logic

but

not conversely.2 In terms of this condition the following sorts of. systems, examples of each of which meet conditions both of relevance and of necessity, can be proved to be stronger than E. The types are not exclusive, or, of course, exhaustive. Type 1. Connexive extentions of E, which result by adding connexive principles such as Aristotle and Boethius to E. Both EA, i.e. E + A f ~A, and EB, i.e. E + A -»• B A ^ ~B, are nontrivial by the following chain matrices:3

'it is not a necessary condition. The system consisting just of ~(A ~A) and rules of modus ponens and adjunction, for example, is much weaker than R, but not contined in it: indeed the addition of ~(A ~A) to R leads to collapse, to triviality. 2

This leaves open questions as to the respective rule structures of the systems. In what follows it can be assumed that the weaker logic has no primitive rules not contained in the stronger logic, and mostly that the logics have the same rule structure.

3

As C. Mortensen pointed out. in the next theorem.

The chain underlying these matrices is specified

242$

3.6 NONIMPLICATIVE

0 *1 *2

EXTENSIONS

OF E SHARE THE VIRTUES OF RELEVANCE

0

1

2

~

&

0

1

2

1 0 0

1 1 0

1 1 1

2 2 0

0 1 2

0 0 0

0 1 1

0 1 2

ANV

NECESSITY

The same matrices show that EA and EB have the Ackermann property, which is a large part (and the important correct part) of the requirement that modal fallacies be avoided. Furthermore EA, which adds a non-implicational thesis to E, contains no implicational theorems not already in E, and therefore meets exactly the relevance and necessity requirements so prized for E. More specifically, and using definitions and results drawn from ABE:Theorem 3.16

(Piano theorem).

(1)

If

^(A. f B.) &...&(A > B ) & t ->•. t i i n n C D then (- C -»• D, where t is some conjunction of theorems of E. E (2) Where EE is an extension of E by some axiom scheme (in E's vocabulary) of the form A f B, i.e. EE = E + (A / B), I- „ D. tit C -»• D iff i- _ E C Proof (2) As E c EE if - (- r C •+ D then f- __ C -*• D. Conversely, if f- __ C -»• D ~~——— — E> £>£. EE then either j- C D already or else instances of A A B are used in the EE Ei proof of C -*• D. Let the instances in the latter case be A, B,, A / B . r l i n n Then there is an E proof of (Aj Bj) &...& (An A B n ) & t C -»• D, where t is some conjunction of axioms of E. This follows in fact from the entailment theorem (ABE, pp.177-78: it may be necessary to star some axioms used in the proof in order to meet star requirements). Hence by (1), „ C •+ D. (1) Consider a Hortensen chain, i.e. an odd element chain with (2n + 1 ) elements, n > 1, ordered as follows: 1 £ follows:vi. vii. viii.

£

N

£

0.

Define metavaluation v on a Mortensen chain, as

for each sentential parameter p, v(p) is some element of the chain; v(B & C) = v(B) n v(C), i.e. it is the least of the values of B and C; v(B v C) = v(B) u v(C), i.e. it is the maximum of the values of B and C;

viv. v(~B) « inverse 'v(B), where inverse, like negation, is given by the following inversion matrix:

w.

1

...

N

0

0

N

...

1

v( B -»• C) = N iff (- B L

C and v(B) c v(C), and v(B —

C) = 0 otherwise.

A wff B is true under such a metavaluation v iff v(B) 4 0 (i.e. all values except 0 are designated); and A is chain metavalid iff B is true on all metavaluations. Then (*) if |- B then B is chain metavalid. The argument for (*) was first devised for the five point chain, called by Anderson Piano: 1

P

A

N

0

243$

3.6 CONNEXIl/E SYSTEM EA SATISFIES FULL RELEVANCE

ANV NECESSITY

REQUIREMENTS

A metavaluation v on Piano was defined by setting v(P) = A for each sentential parameter p; but this assignment proved inessential to the metavaluational argument (such an assignment is used however in show that E and various surrounding systems contain no Maksimova fallacies). Proof of (*) is by cases. Firstly, the rules preserve metavalidity. The case of adjunction is immediate from vii, and that for modus ponens is guaranteed by the requirement v(B) c v(C) of vv (indeed the requirement was imposed just for this purpose). It remains to show, secondly, that each axiom is metavalid. Some examples illustrate the method. ad El. Since |-rA->A + B - » B i t suffices to show that v(A -*• A •*• B) c v(B). — 'E If v(A •* A •*• B) = 0 then the inclusion holds, so suppose v(A •*• A •*• B) = N. Then (- A -»• A -*• B and v(A •*• A) £ v(B). But v(A A) = N, so v(B) C N, whence the desired result. ad E2. Since E2 is provable it is enough to show that v(A -»• B) c v(B C A •*• C), and here it is enough to consider the case where v(A ->• B) • N. Then |- A -»• B and v(A) c v(B), whence by E2 itself j- £ B -»• C -»•. A -»• C. It remains then to show v(B -»• C) c v(A -* C). As the case where v(B •* C) = 0 is immediate, suppose v(B -*• C) = N. Then (- B -»• C and v(B) c v(C); and so f- A -»• C and v(A) c v(C), i.e. v(A C) = N, giving v(B C) c V (A C). To complete the argument of (1), simply apply (*). Suppose -.(A, A B ) £ i 1 &...& (A ^ B ) & t -»•. C D. As it is metavalid, and since v(A. f B.) ^ 0 for n n ' i i each i and v(t) ^ 0, v((A. / B. )&...& (A + B ) & t) ^ 0. Hence, by w , i i n n v(C D) = N always, whence (- gC D. Corollary 3.10 (1) Weak relevance). If ^jA •* B then some variable occurs as an antecedent part of both A and B or as a consequent part of both A and B. Proof. The weak relevance result is established (in ABE, pp.253-4) for E using the 8-valued Mq matrices of 2.11. It may be alternatively and more simply established by the 6 element crystal matrices (presented shortly). By the theorem the same result holds for EA. Corollary 3.11 (2) (Avoidance of Ackermann-Maksimova modal fallacies). If Aj, ..., A^ are Maksimova wff and A is a negative wff then for no B and C is A

1

A

2

" ' ^Sc

A

B

C

P rova b3- e

in

EA

-

Understanding this corollary requires rather a lot of, in fact quite valuable, terminology (some of which we could have profitably used at earlier stages in the text):Antecedent part (ap) and consequent part (cp) in defined by joint induction thus: A is a cp of A; if if of if

~B is a cp[ap] of A then B is an ap[cp] of A; either B & C o r B v C i s a cpCap] of A then both B and C are cps[aps] A; B -* C is a cp[ap] of A then B is an ap[cp] of A and C is a cp[ap] of A.

Truth-functional part of a wff is defined in the expected way: A is a truthfunctional part of A and if ~B, B & C, or B v C is a truth-functional part of A so are B and C. 244$

3.6 POSITIVE,

NEGATIVE

AND MAKSIMOVA

U/FF, AND MORE VIRTUES OF EA

Positive and negative wff are then defined as follows: A is negative iff it has no truth-functional consequent parts of the form B C; and positive iff it has no truth-functional antecedent parts of the form B -»• C. (Useful examples of wff that are negative, positive, both and neither are given in ABE, p.240). Maksimova wff comprise the smallest class of wff closed under modus ponens and adjunction which contains all theorems of E and all wff B C such that B is positive and C is negative. Proof. The result for EA follows, by the theorem, from the corresponding result which is established for E in ABE (circa p.242). However the result may be proved directly rather more simply using the following matrices defined on Mortensen chains: the matrices for &, v and ~ are as before (i.e. are given by join, meet and inversion) and that for -»• is determined by the rule that v(B -*• C) = N iff v(B) c v (C) and v(B C) = 0 otherwise, i.e. the matrix for -*• is as follows:

*1

N

1

N

N \

0

0 N

N

N

:

0

0

v

N

1

Now in some chain matrix with more than 3 values, assign every sentential parameter the middle value, e.g. A in the 5-point chain. By induction, every Maksimova wff A. is nonzero (i.e. takes a value other than 0), and each negative formula A irakes a value not less than the middle value. Thus as B •*• C takes value N or 0, A -*•• B -*• C takes value 0, and accordingly A. ... -•. A ^ -+. A B C takes value 0. Hence since every theorem of EA takes a designated value,for no B and C is the complex formula provable in EA. Corollary 3.12 (3) (EA has no intensional fallacies of modality). J - ^ A ^ B then if A e W ( N £ A ) then B e (W(N EA )).

If

Proof requires some minor reworking of the corresponding theorem for E (viz. theorem 2, ABE, p.251). But Aristotle is of course not a theorem of E, as the 8 valued M~ matrices (§1 or ABE pp. 252-3) quickly show. Indeed the addition of Aristotle to E has a very damaging (interpretational) effect. Theorem 3.17 (1). (MJA, for every wff A, i.e. in EA everything is possible, in E's sense of possible. Even worse, (2) b W A ~ ( B - C) for every B and C. (1) follows from a result established below as to the excessively classical character of E, but it can be proved directly as follows: [B -»• A -»-. B by B A.

B

A ] -+. B

A

A, prefixing the axiom (B

24 5

B) + A + A

OF E RETAIN PR1ZEV

3.6 VARIOUS MOVAL STRENGTHENINGS

B B B B -»• NB (B -+ -A) NB -+. (A ~B) (A - ~B) -> -A)

PROPERTIES

B -+ B -*• A], by Exported Syllogism A -»• A, by previous lines using Rule Syll ~A ~A + -NB

But ~((A HB) -»• ~A) is provable from Aristotle in relevant affixing systems, whence~NB. The proof (within system B of chapter 4) is as follows:B & A -» A, ~A

~(B & A), B

~A

B -»• ~(B & A) by Affixing;

a similar argument leads to B -*• ~(B & A) -*. B & A ~(B & A). Hence, B -*• ~A -*•. B & A •* ~(B & A), whence contraposing M!B & A ~(B & A) ] -*• ~(B •* ~A). And so, by Aristotle, ~(B -»• ~A), and also ~(B C) establishing (2). Thus, in particular, ~((A ->• ~B) •+~A), establishing (1). To sum up, EA is stronger than E, but satisfies relevance and necessity requirements. If strength, relevance and necessity criteria were what really determined choice of systems, if they were jointly sufficient and necessary for correctness, EA should be preferred to E. Yet EA is a seriously defective relevant system which is far from correct. Type 2. Normal modal strengthenings of E. E is a proper subsystem of modal system S3. System S3 is Hallden incomplete (see chapter 5), but may be Hallden completed in two quite different ways, by extension in an S4 direction or by extension in an S7 direction, i.e. by the addition at least of ~ OCIA (i.e. O O A X There are presumably myriads of principles, generated by systems between S3 and S4 and beyond S4, and also by systems beyond S7, which do not upset relevance or necessity requirements (arguments for not rejecting this presumption out of hand will emerge shortly): but certainly there are some. We consider two groups of these principles: Those that belong to normal modal systems, which will be counted as normal modal strengthening of E, and those that do not, which will be counted as non-normal modal strengthenings. Albert. (A B •+ B A) A A, whom we have already encountered, provides a normal modal strengthening of E, conforming to relevance and necessity requirements. Albert is not a theorem of E (see ABE, p.94). Type 3. Non-normal modal strengthenings of E. Relevant logics result by adding principles of non-normal modal logics and of contingency and conventionaloriented modal logics (on these principles see Montgomery and Routley 71). The initial system on which other systems build is E7FA (first entailmental approximation to an S7 version of E) obtained by adding to E the S7 principle ~NNA, i.e. MMA, where MA - D f ~N ~A and, as usual, NA = f (A A) A. Theorem 3.18

(1) E7FA is a strengthening of E; (2) E7FA has the same implicational theorems as E, and hence E7FA, like E, conforms to weak relevance and avoids Ackermann-Maksimova modal fallacies.

Proof. (1) E C E7FA by characterisation of E7FA, but ~NNA is not a theorem of E as the 8-valued matrices, among others, will show. More simply, E c R and ~NNA is not a theorem of R on pain of collapse, since R, NA •**• A, whence ~NNA ~A. (2) One half follows from inclusion of E in E7FA; E extension theorem below.

246$

the other from the

3.6 EXTRAMOVAL

ANV STRONG VISTRIBUTIONAL

STRENGTHENINGS

OF E

A similar theorem follows, proved in a similar way, for the first of the conventionalist strengthenings of E, namely C7FA obtained by adding to E, the principle that all necessity statements are contingent, i.e. CNA, where CA = - MA & M ~A. Conventionalist modal positions have and continue to have considerable appeal and are not so readily set aside as may at first be expected.1 But since such positions maintain that all entailments are contingent and in E, as in strict systems, all entailments are necessary and all necessary statements are entailments - a weaker entailmental base than E is required for a consistent position. Owing to the inclusion in E of the S4 thesis NA NNA (also a theorem of S3 with the E definiition of modality, which accordingly is not appropriate for S7), both NN(B -*• B) and ~NN(B -*• B) are theses of E7FA. Hence also, by a previous theorem, MA is a thesis of E7FA. The non-normal strengthening of E is accordingly E + MA, obtained, that is, by adding the nihilistic thesis of universal possibility to E. E + MA is, like E7FA and C7FA with both of which it coincides, simply inconsistent but nontrivial. Even if E + MA is reckoned incorrect because inconsistent, it cannot be ruled out of consideration if_ the combined selection criteria of relevance, necessity and strength apply: however, as will by now be evident, these combined criteria do not select uniquely anyway. Type 4. Extramodal strengthenings of E result by adding modal-style principles which collapse modal logics, but not relevant logics such as E. An important class of such extensions is obtained by adding to E generalisations or specialisations of the following principle A + B

C •+ D (Anderson)

Extensions of E by such principles have to be taken very seriously if the S5 road is not taken (as we shall argue in discussing the incompleteness of E in chapter 10). Consider the system E7A obtained by adding Anderson to E:Theorem 3.19

(1) E7A is a strengthening of E (2) E7A has the same implicational theorems as E, and hence E7A like E conforms to the weak relevance requirement and avoids Ackermann-Maksimova fallacies. Type 5. Strong distributional (or qualified factorisation) extensions of E and of R. These are the extensions of E and R by principles like (A & C -*• B) & (A •+. C v B) -*•. A -»• B (Strong Distribution of Cut), and, in particular, by interesting (if unreliable) principles which follow from Cut, such as the following: (A B -»• C) & (B -*•• A V C ) B -*• C (A -»• B) & ( B & C - f D ) •*•. A & C -»• D (A •» B v C) & (C •*• D) •+. A •+ B v D

(E-NR-separation, or Mints) (AC-replacement) (DC-replacement)

Distribution, A & (B v C) (A & B) v C, is equivalent in the context of remaining E principles (and indeed of principles of basic system B of chapter 4 and even of weaker systems still) to A & C -»• B, A

CvB-*

A -*• B

(Rule Distribution)

*See again Montgomery and Routley 71. What is not mentioned there is that an important stable of Cambridge philosophers held essentially such a position as to modalities, in particular Johnson, Wittgenstein and Wisdom; e.g. as late as 1929 Wittgenstein 'maintained that propositions of the form "P -J Q" are "about symbolism", and hence contingent' Lewy 76, p.73).

24 7

3.6 E + MINTS RELEVANTLY

EXTENDS E BUT IS INCLUDED

IN R AND NR

Proof of interderivability is in outline as follows (material to be developed in later chapters is presupposed here and in some of the subsequent proofs in this section): ad Rule Distribution. A

C V B -o- A -+ A & (C V B) -ft A (A & C) v B, by Distribution A & C B -fr (A & C) V B B Then use Rule Syllogism. ad Distribution. Let a = A & (B V C) and B = (A & B) V C. Then ( 1 ) a & B -»• B , i.e. A & (B V C) & B -»• (A & B) V C. For A &. (B V C) & B A & B and A & B -»• (A & B) V C. Also (2) a + M B , i.e. A & (B v C) (A & B) v C V B. For A & (B v C) B v C and B v C (A & B) v C v B. By (1), (2) and Rule Distribution, a -»• B, i.e. Distribution. It is Rule Distribution, rather than Distribution, that plays a crucial role in the completeness theorem for E and for other relevant affixing systems (see the extension l^mma 4.3). If strength is an objective it is hard to see why the normalisation of the rule, namely Strong Distribution, is not adopted as a thesis, especially as its inclusion leads to no violation of relevance and necessity requirements. The E-NR separation formula - discovered by Mints and used by Maksimova 73 to show that E is properly contained in system NR (of Meyer and others) and hence that E f1 NR (i.e. "entailment" is distinct from "necessary relevant implication") - follows at once from Cut using Importation, i.e. (A B C) A & B C. Mints, unlike Cut and DC-Replacement, is a theorem of system R: it be proved as follows in R:- Since in R, (A B C) B -+. A -»• C, (A •*. B -»• & (B ->. A v C) B -»•. (A v C) & A -*• C), using ^.-Composition. But, C & (A C) C and A & (A C) -*• C, so by strong v-Composition, (A v C) & (A •* C) ->•. C, whence Mints results by affixing moves. It is immediate since R is weakly rele vant that E + Mints is also. Theorem 3.20 of E.

E + Mints is a strengthening of E, i.e. Mints is not a theorem

That Mints is not a theorem of E may be shown either by the modelling techniques of the next chapter, or lj>uilding on matrix techniques as follows:Firstly Mints is not a theorem of E , the positive part of E, as is shown by the following+matrices (due to Chidgey; see ABE, p.352) which verify all theorems of E but falsify Mints (for assignments to A, B and C respectively of +0, -0 and -0):

-3 -2 -1 -0 *+0 *+l *+2 *+3

-3

-2

-1

-0

+0

+1

+2

+3

+3

+3 +2

+3 -3 +2

+3 +3 +2

+3 -3 -3 -3 +3

+3 -3 +3 -3 +3 +3

+3 +2 -3 -3 +3 -3 +2

+3 +3 +3 +3 +3 +3 +3 +3

-3 all subdiagonal places have value -3

31$

The matrices for connect ives & and v are the lattice-theoretic matrices for the structure M_ derived in §1.

OF STRONGER DISTRIBUTIONAL

3.6 INTERCONNECTIONS

PRINCIPLES

Now E is a conservative extension of E , as is shown in chapter 5 by semantical methods, or alternatively algebraically (i.e. by generalised matrix means) in Meyer 73 (reproduced in ABE, p.288). Hence Mints is not a theorem of E. AC-replacement and DC-replacement are interderivable principles within the framework of systmes such as E which contain Contraposition and Replacement (of coimplicants). Suppose, for instance, AC-replacement is given. Replace each wff component by its negation, and then contrapose: so results B -»• A & (D -»• ~( B & ~C)) D ~( A & ~C), i.e. using the interdef inability of & and v, DC-replacement (after relettering). In systems without Contraposition, e.g. relevant intuitionistic systems and.their extensions (considered in chapter 6), AC-replacement and DC-replacement are not interderivable, and may have significantly different roles. The principles grouped together are all qualified factorisation principles in that they are all readily proved in modal systems such as S2 or S3 using the principles of Factor (and its dual, the principle of Summation). Using Factor, A -»• B -*•• A & C + B & C , AC-replacement is immediate using Conjunctive Syllogism. (DC-replacement comes dually using Summation). Strong Distribution, and so Mints, are derived in S2 as follows: A

B v C •+. •+. A & B C (A B V C)

A •*• A & (B v C) by Factor A (A & B) V C by Distribution (A & B) v C C by Summation and (C v C) C & (A & B C) ->-. (A -*•. (A & B) V C) & ((A & B) V C C) by Composing A •+ C by Conjunctive Syllogism

Since Factor and Summation are derivable in relevant affixing systems such as E only in an identity qualified form, e.g. Factor in the form (A B) & (C •* C) -*. A & C -*• B & C, analogous arguments in relevant logics yield only identity qualified forms of principles, e.g. Strong Distribution is derivable in the form (A A) & (C •*• C) & (A B V C) & (A & Bv -»- C) A C. Strong Distribution is the strongest of the qualified factor principles, yielding AC- and DC-replacement principles as follows: (A & B D) & (A -»• B v D) -*•. A D (A & C -*. B V D) & ((A & C) & B -*• D) A + B-*. A i C B V D B & C ->• D (A & C) & B ->• D (A -*• B) & (B & C -*• D)

A & C -»• D

i.e. AC-replacement results.

rewriting Cut A & C -»• D reorganising the previous line by affixing and simplifying similarly combining and applying syllogism,

Similarly, rewriting Cut,

(A & C (AD and C A V -(B V D)) D. (B But C-^D-*. AB iVCD) + & B' V »• (B V C)-». AA -»• B CV V V D), whence DC-replacement results. Accordingly it follows that if the system ES, i.e. E + Strong Distribution, conforms to requirements of relevance and necessity, so likewise do the extensions of E by any other of the qualified factor principles. Thus we concentrate in what follows on the principle of Strong Distribution, some of whose deductive appeal has already been exhibited.

249$

3.6 STRONG DISTRIBUTIONAL

Theorem 3.21.

(1) (2)

EXTENSIONS

OF E ANV R AVOIV RELEVANCE

AND MOVAL

FALLACIES

ES is a strengthening of E. RS is a strengthening of R, where RS is R + Strong

Distribution. Proof. (1) follows from the previous theorem. Alternatively, Cut is not a theorem of E or of R by the 8-valued MQ matrices. Theorem 3.22. (Weak Relevance) variable.

A

(1)

if h E S

(2)

if|-RSA

B then A and B share a B then A and B share a

variable. Proof.

Since ES c RS it suffices to prove (2)

Consider the crystal matrices defined on the 6 element De Morgan lattice (which is an important extension of the 4 element lattice already considered): b = Nb

The matrix value v(A) of A is determined from the value of components by the rules: v(A & B) « v(A) fl v(B), v(A v B) - v(A) U v(B), v(~A) » Nv(A), and for v(A B) by the table:

Then: (3) Every theorem of RS takes only designated values, i.e. never takes value 0 under the crystal matrices.

*1 *f *a *b *t 0

1

f

a

1 1 1 1 1 1

t a b f 1

a 0 a 1

b

t

0

t 1

1

u b b 1

(Note: as the representation is designed to show, all super-diagonal elements are zero). Proof is by induction on the proof of theorems, for which validating every exiom scheme and showing that the rules preserve validity suffices (a task for which a computer is admirably suited1). ad (2) Suppose A and B share no variable. Assign value a to every variable of A and value b to every variable of B. Then, by induction, v(A) - a and v(B) » b, whence v(A -*• B) « 0. Hence by (3), A B is not a theorem of RS. Theorem 3.23.

(Avoidance of modal fallacies).

If Aj

wf f and A is a negative wf f then for no B and C is A. -*•. A_ B -»• C provable in ES. x

A^ are Maskimova ... A, •+. A

For testing all the results in this section that were verified by computer we are indebted to J. Boland.

250$

3.6 ES SHOULD HAVE BEEN PREFERRED

TO E: E IS SERIOUSLY

INCOMPLETE

Proof is, as before, using the Piano matrices. The only new detail lies in showing that Cut takes a designated value on the matrices, and this it does (as a computer quickly showed). Accordingly ES is a strengthening of E which preserves the main desired features of E. Similarly RS is a relevance preserving extension of R. On the motivation given for selecting E and R, ES and RS should have been preferred to the systems E and R actually chosen: so we shall argue. The argument can take off from Belnap's claim (already noted, from BT, p.15) that a satisfactory theory of entailment will isolate the disease of raw irrelevance, keeping everything of the standard theories that can possibly be kept . The claim is part of Belnap's case for his final requirement on a formal theory of entailment, namely C. 13 A theory of entailment ought to be as strong as is consistent with its fundamental aims. Since ES is stronger than E and keeps more of standard theories while not upsetting the fundamental aims of the enterprise, or for that matter Belnap's other requirements of adequacy, ES should be preferred, on the grounds adduced for selecting E, to E. There are no doubt, on Anderson-Belnap assumptions, maximally preferred systems. These systems should be infinite-valued: for finite-valued relevant systems fail to discriminate sufficiently many distinct proportions (cf. chapter 1). They should be infinite-valued relevant systems and to be suitably maximal they should presumably be such that all their (proper relevant) extensions have finite characteristic matrices. 1 ES is not a maximally preferred system in this sense, as it admits of extramodal relevant extension. Theorem 3.24. (A f B ) i t n n of ES.

(Piano theorem for ES). C

+

(1)

If f- £ S (Aj A Bj) & ... &

D then }-_.., C -*• D, where t is some conjunction of theorems bb

(2)

where ES' is an extension of ES by axiom schemes of the form A A B,

^ES

C

lff

f-ES

C

Proof is as before except that the metavaluation used is defined in terms of ES provability. There is just one new case in the argument for (*):ad Cut. Since Cut is provable in ES, it suffices to show that v((A & C B) & (A -». C V B)) C v(A •* B). If v((A & C -»• B) & (A •». C V B)) " 0 the inclusion holds, so suppose v((A & C B) (A •+. C v B)) • N. Then either v(A & C -*• B) - N and v(A C v B) * 0 or v(A - C v B) • N and v(A & C B) + 0. Since implicational wff only have value 0 or N, the cases are the same. Hence fA & C B, k A-». C V B , v(A & C) C V ( B ) and v(A) c V (C v B). Thus, by Cut, A •*B, and, as v(A) 0 v(C) c V ( B ) and v(A) c V (C) U v(B) by distributive Tattice properties, v(A) c v(B), v(A -»• B) « N, metavalidating Cut. A corollary is that as a theory of formal entailment, E is seriously incomplete. 1

In view of the relevant restriction the desired property differs from the Scroggs (or pretabular) property (defined in ABE, p.447). 34$

3.6 A BUNCH OF RELEVANT

MJNGL1SH

EXTENSIONS

OF R

Type 6. Minglish extensions of R and E. Relevant extensions of R contained in the irrelevant Mingle system RM, i.e. R + A -»•. A A, are obtained by adding to R one or more of the following schemes: Ml. M2. M3. M4. M5. M6. M7. M8.

A

A

A ~A A — . ~A A - ( ~-A A V ~(A ~A & B A & B -«-. A & ~A &

~A

A

:

equivalently A or A

B •*. A v B B A V (A A) V (A & - A B) V ~(A A (~A *• A) V (A (~A ->• A) V (A B B * A

~A

A - A A ~A

B) B) -B) f B) B)

Such principles1 are turned up in the investigation of finite-valued relevant logic (see chapter 12). Though all these principles satisfy weak relevance when adjoined to R, none are correct. Counterexamples are readily designed, which show that relevance is not conserved under application. Consider, for example, the outrageous M8 which leads by detachment to the irrelevance: the Russell class is self-membered confirmed.

the theory of continental drift is

As to weak relevance there is the following result: Theorem 3.25. (Weak relevance). weak relevance.

Each of M1-M8 may be adjoined to R preserving

Proof uses the crystal matrices (CM) and the M_ matrices (BM), both of which serve to confirm weak relevance. The table which follows shows the Validity (/) or invalidity of the indicated formulae in the matrix models CM and BM, and also in model RM84. This can be demonstrated by computer means (e.g. using the program TESTER of ABE, pp.86-7) or directly. Table RTB

j^^Matrix Ml M2 M3 M4 M5 M6 M7 M8

CM

BM

• / /

/ / / / •

X

/ / /

X

X

/ /

RM84 X X X X X X X X

Since each wff is satisfied by at least one of CM and BM, weak relevance may be proved in the usual way. Corollary. The systems R + Ml + M2 + M3 + M5 + M6 + M7 and R + Ml + M2 + M3 + M4 + M5 + M7 + M8 are weakly relevant. The matrices CM and BM likewise show that various wff are not theorems of other extensions, e.g. M8 is not a theorem of R + M2, nor is strong distribution. 'These principles were largely unearthed by Brady, who also established several of the results concerning them.

252$

3.6 THE MJNGLJSH

LIE PROPERLY

EXTENSIONS

BETWEEN

R AND RM

The outcome of testing with a matrix (found by Meyer) for system R is also recorded in the table, because this enables the proof of further results concerning Minglish extensions of R. RM84 is the following 8-valued array (with positive values designated):&

3

2

1

-0

+0

+1

+2

+3

-3 -2-1 -0 *+0 *+l *+2 *+3

3 3 3 3 •3 3 •3 •3

3 2 3 •2 •3 3 •2 •2

3 3 1 1 3 1 3 1

-3 -2 -1 -0 -3 -1 -2 -0

-3 -3 -3 -3 +0 +0 +0 +0

-3 -3 -1 -1 +0 +1 +0 +1

-3 -2 -3 -2 +0 +0 +2 +2

-3 -2 -1 -0 +0 +1 +2 +3

V

-3

-2

-1

-0

+0

+1

+2

+3

-3 -2 -1 —-0 +0 +1 +2 +3

-3 -2 -1 -0 +0 +1 +2 +3

-2 -2 -0 -0 +2 +3 +2 +3

-1 -0 -1 -0 +1 +1 +3 +3

-0 -0 -0 -0 +3 +3 +3 +3

+0 +2 +1 +3 +0 +1 +2 +3

+1 +3 +1 +3 +1 +1 +3 +3

+2 +2 +3 +3 +2 +3 +2 +3

+3 +3 +3 +3 +3 +3 +3 +3

Theorem 3.26. (R proper inclusion). includes system R.

-3 -2 -1 -0 +0 +1 +2 +3

+3 +2 +1 +0 -0 -1 -2 -3

-3

-2

-1

-0

+0

+1

+2

+3

+3 -3 -3 -3 -3 -3 -3 -3

+3 +0 -3 -3 -2 -3 -3 -3

+3 -3 +0 -3 -1 -3 -3 -3

+3 +2 +1 +0 -0 -1 -2 -3

+3 -3 -3 -3 +0 -3 -3 -3

+3 -3 +0 -3 +1 +0 -3 -3

+3 +3 +0 +3 -3 +3 -3 +3 +2 +3 -3 +3 +0 +3 -3 +3

Each extension of R, by MI-M8, properly

Proof is from the rejections by RM84, which verifies all theorems of R. Theorem 3.27. (RM proper containment). ly contained in RM.

Each extension of R by M1-M8 is proper-

Proof. In virtue of the weak relevance theorem which does not hold for RM, it is enough to show that each of the additonal principles is a theorem of RM. This can be done by a variety of means, e.g. decision procedures or tableaux methods for RM, but we choose direct syntactical proof relying on well-known theorems of RM (for which see ABE).

I"rma&

A.

Since

ad M2.

ad M3.

The following sequence of wff furnishes an RM derivation of M2: A •+. A -»• A, ~A ~A ~A, ~A -*•. A -*• A, A -»•. ~A -*• A, A •*. ~A & B A -*•• A i B A V B. A -A & B A and hence j- ^ A Since » 1-RM A A

ad M4.

~A & B Since \-m A

ad M5. ad M6. ad M7. ad M8.

h ^ A

A

ad Ml.

>

•"RM

A V (A + B). ~A -»• A,

(~A -*• A) V (A & ~A -*• B).

A •* B) is a theorem of RM. ~(A B) V (~A ~A & B •+. A B, M5 follows by Addition. Since RM vA A, („., A & B ~A -> A and hence Since I"RM RM l-«A A A) V (A • B). A & ~A & B A B. 'A & B -*•. A Since |B, jRM RM

253$

A & B

A,

3.6 VIOLATIONS

Of RULE y; AMD ALTERNATIVE

DIRECTIONS

OF RELEVANCE

The svstem RK. i.e. R + M6, has interesting properties with respect to the rule y (as Brady discovered): Theorem 3.28.

Proof. — —

(1)

in common.

(1) (2) (3)

y is not an admissible rule or RK; The systems RK + y and RM + y coincide. Hence RK + y is tantamount to RM, since y is admissible in RM (see ABE, p.415).

Let |-DV. ~A and |- v B, for some A and B which have no variables RK RK Since f- ^

V

~A & B ->• (~A Further

A) v (A since

B),

(~A •* A) v (A •*• B) and

A

^RK — » l~ RK ~ "*• ""A), | - R K ~(~A A). Thus if Y were admissible in RK, (A -»• B. Hence as A and B have no variables RK in common, weak relevance would fail for RK. Therefore, as RK is relevant, y is not admissible in RK. (2) It suffices to prove the characteristic RM axiom, A A -*• A, in RK + Y- Given that the rule ~A, B -© A -»• B is derivable in RK + y , the following sequence (from SE I, p.331) establishes the axiom: ~(A B) (A -> A), A - (~(A -*• A) •* A), A - (-A •». A A), A-^. A - A,

A) , A •» (A ->•. -A

But the rule is derivable in RK + y , as the argument of (1) shows. Violation of y is also induced by potentially inconsistent extensions of R, not contained in RM. Consider M9.

(A & ~A

Theorem 3.29.

B)

V

(A & ~A).

R + M9 and RS + M9 properly extend R but are weakly relevant.

Proof is by CM and RM. The classification of extensions of R and E given is not fully exclusive, nor, worse, is it by any means exhaustive.1 It is a classification for the present only, until the state of information as to classes of relevant extensions improves- "What the structure of relevant extensions is, what pretabular relevant systems there are, and so on - all these are rather wide open questions. Equally damaging to criteria of strength are alternative directions of relevance. Good examples are provided by the extensions of system RMf, the positive part of RM, which is weakly relevant, by intuitionistic or minimal negation, but other examples are afforded by alternative connectional logics such as connexive and relatedness logics. Let MRM, minimal RM be the extension of RM+ by constant A with no conditions imposed on A. Theorem 3.30.

1

(1) MRM is weakly relevant (2) MRM only properly overlaps R.

Still to be investigated are, for instance, extensions of R and RS b^ chains of wff» e.g. "Dugundjii formulae", or one or both of wff of the form A A and A n

M™, where A k + 1 - A o A k , with A° = A and B o C * ~(A

254$

~C).

3.7 THE PROBLEMS OF FORMULATING, AND ESTABLISHING, ADEQUATE

RECOVERY

PRINCIPLES

A similar result holds for MEM. Indeed these systems differ from R and E at the first degree.1 A similar theorem also holds for MRM3, since RM3+ is relevant.

§7. Other - sometimes questionable or vacuous - criteria for the choice of system. Although sheer strength is undesirable, something right does emerge from the principle of strength; namely, that it would be desirable to have a reasonably complete cataloguing of correct entailment principles, in particular at the sentential level. Completeness in this sense - which was what Kant and some of his logical predecessors sought - differs rather sharply from completeness in the frequently undiscriminating sense delivered by completeness proofs. There is, however, a lack of precision about the resulting preliminary condition. sl p . An adequate theory of entailment should contain a reasonably complete set of correct entailment principles, and about those it leads to. But this is not entirely reprehensible. The further conditions sl p suggests are those concerning the adequacy of the theory of entailment for investigations in the deductive sciences, especially mathematics. It should not be simply assumed that all of the mathematics, or even all of classical mathematics, is automatically correct. (It may be correct in a conventionalist sense, i.e. according to its own lights, but this is not enough for correctness proper: see Routley 77). Thus the general recovery principle has to be more carefully formulated than it would were the objective simply the derivability of all of, say, classical mathematics. sl . General recovery. An adequate theory of entailment should furnish deductive principles which enable the recovery of correct theses of each deductive science given assumptions of that science. This requirement is also underdetermined and unclear in several respects. Without appropriate clarification it admits of circular satisfaction. Suppose, to illustrate, that the theory of entailment is elevated to determiner of correctness. Or differently, suppose that every thesis that is considered correct is added as a further axiom. Explication leads to competing notions of recovery. But requisite nontrivial senses are exceeding difficult to satisfy (and trying to show that they are satisfied is a colossal undertaking). Even much-vaunted classical logic appears to fail, given only the usual elaboration, since parts of category theory, to take just one example, apparently lie beyond its formal reach, unless it is re-equipped with a different superstructure, which would presumably place other areas of mathematics out of reach. The reason is that mathematics is inconsistent (but perhaps, if equipped with suitable nonclassical foundations, nontrivial). Any consistent cut-down or partition of the inconsistent totality is inevitably incomplete. It is doubtful in fact that any (consistent) classically-based system can be adequate, for well-known limitative reasons (cf. 1.7). While such limitations do not hamper significant nonclassical projects (see EMJB, chapter 11 and Appendix), even beginning to follow out such vast projects would quickly lead beyond the confines of the sentential logics. Even within the confines of sentential logics there are however important recovery, or inclusion, requirements:

1

The question arises whether, since RMA is pretabular (see ABE §29.4), MRM is a maximal relevant logic, in some sense.

255$

3.7 MINIMAL ADEQUACY:

A SYSTEM SHOULD BE JUST AS STRONG AS IT NEEDS TO BE

SI(i). Classical theses recovery. The theses of classical logics should be recoverable; more precisely, all tautologies of S should be theorems of an adequate logic of entailment. This requirement is not a taxing one, but it imposes a further lower bound since it is not satisfied by relevant logics, such as the basic affixing system B, which lack Excluded Middle A V ~A, or Noncontradiction, ~(A & ~A)• It is satisfied by all extensions of FD, of A, and of the minimal such system G of chapter 4 (and thereby hy all systems of eventual interest as analyses of entailment). Theses recovery does not of course guarantee the availability of classical logic, else the general recovery problem would be less vexed. For the rule y of Material Detachment, even when proved admissible for a system, may not hold for extensions of it. A somewhat more demanding recovery principle, already introduced in §5 (though its formulation still lacks the sharpness given to it in Meyer 73) is the requirement sl(ii). Enthymematic recovery. Classical intuitionistic and strict systems should be recoverable enthymematically (i.e. requirement VII above). Mathematical concentration on classical and intuitionistic logics can then be explained in terms of the combination of an implicit consistency assumption (so guaranteeing y) with a marked fashion for short-cuts to results, so that enthymematic implications are strongly favoured. Recovery requirements reveal that some systemic strength is required: otherwise, of course, little could be proved. And proof of recovery properties is useful initial evidence that a system has not become excessively weak. But what is increasingly clear is that great strength at the sentential level is not required, and can be positively very damaging. A system should be strong enough, but no stronger, just as strong as it needs to be (to adapt Belnap). Aristotle's doctrine of the mean applies to choice of the right logic. This means however that the criteria of strength is really replaced by a criterion of weakness, that a minimally adequate logic is sought.1 For not only is minimization, not maximization, a key determinant, but strength drops out in favour of more carefully specified adequacy. For how strong a logic should be is governed not by strength at all, but by other considerations, such as the derivation of this or that, the recovery of this or that, etc. A major reason for minimization is that each strengthening at the sentential level serves to exclude or distort non-sentential applications. None of the other selection criteria often proposed have the selection strength of strength. Generally these criteria can be satisfied, in combination, by a range of systems. But one criterion sometimes proposed that does force undue systematic strength is s2. Normality. The requirement of rule normality is this: where the rule A,, ..., A B is a primitive (or object) rule, then the inference from i n Aj, ..., A q to B is valid, i.e. the conjunction of the premisses entails the conclusion (cf. principle C.10 of Belnap, BT p. 12). Rule normality is one of several criteria for theory choice which appear only in a quite residual form

1

Compare such double optimisation principles from elsewhere as minimax. ually choice of logical system, too, is a matter of optimisation.

256$

Event-

3.7 ASSUMPTIONS BEHIND THE MISTAKEN NORMALISATION

REQUIREMENT

in Entailment1, these having been down-graded from the scene-setting role they play in Belnap*s thesis BT - though they remain of much importance in the matter of choice of a, or the, logic of entailment. There are several rules in common use that violate rule normality, e.g. Substitution (on sentential parameters), Universal Generalisation (UG), Necessitation. To avoid such obvious counterexamples, Belnap distinguishes meta-rules from object rules, accounts the counterexamples meta-rules, and proposes that primitive rules should be restricted to object rules (p.12). The distinction is none too solid, since all rules involve apparatus of the metalanguage in their formulation, and appears to depend damagingly on how a system is formulated (e.g. Necessitation is a meta-rule of S4 under Lewis's original formulation, but an object-rule of later reformulations). However the distinction can be bypassed: it is enough to impose normality on primitive rules. But since this can cause considerable trouble (e.g. consider the lengths Anderson and Belnap are obliged to go into rendering formulations of EQ "normal", i.e. in artificially eliminating UG), the question arises: Why bother? Belnap gives no reasons (in BT) and perhaps given the apparent appeal of the requirement it is unnecessary to do so. However, the appeal of the rule normality requirement may just reflect an inadequacy of symbolic apparatus, and a mistaken equation of systemic provability with truth (at least at the sentential stage), so that what should be represented symbolically as on the left of the table that follows gets whittled down to what appears on the right: Normalising table Symbolic form

Rule Prefixing

B-»C-*A-*B+. A

Suffixing

C

A + B -fc- B -*• C

LB - c A •+ B •+. A •»• C A -

A - C

LB

•*• C

A •+ C LA

Bh

C

A + B+. A + C

(PESyll) B

B

C

A -*• C

(SESyll) A & (A -»• B) •* B (Ass)

A, A -*• B -O B LA

Replacement

B

A

B

A & Modus Ponens 2

Proposed systemic form

A • B, D(A)

D(B)

• B & f- L D(B)

f-

L

A ~ B & D(A) -»• D(B)

D(A)

An adequate formulation of each rule requires at least the availability of a

*It is however indicated as a desideratum, e.g. ABE, p.236. 2 Normalisation of multiple premiss rules is not, strictly, uniquely determined. For example, it can be argued that Modus Ponens should be normalised using implication or, equivalently in stronger systems, fusion o, to yield Identity, A •» B -*•. A -*• B or equivalently in stronger systems (A B) o A -*• B - in contrast to Adjunction which requires extensional conjunction for its normalisation. However, even a weakened normalisation requirement, to the effect that every rule has some systemic normalisation, is in difficulties with rules like Prefixing and Replacement; and the question of the justification of such a requirement remains.

257$

3.7 DIFFICULTIES

IN EXPLAINING

WHAT THE IMPORTANT

REQUIREMENT

OF COHERENCE

REQUIRES

systemic provability functor, |- , for appropriate L (and perhaps also further implication relations since the systemic connective may not provide the required relation). But now L may be quite restrictive, e.g. within sentential logic, and far short of all of deductive science, with the result that the rule says far less than its proposed systemic correlate, e.g. Prefixing is far less committal than PESyll, especially when it comes to applications. In any case, the wholesale deletion of occurrences of • }— " in moving from the symbolic form to its proposed systemic correlate is without justification, and, given modern knowledge of the divergence of truth from provability, appears mistaken. That the requirement of rule normality is mistaken is one of the corollaries of subsequent sections, where it is argued that, in contrast to the rules, all the systemic correlates in the normalising table are not only dubious but false. s3. Coherence. As Belnap says 'The ideal of coherence is not that of formal consistency. What is required is that the parts of the theory "hang together", and in particular that the theorems cohere with the object-rules' (BT, p.13). So far, so good. Coherence is not consistency, but is some sort of hangingtogether of a theory. But what sort precisely, that is the problem. Rather surprisingly, Belnap tries at once to reduce the requirement to some kind of deduction theorem. Specifically, he requires that DE.

|-Aj & ... & A

-»• B

iff Aj

A

B n

(Deduction Equivalence).

Whether such an equivalence holds depends critically upon how the right-hand side is defined. But defined in a straightforward way (as is done in 10.I1 and in ABE, pp.277-8), DE holds for a very large class of relevant, and also irrelevant, logics. In short, the requirement offers little discriminatory power, indeed none that has not already been independently motivated. Belnap tries to inject some selection power by combining DE with normality (i.e. s2, to give system normality); but the resulting power derives from the defective requirement s2. Not only is DE on its own, although desirable, rather discriminating; it seems to have little to do with coherence. A major attempt to explicate the elusive notion of coherence, due to Meyer, is through metavaluations. Roughly, a logic is coherent in this sense, M-coherent, if it can be interpreted in its own metalogic (though extensional connectives and its provability functor).2 To be more precise about the type of

*DE requires in addition to what is provided in 10.1 only the equivalence A , ..., A f- B iff A, &... & A |-B, which Simplification and Adjunction furnish. Thus a corollary is that a theorem of this type is in doubt for systems like connexive and holistic logics which lack a normal conjunction. 2

0n the claims of M-coherence to be a genuine coherence notion, and on its interrelations with truth and other semantic notions, see especially Meyer 72. Meyer and Wolf 76 also adduce some considerations in favour of M-coherence, none of them tight, the best of them being this: 'The requirement that an.acceptable entailment logic be coherent makes precise the notion that it can act as its own metalogic and certify as true the entailments that follow from its axioms'. Even this is doubtful, given the divergence of truth and provability. On the coherence of particular systems, see ABE, p.264ff. and references cited therein.

41$

3.7 METAVALUATIONS,

METACORRECTNESS,

ANV THE REQUIREMENT

OF M-COHERENCE

interpretation in the metalogic envisaged, we define metavaluations. A function v is a metavaluation of a logic L with connective set {&, v, -+} iff it is a two-valued valuation (i.e. it is from wff to L to values 1 and 0) such that, for every wff A and B: v(A & B) = 1 iff v(A) = 1 = v(B); v(A V B) = 1 iff v(A) =1 or v(B) = 1; v(~A) = 1 iff v(A) 4 1; and v(A B) = 1 iff f- A B (i.e. A -* B is provable in L) [and either v(A) ^ 1 or v(B) = 1] . Thus extensional connectives are interpreted, as usual, in terms of their metalinguistic reflections while intensional connectives are interpreted through provability of the corresponding systemic connective.1 The extensional clauses straightforwardly demand correctness of interpretation, while the intensional clause connects entailment with the provability of an implication. The square bracketed clause, initially inserted to validate Modus Ponens, characteristically proves dispensible (see e.g., details of the M-coherence of system E, ABE p.270). A wff C is metacorrect (in L) iff it is true on al] metavaluations i.e. v(A) = 1 for every metavaluation v. ' A logic L is M-coherent iff all its theorems are metacorrect. We advisedly use the term "metacorrect' rather than 'metavalid', since we have already seen, as with FD which was proved metacorrect, that analogues of completeness fail. The one-way connection helps get around the objection that the metavaluational method depends on an implicit equation of truth, or necessity, with provability - equations which seem to underlie its appeal and provide part of its motivating force. As a selection criterion for entailment systems, M-coherence is not that much more discriminating than DE: a great many extensions of FD are M-coherent (as we have begun to see).2 And applied beyond entailmental systems M-coherence is in serious doubt. In particular, stronger modal systems such as system S5 - subsequently argued to be correct for logical necessity - and the Brouwersche system are ruled out as M-incoherent, as are stronger relevant systems exhibiting Ackermann-fallacies such as R. s4 and s5. Naturalness and plausibility are often presented as considerations in favour of a system, a set of principles, or a single scheme. Anderson and Belnap, for instance, appeal to both in various places (e.g. ABE, p.6). And one of the points commonly made in favour of R, and as to why it is interesting, is that it consists of a natural set of principles. In this sense of 'natural', where natural is largely severed from its linkage with nature,3 what is natural differs little from what is intuitively Thus, e.g., for necessity, •, the valuation clause is: v (•A) = 1 iff |- L OA [and v(A) = 1], Many other connectives and constants are readily absorbed in the framework. Note that two-valuedness is inessential: we have already supplied many-valued metavaluations in 3.6. 2

Wolf has proposed strengthening M-coherence to a rationality requirement. This would have advantages from the point of view of effecting a synthesis of criteria since, e.g., a rational sentential logic has no finite matrix (Routley and Wolf 74). Moreover the requirements for rationality beyond M-coherence are fairly uncontroversial. They are that Modus Ponens and Uniform Substitution be rules and Identity, A -»• A, an axiom, and that the logic conservatively extend its {-•, v} fragment.

3

Unless something like Plato's theory of forms is invoked, and logical principles of implication are taken to be among the entities of the universe, part of nature. But such theories are mistaken: see EMJB.

259$

3.7 REFINING

UNVISCRIMINATING

NATURALNESS

ANV PLAUSIBILITY

CRITERIA

acceptable or expected (see Concise OED, senses 1 and 10). Thus unqualified appeal to what is natural is open to precisely the same sort of criticism as was made of unhedged appeal to intuition (in 2.11). For example, thoroughly unsatisfactory principles such as paradoxes of implication, e.g. A B •*• A, especially in the forms A o B -»• A and A, B |- A, to which classical and intuitionistic logicians have become thoroughly accustomed, are often presented as natural principles, and also as plausible. Naturalness and plausibility, so employed, may reflect little more than custom, familiarity, the logical consensus, or the operative paradigm. Unless considerably refined, they are rather useless, and very undiscriminating, as criteria for choice of principles and system. For instance, if a system is natural and plausible, so are many of its subsystems. But, as with coherence, so with naturalness, refinement and explication are perfectly possible,1 and Belnap manages to spin several requirements out of "natural definition", in particular simplicity, separation properties, and adoption of entailment as a primitive. The last, at least, can hardly be obtained: for incompatibility, or inconsistency, has been recognised for more than 2000 years as just as good and natural a starting point in the quest for logical implication. And some magic is also required to obtain the other spinoffs. A start can, however, be made on rendering the requirement of naturalness more precise as follows (cf. Belnap*s C12, BT p.14): A theory of entailment should be formalisable in natural ways. For it is somewhat easier to be specific about, and reach agreement about, natural ways in logical systems. Natural ways are ways that avoid: i) excessive complexity in the formulation of axioms and rules. Good examples of such excessive complexity at the sentential level are afforded by axiomatisations of (first order) incomplete modal logics, or the case of relevant logics by systems, such as Urquhart systems (studied below), taking the Fine rule as primitive (for the systems and the rule, see (6.4).2 A more borderline example is the axiom scheme E7, distributing necessity over conjunction, of system E. Avoidance of excessive complexity only implies comparative simplicity, it does not involve or provide ground for a requirement of simplicity. Moreover, the requirement serves mainly as a backstop, since the systems it would rule out tend to be already excluded on other more direct grounds. ii) excessive circuitousness in derivations. Examples are afforded by devious proofs in sentential settings, for example the proof of ((A -»• B) •+ B) B in modal system S2, and by highly roundabout validations of theorems in semantical analyses, for instance under semantics for Urquhart systems. Other examples come from cases where derivations of a principle involve apparently remote connectives not occurring in the principle itself. On the other hand, to suggest that 'in constructing proofs one should be able to see where one is going and where one has been' (as Belnap does, BT p.14), is to expect too much. With syntactical proofs, even in the archetypically simplistic system S, one often 1

Similarly, plausibility can be refined and a logic of plausibility worked out. Let [I]P symbolise 'It it [initially] plausible that'. Then such principles as A •*• B -fc PA PB appear to hold, whence demonstrably P(A & B) -*• PA, P — A PA, etc. But inference from A •> B to IPA -»• IPB fails, because initially plausible premisses may lead deductively (or in initially plausible ways) to initially implausible conclusions, as in Lewis paradox arguments. For such a duly refined notion of plausibility, it should be required that the postulates of an adequate logic, for instance of entailment, are plausible.

2

The positive Urquhart system corresponding to system R, namely UR+, is investigated semantically in Charlwood 81.

260$

3.7 SPELLING

OUT REF0RMULAB1LITV

REQUIREMENTS

AND THEIR

POINT

cannot see just where on is going or how to proceed. (In a formal system one can always, in a sense, see where one has been.) The requirement of a certain transparency in proofs is not satisfactorily imposed on sentential systems in Hilbert formulation, and were it, E would likely fail; rather what it indicates is that systems should have reformulations where transparency requirements are met. In this way, "natural ways" lead to a further set of requirements:s6. Reformulability. A system of implication should be reformulable or analysable in ways which render proofs relatively transparent, which furnish meanings for expressions, especially connectives, and which reveal mathematical structure. That is, a system should have a. Gentzen or tableaux reformulations or the like, which do render proofs more transparent, and separate out connectives; b.

Semantical analyses;

c.

Algebraic analyses.

and

Showing that these requirements are met for a wide range of relevant logics will occupy a substantial part of the remainder of the text. But the results that these investigations have led to, that so many relevant systems do satisfy in one way or another, all of a, b and c, and variations thereupon, also mean that such reformulability requirements, though very important, especially for acquiring information about relevant systems, do little to discriminate between systems, ruling out only grosser classes of cases, that could anyway be excluded on other grounds.

The requirements of adequacy so far considered are very far from selecting uniquely a logic of entailment (or implication or conditionality): they do not even select a narrow band of systems, far less (what Belnap BT and Kielkopf 77 have claimed) system E. How then are such logical systems chosen? The answer, argued for in CLF, is through a constrained optimisation modelling, by maximising, subject to constraints, a weighted function of factors concerned in the choice. In the choice of logical systems, such as those for entailment and for conditionality, constraints are most important, for they considerably narrow the class of systems remaining in the running, eliminate all the vulgar irrelevant systems, and take us far toward answering the main underlying questions in this book. The fundamental constraint is correctness, or conformity to the facts 1 , which means both (C(- ) inclusion of what is (evidently) true and (C rejection of what is (obviously) false. The constraint accounts at once for several of the adequacy requirements Belnap imposes 2 , and underlies some of those adopted in chapter 1, for instance the no-finite-characteristic-matrix 2

2

As it is described in CLF, where this constaint is distinguished from the factor of accountability to the data, p.82ff.

Also some Belnap does not introduce, but should, in symmetry, impose, e.g. rules of acceptance to match the rules of rejection recorded in C8 (e.g. if (- A •*• B and -|B then -|A), with MP, adj and USubst being the requisite, correct, rules. The remaining adequacy requirements Belnap presents, not already taken up, can all be combined under a requirement of paradox-freedom (and weak relevance).

261$

3.7 ASSESSMENT

OF FACTORS PROPOSED

FOR THE CHOICE OF IMPLICATIONAL

SYSTEMS

requirements, for the reason that there are evidently, and demonstrably, not only finitely many propositions. While incorrectness of some thesis is eventually sufficient for rejection of a system, and correctness, where it can be obtained, is vital, often incorrectness or correctness are not sufficiently evident or really known. Then various of the further criteria for choice of sentential implicational systems tabulated below come into play and become important. Leading factors in the choice of sentential systems roughly classified: Desirable

Widely satisfiable, or indifferent, but nonetheless advantageous when appropriately refined

Undesirable or dubious, unless much refined.

Classical theses recovery

Plausibility

Strength

Naturalness

Normality

General coherence

M-coherence

World semantics

Simplicity

Enthymematic recovery Admissibility of y Relevance and paradoxfreedom

Tableaux methods

Paraconsistency

Gentzen formulation

Hallden reasonableness

Algebraic analysis

No finite matrix Decidability Deduction equivalence ADDITIONS FOR ENTAILMENT

Definition of necessity in terms of entailment

Avoidance of fallacies, e.g. Ackermann fallacies, modal fallacies Analysis as logically necessary implication

It should be noted 1. The listing of factors does not pretend to be exhaustive; rather it is intended to be decidedly indicative. Moreover there is some intermixing of constraints with factors. 2. Many of the "widely or always satisfiable" criteria can be so tightened as to become selectively effective, e.g. by tightening worlds semantics to 'first order world semantics'. But with such tightening the criteria mostly become, at

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3.8 E AND R ARE NOT CORRECT SYSTEMS

REACHED BY INCORRECT

PROCEDURES

the same time, controversial. 3. A great many relevant logics satisfy the criteria cited in the "desirable" column, as we shall see, especially in chapter 5 where criteria such as Hallden reasonableness are further spelled out and demonstrated for main relevant logics. (The main exception to date concerns decidability: see further chapters 5 and 11). So it is perhaps as well that the factors assembled are largely secondary. The primary criteria restraining choice remain those falling under correctness, such as reflection of valid argument and avoidance of suppression and fallacies and immunity to intuitive counterexamples, that emerge directly from the main notions such as implication and deducibility, under consideration. The general question of the choice of systems for entailment, for implication of various sorts, including conditionality, and propositional identity, will be resumed in chapter 15, where too some tentative selections of systems will be indicated. But in order to reach such a choice situation we really need, first, much more information about the main systems concerned, their properties, and their reformulations, and about distinctive logical principles they include and applications of these principles. §8. Counterexamples and counterarguments to excessive principles of T, E and E; and the case against E and R completed. Whatever the motives and procedures for actually arriving at a choice of system, it may still be the case that the system arrived at is correct, or the best choice. A satisfactory system may be reached in unsatisfactory ways or forutitously; and occasionally worthwhile results are achieved without a proper detailed investigation of the options. Thus it does not follow from the fact that the arguments motivating E are unsatisfactory or fail and that the adequacy criteria proposed do not (uniquely) select1 E, that E is not entailment, or is incorrect. The long route to this point has not however been entirely nugatory: there has been some small progress. For if we combine the fact that the arguments we have examined in favour of E and R are unsatisfactory with the prima facie considerations we presented earlier against the stronger relevant systems as correct analyses of entailment and implication, then we have a strong initial presumption against the correctness of these systems. The fact that E, for example, is among stronger relevant systems leads to the presumption that it contains (relevantly restricted) suppression principles, and accordingly does not have the right properties for explicating such notions as logical sufficiency which, so we have argued, are essential for a correct analysis of entailment. A more detailed examination of the characteristics and principles of systems E and R confirms this presumption. Both systems contain many incorrect and defective principles and qualified suppression principles. The conclusion will

x

0r even, as in the case of paraconsistency requirements, militate against systems as strong as E. A major motivating point of which much has been made, especially as regards the rejection of Ex Falso Quodlibet and Disjunctive Syllogism, is that entailment and implication systems are, and must be, capable of accommodating reasoning concerning inconsistent theories and deductive situations. Yet R, and in a lesser way E, trivialises important classes of inconsistent theories, and all the main relevant logics of ABE fail for paradoxical situations much as their more classical irrelevant rivals do: see §6 and §9. 263$

3.8 AN ARGUMENT

THAT'NOT

BOTH E ANV R CAN BE CORRECT

accordingly be: not merely has a good case for E and R not been made out, but there are decisive positive reasons for believing that E is not the correct analysis of entailment at the sentential level, nor R the corresponding analysis of implication. Since entailment is logical implication (see especially the argument of 10.1) but E differs from NR (see also 10), it cannot both be the case both (1) that E represents entailment, (2) that R represents the logic of relevant implication and 1 (3) that necessity meets the requirements of system NR of R with necessity. But the logic of necessity of NR is (so we shall argue in chapter 10) correct, if incomplete; so (3) cannot be faulted, either (1) or (2) has to give. In brief, because E diverges from NR, either E is not the logic of entailment or R is not the logic of implication or (as is the case) both. The choice between R and E turns on the correctness or otherwise of EAssertion A A -*• B -»• B (and its R-equivalents such as Commutation). Moreover correctness implies at least logical necessity; that is, EAssertion is not correct unless it is logically necessary. This closes the one escape that might be attempted under clause (3), namely an argument to the effect that though EAss is true it is not necessarily true. A pure logic of implication, as distinct from an applied one, should contain only theses that are necessary. Since system R contains (as we have already seen in chapter 1) Ackermann and modal fallacies, R fails as a systematisation of entailment. Ackermann fallacies notwithstanding, however, R may still be satisfactory as a systematisation of non-logical implication and/or of conditionality. It is not; for the distinctive principles of R, which separate it from E, namely Commutation, A (B C) -»-. B -»-. A -»• C, and E(xported) Assertion are both false under nonenthymematic interpretations of implication.2 Let us consider Commutation; many of the same points hold for Assertion. Once the folly of Exportation has been seen (something we shall return to when we examine Exported Syllogism), the main justification for Commutation, that it simply represents reversibility of oremisses, is undermined. For without Exportation nested implications can no longer be represented as commutable conjunctions; and only conjunctions satisfactorily represent the traditional concept of interacting antecedents of equal status. Once a clear distinction is made between nested implications and conjoined antecedents, it becomes plausible to say that a statement A may imply that some other statement B is sufficient for C, without A itself being sufficient for C. But given only that B is true, such a case yields a generalised counterexample to Commutation, for the antecedent A •*• (B -»• C) is true (by hypothesis), while the consequent of the consequent, A -»• C, is false, and hence the whole consequent B - (A -»• C) is false if B is true. Instantiation arguments provide an obvious class of propositions satisfying this condition. If for example, we take 'a' as 'Arthur', 'f' as 'is a man' (or a species predicate) and 'g' as 'is an animal' (or as a genus predicate), then it is true that (x)(fx •*• gx) •*•. fa -*• ga, by instantiation. But what we obtain by commuting, fa (x) (fx ga) -*• ga, is not true, for it is not true that

'The conjunction in question can be intensional. Disjunctive Syllogism. 2

The argument need not apply

What follows in this section is substantially based on or influenced by V. Routley 67.

3.8 COUNTEREXAMPLES

TO COMMUTATION

AND TO FAVOURED

RESTRICTIONS

THEREOF

(x) (fx -»• ga) is itself sufficient for ga, although it does imply that something else, viz. fa, is sufficient for ga. This sort of counterexample to Commutation has suggested Necessity-Restricted Commutation as the correct repair, for (x)(fx -+ gx) is necessary, while it is supposed to imply once commuted, ga, is contingent. But if we require that fa be necessary, ga cannot be contingent, since implied by fa, so such trouble cannot arise. So the story goes: but the modal moral is the wrong one. For in fact counterexamples can be produced at least as damaging to Commutation as these supposedly modal ones, but without commission of a modal fallacy. For example, Arthur knows (p •*• p) (p •*• p) is a paradigmatic entailment. But the result of applying Commutation, p -»•. Arthur knows (p ->• p) -•. p, is false. For consider the antecedent. Arthur's knowing the law of identity for some p, say, 'the world is round', cannot be sufficient for p, for of course one may know the law of identity for a great many propositions which are false or dubious. Nor is there a valid argument leading from 'Arthur knows that (the world is round -* the world is round)' as premiss, to 'the world is round' as conclusion; we could not properly deduce that the world is round from the fact that Arthur knows that it entails itself. But since p, which is true, implies the false statement that p is^ deducible from Arthur's knowing the law of identity for it, the main implication, and hence the whole statement which resulted from Commutation, must be counted false. The example works perfectly well against Commutation if p is contingent, but no modal fallacy can then explain the falsity of the consequent, that Arthur knows (p p) p, for both its antecedent and its consequent are then contingent. Furthermore, the same counterexample can easily be extended to apply against Necessity-Restricted Commutation itself, by picking p as a necessary statement, e.g. as p -*• ~ ~ p . For although it is true that (Arthur knows p ~ ~ p -»-. p ~ ~ p ) -*. p ->• -»-. p , it is not true that p -> "-^p -»-. (Arthur knows p -+ -»• p ~~p) p ->• . The consequent of this is just as false as was the consequent of the previous example, and it is false for the same reason. Someone's knowing the law of identity for some statement is not sufficient for the deducibility of that statement, whether the statement be contingent, necessary, or an entailment, as required by the implicationrestricted versions of Commutation adopted by Anderson and Belnap for E (see ABE, pp.26-7) and by McCall for Connexive System CC1 (see 66), namely A (C -»• D) B -»•. C -»• D -»•. A + B, against which this latter example also applies. Since the method of arguing licensed by Commutation is basically an invalid one, leading as it does to antecedents which are quite inadequate for their consequents, the invalidity cannot be removed by restricting application of the principle to a (progressively) smaller class of statements. Whatever is wrong with Commutation - and there is plenty wrong with it - continues to be wrong with it when it is restricted by the requirement that the element commuted forward is necessary or is an entailment. The same sort of examples that counter Commutation also counterexample A A B •*• B, and EAssertion, and EAdjunction, i.e. A -»•. B -»• A & B. A typical counterexample (belonging to the class already noticed) is as follows: P: Q: R:

Everything red is coloured, The rose is red (where the rose is in fact red). The rose is coloured.

Then P Q -*• R is true (upon representing P in the form (x) (fx gx)) and Q is true but P -*• R is false, countering Commutation again. Further Q is true but (Q R) -+• R is false; for while it is true that "the rose is red implies

265$

3.8 THE COWWFALL OF SYSTEM R

the rose is coloured" this does not imply "the. rose is coloured". Also Q is true but R -»• Q & R is false, falsifying EAdjunction. Of course in a theory of conditionals where affixing principles are admitted EAdjunction is easily condemned; for as A & B -»• A, EAdjunction yields A -*•. B A by affixing. Commutation is the locus of other counterintuitive results, such as that for very many pairs of nonequivalent statements A and B, implication is symmetric, A -+ B B -* A:Theorem 3.31 of R style systems: [A -»• B A. A A B] ->. B -*• A A B, where C A D = ~(C -»• D). That is, where A B is consistent (or A and B are connexively consistent), implication is symmetric. Proof (due to S. McCall): 1. 2. 3. 4. 5.

B -»• A -»•. A B A ->• B ->-. A B, by exported syllogistic principles B + A + . A + B + . A M + . A f B , from 1, by contraposing B A A H A B A / B, from 2 by commuting B ->- A [A -*• B A- A A B] A •*• B, from 3, by contraposing [A B A A B] B A -»•. A B, from 4, by commuting.

Corollary. The result also holds for systems with Restricted Commutation which permit permutations of negated entailments, e.g. for systems such as ES5 (i.e. E with an S5 modal structure) subsequently considered. Not only does the result contradict Anderson's appealing principle, A A B A. C D, that a nonimplication never implies an implication (discussed in 10.6); often the result leads from truth to falsehood, and so can be straightforwardly counterexampled. It is enough to consider examples where A B is consistent, and perhaps B A holds true, e.g. A (1) (2)

This metal is gold The table is brown

B This metal dissolves when immersed in aqua regia The table is coloured

Though A •> B is consistent, A does not imply B; hence, since either B -»• A is true or B A holds at some logical worlds, 5 is false. Like much damage to R and analogous systems, the damage spills over, though perhaps in attenuated form, to E. For it is argued (in 10) that the necessity of E, being that of logical necessity, ought to be of S5 strength; that is, E ought to be extended to the system ES5. Then however the above corollary applies; and principles of the duly extended system are opened to obvious counterexamples. Because, in particular, of the commutation principles it guarantees (most notably Commutation and EAssertion), R does not succeed in capturing, or even approximating well, either (nonanalytic) implication of conditionality - whatever its other, and considerable, interest as a formal logic and as a relevant logic. System R also fails as an explication of implication and of conditionality for other reasons. The criticism we shall make of weakening principles and of exported syllogistic principles as implication principles applies against R as well as against E and T. Since further R, again like T and E, contains full contraposition and augmentation principles, it cannot serve as a comprehensive logic of conditionals. Indeed R is disqualified even as offering a more restricted theory of conditionals, e.g. of contraposible augmentable conditionals; for it would have it that every conditional is implied by the principle of Identity, the Leibnitz fallacy. Finally, R violates stronger paraconsistency requirements (imposed in 1.7).

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3.8 COUNTEREXAMPLES

TO THE RESTRICTED

PERMUTATION

SUPPRESSION

THESES OF E

Theorem 3.32. Any extension of R closed under Modus Ponens, Ad/junction and Substitution on sentential variables, which has as theorems both A and —A for some wff A, is trivial. Proof is as ABE, p.462. In short, inconsistent extensions of R closed under substitution are trivial. The failure of R does not immediately disqualify E as an analysis of logical implication, because, among other things, it is the presence of Commutation and EAssertion in R that is responsible for the breakdown of the connection of E with NR. However the counterexamples to the distinctive principles of R are readily adapted, as we have seen, to apply against the qualified permutation principles of E. The permutation principles hold in E provided the permuted component is an entailment; so in particular the following principles are theses of E, where p is an implication, i.e. of the form C D: (A -+. p C) p P -»>. (P -»• C) C ((A A) -> B) -»• B

A

C

Restricted Commutation Restricted Assertion Suppression

In the context of the other postulates of E, Suppression, El provides all the suppression that is required to yield such principles as Restricted Commutation and Restricted Assertion. Counterexamples of the type already exhibited apply against the latter restricted principles. Let P be some logical implication, e.g. r e.g. That wood is good entails that is is not the case that wood is not good. And Sheim knows (P P) P -*• p. But p does not entail Sheim knows (P -*• P ) entails p , since the latter is false. Or consider the counterexample to Restricted Commutation given by the following assignments: A is 'Every true implication implies every true statement', p is any true implication and C is ' 2 + 2 = 4 ' . There are also important objections of a different cast to principles such as Restricted Commutation, namely, the heavy dependence on the form of a sentence as to whether it can be commuted. Whether certain sentences can be commuted, then,will turn on how they are formalised. The result is excessive reliance on specific linguistic forms, and insufficient stability in reexpression of form. For counterexamples to Suppression we have to go a little further afield. A leading feature of the Peripatetic theory of implication (developed in chapter 11), which aimed at giving an account of what may be called progressive reasoning, was rejection of the decidedly non-progressive principle of Identity, A A. Thus it is true that (1) If A A then (as a matter of logic) the Peripatetic theory of implication is wrong: i.e. (A -*• A) -*• B for the wff B given. But (1) does not imply that the Peripatetic theory is wrong, i.e. (1) B is false. (As the counterexample plainly shows the law of Identity is illegitimately suppressed.) Identity has been taken not merely as true, but as unchallengeable, as not open to rejection. The fact that this is so emerges as soon as we consider a class of people who reject or doubt the principle (e.g. Meyer or Martin to consider some contemporary examples). Even if one considers that the principle is necessarily true, it should not be treated in this omissible fashion, certainly not if an account of sufficienty is to be obtained. No proposition, not even the law of Identity, should be so protected from questioning and doubt. Of course this sort of protection is just the shielding suppression affords. The method illustrates a general method for revealing

267$

3.8 SCRAPPING

THE NECESSITY

THEORY OF E, AND GOING DOWN TO T

this sort of suppression of necessary truths, namely consider a class of people who doubt them. Similar counterexamples work against E axiom E7, and against Ackermann's rule (6) of commutation (which yields Suppression). The examples and theory given, if correct, destroy the suppression and qualified permutation features of E and therewith, what is acclaimed as a prime virtue of E, the theory of necessity built into it. Without Suppression and its mates necessity can no longer be defined through entailment (specifically A •* A •* A) and have the required logical features. Nor can the theory of necessity be restored by simply grafting independent axioms for necessity on to E, i.e. by adding a modal superstructure. For to recover necessity a la E, a permutation principle has to be worked into the theory, most obviously a principle such as DA •>. A -»• B DB (cf. ABE, pp.29-30). But such principles are open to counterexamples like those already advanced; e.g. since D(A -»• A) the above principle yields ((A A) -*• B) -*• DB, which the Peripatetic example refutes. Nor is the loss of such an account of modality in terms of entailment, as happens with the abandonment of restricted permutation principles (see ABE, p.49), any serious loss. The attempt to reduce modality to entailment is a mistake (as we argued p.40: see also chapter 10), a mistake E embodies. Rejection of the permutation features of E takes us down to system T of "ticket entailment", a system which also stars, though not in a major role, in ABE.1 T is not the thoroughly reliable stopping point we are seeking. For E contains other objectionable features than qualified permutation, most conspicuously the principles of Exported Syllogism (ESyll), which are transferred to T. For the present we do not differentiate prefixing and suffixing forms of ESyll. Only later (in 4 and especially in 11) do we sharply separate S(uffixing), A -+ B -+. B C A -»• C, and P(refixing), B C A -»• B A ->• C. As far as entailment is concerned - and also a very common form of the conditional, namely contraposible conditionals - the forms stand or fall together. ESyll is frequently argued for on the ground that entailment must be transitive, and it is cometimes misleadingly called 'transitivity* (as e.g. in ABE). It is indeed hard to see how the deducibility relation could be other than transitive (cf. the argument of 2.1). For transitivity seems to be a well established feature of pre-formal deduction and reasoning, ranking with Leibnitz's Praeclarum principle; and it is crucial both for the logical sufficienty notion of entailment and for important associated interpretations such as inclusion of content. What is far from obvious, however, is that transitivity and only transitivity is what is taken up by ESyll. By transitivity of entailment is ordinarily understood the following rule-like connection: Where A entails B and B entails C then A entails C (Rule Syllogism), which has its direct (and normal!) systemic formulation Conjunctive Syllogism: (A + B) & (B

C)

(A -* C)

CSyll.

ESyll plainly goes well beyond these principles, yet little justification has been offered for so doing.2 In particular, a justification of transitivity System T (formerly called P) is formulated syntactically in §3 above. tics for T are given in chapter 4. 2

Seman-

Thus the use of the term 'transitivity' to refer to exported forms of CSyll is misleading. This accounts for the different choice of terminology here adopted.

268$

3. S ONLY BLATANT SUPPRESSION,

BY EXPORTATION,

TAKES CSyU

TO ESyU

of entailment provides at most a justification of CSyll, and provides 110 (clear) warrant for accepting ESyll. Indeed one's suspicions of ESyll should be aroused by the duplication of systemic forms of transitivity: recall the situation in strict implication where modally-qualified exported duplicates of principles occur. The contribution of the exported forms of Syllogism is not in fact to inferential power, but in allowing various kinds of unacceptable truth suppression. Why do we need ESyll as well as CSyll, and what does the additional form contribute to a theory of deducibility? ESyll can in fact be obtained from CSyll by Exportation. It helps in understanding it therefore to understand the operation and effects of Exportation, even though Exportation is not, in its unrestricted form, a principle of any relevant logic. The reasons for this are clear - Exportation is an extremely drastic suppression principle allowing the general suppression of theorems, and in interpreted systems the general suppression of truths. It therefore produces irrelevance in rapid and classic style when combined with principles such as Simplification. Consider the following prize winner (non-unique) for short paradox proof: CSimp. Exportation

A & B A. A & B -»• A -*•. A B -*• A A B -+ A So, where A is a theorem by detachment, B anything implies a truth.

A.

So, in an interpreted system,

The mechanism for producing the irrelevance is clear in this proof. Exportation allows first the separation, and subsequently, the detachment, of one of an originally joint set of premisses or antecedents (in A & B -*• A). The premiss of the original joint set which Exportation allows us to drop, A, is in fact the one which is relevant to the conclusion A. The result of dropping it is therefore irrelevance. Exportation is perhaps the most general and blatant suppression principle of classical logic. Given Detachment, what it tells us quite directly is that a statement is omissible from the premiss set when it is true, or that when we have two statements A and B which jointly imply some statement C, B alone is sufficient for C if A is true, and A alone is sufficient for C if B is true. But if A alone is sufficient for C, B is essential. Thus statements are, according to Exportation, always omissible wherever they occur as conjoined premisses in a deduct ion. Such a principle clearly destroys a sufficiency interpretation and permits general truth suppression. It also denies the rather intuitive principle of the joint force of premisses, that two true statements may, taken together have consequences which neither has on its own that is, in true Socialist fashion, they can co-operate to produce something which neither can produce alone. For every proposition p there is some other q such that p and q are jointly sufficient for r but neither p nor q on its own is sufficient for r. Formally, the Joint Force Principle says: (Pp)(Pq) (Pr)(p & q r & ~(p -*• r) & ~(q -»• r)). It tells us that the joint consequences of propositions may be more than the sum of the consequences of each.1 Such joint consequences are not included in the meaning of either of the premisses - as with the generation of theorems from an axiom set. 1

Exportation, together with Simplification, Contraposition, Rule Factor (A B -fr A & C -»• B & C) and Rule Syllogism implies the classical law p & q -»• r -*•• p •+ r v. q -*• r which for every proposition denies the Joint Force Principle.

269$

3.S JOINT FORCE, ANV CLASSIC COUNTEREXAMPLES

TO EXPORTATION

It is easy enough then to find convincing counterexamples to Exportation. All we need are two true joint premisses, both of which are essential for obtaining the conclusion, that is, the conclusion is a consequence of the premisses taken jointly but not of either alone. This is in fact the normal case in progressive reasoning, where the conclusion is not included in either premiss separately, e.g. a traditional syllogism. According to Russell (19, p. 177) (3x) (fx & gx) & E! Txfx jointly imply g(')x f x), but neither of the conjoined propositions on their own implies the conclusion.1 Picking the first predicate 'f' as 'is an Emperor of Ethiopia', the second predicate 'g' as 'keeps giant tortoises in his garden', we obtain from this the very plausible implication: There exists an emperor or Ethiopia who keeps giant tortoises in his garden (p) & There exists exactly one emperor of Ethiopia (q) -*•. The emperor of Ethiopia keeps giant tortoises in his garden (r). But applying Exportation, since p & q ->• r, p -*• (q -»• r). Now since p is in fact true (or at least was in 1967 when this example was written down), and since '->' is truth-preserving, it must be true that q -»• r i.e. that there exists exactly one emperor of Ethiopia •* The emperor of Ethiopia keeps giant tortoises in his garden. But surely Russell is right, and so seriously wrong in his choice of underlying sentential logic:2 q on its own does not imply r; r is the consequence of both p and q, and not of either separately. This example illustrates how Exportation (and suppression) repudiates deducibility as a meaning relation, by refusing to distinguish between a proposition's being true and its being stated as a premiss. But it is only if p is actually stated as a premiss that it is able to supply the needed meaning connection between q and r. The mere truth of p cannot supply such a meaning connection. The truth of p does not compensate for its absence as a premiss because p itself must be explicitly stated if the premiss is to state sufficient ground for the conclusion. Exportation often allows the replacement of a premiss which is fully sufficient for the conclusion by one which is only partially sufficient. To see this replace •* by 'is sufficient for': then Exportation can be read: A and B are jointly sufficient for C implies that A is sufficient for (B alone is sufficient for C) - which is clearly false. It therefore has a disastrous collapsing effect on lawlike implication. For, as in the following example, a statement which is initially lawlike may cease to be lawlike upon the application of Exportation. A classic example of catalysis illustrates the point: The contents of the smooth-walled jar at Sjtj are brought into contact with a rough surface (p) & The smooth-walled jar at S^t^ contains hydrogen peroxide ^ ^ f ^ ) (q) *»•. The contents of the

1

2

This classic counterexample to Exportation is suggested by Nelson 30, p.445, note 1.

Russell considers (for example in 73, p.86, fn.2) the possibility of modifying 'the' (classical)theory of implication to avoid the hard paradox of implication (and the difficulties this makes, as Russell points out, for Meinong's theory of objects), but adds (in 73) that 'it seems probable that any modification adequate for this purpose would be inadmissible on other grounds'. Unfortunately, Russell does not elaborate.

270

3. S BECAUSE

EXPORTATIVE, ESyU

CANNOT BE GIVEN AUTOMATIC

ACCREDITATION

smooth-walled jar will decompose to 21^0 + (r). Here p and q are jointly sufficient for the result r, i.e. p & q r. Now consider a case where p is true; then, as before, q -*• r must be true. But that the smooth-walled jar contains hydrogen peroxide is certainly not sufficient for the decomposition of the hydrogen peroxide; for hydrogen peroxide in a smooth-walled jar will keep almost indefinitely. And this resulting statement, q ->• r, similarly is no longer lawlike. Further counterexamples to Exportation interpreted as a principle of lawlike implication may be obtained by choosing conditions which are only jointly sufficient for some result, not individually sufficient (cf. the way it is faulted as a conditional by the switches paradox of 1.2). The fact that ESyll is an exported form of CSyll should, if anything, increase our suspicions of it. For Exportation allowing as it does for the dropping of true premisses (by exporting and detaching) and hence for the contraction of a sufficient set of premisses to an insufficient set, is a basic suppression principle. It seems that ESyll allows a similar reduction of the premiss set in the case of the conjoined implicative premiss of the relevant form (i.e. of CSyll form). 1 The contribution ESyll makes over and above CSyll would appear to be that of allowing the suppression of certain true implications in certain positions. This is a very limited form of suppression and it is not one which produces spectacular effects, like more wholesale suppression, in the form of irrelevance. But if we are attempting to explicate the concept of sufficiency, it must be rejected for just the same reasons as the more obvious and general suppression principles and resulting irrelevant forms. ESyll principles provide good examples of those sorts of principles we alluded to earlier, namely those that allow a certain degree of suppression but not so much as to produce irrelevance. Remember that relevance is only a necessary condition for an adequate system; so the fact that ESyll can be added to certain systems without inducing irrelevance provides on its own no justification for ESyll. Normally we would reject only only Exportation but exported forms of principles, because exported forms allow suppression, even if of a limited sort. A good example is provided by the principle of Factor, A -*• B -*•. A & C •*• B & C, which results by exportation and detachment of a case of Identity from the principle (C C) & (A ->• B) -»•. A & C •* B & C, an impeccable special case of Leibnitz's Praeclarum principle. In the context of affixing principles, Factor does in fact lead to higher degree irrelevance. Yet compare Praeclarum (or the speical case cited) with CSyll. Anderson and Belnap and Co. welcome the exported form in the second case, but not in the first. Why? No decent reason is to be found in ABE except that together the exported forms would lead to irrelevance. But separately neither need induce irrelevance: so it seems that if one is blackbanned, both should be. It is evident however,that Anderson and Belnap think that there is a case for including ESyll, as distinct from CSyll, as a principle of entailment. Their case for ESyll is brief: Anderson and Belnap reject Lewis's criticism of the principle, and accept Ackermann's restatement of the principle, and adduce one example favouring the principle (ABE, p.9): ...Ackermann 1956 is surely right that "unter der Veraussetzung A B ist der Schluss von B -»• C auf A ->• C logisch zwingend". The mathematician is

System E admits qualified forms of Rule Exportation, e.g. (A A -»• C A -> B B -»• C A-+-C.

271$

B) & (B -»• C) -*-.

3.8 ACKERMANN'S

ARGUMENT

FOR ESyll TURNED INTO A

COUNTEREXAMPLE

in no ellipsis in arguing that "if the lemma is deducible from the axioms, then this entails that the deducibility of the theorem from the axioms is entailed by the dedcuibility of the theorem from the lemma". But this sample mathematician's argument is not clearly a case of Exported rather than Conjunctive Syllogism. For precisely the difference between the two principles lies in our ability to drop from the antecedent set, in the one case, but not in the other, the implication A ->• B, in this case the information that the lemma is deducible from the axioms. But we are prevented from dropping this implication in the example by the use of the word 'lemma', for 'lemma' means something which is deducible from the axioms. Once this implication, included in the meaning of the word 'lemma', is explicitly inserted, the consequent becomes, not the consequent of Exported Syll, but a case of CSyll. And if we restate the example in neutral terminology which does allow this implication to be dropped, such as by use of terms such as 'conjecture' or of proper names, we arrive at a counterexample to ESyll. That is, in their example allegedly supporting ESyll, Anderson and Belnap have generously provided us with the basis of a counterexample to the principle. If the omission of A ->• B allowed by ESyll were correct, we could correctly argue in the following way:- Let James be some conjecture implied by some conjunction of assumptions S. Let Joan be some further conjecture which is deducible from James. Then by ESyll, S -»• James •*. James •*• Joan •*•. S -»• Joan. If James is in fact a theorem of system S, then by detachment or the truthpreservation of James -»• Joan S •*• Joan. But that Joan is deducible from James is not sufficient to show that Joan is a theorem of S. The fact that James is a theorem of S is crucial to this conclusion, for if James were not a theorem it might be true that Joan was deducible from James and false that Joan was a theorem of S. But our antecedent, supposedly adequate, now tells us nothing at all about the status of James. And it is just this crucial fact, that James is a correct conjecture, a lemma, that ESyll allows us to omit, but that CSyll correctly forces us to retain. Some of these (and other) problems with ESyll Lewis saw: I doubt whether this proposition should be regarded as a valid principle of deduction; it would never lead to any inference p -J r which would be questionable when p q and q -3 r are given premisses; but it gives the inference q -$ r. -3 . p -3 r whenever p -3 q is a premise. Except as an elliptical statement for "p q. q -J r: -3 . p -J r and p 4 q is true," this inference seems dubious (Lewis and Langford, 32, p.496).2 *And Ackermann's claim as to logical compulsion is undermined shortly. 2

Lewis started out accepting ESyll, but subsequently came to reject it conclusively: I soon became dissatisfied also with another postulate of this system as amended (now called S3), believing that certain consequences of this assumption are not in accord with the logical facts of what is strictly deducible from what. Objections to ESyll on the grounds that is is a suppression principle, that the antecedent does not state a sufficient ground for the consequent, are also implicit in Moore 62, p.317 and p.289, and in several more recent writers.

272$

3.S ELABORATION

OF LEWIS'S CRITICISM

OF ESyil

Lewis was surely right in so doubting the exportative aspect of ESyll - the objection being essentially the same as the objection previously lodged to (Rule) Exportation itself, that in A B B C A -+ C, as opposed to CSyll, B -»• C may well not be sufficient on its own for A -*• C even where A B is true. Lewis, in doubting the sufficiency of B •*• C for the consequent, is evidently raising an objection to these suppression features. Just as with Exportation itself, so with ESyll, the suppression feature of the separable joint antecedents and premisses occurs; and it is after all the ability to export one of the joint antecedents which makes the difference between ESyll and CSyll. So it is that by ESyll (but not CSyll), where A -»• B is true, B -»• C is sufficient for A C. Just as we can find cases with Exportation where antecedents are used jointly but the consequent does not follow from either antecedent alone, so we can find cases where though the antecedents of CSyll are sufficient for the conclusion neither antecedent on its own is. This certainly happens where the omitted antecedent is true. Lewis does however put his point in a way which may make it look as if he is attacking the derived S2 Affixing rule and which accordingly leaves him open to a misinterpretation, a misinterpretation Parry makes (68, pp.136-7; our notation): However Lewis's argument against S3 cannot be accepted as an argument for S2. For though S2 lacks 31.31 [ESyll] it has a Metatheorem . . . A H B -fc B -3 C . A C. ... I see no objection to the Metatheorem as a principle of strict implication. But according to the intent of the system, if Q is deducible from P, P Q. Hence if the Metatheorem holds, 31.31 must be true. ... So if S2 states correctly the properties of real implication (the converse of deducibility) as far as it goes, S3 does so more adequately. The derived rule of Suffixing of S2 does not mean that there is a valid mode of inference from premiss to conclusion, from which it can be inferred that premiss entails the consequent. Rule Suffixing (as a derived rule) tells us no more than that where A -i B is provable in S2, so Is B -J C -J . A -4 C. Contrary to Parry, the Metatheorem does not state a deducibility connection.1 Indeed it gives no warrant to move from the truth of A -•? B, or the hypothesis that A -J B, to the result that B -i C -J . A -F C - which is what ESyll licenses. It is accordingly quite invalid to move from the derived rule, as Parry does, to the truth of ESyll. There is an important line to be drawn between implicational statements and derived rules, evident enough from such rules as Generalisation A(x) -fr (x)A(x) and Necessitation A -t> DA (cf. §7 on normality). To see the deficiencies of Parry's argument against Lewis, simply replace ESyll, i.e. 31.31, by A -3 DA, S2 by S4 and the Metatheorem by A -» DA. It is important in assessing the case against and for ESyll, to distinguish clearly between ESyll on the one hand and the corresponding systemic rule, Affixing, on the other. The principles by no means stand or fall together, as we shall further see, but as is already evident from the differences between modal systems S3 and S2. But the availability of Affixing in a system serves

*It is doubtful (there appears to be no published evidence resolving the issue) that the intent of strict implication systems was to have all derived rules yield deducibility relations, or to have derived rules yield strict implications. Certainly substitution rules and derived necessitation rules do not yield strict implicational principles.

273$

3.8

WHAT ESyU

DOES THAT CSyU

AW AFFIXING VO MOT

to meet much of the pragmatic argument that may be put up for the adoption of ESyll, for example loss of deductive power without ESyll; failure of the derivation of the rule of Substitutivity of Coimplications without ESyll, etc. For practically all the deductive power required can be supplied by Affixing: Affixing can replace ESyll in the derivation of Replacement and in many other places, in particular, what is perhaps the main deductive role of ESyll, replacement of antecedent and consequent parts can be accomplished by Affixing. True, there are principles that ESyll helps deliver that Affixing cannot furnish. But for the most part these are principles that are decidedly undesirable. Smiley's horror, A B B -*• A B A, and its mate, B + A -*•. A -*• B B -»• A, are examples.1 Such unlikely consistency results as A -*• B (A ~A) -y a ~B) and A -> B -»-. A 0 A + A 0 B (discussed in 5.2) and perhaps (A •* B) & (A 0 A) A 0 B are other examples. But what ESyll principally does, that Affixing does not, is to provide affixing in all extensions of the logic, in all theories containing the logic. But it is just through such cases that ESyll is open to criticism and to counterexample. The Joan-James counterexample admits of generalisation. Let T be a regular theory, i.e. a theory which contains all theorems of its underlying logic. Then, unless the logic already includes ESyll as a theorem, A -*• B may well belong to T without B -*• C A -*• C belonging to T. In the next chapter we will encounter several examples of such theories (prime regular extensions of L-theories in unreduced modellings). The features of such theories undermine Ackermann's claim. For let T be our set of assumptions. Then though A B € T, B ->• C A C jf T so the inference in T from B -*• C to A •* C is far from compelling. It can reasonably be objected that the general point on its own is not telling as the same sort of objection can be made to CSyll: there are regular theories where though A -»• B and B -»• C hold A -*• C does not. Such theories however violate sufficiency requirements (by the arguments of 2.1), whereas those countering ESyll do not: so at least we shall argue. Thus CSyll should be among the theorems included in T, while ESyll need not be. The arguments for CSyll as a sufficiency principle - that, for example, it is logically necessary that if A is logically sufficient for B and B for C then A must be for C, or, correspondingly, that if C is deducible from B and B from A then C must be from A, simply as a matter of putting the deductions or sufficiency argument together one below the other - do not extend to ESyll, without an impermissible reduction of ESyll to CSyll. Even if there is a deduction of

1

The blame for these principles, unsatisfactorily discussed in ABE, p.32, cannot be placed on qualified permutation principles. For the Smiley horrors can be derived in systems like T which lack such principles, for instance as follows: A -»• B B A B ->-. B

A ->-. B -»• B, B -»• B ->•. B A B ->• A ->-. B A, A B

A B

B ->• A, A B

A (by contraction).

In fact the result obtained before Contraction is applied is damaging enough, and derives entirely from application of ESyll (together with Modus Ponens). The blame is therefore properly assigned to ESyll.

274$

3.S CONTRARY

TO ESyU,

IDENTITY

IS NOT SUFFICIENT

FOR ANY TRUE

ENTAILMENT

B from A it is not necessary that from the deduction of C from B there is a deduction of C from A; for we might not have access to the deduction of B from A 1 or be able to transfer it to reduce the case to one of CSyll.

It is not merely that ESyll fails where T = Tr, the class of truths: ESyll says more than that where A •* B is true so is B ->• C -»•. A -*• C; it also asserts that were A -*• B, true, B C -*•. A -> C would be also. ESyll is vulnerable on the first score; it is naturally even more vulnerable on the second. As to the first, let A B be any true entailment. Then by ESyll A B B + B A B; and by truth-preservation of '-•' , B -*• B A B, since A •* B is true. But (p) (p -*• p) B -*• B, and so by ESyll, (p) (p p) -»-. A B, that is the general law of Identity is sufficient for any true entailment.3 This may indeed have been Leibnitz's view - that Identity is an adequate base for all reasoning but it is commonly regarded as discredited, particularly by the modern axiomatic method. It is just not true that Identity would be regarded as sufficient basis for asserting each and every true entailment;1* and this for such good reasons as that mostly many other principles are required for arguments to true entailments. Similarly, to enlarge on the damage ESyll does, the principle would allow the elimination of all meaning postulates. Suppose A ->• B is some requisite meaning postulate (that is to say B holds every situation in which A holds); then by ESyll A A -*•. A -»• B; so A -*• B should be deducible just from A A, and is inessential. It is important to note, once again, that points like this do not apply against Affixing rules, such as that if A •* B is a theorem of a given system then so also is C -*• A C B. For these rules do not allow general detachment from truths in applications of the system, they are essentially system-bound because they are tied to systemic provability. The same point applies as regards the next group of examples. Some of Lewy's arguments to show that the intuitive notion of entailment is incoherent can be converted into counterexamples to ESyll. This is perhaps not so surprising given that the arguments have been viewed, as by Lewy himself, as bringing down the transitivity of entailment. Reconsider an adjusted version of Lewy's paradox 2 (discussed in 2.11), Recall that the defect of Lewy's argument

With Affixing, where the deduction is proven in the same system, we do have appropriate access. Syntactical transfer of implications into subproofs by reiteration may be impermissible, because involving higher degree suppression (see §5). And transfer may correspondingly be blocked semantically with non-transitive modal worlds or tableaux, where implications cannot, in contrast to S3 and S4 cases, be transferred to accessible worlds. 3 %

We almost certainly owe this objection to ESyll to H. Montgomery.

0n this point Anderson and Belnap's persuasive objection to the contention that the 'if ... then' of mathematics is the material-implication (ABE,p.17) can be parodies. The mathematician concludes his paper with the note: 'This conjecture has connections with the fundamentals of logic which might not occur to the reader, even the extraordinarily rare reader who is conversant with relevant logic. For example, if the conjecture is true then it follows immediately from the tautology that everything implies itself ...'. Similarly Goldbach's conjecture if correct is deducible from the truth that every even number is an even number.

275$

3.8 COUNTEREXAMPLES

TO ESyll DERIVED

FROM LEW'S

PARADOXES

was that it relied upon suppression principles. That reliance can be replaced by appeal to ESyll. At the same time, to make the argument sound, material equivalence in Lewy's argument has to be replaced by an equivalence based on a good implication. The argument looks at the co-implications (A) c rb, (B) rb rs and (C) c •**• rs, where c, rb and rs represent, respectively, 'Caesar is dead', 'Russell is a brother' and 'Russell is a male sibling'. Lewy's claim is that (B) is true, indeed necessarily true, but that the conclusion of (an adaption of) his argument, (A) (B), i.e. that Caesar is dead Russell is a brother entails that Russell is a brother •**• Russell is a male sibling, is counterintuitive. It certainly is. But it can be reached by a slightly different route from Lewy's using ESylL For by applications of ESyll, rb ** rs c ** rs, whence since (B) is true, (A) (C). Note that this result cannot be obtained by Affixing rules (or by Replacement) since (B) is not systemically provable. Now by Composition (A) (A) & (C) and by CSyll (A) & (C) - (B), whence by Rule Syll, (A) - (B). Other of Lewy's "paradoxes" are amenable to similar redirection against ESyll. For example, paradox 3, which improves on paradox 2, leads, by an exactly similar argument to the counterintuitive conclusion that Caeser is dead ** this has shape ->•. this has size **• this had shape. A little quantificational logic can make this look even more ridiculous, for instance, generalising and distributing it follows that (z) (Caesar is dead ** z has shape)entails that everything has size everything has shape.1

The more telling case against ESyll surfaces where the theory T is distinct from Tr, where T is one of the merely hypothetical theories that have to be taken into account in assessing deducibility. It may be that the last biconditional involved, this has shape ** this has size, already affords an example of this type, since there may just be cases where shape and size part company. The general form of the amended Lewy paradoxes can be reduced to the following form where antecedent detachment is not warranted: L.

(r ** s) & t

c «* r -+. r

s

(where 's' shortens 'rs' and 'r' 'rb'). Here the tautology t amounts to but an instance of Identity, c **• r -+. c ** r, which can be further reduced to instances of Identity like (c •**• c ) & (r r). The claim is that L, for which the operative principle is ESyll, is false. Typical counterexamples take the following shape: Let c be some statement entirely irrelevant to r and s, e.g. that the Amazon forests will not survive the despoliation integral to advanced capitalism, where r and s are respectively the statements that moonrock R27 is coloured and this moonrock R27 has weight. Now restrict consideration to worlds where all and only coloured things have weight (not entirely remote scenarios). Then in all situations the antecedent holds, but the consequent certainly appears to fail since c ** r hardly makes r ** s true, is hardly sufficient for r ** s, involving as it does the irrelevant c and not involving s at all. Some of the points made can be reworked directly in terms of ESyll. Let A -*• B be a true entailment that is not a logical law, for instance, a is red -»• 'it already follows by the original argument that the proposition that Caesar is dead •**• everything has shape, entails this conclusion. The counterexamples are not entirely remote from those remarked on, in UL, to Meyer's "relevant" arithmetic R#, where, e.g. 2 « 1 + 1 -*>. 2001 = 1998 + 3. Arithmetic based on E would lead to similar anomalies.

276$

3.8 PART OF THE MORE THEORETICAL

CASE AGAINST

ESc/U

a is coloured, or Tom is a batchelor -*• Tom is a male sibling. Now consider a situation d where A •*• B fails to hold: there would be many such impossible situations even were A -»• B a logical law. But as A ->• B is not a law, d may be some regular situation distinct from the actual one. Given that CSyll is correct, where both A -+ B and B -»• C hold in d so does A C. But A ->• B does not hold in d, and there is nothing to prevent assignments such that B -»• C holds but A -»• C does not. To show that they can be made it is enough to discern worlds b and c, appropriately counterexample-related to d, such that B C holds in b but A •* C fails in c. Take C as 'a is heavy'. Now let be be a (not very strange) world where everything that is coloured is heavy, and c a world where there are (or are accessible) some lightweight red birds, so falsifying A -*• C. Then B C -*. A D does not hold in d, completing an elementary countermodel to ESyll. The dirt on ESyll is not confined to syntactical and elementary theoretical considerations. ESyll fails the depth relevance requirement (see 15). Semantically too ESyll is rather demanding, far more so than the CSyll, as modal logic semantics help reveal. To reach modal systems S3 and S4, which result from S2 and T respectively essentially by the addit ion of ESyll, it has to be required that the accessibility relation is transitive - no such condition is required for CSyll which is rather automatically accommodated. Moreover the transitivity requirement corresponds (as semantic tableaux methods reveal) to allowing the transfer of A •* B (i.e. of arbitrary strict-implications) to accessible (R-related) worlds. The transfer requirement corresponds syntactically to reiteration of A B into subproofs, i.e. to limited suppression. The transitivity and transfer requirements have, as we have seen, little warrant. In the case of semantics for relevant logics nothing so simple as transitivity will do: the guarantee of ESyll principles in the semantical analysis of relevant logics requires the introduction of complex, pretty ad hoc, and eminently deletable semantical postulates: namely, Pascal and its mate (q3 and q4 of 4.3). So far no convincing case for accepting these unobvious postulates has been forthcoming - other than appeal back to the ESyll schemes they model. Nor (despite the name 'Pascal') have nice interpretations for the postulates yet been found elsewhere, e.g. in geometric theory, where 3-place relations do figure. Of course it may be that with more work or better insight appealing geometrical interpreations will emerge. Even better, more natural or plausible modellings, which avoid or supplant Pascal, may be unearthed. While such interpretations and modellings are certainly well worth seeking 2 , they will have only limited value as regards vindicating ESyll. Some of the reasons are of the following sort: A leading idea in arriving at semantics for relevant logics was that the semantics should supply situations which counter instances of the law of Identity, i.e. situations such that schemes like A -*• A do not hold in them. This is essential if such undesirable schemes as B A -»• A, A •*•. A -»• A, B -*• B A -»• A and B ->• A A A are to be refuted (given the theory of deducibility already defended in 2.1). But having thrown out these thoroughly defective schemes, what does ESyll do but restore those that show appropriate connections? For instance it restores B A •*•. A •*• A where

1

Setting down relational diagrams of the modelling conditions for ESyll helps reveal their unobviousness and noncompulsariness, and its questionability.

2

The quest for such modellings is an important part of the search for decidability results for E, R and T. There is presently no solid evidence ruling out such modellings.

277$

3.9 GROUNDS

FOR DOUBT ABOUT MINIMAL CONTRACTION, A&&

A B is true, it restores B A -»-. A -»• A in situations where A -*• B, holds, so ensuring Smiley's horror: A -*• B B -»• A -»-. B A, which is of the form: a p -*• p - a relevance restricted paradox. And so on. But there is little case for this partial, relevance controlled, restoration, and indeed it runs counter to the whole idea of removing suppression of the type most outrageously exhibited in the paradoxes in toto, and in such principles as Exportation. §5. Contraction and Reductio principles, and the failure of Contraction in paradoxical situations and theories and where self-reference features. System T is in trouble not only because it contains ESyll: it is also open to question because it includes two further types of principles that are perhaps even more difficult to evaluate than ESyll, namely, firstly, contraction or weakening principles - Contraction (or W), A -»• (A -*• B) -*. A ->• B, is an axiom of standard formulations of T - and, secondly, reduction principles - Reductio, (A ~A) •*• ~A, is an axiom of T. Not that these seemingly different types of principles are by no means independent; in fact in the framework of systems like R-W (i.e. R without Contraction) and S2° - and for that matter in various subsystems of these systems - the principles are interderivable. However the principles do not stand or fall together (despite their conflation in modal logics, where they serve as modal reduction principles, and in commutative logics like R); they are appropriately independent in basic system B (of chapter 4) and important extensions thereof such as deeper (D) relevant logics. These not-excessively-powered systems also discriminate various different Contraction and Reductio principles which merge in systems such as R. For example in system R, the different absorption principles W and Ass, i.e. A & (A-+-B) -*- B, are interderivable. Initial grounds for doubt about the weakest of the Contraction principles, Ass, derive from a comparison of Ass with Disjunctive Syllogism, grounds that are admittedly esoteric since the fallaciousness of DS is not yet widely appreciated. Consider the paired forms in the following table: Straight form DS Ass

Contraposed form

A & (A ^ B) -»• B A & (A -»• B) B

~C -»•. ~(D C) V ~D ~C ~(D -»• C) V ~D.

As the straight forms show, Ass improves the antecedent connection of DS from a material-implication to a genuine implication, which does supply an intensional connection between A and B. Is this enough to break the comparison? Yes and No. Yes, because of the connection. No, because, as we shall see, both principles have a similar role in facilitating strictly incorrect paradox arguments. Does the bottom right entry even begin to look plausible? Why should the negation of C guarantee the negation of an arbitrary D or else that D doesn't imply C. Well, semantically it does not guarantee it, without further conditions being imposed on worlds. This is so whether implication is evaluated modally or relevant. Applying the first degree semantics, Ass is valid only if there is no world a in any model such that A and A B both hold at a but B does not, i.e. such that I (A, a) = I (A -»• B,a) = 1 ^ I(B,a). If the model is not to provide a countermodel to Ass, I (A -»• B,a) » 1 must ensure that when I (A,a) = 1 then I(B,a) = 1. But why should it? To make sure that it does a further reflexivity condition has to be imposed both modally and relevantly. The modal case is sufficiently representative. According to the modal rule - to be much studied subsequently - I (A -+• B,a) = 1 iff, for every world b such that Rab, such that b is (modally) accessible from a, when I(A,b) = 1 then I(B,b) = 1 . In order for the rule to do the right thing, to rule out countermodels to Ass, it has to

278$

3.9 CRITICAL ROLES OF PS AMP A&6 IW PARAPOX ARGUMENTS

be true generally that Raa, i.e. that the world accessibility relation is reflexive. Similarly under relevant modellings the 3-place relation R has to be reflexive, i.e. Raaa must hold for every a. Such reflecivity requirements in fact impose serious restrictions on the class of situations admitted in semantical evaluation. They have the effect of ruling out various paradoxical situations. The simple relation Raa includes various paradoxical worlds in rather the way that the simple equation a = a* excludes inconsistent worlds. Both DS and Ass have critical roles in paradoxical arguments: they enable the logico-semantical paradoxes to do their damage.1 They achieve this by narrowing the class of worlds admitted, to exclude those where the paradoxes could run their course without damage. Consider first a simplification of the Epimenides paradox which shows how DS operates. According to one version of the paradox a Cretan said that whatever is said by a Cretan is false. In a context where there are no other Cretan statements or all other Cretan statements are true, this leads classically to contradiction and triviality. The version reduces when quantification ('whatever') is contextually eliminated to the following situation: s

Cs Cs

3

Fs

(i) (ii).

That is, for some situation a, which can occur, there is a statement s, made by a Cretan which says of itself that either it is not made by a Cretan or else it does not hold in the situation (or under different construals of F, is absurd or is trivial or false, either in the given situation or absolutely). Now suppose s holds in a. Then Cs D Fs holds in a, so as Cs holds in a, Cs & (Cs ^ Fs) holds in a. Hence, by DS, Fs holds in a. Thus s does not hold in a. Hence, irrespective of the initial supposition s does not hold in a (by Excluded Middle). Therefore, Cs D Fs does not hold in a. Accordingly, given the restrictions DS imposes on situations, to exclude inconsistent ones, Cs & ~Fs holds in a, whence ~Fs holds in a, from which it emerges that s does hold in a, which is impossible. Where a is the factual situation T, s is both true and also not true. Without Ds (or an equivalent), derivation of the paradox breaks down at two critical points. There is no guarantee that, given that Cs & (Cs D Fs) holds in a that Fs holds in a: all that ensues from the antecedent is that Cs holds In a and that either Cs does not hold in a* or that Fs holds in a. There is no guarantee either that when Cs ^ Fs does not hold in a that ~Fs does hold in a, which again depends upon equation of a* with a. It should be observed that the paradox argument depends on DS, not on Material Detachment (Y). Material Detachment operates only as regards situation T, and it permits detachment only from theorems of the base logic. Wherever contingent truths or nontheorems occur, DS has to be used if an inference based on both connections is to be made. Similar points apply as regards the role of Ass in paradoxes. Just as Ds cannot be supplanted by MD, so, for precisely similar reasons, Ass cannot be replaced by Modus Ponens. An analogous paradox argument using a relevant implication in (i) in place of material-implication, and Ass in place of DS, breaks down, not in the positive part, but because there is no such simple way to get from a nonimplication to a counterexample in the same (or a sufficiently related) situation. However it is certainly the case that there are more sophisticated paradox arguments in which Ass has a crucial part. Consider first an argument (really a version of the Curry paradox) which shows how Ass operates in paradoxical situations to induce triviality. Let r be a statement which says of itself that it implies 1

DS appears to have a similar critical role in Rosser's important modification of Godel's theorem. 279

3.9 LOGICALLY

FUNDAMENTAL COUNTEREXAMPLES

triviality (or the absurd, the Bad, etc). r **. r

triv.

TO A&6

Hence (iii)

Surely one can have - nothing in English stops one from having or talking about - a statement which says of itself that it implies triviality, without triviality, just as one can have a statement that says of itself that it is not true, without triviality. But, using Ass, and little other logical apparatus, r leads quickly to triviality. We present the argument more semantically first, then syntactically. By (iii), for arbitrary situation c, r holds in c iff r - triv holds in c. If r does hold in c then r -»• triv holds in c, so r & (r -* triv) holds in c. Hence by Ass triv holds in c. Since these connections (i.e. if r holds so does triv) hold for every c, r - triv. Then by (iii) since (r -+ triv) -»• r, r, and so triv. Syntactically, the argument proceeds as follows: 1. 2.

3. 4.

r r -»• triv r -+ triv r r -> r

from (iii).

_

J

6.

r -»• r & (r •* triv) , r & (r •*• triv) -»• triv r triv ,

7.

r

,

8.

triv

,

5.

from 3 and Ass from 4 and from 6 and from 7 and

1

5 2 6

Abstraction axioms - for propositions, properties, classes, and so on supply instances of statements like r. Consider, to take a simplest example, the class determined by the condition that x e x -»• triv. By the Abstraction Scheme, for some particular class w, for every x, x e w * * . x e x triv. Thus instantiating, w e w w e w triv, i.e. (iii) holds where r is w e w. (The full argument to triviality from the Abstraction Scheme is given in 4.3). A simple modification of the argument produces triviality from the Truth Scheme, i.e. from the equivalence, a is true iff p, where a is the name of p, i.e. symbolically: Tr(a) ** p. 1 Now consider the sentence (p), ' if this statement is true then triv holds' i.e. Tr(p) ->• triv. Then by the truth scheme, Tr(p)

Tr(p)

triv,

i.e. Tr(P) provides r in (iii). Thus Ass also trivialises the Truth Scheme given that scheme is, as it certainly seems to be, universally valid, and so correct for self-referential statements. Or consider, to adapt Pseudo-Scotus, the argument: This argument is valid; therefore triv.2 Then a statement of the form r **. r -»• triv follows, where r is "This argument is valid". For the argument is valid iff the entailment J

For this modification of the argument, and in some of the ensuing discussion, we borrow from Priest 80, p.431ff. Adoption of parts of this article implies of course no blanket endorsement: on the contrary the article contains serious inaccuracies and misattributions and is, in our view, mistaken on substantive issues concerning the interconnection of entailment and truth preservation and concerning worlds semantics for relevant logics. Nor is the "sense" semantics offered much more than older content semantics in a more algebraic but still familiar dress.

2

There are many variations on arguments of this sort. sound is argued in Priest and Routley 82.

UO

That such arguments are

3.9 A&.i RULES OUT PARADOXICAL

r

SITUATIONS

ANV LEGITIMATE SELF-REFERENTIAL

REASONING

triv is true.

Of course all these arguments depend upon logically admitting instances of self-reference, in particular those supplied in Abstraction and Truth Schemes. But there is nothing wrong with that, and all attempts to block logical reasoning involving self-reference are seriously wanting (the points are argued in detail elsewhere, e.g. EMJB). There are, moreover, independent grounds for doubting Ass. One independent ground for doubt is that it is straightforward matter to design countermodels to Ass (e.g. in terms of the semantics presented in chapter 4 or within modal logics) and that its validation requires an additional modelling condition that there is no compelling reason to accept. Another is that on other interpretations there is no decisive reason to suppose that Ass holds; for instance, it is doubtful that the content of A & (A B) includes that of B (and on modellings like that of UL and Priest 80 it need not or does not). But such an argument has its dangerous aspects, especially as we can now countermodel virtually any logical principle. Accordingly a general objection to semantics which falsify entrenched principles like Ass takes the following turn: 'Any theoretical account of validity must be assessed against the facts, the facts in this case being derived from our intuitions concerning logical truth and validity. But we are more sure of the truth of Ass than of any theoretical account of logical truth. Hence any account according to which Ass is not logically true is thereby refuted. The objection, however, is somewhat naive methodologically. It is well-known that a good theory can successfully overturn incompatible "facts". Moreover, in the context of a debate concerning which of two "obvious" claims has to go, it clearly begs the question' (Priest 80, paraphrased). Quite apart from semantical countermodels. observing the damage Ass wreaks in paradoxical situations, is enough to shake confidence in it, and to begin to shift the onus of argument. Why retain such a principle? What justifies it? Justifications commonly appeal, in one way or another, to the rule of Modus Ponens, which is taken to be entirely impeccable. But Modus Ponens no more supports its "normalisation" Ass than Material Detachment licenses Disjunctive Syllogism. Like all theses Ass does much more than the corresponding rule. As Whitehead and Russell assert (PM, p.110) ' it differs from [the rule of Modus Ponens] by the fact that it does not apply only when A is really true, but requires merely the hypothesis that A is true'; in short it applies in situations far beyond the actual one. What it does in each world is to rule out intraworld counterexamples to A B should it hold at the world; that is, it excludes an important class of inconsistent situations, those paradoxical situations where A -»• B holds but though A holds B does not. Much as Disjunctive Syllogism illegitimately rules out inconsistent situations, so Ass rules out a range of paradoxical situations. Similar points apply against attempts to reinstate Ass as simply encapsulating the truth that were A is true and there is a valid argument from A to B of course B is true. The attempt appeals to applications of Modus Ponens, whereas what is required is much more questionable: for instance, that were A to hold in, and were a deduction of B from A correct in, a paradoxical situation d, then B would hold in d. Furthermore Ass yields syntactically, even in system A, a rule much stronger than Modus Ponens; namely the Contraction Rule: A-* (A B) A+B, which is a source of Curry paradoxes in dialectical set theory. The derivation of the rule is in outline as follows:

281$

3.9 GROUNDS

A

TOR DOUBT ABOUT REDUCTIO,

AS DISTINCT

FROM LEM

(A + B) -t. A & A A & (A-vB) -t> A -»• B, using Ass, Tautology and Rule Syllogism.

Where Affixing rules are available the converse derivation, which also ties in Rule Importation as a further equivalent, is as follows: A

B+C-oA&B-*. A & B C, by Simplification and Affixing -0 A & B C, by Rule Contraction. Thus as A + B A B, (A B) & A B, whence commuting A & (A -»• B) -»• B. If Ass is in doubt or incorrect, strong contraction principles such as Absorption and Importation, which strengthen the rule Ass supplies to an entailment, are in at least a bad a shape. There are further reasons for suspicion of these stronger principles. The modelling conditions, for instance, are more complex and remote. In addition, the principles interfere seriously with decidability arguments which hold for systems which dispense with Absorption. Reductio, A ~A ~A, does not exclude situations in the same way or to the same extent as Ass (though it makes some exclusions: see the later part of DSLM). For that reason we shall for the present, say much less about it. But in systemic investigations we shall generally carry Reductio as an optional extra. It is important to distinguish Reductio from LEM, A v ~A, and the grounds for doubt about this form of reduction from intuitionistic criticisms of certain sorts of reduction proofs and of LEM. On the one hand in normal settings Reductio is much stronger than LEM, and is tantamount in basic system B (of 4) to the counterexample principle A B •*•. ~A v B, which together with Identity implies LEM.1 LEM in itself is tantamount, in FD for instance, to a rule form of Reductio, namely A -+ ~A ~A. Thus arguing for Reductio on the strength of LEM would involve the same sort of error as arguing for Ass from Modus Ponens. Nor has Reductio been conceded in granting as much as LEM (or LNC) in that of a framework such as system A. What can be shown, by Composition and Rule Contraposition is (A •+ B) & (A •*• ~B) ~(B & ~B) -»• ~A. But without a Commutation Rule, such as 6, there is no way of suppressing ~(B & ~B) in its internally nested position and arriving, as in system E, at (A -*• B) & (A ~B) ~A. On the other hand, Reductio, unlike LEM, is a thesis of intuitionistic logic (and similarly (A -»• B) & (A •* ~B) ->• ~A). Indeed in minimal logic it is a case of Absorption, viz. A -> (A -*• A) -»•. A -»• A, and open to similar objections in terms of what it does, for instance in paradoxical situations. One of the worries about Reductio in a paraconsistent setting, is that the argument - at least as often presented - seems to depend upon the rejection of contradictions, when however given paraconsistency contradictions are no longer always rejected, only mostly since not true but seriously false. The argument revealing the source of the worry is roughly paraphrased: A leads to a contradiction, say C & ~C, but that's bad, we can't have that, so we can't have A which lead to it, so ~A. But in fact this argument is essentially a form of Contraposition which makes use of LNC, both of which relevant paraconsistent logics can perfectly well include, and should. So this sort of worry is unwarranted. Paraconsistency alone need not undermine Reductio. However

1

The admission of evidentially incomplete situations, highlighted by intuitionism, does not undermine LEM, since the factual world T may not be such a situation. The further intuitionistic case against LEM strikes us as seriously flawed; cf. Russell, 37, Introduction.

282$

3.9 STRONGER

RELEVANT

SYSTEMS

ARE AN INCOMPLETE

REFORM OF STRICT

IMPLICATION

further considerations drawn from paraconsistent reflection may, for instance the avoidance of gratuitous limitative results (see DLSM). And Reductio certainly does limit the class of reasoning situations that can be deductively investigated (see 4). Supporters of E can of course put the counterexamples and counterarguments adduced against principles of E aside as merely showing the unreliability of (even mature) intuition, rather as stricters put aside the paradoxes; but in doing so they will substantially weaken their own case. For they are then playing the opposition game, the stricters' game, and the same objection can be made:- Why put up with such intuitive results when alternatives which lack them are clearly available. The pragmatic answers - power, strength, etc. are also exactly those stricters often produce to justify their (competing) systems; and if they can be used to "validate" E, it is difficult to see why similar considerations cannot be used to "validate" strict implication. Much as strict implication is an incomplete reform of material implication, so systems E and T are incomplete reforms of strict implication. Though E, for example, avoids many of the faults of strict implication, and undoubtedly represents a substantial improvement upon strict theory, it still tolerates relevancepreserving suppression principles. Although these are not so blatant in their operation as to issue in the obvious defect of irrelevance, they are nevertheless very damaging, for all that, both to the core notion of entailment as logical sufficiency and to associated interpretations and roles of entailment.

283$

4.0

REASONS

FOR THE SWEEP OF THE

INVESTIGATION

CHAPTER 4 THE SEMANTICS OF ENTAILMENT AND SUFFICIENCY

CONDITIONALS.

RELEVANT AFFIXING SYSTEMS WITH NORMAL CONJUNCTION, DISJUNCTION AND NEGATION, AND THEIR

EXTENSIONS.1

The logic of entailment - which we have good grounds for believing is the same as the logic of necessary, or logical, relevant implication? has, we expect, a logical structure very like that of implication; and the logic of relevant implication furnishes what we take to be the logic of an important sufficiency conditional. It is not yet evident, however, that there.is a uniquely determined (correct) logic of entailment, or of implication - there is certainly not just one common logic for entailment, natural implication and conditionals. Accordingly it is reasonable to begin, as we prefer to do, by considering a very large class of relevant logics, logics which furnish a_ class of prime candidates for entailment, natural implication and sufficiency conditionals. Later we consider other classes of candidates and the connections between such systems and the notions to be explicated, and we further consider the issue of the different constraints and conditions of adequacy on the notions of entailment, of natural implication, and of conditionals. We examine first a central class of higher degree entailment and implication systems characterised by the features (i) that the implication connective satisfies the rule of affixing, i.e. A -»• B, C -»• D-*B •*• C A — D, and (ii) that the extensional connectives &, v, and ~ are normal. Thus the systems have at least connectives &, v and The weakest system B in this class of systems is in fact still classical in its negation part inasmuch as it includes double negation postulates and all forms of contraposition in rule form; but it is not entirely classical inasmuch as it may lack the law of excluded middle A v ~A and other classical tautologies. In a later chapter however, we provide semantics for enlargements of (positive) entailment logics by non-classical negations such as intuitionistic negations. l This chapter and parts of succeeding chapters 4-8 take up and improve the unpublished monograph, The Semantics of Entailment V (Routley and Meyer, 1972). In this monograph we extended the semantics provided in The Semantics of Entailment I-IV for relevant logics E, R, NR and a class of positive entailment logics, in several directions: to include more positive systems, to include a variety of negation postulates, to include non-normal entailment logics corresponding to non-normal strict implication logics, and to include various alethic, deontic, chronological and epistemic enrichments of entailment logics. We applied the semantics to establish such desirable properties as Hallden reasonableness, the admissibility of the rule of material detachment, and separation results. The presentation here is, however, independent of our earlier papers on the semantics of entailment, and can be read (apart from a few minor details which are easily supplied) without reference back to these papers. The chapter is also largely independent of previous chapters. 2

See chapter 10, where the equation of entailment with logical relevant implication is defended.

2 84

4.1

STRUCTURE

OF THE SENTENTIAL

LANGUAGES

Though we consider a very substantial class of relevant logics and irrelevant extensions thereof, we appear to be only at the beginning of plumbing the wealth of propositional logics that can be analysed by the class of methods presented and minor variations upon them. Not only can indefinitely many further axiom schemes beyond those we treat be given semantical correlates (though not every new axiom scheme can be treated without adjustments, as later sections reveal); but further the (style of) semantics works also for non-normal sublogics of the weakest system B that we analyse in this chapter. Nonetheless B is a natural minimal system among systems with a normal negation in terms of the relational semantics given for the class of systems and the methods used for establishing completeness. §1. Syntax of the sentential systems. The sentential language SL is represented by the triple , where IWff is the set of initial well-formed formulae, Imp is the set whose members are the improper primitive symbols, in this case the set consisting of the one-place connective ~ and the two-place connectives &, v and and Wff is the set of wff built up as usual from the elements of IWff and the connectives in Imp. The set IWff is usually taken as the denumerably infinite set p, q, r » P > q » r » P > etc., of sentential parameters. But in fact IWff may be any sufficiently large well-ordered set of symbol complexes (which are not subject to further logical analysis); in particular IWff may be the set of descriptor-, quantifier-, and connective-free wff of structure logic or of quantification logic, i.e. the so-called elementary or zero order wff of quantification logic. Thus a sentential language need not consist only of (what are interpreted as) sentential components; it is simply that logical analysis of these initial structured components with non-sentential parts is not mandatory. It is quite misleading to say of course, what is commonly enough said, that a sentential language is an ordered set, because languages are not ordered sets though they may be represented by way of certain ordered sets. A representation is not an identity (cf. Grossman 73); what is less misleading is to equate the sentential language SL with its wff, from which the triple representing SL can be unscrambled. The sentential language SLt, which we consider subsequently, differs from SL in representation only in adding the sentential constant (or zeroplace connective) t to the set Imp of improper symbols. Alternatively, since t alone is a wff, t could be added to IWff, but to match algebraic analyses it is preferable to add t (simply) to Imp. The question naturally arises as to whether Imp should be broken up into various further sets, in particular whether connectives should be separately classified from constants, and also whether brackets should be included. Further analysis of Imp is never needed in what follows, so a further distinction would be otiose; and though brackets could be included, it is easier for comparisons with algebraic and linguistic analysis to omit them. In fact 1

The basic system B is not simply a negation extension of the basic positive system B+ of SE III, but deletes, what is known to be inessential in B + , sentential constant t and its associated axioms: the axioms on t turn out, like many other axioms, to be optional and not necessarily desirable extras. The system B of AL I is here called G.

285$

4.1

WELL-FORMED

FORMULAE,

DEFINITIONS,

BRACKETING

CONVENTIONS,

ETC.

to avoid problems oyer the status of brackets, brackets are simply introduced definitionally in terms of wff built up in a standard bracket-free way as follows:i) ii) iii) iv)

Every element of IWff is a wff. Constant t is a wff. If A is a wff, so is ~A. If A and B are wff, so are &AB, vAB and -+AB.

In the case of SL clause ii) is of course omitted from the definition of Wff. Symbols A, B, C, A 1 , etc., are used as wff variables, i.e. they hold places for arbitrary wff including defined wff. Transformations which place two-place connectives between the wff they link and introduce accompanying scoping brackets are provided among the definitions: Dfl.

(A & B) = D f &AB.

Df2.

(A v B)

=

Df

Df 3.

(A + B) = D f -AB.

Df4.

(A a B)

=

Df

=

Df

Df5.

(A = B) = D f « A => B) & (B => A)).

Df6.

f

Df7.

(A o B) = D f ((A

Df8.

(A A B)

Df9.

(A V B) = D f (~A - B).

B) & (B + A)).

~Df

vAB. (~A y B). ~t. ~(A + ~B).

Df8 defines intensional conjunction, a relation distinct from fusion and consistency (symbolised ° and O respectively) except in very strong distinction-smudging intensional logics such as system R of relevant implication. Df9 defines intensional disjunction, a connective deployed in the logic of entailment by C.I. Lewis as long ago as 1912. Constant t, already well-known from chapter 2, may be construed as the conjunction of all theses (or truths). Logics with constant t are studied in chapter 5. Standard conventions are adopted for omitting and restoring brackets, namely outer parentheses are omitted, two-place connectives are ranked &, v, in order of increasing scope (i.e. & binds more strongly than v, v than -*•, etc.), and otherwise association is to the left except as qualified by Church's convention for the use of a heavy dot (Church 56, pp. 74-5), that where in omitting a pair of brackets in favour of a heavy dot in the left bracket position the convention in restoring brackets is (instead of association to the left) that the left bracket, (, replaces the heavy dot and the right bracket,), goes in immediately before the next right bracket which is already present to the right of the heavy dot and has no mate to the right of the heavy dot or failing that at the end of the expression. We shall also subsequently consider languages which add to SL other connectives and constants, in particular the two-place connective ° and the constant F. The bracketing conventions are implicitly elaborated to cope as further connectives and constants are introduced. A logic on a language is given as usual by sets of axioms and of derivation rules (or a consequence relation). More algebraically, a formal sentential logic FL may be represented as a triple <SLg, Ax, RL>, where SLg is a sentential language, Ax is an effectively specified set of wff distinguished as axioms (or a set of axiom schemes which encapsulates such a set of wff) and RL is an effectively specified set of (primitive)

2 86

4.1

AXIOM SCHEMES

A W RULES OF THE BASIC AFFIXING

SYSTEM B

transformation (or derivation) rules. Then a proof and a theorem may be defined along standard lines (except that non-existential quantification is deployed, since proof sequences for many theorems do not exist), respectively, as a sequence of wff generated from axioms by applications of transformation rules, and as a wff of which some sequence is a proof. However to avoid likely confusion about the intended role of transformation rules in weaker implicational logics it is perhaps better to view Ax as the set of initial theorems and RL as a set of rules for generating theorems from theorems. From this viewpoint FL is better represented alternatively as a quadruple <SLg, Ax, RL, Th>, where Th is the set of theorems derivable from Ax using zero or more applications of rules from RL. This representation has the advantage of enabling us to follow the common practice of equating distinct logics which have the same language (possibly after definitional introduction) and the same set of theorems; where FL and FL 1 so coincide we write FL = FL', even though FL and FL' may differ in axioms or rules. The addition of Th to the structure representing FL also has the advantage of meeting a criticism of circularity in the English transcription of transformation rules we favour. For example, the rule of Modus Ponens, symbolised: A, A B is read: where A and A -»• B are theorems so is B. Generally the rules are not, in what follows, intended to reflect entailment relations, or inferential connections such as that expressed by 'As ...; therefore... '; they are simply schemes for generating theorems from theorems. But how can the rules generate theorems from theorems when theoremhood is defined in terms of the rules themselves? The puzzle can be resolved either by a recursive definition of theoremhood or, as we have done, by adding the class of theorems to the structure of FL and equating Th with the smallest class of wff containing Ax and closed under RL. The basic logic B has language SL and the following axiom schemes and rules: Al.

A •*• A

A2.

A & B •» A

A3.

A & B -»• B

A4.

(A -»- B) & (A -*> C)

A5.

A -*• A v B

A7.

(A •*• C) & (B -»• C) -»-. A v B

A9.

—A

Rl.

A, A

R2.

A, B A

X

A6.

B -»• A v B

A8.

A & (B v C) -»-.

(A & B) v (A & C)

A B->B

(Modus Ponens)

& B, i.e. where A and B are theorems so is A & B (Adjunction)

R3' . A + B, C + D - f B + C-*. A R5.

C

A -»- B & C

A -*• ~B - » B -*• ~A

D

(Affixing) (Rule Contraposition)

The customary renditions of derivation rules in logical work are decidedly unsatisfactory. Consider, e.g., such renditions as 'From ... one may infer ...', 'From ... to infer ...' and 'From ... infer ...'. As to the first suppose one doesn't feel like it, or is opposed to doing so. The second form is scarcely grammatical: what is it supposed to mean? What sort of order is the third: is one to go through all the inferring schematically prescribed? For more on rules see 15.

2S7

4.1

OF EXTENSIONS OF SYSTEM B

A WEALTH

Alternatively Affixing may be broken into the pair of rules: R3.

A + B-*B-*C->-, A + C

(Suffixing)

R4.

A + B-frC + A ^ , C

(Prefixing)

B

The list of axiom schemes may be shortened by defining v thus: A v B = D f ~(~A & ~B); for then A5-A7 are derivable. Additional sample schemes drawn from the following lists may be added to basic system B (or to subsequently introduced systems Bt, B° and Bt°) singly or in combination to yield a wealth of stronger systems:Bl.

A & (A -*• B) -»• B

B2.

(A + B) & (B

B3.

A

B -»-. B

C)

A + C

C -»-. A

C

B4.

A -*• B

B5.

A

(A -* B)

B6.

A

A -*• B

B7.

A -»• (B

B8.

A -»• (B + C) -»-. A

B9.

A

BIO.

A ->•. B ->• B

Bll.

B

B

C -> A

C ->• B

A •*• B; equivalently, B5'. A -*• (B

C

B

C) -»-. B

A

C) -»-. A & B

A B

C A -»• C

(B -»• C) -»•. A + C

A -»• B

B12.

A

B -*•. C -»• A

B13.

A

B

B14.

A -*• B -•. A->C->. A + B i C or, equivalently,

B14' . A

A & B

C

B + C

A v B + C

B15.

A & B - ^ C ^ - . A->. B - > C

B16.

A v. A ->- B

B17.

A -»• B v. B •*• A; equivalently B17' . A

B17". A v b - * A V . B18.

A

B19.

A v B

B20.

A & B

A & B v. B

A & B

AVB->B

A -> A A + B C

A

B C v. B

C

For a good entailment or implication relation most of these additional schemes, and certainly every scheme from BIO on, will of course be rejected. Dl.

A & B

D2.

A v ~A

C

A & ~C -> ~B

D3.

A

D4.

A -* ~B ->. B •*• ~A

~A -*•. ~A; equivalently, D3'.

A -*• B -»-. ~A v B

D5.

B -»-. A v ~A; equivalently, D5'.

A & (~A v B) -»-. B

D6.

A •*•. ~A -*• B; equivalently, ~(A

B) -»• A

*In later chapters, especially 5, many further schemes and rules are semantically analysed, and limitations of the methods are explored. 288$

4.1

D7. D8.

SOME NOTEWORTHY

~(A -»• B) B -*• A A ~(B -> C) ~B

EXTENSIONS

OF SYSTEM B

~A

We also consider sample rule extensions of the systems, though in view of the strength of B the rules do not generally lead to systems which cannot be represented as axiomatic extensions. BR1.

If A is a theorem, so is (A -*• B) •*• B

BR2.

If A

BR3.

If A & B -*• C is a theorem, so is A & ~C

B is a theorem, so is ~A v B ~B

Extensions of B by any selection of postulates Bl-9, D2-4, BR1-2 are called relevant extensions. Naturally these are not the only extensions that satisfy Belnap's weak relevance criterion (see 3 above). Among the systems formulated are the following:G: B + D2. G, basic classical logic, is the weakest of the affixing systems formulated which includes all classical tautologies (in connectives {&, v, ~}) as theorems, a fact which may be proved either syntactically, along the lines of Belnap 60a or semantically from the normalised analysis provided in chapter 5. DA: B + {Bl, B2, D3, D4>. DA used to look a promising choice, among the affixing systems formulated, for "the" theory of entailment; but it is hardly uniquely determined, for one can all too easily contemplate, for example, deleting Bl or strengthening it to B5, or weakening D3 to D2. System DA of course includes G. More promising than DA are the following two systems (of UL and elsewhere): DK: G + {B2, D4} DL: DK + D3; i.e. DL results from B upon replacing R5 by Contraposition (D4) and adding Conjunctive Syllogism (B2) and Reductio (D3). C: B + {Bl, B3, D4}. System C and various of its extensions are important because of their reduced semantical modellings, and owing to the fact that they have so far resisted all efforts to settle their decidability status. C is symmetrical as regards prefixing and suffixing principles: B4 may be derived from B3, or B3 from B4, using D4. N: B + {B3, B4, D4, D4}. N, now called T-W, is of interest because of certain conjectures, notably Belnap's conjecture, now settled affirmatively, that for the pure implicational part of N, where A and B are distinct wff, at most one of A -*• B and B A is a theorem; and for the conjecture that the "ideal" set-theoretical extension of N is non-trivial, i.e. absolutely consistent. The ideal set theory concerned differs from that discussed in Fraenkel Bar-Hillel 58 only in that the underlying logic is a relevant logic and that the axiom schemes of comprehension and extensionality are formulated with connectives *»• and ** appropriately replacing the classical connectives => and =. (There are minor problems however about extensionality; in a relevant framework the axiom needs qualification, as explained in UL.) In particular, the key scheme of comprehension (or set abstraction) is symbolised: ABS.

(Pw)(x). x e w

A(x),

where A(x) is a wff containing x (but perhaps not w) free. In relevant logics, which lack such spread principles as A, ~A -frB and its equivalent y (i.e. A, ~A v B -»B) the Russell paradox (according to which both wi e wj and ~(wi e wi) hold for the Russell class wi of all non-self-membered classes, i.e. for w x « {z: ~(z e z)}, and its variants do not lead to absolute inconsistency, and are unproblematic for the dialectician who is prepared to accept, or even welcome, the isolated contradiction (see Routley-Meyer DCL). Variants of the Curry paradox are more bothersome. Any relevant ideal set theory in which Rule Absorption: A •*• (A -*• B) -*• B, is derivable is absolutely inconsistent, that is to say trivial, as the following version of the Curry paradox shows:

289$

4.J

IDEAL RELEVANT SET THEORY, AND FAMOUS RELEVANT

LOGICS WHICH TRIVIALISE

(1)

(x). x e W2 **. x e x -*• p, by ABS, where w 2 is the chosen set and p is an arbitrary sentential variable.

(2)

W2 e W2 **. W2 e W2

(3)

w2 e w2

(4)

W2 e W2 •*• p, by Rule Absorption.

(5)

W2 € W2

(6)

W2 e W2

(7)

p

IT

p, by Instantiation.

w 2 e w 2 •*• p, from (2) by Simplification.

p

w 2 € w 2 , from (2). , from (4) and (5).

, from (4) and (6) by Rule Syllogism.

Hence B, for every wff B, by substitution or a parallel argument to (7). It follows that ideal set theory based on B + {B5} is trivial, and, more surprisingly, that the theory based on B + {Bl} is trivial, since Rule Absorption can be derived» thus:Suppose |- A -*•. A -»• B. Then, as A -*• A, by Rule Composition, |- A -*•. A & (A -»• B). Hence given A & (A •*• B) -»• B, i.e. Bl, [- A -»• B, by Affixing, indeed by Rule Syllogism. Hence too, any ideal set theory, or abstraction theory, based on a logic with just the following - at first sight rather unexceptional principles - Identity, Rule Composition, Rule Syllogism, Conjunctive Simplification, Ass (i.e. Bl), and Instantiation - is trivial. Despite this alleged set-back for the relevant dialectical position which is prepared to grasp the set theoretical paradoxes in isolation1, it remained until very recently an interesting and important open question whether ideal set theories based on relevant logics, such as N or, without Bl, are absolutely consistent. The question has been resolved affirmatively (see PL, especially Brady's work). A related open question, also resolved affirmatively, is whether ideal set theories based on relevant logics without Bl and also without D3, or more important, D2, which leads to a rule form of D3, are simply consistent. Historically more important systems than N are these systems which trivialize ideal relevant set theory:T: N + B5. T is Anderson and Belnap's system of ticket entailment (formerly known as system P). R: T + B6. without too much in the way of justification, R has become known as "the system of relevant implication". E: T + BR1. In short E adds to T a rule form of the axiom scheme R adds to T. IT': E + y, That is, Ackermann's system II' results upon adding rule y to E, but in fact E and II' have the same theorems, as will be proved below. S4: E + BIO, i.e. E + BIO, or T + BIO. For since in Lewis system S4, A •*• A B B is a theorem, t may be defined as A -*• A. Systems, II' and S4 are studied in detail in SE IV, appended. E is the extension of E by t. CR: B + {B3, B5, B6, D5}. Thus CR, which is treated in CR I, adds to the axioms of B and R + the paradox B A v ~A or its B-equivalent A & ~A -*•• B. S: R + D5, i.e. CR + D4. There are of course several other routes to classical sentential logic S. Naturally many of those systems have much neater axiomatisations, since the addition of further axioms can render postulates of B redundant. System R, for example, may be reaxiomatised using just the rules of Modus Ponens and Adjunction and the schemes: Al, B3, B5, B6, A2, A3, A4, A8, A9 and D4. 1

For the allegation see Meyer, Routley and Dunn 79, and for an initial response to the allegation, UL.

290$

SOME THEOREMS AND EQUIVALENCES

4.1

OF SYSTEM 8

The following theorems of B and extensions are applied in establishing completeness (to the right of the proof sequence main axioms and rules applied in the derivation are cited). T3.1 Proof.

~(~A & ~B) -*. A v B ~(A v B) ~A

A 5 , R5

~(A v B) -»• ~B

A6, R5

~(A v B)

A4

~A & ~B

The result then follows by R5. T3.2 Proof.

A v B -»• ~(~A & ~B) A -»• ~(~A & ~B)

A2, A9, R5

B

~(~A & ~B)

similarly

B •» ~(~A & ~B)

A7

A

T3.3 Proof.

v

~(A & B)+. ~A v ~B ~(—A & — B ) ~A v ~B

T3.1

—A & —B

A9, A2-A4

A & B

~(A & B) -> ~ ( — A & — B )

applying R5, A9

Result by R3, Rl. T3.4

~A & ~B -f ~(A v B)

T3.2, R5

DR1. A <*• B - » D ( A ) D(B), where wff D(B) results from D(A) by replacing one or more occurrences of B by A. (Substitutivity of Coimplicants). This rule, which is proved by induction, is, along with Affixing, a key (and not indisputable) feature of normal affixing systems. T3.5 T3.6 Proof.

~~A ** A

A9: R5, Al

(A B) & (C •*• D) A + B-*. A + B v D

A v C -»• B v D A5, R4

C -»• D ->•. C ->• B v .D (A

B) & (C + D)

(A •*• B) & (C (A

A6, R4 A + B v D

A2, R3

D) -»-. C -»- B v D

A3, R3

B) & (C -»• D)

(A -»• B v D) & (C

(A -»- B v D) & (C

B v D)

B v D) •*•. A v C -»• B v D

A4 A7

Result by R3 or R4 with R2. T3.7 (A -»• B) & (C -*• D) Proof, dual to T3.6. T3.8 Proof.

A & B -»• C & D

(A B) v (C + D) +. A i C + B v D (A -*• B) v (C D) (A -»• B v D) v (C B v D) A5, A6, R4, T3.6 (A B v D) v (C B v D) -*-. (A & C •*• B v D) v (A & C -»• B v D) A2, A3, R3, T3.6 E v E -»• E A7, Al

Result by R3 or R4, and R2. DR2. A & B - » - C , B + C v A - o B + C Assuming the premisses, C v (A & B) C v C, by T3.6, whence (C v A ) & (C v B) -»• C. Hence as B C v A and B -*• C v B, B

C.

291$

4.2

REMOVING

AFFIXING

RULES FROM B

The equivalence of certain postulates, already stated to be interderivable in the context of B, is also a straightforward derivational matter. i) B5 is deductively equivalent to B5' (in the context of B). Given B5 1 , B5 follows trivially. As to the converse, by A2, R3 and R4, (A B -»• C) A -*•. A & B C. By A2, R3, (A •*•. A & B -f C) •*•. A & B A & B -»• C. Hence by B5 and R2, R3, A (B -»• C) A & B -»- C. ii)

iii)

D3 is deductively equivalent to D3'. D3 follows trivially, given D3'. Conversely, A & ~B -»• ~B by A2. Hence by R3, ~B + ~(A & ~B) (A & ~B) •> ~(A & ~B); so, by D3, ~B ~(A & ~B) ->. ~(A & ~B). Since ~A ~(A & ~B), A2 and R5, ~B + ~A ~B ~(A & ~B) by R3. Hence ~B ~A -»•. ~(A & ~B) , so — A -»• ~B ~(~B & — A ) . The desired D3' follows by DR7 and T3.5. D5 is deductively equivalent to D5'. D5 comes from D5' by Lewis paradox arguments. D51 comes from D5 by R5, A3, and A8.

For a number of systems the parts are also of interest. Where L is one of the logics introduced, L^ is the implicational part of L, obtained by deleting connectives &, v, ~ and associated postulates concerning these connectives; L-=r is the implication-negation part; L. is the implicationX a conjunction part; and L is the positive part, obtained by deleting negation, and its associated postulates. We use the system label L for an arbitrary one of the extensions of B (and Bt) and their parts introduced in the text.

§2. The preferred formulation of B: the removal of affixing rules and distribution. From the point of view of the choice of a basic relevant

logic it is of much interest that B and its extentions can be reaxiomatised without the not uncontroversial rules of Prefixing and Suffixing. These rules are replaced by the rule of substitutivity of coimplicants DR1 and at least one specific axiom scheme instancing an application of Affixing, e.g. C A & B C + B or A + B (C & A) B. One such reaxiomatisation of system B has the following postulates Al, A2, A4, A6, A7, A8, A9, Rl, R2, R5, and DR1, together with A3'. C •*• A & B -»-. C -*• B, and A5'. A v B C A -»• C. More symmetrically, the main features of conjunction can be captured in the schemes: C + A & B-+. C + A C A & B -»-. C -»• B (C -*• A) & (C -*• B) -»-. C -*• A & B; and similarly for disjunction. It is immediate from foregoing theorems that B as reformulated is included in the original axiomatisation. That the original formulation does not outrun the reaxiomatisation may be proved as follows:DR3. A + B " » A & B A One half is immediate from A2. For the other, since A B and A -*• A, A -»-. A & B by A4. In fact DR3 can be strengthened to DR4.A->B^A&B^A using A3 or A3', but it cannot be further strengthened to the biconditional A-*B«*. A & B - ^ A without inducing paradoxes of implication (see the discussion in chapter 13). DR5 (Prefixing). A - * B - * A & B - » " A , C + A & B + . C + B, by DR3, A3' -f C A -»•, C -»• B, by DR1.

292$

4.2

ELIMINATING

DISTRIBUTION

FROM 8

DR6. A B B ** A v B. Proof is like that for DR4. DR7 (Suffixing). A + B h - B « A v B, A v B + D A D, by DR6, A5' B D ->-. A ->• D, by DRL. When Contraposition, D4, is available, v may be defined in the usual way, A5'-A7 derived, and DR7 proved alternatively thus: A + B - » ~ B + ~A, , by Contraposition -i> ~D ->• ~B -*•. ~D •*• ~A, by Prefixing -o B D ->-. A -»• D, by DRl and Contraposition These reaxiomatisations of B and its extensions focus the issue as to the correctness of the affixing rules on some special cases concerning & and v. Thus if C A & B C ->- A is a correct principle for sufficiency or conditionally 1 then Prefixing is admissible. If, however, this principle really has A & B -*• A as a further premiss and is obtained by suppression of this premiss from the correct (C A & B) & (A & B -f A) -*•. C + A, then the principle is not a genuine sufficiency one and should be rejected as an entailment principle. In this event one will turn to a relevant logic which does not contain system B for a formalisation of entailment (such systems are studied in later chapters). The axiom scheme A8 of distribution can also be eliminated as a separate scheme by appropriately strengthening the scheme A7 of v-Composition (in precisely the way that has already been argued for in chapter 2 with Distributive Lattice Logic) to the following A7\ (D & A -»- C) & (D & B C) D & (A v B) -»• C. We show the deductive equivalence of A7' to A7 and A8 in the framework of the remaining postulates of system B. ad A7*.

C) -»•. D & A v D & B C, D & (A v B) C, by application of A8 and Affixing. ad A7.

(D & A •*• C) & (D & B

by A7

(1)

(A v B) & A «*• A, (A v B) & B

(2)

((A v B) & A -»• C) & ((A v B) & B -»• C) +.(A v B) & (A v B) -»• C } A7'

(3)

(A

(4)

A & A «*• A

B, from A4, A5, A6.

C) & (B -»- C) -»-. (A v B) & (A v B) + C, from (1), (2), A4. , from A3, A4.

A7 then results from (3) and (4). ad A8. A & B (A & B) v (A & C) , by A5 A & C -*•. (A & B) v (A & C) , by A6 The result then follows upon detaching from A7'. The objection is sometimes made, to eliminating A8 in this pleasant way, that v-Composition, A7', then becomes inextricably bound up with conjunction. But weaker v-Composition A7, is already bound up with conjunction, and Adjunction is presupposed when v is introduced. Indeed in stronger relevant logics & and v cease to have the simple symmetry they exhibit in positive logic, for example, where either conjunction or disjunction can be added to the implicational fragment to give a separate fragment. 1

Observe that one of the reduced affixing principles, A-»-C-»-. A & B + C, is rejected by modern theories of counterfactuals, on the ground that the addition of B may undermine A's support for C. Once again, the theories are not concerned with a genuine sufficiency notion and have, perhaps illicitly, smuggled in connexivist deletion assumptions. 293$

4.3

METAPHYSICAL

PRELIMINARIES TO THE FORMAL

SEMANTICS

For the record the preferred formulation of B that emerges is as follows: Rules: Modus Ponens, Adjunction and Replacement, i.e. the standard rules of replacement systems, together with Contraposition. Axiom schemes: A -*• A , •—A ->- A, C -> A & B C B, A & B A, (A -»- B) & (A -*• C) A B & C A v B -»• C ->. A C, B A v B, (D & A -»• C) & (D & B -»- C) D & (A v B) ->- C. §3. Formal semantics for B and extensions. The semantics of these systems are studied in a predominantly formal way, as part of the larger theory of relations. In view of the prominent part a 3-place relation R (reducing to a 2-place relation in the case of irrelevant logics) plays in the semantical analysis, the semantics are also called relational semantics - in contrast, for example, to the operational semantics later developed where R is replaced by a pair of 2-place operations. At various appropriate points we sketch some of the motivation for the semantics and also intuitive interpretations of the semantics. But before we present the formal semantics it is important to clarify several features - both notational and interpretational - concerning the metalogic in terms of which the semantics is formulated. Firstly it is important in accounting for the choice of terminology and symbolism adopted in expounding the semantical analysis, to clarify the underlying metaphysics. All the semantics developed are in terms of worlds or situations - not merely situations which are fragments of the actual situation, nor merely possible worlds, but a wider class of worlds which include inconsistent and incomplete worlds (as explained in FD). In talking about and appealing to such worlds we make however no ontological assumptions as to their existence or even as to their possibility. None of these worlds exist (including the real or actual world considered as the totality of what is the case: the fusion of things that exist is different), and some could not possibly exist. Hence our deployment throughout of the ontologically neutral quantifiers P (read 'for some') and U ('for every') of GR. Connected with this is our use of the term 'holding at (or in) a world' instead of 'true at a world'. The only world that determines truth is the factual one, T. Other worlds need not resemble the factual world in any way, and what holds in them does not provide some simulacrum of truth, nor, in terms of what these are about, of existence. There is only one way of existing (and of being true, and so no degrees of truth), and this way is determined by the factual world. Although we have retained the now standard term 'world' - deleting only the customary, but undesirable and ambiguous, qualification to 'possible' - the terminology is none too felicitous: for it suggests existing things where there may be none, it suggests completeness where there may be none, it may even suggest consistency though there need be none. A less misleading and more suggestive term than 'world' is the term 'situation', which we sometimes prefer.1 Apart from the neutral quantifiers, the other piece of metalogical terminology that calls for further connnent is the implicational symbol > (also read 'if...then...'). > can, at bottom, be construed as material implication, i.e. the metalogic can be taken as classical, extensional, logic. In this way at least, through a semantical translation into an extensional metalogic, we may succeed in convincing some classical diehards that relevant logics do make sense and are intelligible. But there is no need to construe > materially: it can, for example, be alternatively, and better, construed as the strict implication of Lewis's system S2, or as a STrong implication of a classical relevant logic, or as the 3-valued connective of GR. Ideally 1

The -underlying theory and metaphysics of worlds is the neutral theory of HUB, p.202 ff. 294$

4.3 A BOGUS DILEMMA AS TO THE CHOICE

OF

METALOGIC

however > should be replaced by a relevant implication; for relevant logics should supply their own metalogic, their own semantical explanation of their connectives, and their own proof standards - in precisely the way intuitionism requires. Though the main metalogical analysis which follows does not measure up to these standards, we have long believed that it should be a practical, if tedious, matter amending the presentation so that it does. (A beginning on such a task was made in GR where the semantics for a 3-valued significance logic are presented in terms of first-degree relevant logic.) Since the question of relevant semantics for relevant logics has become a rather more pressing matter, we proceed in chapter 15 to replace the metalogic used by a relevant metalogic. However, it is worth exposing here and now certain bogus dilemmas relevant logicians are sometimes confronted with as to the choice of their metalogic; for these dilemmas, though bogus, reveal points of far-reaching interest for the whole semantical enterprise. The first dilemma goes as follows:- Either the metalogic is classical or it is not. If it is classical then the relevant enterprise presupposes classical logic, is parasitic on it, and the whole semantics should be redone relevantly. If however it is not classical, then the semantics is not (extensionally) intelligible and so hardly merits bothering with. In any event there is little point in struggling intellectually with such an esoteric subject, since when intelligible it presupposes good old extensional logic anyva>. The dilemma is bogus because, as we have observed, the metalogic chosen can be either classical or non-classical (e.g. relevant, though not necessarily identical with the logic) and in either case yield valuable results concerning the logic under study, while in neither case destroying or prejudicing the relevant enterprise. It should be clear from the wealth of semantical studies that semantics, whether done classically, or, in the intuitionist case, intuitionistically, not only can furnish understanding of a system and its operations and new insights regarding a system1, but also can provide easy routes to many worthwhile formal results, e.g. results concerning decidability, compactness, separation properties, conservative extensions, interrelations with other systems, and so on. But if these results are established classically doesn't that presuppose the correctness of classical logic - something already denied in chapter 1? No, it is not assumed that classical logic is correct; it is only assumed that classical logic can be correctly applied in a very limited area, that of the applied predicate logic of the metatheory, which adds only a tiny fragment of orthodox set theory to classical predicate logic - apart from infinitely many individuals in the shape of wff, little more in fact than the virtual theory of sets. Our assumption - let us call it the classical assumption - is, in effect, that this so far unformalised logic, Meta-L, is negation consistent.2 The dilemma, though bogus, has some bite then; for on the classical option it forces us - for reasons we now try to explain - into an assumption we should much prefer to avoid. Suppose, what appears false but highlights the problem, that the classical assumption were tantamount to assuming the consistency of classical first order arithmetic - a matter which is hardly settled definitely 1

As Meyer explains in 'Reply to Professor Kielkopf', paper presented to the MidWestern Division of the American Philosophical Association, St. Louis, 1973.

2

There is nothing very unusual about (implicit) metalogical assumptions being made in metalogical proofs. But they are not often exposed, despite their importance in some cases, e.g. that of limitative theorems (on which see Routley 66a).

295$

4.3 WAYS OF AVOIDING

THE CLASSICAL ASSUMPTION OF METALOGICAL

CONSISTENCY

(from a relevant viewpoint) by classical consistency proofs which themselves presuppose classical correctness for a larger area than that for which the proofs establish it. Then, since Meta-L is strong enough for classical arithmetic, classically at least, by Godel's consistency theorem, our assumption is irremovable. It can only be traded in for equal or larger ones. It might be thought that, because stuck with this assumption while we use a classical metalogic, we are worse off than the classical logician. This is not so: the same assumption has really to be made classically. For suppose that a metalogic were inconsistent; then it would enable anything at all to be proved about the logic, including results that may not hold of it: without consistency any guarantee of correctness is, classically, lost. Accordingly the semantical arguments that follow are, classically at least, undamaged through the assumption. Unlike the typical classical logician, however, we resent having to make the classical assumption - though we believe it to be true - and resent having later results, which apply the semantics, depend implicitly upon it. So we are interested in finding ways of avoiding it. The prospects for doing this are good. One approach-is to start by reformulating Meta-L as a relevant logic, as R-Meta-L say (this involves judicious replacements of occurrences of > by either or a combination, which being determined in large measure by whether Modus Ponens is required). With R-Meta-L available we have at least two choices as to how to proceed. On th6 first choice we carry out the semantical and metalogical arguments relevantly simply avoiding incorrect classical assumptions. We have already claimed that this can be done (the claim is elaborated and substantiated in chapter 15). This procedure has some disadvantages, but they are not nearly as serious as the dilemma, tries to make out. Consider, in particular, the matter of intelligibility. It is a myth, put about no doubt by some extensionalists, that extensional discourse is intelligible but intensional discourse is not. Intelligibility knows no such boundary.1 The most hardened extensionalists can learn English, and understand, often perfectly well, the highly intensional discourse of urbanised daily communication of cotmnerce, television and newspapers. Extensionality is neither necessary nor sufficient for intelligibility; In particular intelligibility does not presuppose extensionality - which is as well since any sort of discourse can be given extensional expression (see Routley 75). And the price of extensional intelligibility is large, namely assumptions like the classical assumption. The other jibes used in trying to strengthen the dilemma go the same way as the intelligibility myth. Relevant logics are not particularly esoteric: no more so than many reaches supposedly of extensional logic (for examples consult almost any recent copy of the Journal of Symbolic Logic). Which of the subjects, if any, one chooses to study or work on turns on a range of factors; but esotericness is not a very important factor among these. As we have tried to show (both here and in other publications), there are substantial reasons for studying relevant logic, and this one can intelligibly do without any classical backing at all. The second choice, to return from warranted sounding off, is once again a more classical strategy. The idea is to prove that the rule y of material detachment is admissible for R-Meta-L. If so the Meta-L can be used without the classical assumption. Since proof of y would establish the classical assumption, the classical response is bound to be that once the metalogic is fully formalised Godel's second theorem will intervene once again to stop y being established 1

But in view of the classical insistence that the only intellibible operators are extensional ones, there is a serious practical point in offering extensional semantics for non-classical logics, namely that of making the logics classically accessible and less easily avoidable.

296$

4.3 A SECONV VILEMMA,

REVEALING

THE WORTHLESSNESS

OF THE SEMANTICAL

ENTERPRISE

in a non-question-begging way. Within a relevant framework like R-Meta-L this is, however, by no means so obvious (see the discussion of consistency arguments in UL, §10). Sadly this issue too will have to be left open at least until chapter 15. For though y has been shown admissible for relevant type theory, the issue as to whether y is admissible for relevant logics which make appropriate infinity assumptions has so far proved recalcitrant. Nonetheless we are moderately confident that y is admissible and have some confidence that its admissibility can be proved without use of y itself, and so we are prepared to replace the classical assumption by the stronger assumption that y is admissible for suitable R-Meta-L. All this leaves things for the present in a less than satisfactory shape. We are stuck either with the presumably routine, but laborious, task of reworking the semantical arguments within a relevant framework - something we have postponed doing - or of proving v in such a framework - something we may not, in the end, be able to do if too much of the metalanguage is written into R-Meta-L-or of accepting at least the classical assumption as a metalinguistic principle - an assumption yet to be established in an unproblematic way. So although the first dilemma is bogus, it leads to some considerable, if temporary, embarrassment. Fixing up the classical mess is, however, something that we prefer to leave until last. A second dilemma is sometimes adduced with a view to showing that the enterprise of furnishing semantics for non-classical logics is hardly a worthwhile activity, and accordingly that semantical analysis has little logical point. Since the conclusion is false - we have already seen something of the range of important results deeper semantical analyses throw out as corollaries - the argument is bound to be defective. The argument takes similar lines to the first dilemma. Either the metalogic used for the semantical analysis is the same as the logic studied, or it is different. If it is the same, as it should be to be faithful to the intuitions of the logic, then the semantics will be un— informative, simply a trivial mapping of the connectives into themselves. But if it is different, then the proof standards of the logic have been abandoned, so that results achieved metalogically cannot claim correctness. The intuitionistic case is put up as paradigmatic of the failure that is supposed to occur: a genuine intuitionist who rejects classical arguments on intuitionistic grounds ought not to accept results established semantically using a classical metalogic. Similarly as thoroughgoing relevant logician has abandoned the relevant enterprise if he accepts results established non-relevantly.1 According to the dilemma then, the semantics will be either uninformative or incorrect (cf. the paradox of analysis argument); so furnishing semantics is hardly a worthwhile enterprise. Once again some truths and some awkward facts lie behind the argument, but it is easy to see that the argument does not establish its conclusions. Firstly, where the metalogic is the same, as is typically the case in semantical treatments of classical logic, it is just not true that the semantics is uninformative, since much information flows from them once they are proved adequate. But the criticism has a point, that is frequently made, that the semantical rules provide little or no clue to the meaning of the connectives and quantifiers, since they are circular: for example, only if the sense of 'and* (or '&') has been grasped is the conjunction rule intelligible, and then it is unnecessary, so the argument goes. However, just as a grammar of English in English can be enlightening and informative, so a 1

So Kielkopf contends in 74a and 75.

297$

4.3

GETTING VOWN TO VETA1LS:

MOVEL STRUCTURES

FOR B

semantical analysis of classical logic within classical logic can also be. The grammatical analogy is useful too in getting to grips with the second horn of the dilemma. A German linguist who writes a grammar of German in English has not thereby abandoned the German enterprise (whatever that means) or admitted the superiority of English as against German. Somewhat similarly, as we tried to explain in the case of the first dilemma, a relevant logician who uses a classical metalogic for semantical investigations has not thereby conceded the correctness of classical logic (but only of classical logic in a given classical context). Nor has he merely and perversely established results that are relevantly unacceptable; for the results stand though they may not be assumption-free (indeed every such result depends, in one sense, on the methods of proof metalinguistically admitted). Correctness is not lost then; but correctness may involve some assumptions, such as the classical assumption. Likewise the relevant enterprise has not been abandoned; but hoy it can proceed, with assumptions rather more in the open, has, hopefully, been clarified. With the obstructive argument that our enterprise of furnishing semantical analyses for relevant logics - is not a worthwhile activity rebutted, we can return to details of the semantics. A B model structure (B m.s.) is a structure M = <0, K, R, *> where K is a set (of worlds), 0 is a one-place relation on K which is exemplified (or a non-null subset of K), R is a three-place relation on K and * a one-place operation on K such that the following definitions apply and postulates hold for all a, b, c, d e K: dO.

T

£x0x, i.e., T is an arbitrarily selected element of 0.

dl.

a < b = D f (Px)( x & Rxab), i.e. R ab,

where R[0] = D f (Pxe0)R[x], with R[x] signalling that x occupies some place in relation R. It follows from dl that 0a & Rabc >. b < c; where RTab, a < b, but not conversely. pi.

a < a, i.e. for some x for which Ox, Rxaa.

p2.

a < d & Rdbc >. Rabc

p3.

a = a**

p4.

a < b > b* < a*, i.e. if a < b then b* < a*.

and that

Use of both Hilbert's epsilon operator, £, and definition dO may be eliminated either by redefining validity in terms of holding at each element of 0 or better by adding a base element T, with T in 0, to the m.s., that is, in effect, an element T subject to the postulate p0.

0T.

The original m.s. implicitly assumed, what is equivalent, (Px)0x. A type 2 B model structure (B2 m.s.) is accordingly a structure M = , with K a set of scenarios, worlds, indices, points, language games, situation-contexts, or whatever. Correspondingly under the worlds interpretation 0 is a substitute for the set of (complete) consistent worlds, and the base T is the factual world.1 B m . s . and B2 m.s. are isomor1

Under the worlds interpretation the arbitrary selection of T may be suspect. Leibnitz, for example, thought the selection was not arbitrary, but of the best, thus assuming a ranking of possible worlds with a unique maximal element. Leibnitz's assumption of course imposes more (cont. on next page)

298$

4.3

VARIATIONS

phic structures. isomorphic L m.s.

ON THE HOVEL STRUCTURE ANV REVUCEV MODEL

STRUCTURES

Commonly we shall employ 'L m,s,' to refer to any among

The model structures differ from the m.s. used for systems R and NR and the positive entailment logics (in SE I-SE III) only in replacing the single element, called there 0, by a set of elements 0, the regular situations in which all theorems hold. The point of the change will be explained in the course of and following the completeness proof (theorem 4.10 below). The essential point is that the class of m.s. adequate for positive logics, reduced m.s., turns out to be insufficiently general to model weaker implication relations in the presence of negation. For systems T, R and M where schemes B3 and D4 hold 0 can however be reduced, as in the case of positive systems, to a singleton T. Thus a basic C m.s., for system C, i.e., B + {Bl, B3, D4} is a reduced structure M = . In reduced structures since T is the sole element satisfying 0, dl reduces to a < b = D f RTab (as in SE III and SE IV). But for weaker systems such as G this reduction is impossible: reduced structures alone prove to be inadequate (see chapter 5). Further ordering postulates such as p5.

a < b & b < c > .

a
and indeed postulates requiring that < is a partial ordering, and further monotonicity requirements such as p6. a < d & Rbca > Rbcd could be added to the requirements on a B m.s. without affecting the class of theorems of logic B, as the adequacy proof will show. There is evidently a class of isomorphic (and so equivalent) model structures for system B. Other variations worth recording are that B m.s. may be made fully monotonic by requiring as well as p2 and p6 p7.

a < d & Rbdc > Rbac

;

and that R and < may be introduced as independent relations subject to the conditions, beyond pl-p5, p8.

Oa & Rabc >. b < c, i.e. RObc > b < c, and

p9.

(Pd)(Od & Rdaa), i.e. ROaa.

Such L m.s. are called L3 m.s. We have given an intuitive interpretation of some of the semantical elements, namely of K as a class of situations, of 0 as the subclass of regular situations, and of T as the real,factual, or base situation. Analogous interpretations1 of the remaining elements of L3 m.s. are as follows:< is a relation of situational inclusion, i.e. a < b iff situation a is included in situation b. Rabc iff b and c are pairwise accessible from a, or, to take a more revealing modal analogue, iff a and b are compatible (footnote 1 from previous page cant.) structure than we need for purely implicational purposes, and indeed more than is desirable (see Routley 75). In fact for weak extensions of B the arbitrary choice of T cannot be given up, as lemma 4.5 below makes plain. 1 The interpretation of relation R and operation * are along modal lines, to show that these notions are neither less nor more intuitively intelligible than their modal counterparts. But subsequently more revealing intensional interpretations will be offered. On interpretations of the elements of L m.s. see also SE I, p.200 ff.

299$

4.3

MODELLING

CONDITIONS

FOR EXTENSIONS

OF SYSTEM B

relative to c, or conversely iff c is compatible with a and b. The last construal is intended to correspond to the modal interpretation of Rac (with two-place accessibility relation R) as c is possible relative to a. And just as the modal connection reduces, canonically, to: for every wff A, if A holds in situation a then O A holds in situation c, so the relevant connection reduces to its direct generalisation, namely: for every A and B, if A holds in a and B holds in b, then the fusion of A and B, A o B, holds in c. (The connective <>, of intensional conjunction, will be investigated in detail in chapter 5; but here it is worth recording that fusion coincides in system R with Lewis's fundamental consistency connective, o , for which, analytically, A <> B ~(A ~B) i.e. A entails B iff A is inconsistent with the negation of B. Modally, consistency, O , is of course the obvious twoplace analogue of possibility, 0 .) Finally, * is a reversal operation (of 2), i.e. a* is the reverse of a. (Topologically, where K is construed as a set of spatial objects, * is the ordinary reversal operation of turning inside out. Then T, in case T = T*, is an object like a Mobius strip which is unchanged by being turned inside out. *) The following postulates are added to B m.s. (or to subsequently introduced Bt or B° or Bt° m.s.), singly or in combination, to provide modellings for the wealth of further systems of sentential logic introduced in the previous section. The postulates we select are only some among equivalent postulates that will suffice for the corresponding axiom. Many of the postulates can be simplified using the definitions: d2.

R 2 abcd =

d3.

R2a(bc)d -

Df

(Px) (Rabx & Rxcd); (Px) (Rbcx & Raxd);

3

(Px) (R2abxe & Rcdx)

d4.

R ab(cd)e =

ql.

Raaa

q2.

Rabc > R 2 a(ab)c

q3.

R 2 abcd > R2b(ac)d

q4.

R 2 abcd > R2a(bc)d

q5.

Rabc > R 2 abbc

q6.

Rabc > Rbac

q7.

R 2 abcd > R 2 acbd

q8.

R 2 abcd > R3ac(bc)d

q9.

R 2 abcd > R3bc(ac)d

qlO.

Rabc > b < c

f

The postulate T < a, which implies qlO, and can stand in its place in positive logics (see SE III), is too strong in the presence of negation and postulate si below. Then, since a = a** < T* < T, the postulate leads to classical logic S, with its single situation T.

1

qll.

Rabc > a < c [or, equivalently given s2, in reduced m.s., T < a].

ql2.

R 2 abed >. a < d

The modal rendering of three-place relation R was suggested independently by Dunn; this interpretation of * was proposed by V. Routley, who was also one of the main architects of the underlying theory of impossible worlds. Other interpretations, and also derivations, of * are given in 2., 3 and 13, and of R in 13.

300$

4.4

VALUATIONS

AMD

INTERPRETATIONS

ql3.

Rabc >. a < c & b < c

ql4.

R 2 abcd >. Racd & Rbcd

ql5.

R 2 abed > (Px) ( b « x & c < x &

ql6.

a < b & 0 x >. a < x [or, equivalently, in reduced m.s. a < b >. a < 0].

ql7.

a < b v b < a

ql8.

Rabc >. a < c v b < c 1

ql9.

Rabc >. Rbac & a < c

Raxd)

q20.

Rabc & Rade > (Px)(b < x & d < x &. Raxc v Raxe)

si.

Rabc > (Px) (b < x & c* < x & Raxb*)

s2.

x* < x for x e 0 [or, in reduced m.s., 0* < 0].

s3.

Raa*a

s4.

Rabc > Rac*b*

s5.

a* < a

s6.

Rabc >. a < b*

s7.

Rabc & Ra*de >. d < c v b < e

s8.

Rabc > (Py)(Rac*y & (d,e)(Ry*de > d < b*))

drl.

RaOa, i.e. (Px)(0x & Raxa);

dr2.

Ox > x* < x

;

i.e. s2.

dr3.

a* < a

;

i.e. s5.

cf. r2 of chapter 5.

Thus the rules selected really add nothing new (as previously remarked). The postulate corresponding characterised as follows: qi

corresponds to Bi ;

si

corresponds to Di;

one-one to a given axiom (or rule) is

dri corresponds to DRi

The class of correspondences is enlarged as we proceed. Where L is a logic obtained by adding axiom schemata from those cited to B, an L m.s. is a model structure M obtained by adding the postulates corresponding to the additional axiom schemes of L to a B m.s.

%4. Valuations, interpretations, validity. Where M = (or an isomorphic structure) is an L.m.s. for logic L, a valuation v of SL (or SLt) in M is a function which assigns a holding-value from II = {1,0}, i.e. {on, off} to each element (sentential parameter) p in IWff at each set-up a e K subject to the restriction that it respect relation <, i.e. that for every a, b e K (1) a < b & v(p,a) « 1 >. v(p,b) = 1 Thus valuation function v is such that v(p,a) = 1 or v(p,a) = 0 for every initial wff p, subject only to constraint (1). The interpretation I associated with valuation v in % is a function SL xK in II t satisfying the following conditions for every wff in SL 1 This condition was subsequently found independently by Dunn: see Dunn 76.

301$

4.4

TRUTH, VALIDITY,

MODELS, AND L-IMPLICATI0M

and for every a e K: i.

I(p,a) =

v(p,a).

ii.

I(A & B,a) = 1 iff I(A,a) = 1 and I(B,a) - 1.

iii. I(A v B,a) = 1 iv.

I(A

v.

I(~A,a) - 1

iff I(A,a) - 1 or I(B,a) = 1.

B,a) = 1 iff, for every b, c e K, if both Rabc and I(A,b) = 1 then I(B,c) - 1. iff I(A,a*) ^ 1.

A wff holds on a valuation v, or on the associated interpretation I, in L m.s. M at an element a e K (or in world a), just in case I(A,a) = 1. Wff A is true on v, or on the associated I, in L m.s. M just in case I(A,T), = 1; and otherwise A is false on v, or on I, in M, and v or I falsifies A in M. A is ensured on v, or I, iff A holds on v at every element of 0. A is valid iji L m.s. M just in case A is true on all valuations therein; and otherwise is invalid in H. Finally, A is [reduced] L-valid just in case A is valid in all [reduced] L m.s.; otherwise A is [reduced] L-invalid. Equivalently A is L-valid iff A is ensured, i.e. holds at every element of 0 for every L m.s. (without a distinguished base element T). An L model M^ is a structure <M,v> (or, just as well, in the case of L2m.s.) where M is an L m.s. and v is a valuation in M. Truth, validity, etc., are defined for models just as before. An L-model My verifies every member of set S of wff of L just in case each member of S is true in M (i.e. true in M on v); otherwise M is a countermodel to S; and M^ falsiiYes every member of S just in case ev¥ry member of S is false in M^. §5. Semantic implication3 truth and soundness. Where v is one of the valuations introduced in L m.s. M, A implies B on v, or on I (or A v-implies, or I-implies, B) just in case for every a e K if A holds on I in M at a then B holds on I in M at a, i.e. [A] £ [B] where [A] {a e K: I(A7a) = 1 } . A implies B iji M" just in case A I-implies B on every valuation in M; and A L-implies B iff A implies B in every L m.s. The lemmata which follow simplify the proof of soundness. Lemma 4.1. Where I is an interpretation of SL in L m.s. M, then, for every wff A of SL and every a, b e K, (2)

if a < b and I(A,a) = 1

then I(A,b) = 1.

Proof. The argument is by induction on the length of A, with (1) as induction basis. In the induction step there are routine cases, which we illustrate, for each connective (and constant). ad -»•. A is of the form (B C). Suppose a < b and I(B -»• C,a) = 1. Then whenever I(B,c) = 1 and Racd then I(C,d) = 1. But where Rbcd and a < b then Racd, by p2. Hence I(B -»• C, b) - 1, as required. Lemma 4.2. Where I is an interpretation of SL in M, for every wff A, B of SL, (3)

if A implies B on I then A -»• B is true on I, and, where M is a reduced m.s., A I-implies B iff A + B is true on I;

(4)

if A implies B in M then A -*• B is valid in M, and, where M is reduced, conversely;

(5) A L-implies B iff A ^ B is L-valid. Proof: As to (3), suppose I(A,a) = 1 and RTab. Since OT, a < b, and so by (2), I(A,b) - 1. Hence since A I-implies B, I(B,b) - 1; whence A •* B is

302$

4.5

PRELIMINARIES

TO, ANV PROOF OF, SOUNDNESS

OF L SYSTEMS

true on I. The converse fails where M is not reduced because there is no guarantee that RTaa holds. Where M is reduced, suppose A B is true on I, i.e. I (A -*• B,T) = 1. Then, since for every a e K, RTaa, if I(A,a) = 1 then I(B,a) = 1, i.e., A I-entails B. (4) and one half of (5) follow from (3) by quantificational logic. It remains to show that if A ^ B is L-valid then A L-implies B. Suppose then A + B is L-valid. Let M be an arbitrary L m.s. with base T say and v be any valuation in M. Suppose further for arbitrary a e K that I(A,a) = 1 where I is the interpretation associated with v. It suffices to show that I(B,a) = 1, for then the desired result follows by quantificational logic. Now by pi for some element, d say, of K, Od and Rdaa. Consider an L m.s. M' which differs from M simply in change in base, by selecting d as base in place of T, i.e., where M = , M' = . M is an L m.s. since no modelling conditions depend on the choice of T as base. Define an identical valuation v' in M' thus: v'(p,b) = v(p,b), for every initial wff and every b e K. (Since K' = K there is no need to distinguish elements.) Then by a trivial induction, (a) I'(A,b) = I(A,b), for every b e K and every wff A. By (a), I' (A,a) =1. As A ^ B is L-valid, I* (A -*• B,d) = 1. Since Rdaa, it follows that I'(B,a) = 1 by the valuation rule for -»•. Hence by (a) again, I(B,a) * 1. Thus though the converses of (3) and (4) do not hold generally that of their generalisation (5) does, since although there is no one element to track back to in every case as central, pi ensures that in each case there is some appropriate element to go to as base. By taking the class of appropriate elements (3) and (4) may however be improved upon, thus: (3') [A] £ [B], i.e. A I-implies B, iff for every x € 0 I (A -»• B,x) = 1, i.e. I (A -*• B,T) = l.for every choice of T. (4') A implies B in M iff I(A B,T) - 1 for every valuation in M Proof: (4') follows~from (3'), and the new half of (3') is proved like (5). Theorem 4.9. If A is a theorem of L then A is L-valid, and therefore reduced L-valid. Proof. As usual we show that the axioms schemes are valid and that the rules preserve validity. (i) the rules preserve validity. Only R2 is immediate; it follows in virtue of the valuation rule for the connective &. ad Rl. Suppose, for arbitrary L m.s. H and valuation v in M, I(A,T) = 1. It suffices to show, given the L-validity of A -*• B, that I(B,T) = 1. But as A ^ B is L-valid A L-implies B by (5), whence the result. ad R3. Suppose B -> C -»-. A -»• C is not L-valid. Then for some valuation v and associated interpretation I in L m.s. M, I(B C A -»• C,T) = 0. Hence for some a, b e K, RTab and I(B -»• C,a) = T and I(A + C,b) = Q. Since OT and RTab, a < b; whence by lemma 3.1 I(A -»• C.a) » 0. Hence for some c, d e K, Racd and I(A,c) = 1 and I ( C , d ) 0 . Thus, since I(B C,a) - 1, I(B,c) = 0. Since I(A,c) = 1 but I(B,c) « 0, A does not L-imply B. Hence, by lemma 4.2 (5), A -»• B is not L-valid. The case for R4 is similar.1 1

The variation on the neater - overneat - method by which SE III shows that the rules preserve validity is forced by the failure of the converses of lemma 4.2 for unreduced models. The slicker methods can be reinstated for systems like C where reduced m.s. suffice. These methods in effect depend on the fact that material implication analogues of the rules hold, e.g., the verification of R3 in SE III in effect vindicates in passing the objectionable A -*• B =>. B -»• C -»•. A -»• C : see further chapter 5. 303$

4.5 GORY DETAILS

OF CASES IN THE ARGUMENT

FOR

SOUNDNESS

ad R5. Suppose for an arbitrary valuation v in L m.s, M and a e K, I(B,a) = 1. It suffices, given that A -*• ~B is L-valid, to show that I(~A,a) = 1, i.e., I(A,a*) = 0. Suppose, on the contrary, I(A,a*) = 1. Since A ~B is L-valid, A L-implies ~B, by lemma 4,2 (5); hence I(~B,a*) = 1. Thus by the valuation rule for I(B,a**) » 0, i.e., by p3, I(B,a) = 0, contradicting I(B,a) = 1. Hence I(A,a*) 4 1. (ii) The axiom schemes of the basic logics are valid. All except A9 are vindicated as in SE III, applying lemma 4.2. The vindication of A2 will illustrate the quite elementary method. ad A2. Suppose for arbitrary I in M that, for arbitrary a e K, I(A & B,a) = 1. Then by the rule for &, I(A,a) = 1. Hence A & B I-implies A; whence by lemma 4.2 (3), A & B ->• A is true on I. ad A9. Suppose I ( — A , a ) = 1; by v., I(A,a**) = 1, and by p3, I(A,a) = 1. The result follows by lemma 4.2. (iii)- The axiom schemes of L are valid in L m.s.: it suffices to show that where a postulate holds its corresponding axiom is appropriately valid. (Several of the B axioms have already been treated in SE III.) We illustrate the method, which again applies lemma 4.2. ad Bl. Suppose for a e K, I(A,a) = 1 = I(A -*• B,a); to show I(B,a) = 1. By ql, Raaa; hence by the interpretation rule for -»• if I (A,a) = 1 then I(B,a) = 1. ad B2. Suppose for arbitrary a in K, I(A -»• B,a) = 1 = I(B -»• C,a). In order to show I(A C,a) = 1, suppose Rabc and I(A,b) = 1. It has to be shown that I(C,c) = 1. Since Rabc, by q2, for some x in K, Rabx and Raxc. Then, unpacking assumptions by the -*• rule, I(B,x) • 1 and so I(C,c) = 1, as required. ad B3. Suppose for arbitrary a in K, I (A -»- B,a) = 1. To show I(B -*• C ->-. A •*• C,a) = 1, suppose Rabz and I(B -*• C,b) - 1. To show then that I(A -»- C,z) = 1, suppose Rzcd and I(A,c) = 1. It suffices to prove I(C,d) = 1. By q3, for some x in K, Racx and Rbxd. Then unpacking assumptions, I(B,x) • 1 whence I(C,d) = 1. ad B8. Suppose for arbitrary a, b, c, d, z I(A -*. B -»• C,a) « 1 = I(A -»• B,b) = I(A,c) and Rabz and Rzcd. If we can show I(C,d) = 1, then by the rule for and lemma 4.2, i(B8,T) = 1, as required. But by corresponding semantical postulate q8, for some x and y, Racy and Ryxd and Rbcx. First since I (A -»• B,b) = 1 = I(A,c) and Rbcx, I(B,x) = 1; then as Racy and I (A B C,a) = 1, I(B ->• C,y) = 1; and finally, as Ryxd, I(c,d) = 1. ad BIO. Suppose I(A,a) = 1 = I(B,b) and Rabc. Since by q9 b < c, I(B,c) = 1, by lemma 4.1. Hence by lemma 4.2, I(BIO, 0) = 1. ad Dl. Suppose, for a reduction, I(A & B -*• C,a) = 1, Rabc, I(A & ~C,b) = 1, but I(~B,c) 4 1, i.e. I(B,c*) = 1. By si, for some x in K, b < x, c* < x and Raxb*. Also I(A,b) = 1 = I(~C,b), so I(C,b*) i 1. Using ordering properties, I(A,x) = 1 = I(B,x), whence as I(A & B -»• C,a) = 1 and Raxb*, I(C,b*) = 1, giving a contradiction. ad D2. Either I(A,x) = 1 or I(A,x) i 1. Since for x e 0, x* < x, I(A,x*) + 1, whence I(~A,x) - 1. Thus I(A v ~A,x) = 1, for x e 0, whence I(A v ~A,0) = 1. ad D3. Suppose I(A -»• ~A,a) = 1; to show I(~A,a) = 1, i.e. I(A,a*) = 1. By s3, Raa*a, so if I(A,a*) = 1 then I(~A,a) = 1; hence either I(A,a*) + 1 or I(A,a*) = 1, i.e. I(A,a*) = 1. ad D4. Suppose I(A ~B,a) = 1, Rabc and I(B,b) = 1; to show I(~A,c) = 1 i.e. I(A,c*) 4 1. By s4, Rac*b*, so if I(A,c*) = 1 then I(~B,b*) = 1, i.e. given I(B,b) = 1, I(A,c*) 4 1. ad D5 1 . Suppose I(A &. ~A v B,a) = 1. Then I(A,a) - 1, I(A,a*) = 0 or I(B,a) = 1. But a < a*, whence I(B,a) = 1.

304$

4.6 AN OUTLINE

Of THE MAIN ARGUMENT

USEV TO PROVE

COMPLETENESS

ad BR1. Suppose I (A B - B,0) = F, Then for some a, I(A + B,a) = 1 and I(B,a) = 0. Applying drl, for some x, Ox and Raxa. Hence, as Raxa and I(A,x) = 1 imply I(B,a) = 1, I(A,x) = F where Ox. But this is impossible when A is a theorem and so valid in every m.s.

§6. Semantic completeness: argument. The structure

synopsis

of the argument

and the

detailed

of the argument is as follows: i. Given a non-theorem A of relevant logic L, we construct a suitable theory T which includes L but to which A does not belong. A theory is always a linguistic theory, a set of wff closed under certain operations. ii. In terms of T and L we of two ways:

design a canonical model structure in one or other

iia. for stronger relevant systems we form a class of theories based on T; iib. for weaker systems, and more generally, we consider a class of theories, based on L, and including T. The class of suitably closed theories forms the class K of worlds of the canonical model M and theory T is the base T of M . In terms of the c c c theories of K c , which are simply certain sets of wff, remaining components of the canonical structure are defined: T is defined as that subset of K such that for each a e T . L c a , i.e. a c c c — contains all theorems; for a, b, c in K , R abc holds iff whenever A -*• B e a c c and A e b then B e e , and for a in K ; a* is the set of all wff A such that c ~A is not in a. iii. It is shown that the canonical model so defined for L is indeed an L model structure. This includes showing, for example, that * is welldefined, and that relation R £ satisfies requisite modelling conditions. iv.

A canonical valuation V £ in M £ is defined in terms of membership:

specifically, for initial wff p, v c (p,a) = 1 iff p e a for a in K . It is shown that v is indeed a valuation in M . c c v. It is proved that for the canonical model holding coincides with membership, that is to say, for every wff D, I(D,a) = 1 iff D c a, for every a e K c > vi. The preceding details are drawn together to show that non-theorem A is not valid. For as A i T, I(A,T ) i 1, that is A is false in the canonical model <M ,v>, which is an L mode?. c The argument is a very substantial elaboration and refinement of the argument of Kalmar and others for the completeness of classical sentential logic (which was essentially an argument for the first degree fragment only). The argument can be seen as building on and varying the extension of the classical argument to modal logics. Filling out the sketch, however, requires much work, and involves two features of importance (which turn out to have wide applications in semantics), namely, first, the use of linguistic theories in defining the semantical model and, secondly, the cutting down of a wider class of theories to a narrower class with further desired properties in order to accomplish steps iii and v.

305$

4.6 BASIC NOTIONS CONCERNING

THEORIES

ANV THEIR KINVS USEV IN PROVING

COMPLETENESS

Where L is any one of the logics under examination, an L-theory T is a set of wff of L closed under adjunction and provable L-implication? i.e. T is an L-theory iff whenever A, B e T , A & B e T and whenever |- A ->• B, i.e. A -> B is a theorem of L, and A e T then B e T , for all wff A ana B of L. An L-theory T is regular iff all theorems of L are in T; prime iff whenever A v B e T either A e T or B e T; consistent iff the negation of some theorem of L does not belong to T; normal iff regular, consistent and prime. Let K^ be the class of L-theories and K^ the class of prime L-theories; for every a, b, c e K^, R^abc whenever A - B e a and A e b, B e c; 0 a iff a is regular; and where a is prime (v)a* = (A: ~A i a}. Finally where (as in SE III) L also represents the set of theorems of logic L, select T^ as L itself. Define the wide L m.s. M as the structure R ^ with base T^ = (with the relations R^ is the

L; and the canonical L m.s. M^ as the structure < 0 L » K^J R^J * > base T^ some appropriately chosen element of 0^), where the unbarred are the restrictions of the barred relations to prime theories, e.g., restriction of R^.

In order to prove the completeness with respect to reduced m.s., we restate all these definitions for a wider class of theories.1 Where T is any regular L-theory, a T-L-theory a is any set of wff of L which is closed under adjunction and under T-implication, i.e., whenever A B e T (also written J- A -> B) and A e a then B e a for all wff A and B of L. A T-Ltheory is an L-theory, since T is a regular L-theory; but the converse does not hold generally. A T-L-theory is regular iff all theorems of L are in a; prime iff whenever A v B e a either A e a or B e a; etc. A T-L-theory (or an L-theory) a is null iff no wff are in a; universal iff every wff is in a. Null and universal theories are degenerate. Where T_is a prime regular L-theory we define the following T-relations on T. Let be the class of T-L-theories and K^ be the class of prime T-L-theories. For every a, b, c e K^, R^,abc iff whenever A ->- B e a and A e b, B e c, i.e., (A,B) ( A - > - B e a & A e b > . B e c); T a iff a is regular; and where a is prime a* = {A: ~A f a}. Define the T-wide L m.s. M as the structure K^, R T >, and the T-canonical L m.s. ^

as the structure .

Needless to say

T subscripts are often dropped, especially from T, when the context is clear. If degenerate theories do not belong to K^, M^ is non-degenerate (n.d. for short). l The reader who prefers a simpler treatment and is not impressed by reduced L m.s. can disregard the complications of T-L-theories and can generallyread 'L' for 'T-L* throughout.

306$

4.6 FURTHER COMPLETENESS

PRELIMINARIES:

L-VERIVAB1L1TY

ANV

L-MAX!MALITV

For proper extensions of the basic m.s., a L m.s. fits, or an m.s. is fitting for L, iff for each of the axiom or rule schemes of L which extend it beyond a basic system the corresponding semantical postulate holds. In the proof details that follow we shall distinguish these cases: i) the case of general, or unreduced, modellings where we consider only L-theories; ia) the case of reduced modellings where T is only required to be a prime regular theory. We consider the ia) case separately partly in order to connect the proof details with those given in SE I and SE IV for such systems as R and E. The ia) case succeeds only for system C, i.e., B + {Bl, B3, D4}. and its extensions; however the extensions include such illustrious systems as E, R and T. In showing that the canonical L m.s. are indeed L m.s. the following preliminaries help.1 Where L is any (relevant) logic, a set T of wff is L-derivable from a set S of wff, written s l - T , iff for some A,,...,A in L 1' ' m S and some B, ,... ,B in T, 1- T A, &... & A B v.. . v B . An L-derivation of 1* 'n ' ' L I m 1 n A from set S of wff, written s j - ^ A , is a finite sequence of wff with A^ = A such that each member of the sequence either belongs to S or is obtained from predecessors in the sequence by adjunction or a provable L implication (i.e. in the latter case A^ is obtained from A^ since •*• A^) . An L-derivation of T from S is an L-derivation of some disjunction B, v...v B of wff B,,...,B of T from S. Hence of course T is L-deriv1 n I n able from S iff there is an L-derivation of T from S, and a set S of wff is an L-theory iff S is closed under L-derivation. A pair of sets of wff <S,T> is L-maximal2 iff (1) T is not L-derivable from S; (2) S u T « the set of all wff of L, i.e. every wff belongs to either S or T but not both. Where <S,T> is L-maximal, (i) S is an L-theory, and (ii) S is prime. ad i)

Suppose A, B e S, A & B i S.

Then by (2), A & B e T, whence S|- L T

since f ~ L A & B - » - A & B , contradicting (1). Then by (2) B e T, and so s|-

Suppose A e S and |- A -»• B but B I S. L contradicting (1). ad ii)

Suppose A v B e S , A ^ S , B ^ S .

T, L

Then A, B e T, and so S ]- ^T.

Lemma 4.3. (Extension lemma). .Let S and T be sets of wff of L such that T is not L-derivable from S. Then there is an L-maximal pair <S',T'> with S c S' and T £ T'. Hence too S is a prime L-theory. Proof: Let the wff of L be enumerated: A,,A-,...,A Define sets 1 2 n

1 The preliminaries are presented for case i): they can simply be reworked for case ia) - a claim finally made good In chapter 11. 2

These notions and the extension lemma below are adapted from Belnap 75: advantage has been taken of Dunn's way of setting out these results. 307$

4.6 PROOF OF THE FUNDAMENTAL

EXTENSION

LEMMA

S^, T^ recursively, for each non-negative integer i, as follows:S Q = S and T Q = T.

Given that S i and T ± have been defined, then S i + 1 and

T... are defined thus:l+l (i) Suppose S± u {A i + 1 > h L T ± «

Then

s

= i+1

s

and T ±

(ii) Suppose T^ is not L-derivable from S.^ u S ^

" T± u

i+1

U ^ } .

Then T i + 1 = T^ and

- S ± u {A i + 1 }.

S* = i US.i and T 1 = uT.. i i

(iii)

<S',T!> is L-maximal. Condition (2) for L-maximality is automatically fied since each wff A^ is put into one of S^ or T^ according to the satisfied specification of the construction. ^ruction. ad (1) Suppose on the contrary S'j-^T'. S' i say with i ^ 0, such that ^ b ^ * S

Then there is a smallest integer,

T h u s ' T ^ is not L-derivable from

i-1'

Suppose further S. u {A } |- T . Then too S = S. n , and 1—1 1 L 1—1 1 1—1 = T. u {A,}. Hence S. n |-_T. . u (A }. Let the wff of S, - and i l-l i l-l 1 L i-1 i i-1 T. . in this derivation be respectively B,,...,B and C,,...,C . Then i—x l m l n |-T B. &...& B C, V...C v A., i.e. I-B C v A. for short where L I m i n i ' i B = B. &...& B and C = C- v...v C . Since, however, S., , u {A.} 1U _ T . ., 1 m i n ' ' i-1 x L l-l for some conjunction B' of elements of S^ ^ and disjunction C' of elements

T

of T. i-l' f - T B" & L i Hence T

U_ B' & A C\ Thus where B" = B & B' and C" = C v C', ' L i ' A. C" and |-_ B" -»-. C" v A, , whence by DR2 |-T B" -»• C". L i L ^ is L-derivable from S ^ ^ , contradicting assumptions.

Thus it is not the case that S and S^ = Si.

u {A^}.

Then \- L B & A ±

non-derivability.

1"~1

u {A } |- T . 1 Li l —1

Then T

1

By assumption there is a derivation of T C.

But then S ^

u {A ± } b L T ^ ,

= T i~~l from

contradicting its

Thus (1) follows.

Lemma 4.4. (Priming lemma) Let T be an L-theory and U a set of wff disjoint from T which is closed under disjunction, i.e. for distinct wff A and B if A, B e U then A v B e U. Then there is an L-theory T' such that (i) T _c T', (ii) T' is disjoint from U, and (iii) T' is prime. Proof. U is not L-derivable from T: For suppose otherwise for some A. ,... ,A e T and B. ,... ,B e U, |- T A. &.. .& A B. v.. .v B . Then since i m i n L I m l n T is an L-theory B^ v...v B^ e T, but as U is closed under disjunction B- v...v B € U, contradicting disjointness of T and U. 1 m By lemma 4.3 there are sets T' and U' with U £ U' and T 1 an L-theory satisfying (i) and (iii) and such that U' is not L-derivable from T'. Since U' is not L-derivable from T', U ? and T' are disjoint whence (ii) follows.

308$

4.6 DETAILS OF COROLLARIES

TO THE PRIMING LEMMA

Corollaries 1. Where A is a non-theorem of L there is a prime regular Ltheory T such that A i T. 2.

For every a, b e K^ and c e K^, if R^abc then, for some

x e K^, b £ x and R^axc. 3.

For every a, b e K^ and c e K^, if R^abc then, for some

x e K^, a £ x and R^xbc. 4.

For every a, b e K^ and c e K^, if R^abc then, for some

x e K^ and y € K^, a £ x and b £ y and R^xvc. Where a, b, c 1 e Kj^, j^abc' and C t c 1 , then there is a

5.

c e It^ such that R^abc and C I c. Where a, b', c 1 e

6.

R^ab'c' and C i c', then there are

b, c e K^ such that b* £ b for which R^abc and C i c. Proofs 1. Take T in lemma 4.4 as L and U = {A}. Then T is an L-theory and A i T. Finally take T in 1 as the maximization of L. 2. (i)

Set U = {A: (PB) (A ^ - B e a & B ^ c ) } .

Then

U is closed under disjunction.

(ii) b (with b = T) is disjoint from U. Hence by (i) and (ii) lemma 4.4 applies to provide an appropriately prime theory x with b £ x. Also (iii)

Rjaxc.

ad (i)

Suppose A^, A 2 e U.

B^ i c, B 2 i c.

Then for some B^, B 2 , A^ -»• B^ e a, A 2 -»- B 2 e a,

Since c is prime B^ v B^ k c;

by T, A x v A 2

B 1 v B 2 e a.

and since a is an L-theory,

Hence by definition of U, A][ v A 2 e U.

ad (ii). Suppose^ otherwise A e b and A e U. Then for some B, A and B i c. But R^abc, whence B e e , giving a contradiction. ad (iii). Suppose A -»• B e a and A e x. A i U , i.e. whenever A -»• B € a, B e e . 3. (i)

Set U = {A: (PB,C)(|~ L A

Then since x is disjoint from U, B + C& B e b & C ^

(ii)

a (with a = T) is disjoint from U. RjXbc, where x = T*.

ad (i).

C

2 1

h V

c)}.

U is closed under disjunction

(iii)

A

B e a

L C

Suppose A ^ B 2 "" C 2 '

2 *

C

But by T8 h

B

since

L

Bx

e b

1 c

A 2 e U. '

B

2

€ b

'

Then for some B^, B 2 , C ^ C

1 ^

is

Prime5

and

^

v. B 2 -*• C 2

Cj

C

2 ^ ° '

T3.8, A 1 v A 2 B ± & B 2 -»•.

f~ L B 1 & B 2 C 1 v C 2 ; whence A 1 v A 2 € U. ad (ii). Suppose otherwise D c a and D e U. D p L B -»• C, B e b, and C d c. Since D e a, B

C 2 > A^ | - L B 1

C^

Hence B^^ & B 2 e b;

v

I"l B 1

C

1

V

Hence

*B2

C

^

v A2

2*

Then for some wff B, C e a, whence since I^abc

and b e b, C e c. ad (iii). Suppose B -»• C e x. Since x = T' is disjoint from U , B •+• C d U. Hence for every B', C' for which (- L B -»• C -»-. B* •*• C', if B 1 e b then C 1 e c.

309$

PROOF OF THE PRIMING

4.6 AN ALTERNATIVE

Thus, by identity, if B e b then C e c. 4.

LEMMA USING ZORN'S

LEMMA

So Rxbc as required.

Combine 2 and 3.

5. Take T in lemma 4.4 as c' and U = {C>. the result follows.

Then with c = T'

Combine 2 and 5. 1

6.

Lemma 4.4 and corollaries.2'

Alternative proofs using Zorn' s lemma.

The following form of Zorn's lemma (proved equivalent to the axiom of choice e.g. in Mendelson 64, p.198) is applied: Any non-null partially-ordered set, in which every chain (i.e. every totally-ordered subset) has an upper bound, has a maximal element. Proof of—lemma 4.4. Consider the class Y of L-theories containing T and disjoint from U. Y is a non-null set, by the hypotheses of the lemma; Y is partially ordered by set inclusion; and subsets of Y which are totally ordered by set inclusion are bounded by their union. Hence Zorn' s lemma applies, and delivers a maximal L-theory T' disjoint from U. It remains to show that T' is prime. Suppose, on the contrary, B v C e T ' , B ^ T ' , C i T', for some wff B and C. Form the supersets B & T' and C & T 1 of T' where B & T' = D f {D: (PA e T'). |- B & A + D } . AS B & T' and C & T' are L-theories containing T, they are, by the maximality of T', not disjoint from U. Thus for some wff A^ and A 2 , A 1 e B & T', A 1 e U, k^ e C & T' and A 2 eU. Hence for some B- , C X

X

e T', |-

L

B & B

X

+ A X

and |-

1J

C & C

X

-»• A_. Z

Hence where

D - B. & C-, |-T (B & D) v (C & D) -*-. A v A„. Thus j—, D & (B v C) -*•. X X L X Z Li A x v A2But D & (B v C) e T' hence A^ v ^ £ T'. But also A x v A 2 e U, by the disjunction closure of U, contradicting the disjointness of T' and U. Proof of corollary 1 by direct application of Zorn's lemma. The set of theorems of L is a regular L-theory without A. Order the set of all regular L-theories without A by set inclusion. The set is non-null and is partially ordered by set inclusion. Furthermore every chain in the set is bounded by its union. Hence Zorn's lemma applies to give a maximal regular L-theory T, without A. Further T is prime; B v C e T, B i T, C I T. [T,B] [T,C] (1)

= =

for suppose otherwise that for some wff B, C of L, Form the sets

{D: (PA e T). {D: (PA e T).

A & B •*• D e T} A & C D e T}

[T,B] is closed under adjunction.

Then for A, A 1 e T, | "

T

A&B^D,

ponens | ~ t A & A ' & B + . D & E .

Let D, E e [T,B].

|~T A ' & B -»• E.

Hence by A2-A4 and modus

Since T is closed under adjunction A & A' e T,

whence D & E e [T,B]• 1

This neat way of replacing theories by prime theories borrows from Fine 74.

2

The point of these text with those of weapons than those problems left open

alternative proofs is to link the proofs given in the SE 1-SE V, and to provide alternative, more powerful, furnished by the extension lemma for the attack on in subsequent chapters.

Note that whereas the extension lemma adds wff one at a time, Zorn's lemma delivers the whole set at once.

310$

4.6 ALTERNATIVE

PROOFS OF PRIMING

COROLLARIES

(2) [T,B] is closed under T-implication, and hence, since T is regular, under orovable L-implication. Suppose D e [T,B] and D -*• E e T (or, in the other case V l D •* E). Then for some A e T, (- A & B -* D; so |- A & B -»• E and E e [T,B]. Thus [T,B], and similarly [T,C], is an L-theory. (3) T c [T,B], and so [T,B] is regular. [T,B] properly includes T since B e [T,B] but B k T.

Similarly for [T,C].

Since [T,B] and [T,C] are regular L-theories which are supertheories of T and T is maximal, A e [T,B] and A e [T,C]. Thus for F, F' e T, h T F & B •*• A and (- t ~F' & C •*• A. Thus where H = (F' & F) € T, |-T H & B -> A and |- T H & C by A8 and R3, J-

A.

Thence by A7, |- T H & B v H & C •*• A;

H & (B v C) •* A.

is impossible.

and

But H & (B v C) e T, whence A e T which

Accordingly T is a prime theory.

The lemma cannot be extended generally to theories closed under the rules of weaker logics such as B and G, e.g. it is false that where ~|~ r A there is a prime regular G-theory T closed under the rules of G (or just Affixing) such that A i T. _ Proof of 3, i.e. RjXab > (Pz e K ^ (x £ z & R ^ a b ) for x,a e and b e K^. Suppose Rj^xab, for x, a e K^ and b e K^.

Let Y be the set of all L-theories

y such that x £ y and R^yab. Y is a non-null set partially ordered by set inclusion, totally ordered subsets of which are bounded by their union. Hence Zorn's lemma applies and provides a maximal L-theory z such that x £ z and R^zab. It remains to show z is prime. Suppose, on the contrary, B v C e z, B e z, C € z for some wff B and C. Form the supersets B & z and C & z of z where B & z {D: (PA e z ) B & A -»• D}. B & Z and C & z are L-theories, both of which properly include z: hence by maximality of z, R^(B & z)ab and R ^ C 6> z)ab both fail. It follows that for some wff, Aj, A^ y B ^ B 2 » A^ e a, A 2 e

a, Bj^ / b, B 2 ^ b, A^

Hence for some A 3 > A^ e z, |-B & A 3

Bj^ £ B & z, A^ ^ B 2 £ C & z. A

-*• B ^

Hence by A2, A3, T6, |- (B & A) v (C & A) is (A 3 & A 4 ).

Thus using A8,

(A^ -> B ^

(B v C) & A -f.

Hence since B v C, A^, A^ £ z, (A^ -»• B^) v (A 2 A^ & A 2

B 1 v B 2 e z.

(- C & A^

(Ax

A2

v (A 2

B2. B 2 ), where A

B][) v (A 2 -»• B 2 ) .

B 2 ) e z.

So by T8,

Since, then R ^ a b and A^ & A 2 e a, B 1 v B 2 e b.

But b is prime, so B^ e b or B^ e b, contradicting B^ I b and B 2 I b.

Thus

z is prime after all. Proof of a £ b > (Pz e 1^)(O^z & R^zab), for a, b e K^. a, b e K^.

Then R^Oab, and also 0^0.

and R^zab.

Since 0 £ z, O^z.

Suppose a £ b for

Hence by 3, for some z e K^, 0 £ z

Proof of 4, i.e. R^abc > (Px,z e" K^)(a £ x & b £ z & R^xzc), for a, b e K^ and c e K^. Suppose R^abc for a, b e K^, c e K^. RjXbc.

By 3 for some x e K^, a £ x and

Now much as before let Y be the set of all L-theories such that

b £ y and R^xyc.

Zorn's lemma supplies a maximal element z with the right

311$

4.6 PROOF THAT THE BASIC CANONICAL

HOVEL STRUCTURES

ARE INVEEV MODEL

STRUCTURES

properties, provided z is prime. Suppose otherwise B v C e z , B ^ z , C ^ z , and form B & z and C & z as before. Then for some wff A ^ e B & z , A 2 e C & z, B^ i c, B 2 i c, A^ -»• B^ e x, A ^ |- C & A^ -> A 2 -

Thus

B^ e x;

hence for A^, A^ e z,

f- (B V c ) & (A 3 & A^)

Hence, since by T6, A^ v A 2

A1 v A ^

B & A^ + A.^,

so A 1 v A 2 e z.

B^ v B 2 e x, and since RjXzc, B^ v B 2 e c.

But c is prime so B^ e c or B 2 e c, which is impossible. Leimna 4.5.

(I)

The canonical B m.s. is a B m.s.

(IA) Where T is a prime regular L-theory for the bottom system C, the T-canonical L m.s. M^ is a reduced C m.s. (II) For each extension L of the basic systems, the [non-degenerate] canonical L m.s. fits; thus the m.s. M^ is an L m.s. (IIA) Where T is a prime regular L-theory for an extension of a bottom C system, the [non-degenerate] T-canonical L m.s. fits. (We pick up the qualifications, where they are needed, to non-degenerate L m.s. in the course of the proof.) Proof of (I) and (IA). As a preliminary to establishing the postulates, it is shown (vi) a < b iff a £ b, for a, b e K^, where < is defined in terms of C>L and

and £ is set-inclusion.

but also a £ b > R^Lab; a < b.

Since O^T » R j ^ a b > (Px) (0 L x & RjXab) ;

and hence a £ b > a < b.

Then for some x, x is regular and R^xab.

whenever A e a, A e b, i.e., a £ b . though a < b

For the converse suppose Hence since A -»• A e x,

Thus (vi) holds for a, b e K^.

>. a £ b for a, b e K^ since the formula

Now

can be put into

prenex universal form, and thereupon appropriately restricted to prime theories, for the converse, a £ b >. (Px e K^) (0 L x & R^xab), with a, b e K^, it has to be shown that x can be replaced by a prime theory z of K^. But that it can be is immediate from corollary 3 to the priming lemma. Proof of (vi) in the case of a, b e Kj, for (IA), follows similar but even briefer lines. For since, in the case of a reduced m.s., the postulates and (vi) hold in prenex universal form they also hold automatically for the restriction of elements to K^, and for restrictions to non-degenerate elements. Given (vi)» postulates pi and p5 are immediate, and p2 and p6 follow using the definition of R^, or R^,. To complete the proof of (I) and (IA) it remains to show (a) that * is a well-defined operation on K^, or on K^, as the case may be, and (b) that p3 and p4 hold. We provide the details for (IA): those for (I) are similar. ad (a). Let a be a prime theory in K^,. (ai) a* is closed under T-implication. Suppose A e a* and A B Then ~A i a and |- ~B ~A, by R4. Hence ~B i a, whence B e a*. (aii) a* is closed under adjunction. Suppose A, B e a*; then ~A I a and ~B i a. Hence since a is prime ~A v ~B I a. Hence since |-~(A & B) -*-. ~A v ~B (T4.3), ~(A & B) i a, i.e. A & B € a*, (aiii) a* is prime. Suppose A v B e a*. Then ~(A v B) i a. Since |- ~A & ~B ~(A v B) (T4.4), ~A & ~B i a. By

312$

4.6 SYSTEM-EXTENDING

AXIOM SCHEMATA

ENSURE CORRESPONDING

SEMANTICAL

POSTULATES

adjunction either ~A i a or ~B i a, i.e. either A e a* or B e a*, (aiv) a* is non-degenerate where a is non-degenerate. Suppose a* is universal [null], i.e. every [no] wff belongs to a*; then no [every] wff belongs to a, contradicting a's non-degeneracy. (av) * is an operation. The result is now immediate from its set-theoretic definition. ad p3: immediate from T4.5 and the definition of *. ad p4. Suppose a £ b and, for arbitrary A, A e b*. Then k i b, so ~A I a, so A e a*. Hence a £ b >. b* £ a*, and p4 follows by (v). Proof of (II) and (IIA). We assume one axiom scheme at a time and prove its corresponding semantical postulate (or postulates). ad ql. For a e K^, suppose for arbitrary wff A, B, A -*• B e a and A € a. By adjunction A & (A -»• B) e a, and by assumption of Bl, A & (A B) -*• B e T; hence by T-implication B e a. Thus, by definition of R,j,, I^aaa. Proof that R^aaa, for a e K^, is the same. The stymie of proof follows similar lines for most postulates that can be put in prenex universal form. For those that cannot_we proceed to establish the required postulate formulated in terms of R^ or R^; then we cut down to prime theories, non-degenerate theories, and to relations R^, by applying corollaries to lemma 4.4 or by applying Zorn's lemma directly. (The treatment of q2 in SE II, lemma 6, typifies this procedure: q2-q6 are already vindicated in SE III and q7-q9 are similarly established.) We sketch the procedure in the case of q2, q3 and q7. ad q2. Suppose that for a, b c e K^, R^abc. Define x as {C:(PB)(B -»• C e a & B e b)}. Then (a) x is a K- theory, (b) FLabx, and (c) ILaxc-, (b) is L L L immediate. ad (a). »2

Suppose

C^, C^ e x.

Then for some B , B 2 > B.^

C 2 £ a, B 1 £ b and B 2 e b.

& C 2 e a, and also B^ & B 2 e b. and l - !^! h

C2*

-»• C^

ad (c). B^^ e b.

Bj^

Then for some

B

C 2 , hence B^

Thus C^ & C^ e x. 1» B1

By Prefixing

C^e a, as a is a I^-theory.

Thus C

As A e x, for some B^, B^

A) & (A -*• D) e a, B^

C^

Suppose next C^ e x

C 1 £ a and B^ e b.

Suppose A •*• D e a and A e x. As (B 1

C^ e a,

Hence, by Praeclarum (T4.7), B^ & B 2

D e a, by B2.

e x.

A e a and

Hence as Rj^bc and

B^ e b, D e c, as required. Finally, by corollary 3 to lemma 4.4, there is a z e K^ such that x £ z and R^azc. Since x £ z, R^abz, whence q2 follows. ad q3. Suppose that a , b , c, d, z e l ^ , R^abz and R^,zcd. As in the case of q2, define x as {C:(PB)(B -»• C e a & B e c)}. Then, similarly, x £ ^ ^.acx, and R^bxd. Only the last is somewhat new:- Suppose A -*• D e b and A e x. Then for some B^, B^ -»• A e a and B^e c. Since B3 e T, A -»• D B -»• D e a, whence as R^abz, B -»• D e z.

Since further R^zcd, D e d, as required.

It remains to be shown that x can be replaced by an appropriate y e K^. Since R^bxd, by the argument given for corollary 2 of lemma 4.4, for some y £ K^,, x _c y and R,j,byd. Thus too, since x £ y and R^,acx, Rj,acy. ad q7.

Suppose for a, b, c, d,.y e K , R aby and R ycd. Define x It I» -L (A -»• B e a & A e c)}. Then x is an L-theory and Racx. To show suppose C -*• D e x and C e b. Then for some wff A, A (C D) e

313$

= {B: (PA) — R^xbd, a and A e c.

4.6

MORE PROOFS THAT

AXIOM

SCHEMES

VO VERIFY

CORRESPONDING

MODELLING

CONDITIONS

so by B7, C (A -+ D) a._ Hence_as Raby, A D e y; and as Rycd, D e d. Hence R^2abcd > (Px e K^)(R^acx & R^xbd); then application of lemma 4.4 yields a prime x in place of x e K^. To vindicate postulates such as qlO and qll we appeal, however, to the non-degenerate canonical model: we illustrate for the T-canonical model. ad qlO. Suppose R,j,abc for a, b, c e K^, and let B be a wff in a. Then by B9, A ->• A e a, whence if A e b then A e c; in short b £ c. ad qll. Suppose R^abc, and B e b for some B. Then by Bll, if A e a, B A e a; s o a s B e b , A e c ; whence a £ c. ad ql2. Since, by B12, B C A -*• A, C -»-. A ->• A, so it can be assumed that situations are non-degenerate. So suppose R^abx, R^xcd, A £ a, B e b, C e c for arbitrary wff A and for some specific B, C. Then applying B12, A e d. ad ql3. B13 is equivalent to B9 and BIO. ad ql4. Straightforward from the definition of R^ ad ql5. Suppose R^abz and R^zcd with a, b c, z e K^ and each non-degenerate. Non-degeneracy can be assumed because B15 yields Bll. Define y = {A: (PD e b, E e c) J-^D & E -> A}. Then y is an L-theory. Further b £ y and c £ y, using non-degeneracy. Suppose A B e a and A e y. Then for some D € b, E e c, |- D & E A; s o ( - A - > B ^ - . - D & E ^ B , and D & E -* B e a. Hence by B15, D E -* B e a. Thus by R^abz,_E B e z, and, by R^zcd, D e d. Reassembling (Py e K^)(b £ y & c £ y & R^ayd). Now apply lemma 4.4 or Zorn's lemma to replace y by a prime x e K^. The proof in the case of reduced m.s. is similar. ad ql6. Assume non-degeneracy and suppose B I b. Now suppose a £ b and and A e a, for a, b e Since A v. A -»• B e Tj» A e T L or A B e T^. If A B e T^ then as A e a, B e a, s o B e b contradicting an assumption. Hence A e T , as required. ad ql7. Suppose ~(a £ b) and A e b; it suffices to show A e a. For some wff C, C e a and C I b. Hence A C £ T^; and so, by B17, C A e T^. 1 But C e a, so A e a. ad ql8. Suppose R^abc and a £ c and A e b. Then for some wff B, Bfc- a but B i c. Since B e a , A v B e a and, by B18, A v B + A v B e a. Hence since A v B e b and R^abc, A v B e c. But c is prime and B I c so A e c, as required. K^, may be degenerate. ad ql9. It follows from B19, b L A A -»• B ->• B and B A B B. Hence as in the case of q6, R^abc > R^bac. To show R ^ ^ c > a < c, suppose R^abc, A € a, and for non-degenerate b, C e b. Since |-A ->-. (C A) -*• A -*• A, by substitution in B ->•. A B + B, (C->-A)->A->-A€a; but (C -*• A) -> A e b; hence A e c, as required. ad q20. Suppose R^abc and R^ade and define x = {D: (PA e b, B e d) |-^A & B D}. Then x e K L and b < x and d < x, since j-jA & B -»• D iff j~L A -*• D v. B D, using B20. Now suppose C -* D e a and C e x; it remains first to show D e c or D € e. Since C e x for some A^ e b, B^ e d, (- A^ & B^ C. Hence f- C -»• D A^ & B^ D, whence A^ & B^ D e a. Thus by B20, A ->• D v B. -»• D e a, and as a e K , A.. ->- D e a or B 1 -»• D e a. Suppose J. J. Li X X A^ -*• D e a, then as A^ e b, D e c, as required.

The remainder is proved

by lemma 4.4 or Zorn's lemma. ad si. Define y = A:{(PB e b, C e c*) |- B & C -*• A}. Then it is immediate, using non-degenerate theories, that b £ y and c* £ y. And as in the case

314$

4.6 THE MAIN STRONG COMPLETENESS

THEOREM

STATEV

of ql5, y is an L-theory. It remains to show R^ayb*, for then lemma 4.4 can be applied. Suppose, for reductio, A D £ a, A £ y and D i b*, i.e. ~D e b. Since k e y , for some B e b and ~C I c, |-B & C ->• A. Since then, |- A D -*. B & C -»• D, B & C - > - D e a , whence by D9, B & ~D ~C e a. But B & ~D e b and R^abc, so ~C e c, contradicting ~C I c. ad s2. In the reduced case, since A v ~A e T, A e T or A ^ T*; thus T* £ T. Similar in the general case where A v ~A e x^ for x^ e 0^; then x^* c x^T ad s3. D3', ~A ad s4. B i b*. ~B e b,

Suppose A -*• B e a and A e a*; to show B e a. Then ~A I a and, by v B e a. Thus as a is prime, ~A e a or B e a, whence B e a. Suppose for a, b, c e K^ (or K^) , ^ a b c , A B e a and A e c*, but By D4, ~B ~A e a, so as F^abc, if ~B e b then ~A e c. But so ~A e c, contradicting A e c*.

ad s5. Suppose a is non-null: then for some wff B, B e a. Hence since B A v ~A, A v ~A e a, whence a* < a. If a is null, however, a* < a is false, and the argument fails. Thus in the completeness argument for S5 null situations have to be excluded, as is done in 5.4. But a neater repair is to represent the null situation e within the semantical theory, and to specifically exclude the case where a = e. ad s6. Suppose, in the non-degenerate case with B i c, that R^abc, A e a and ~A e b. By D6, ~A -*• B € a, so B e c, a contradiction. Hence when R^abc if A « a then ~A k b, i.e. A e b* as required. ad s7. Suppose I^abc, I^a*de, A e d , B e b , A ^ c , B ^ e . Then A B i a*, so ~(A + B) £ a; hence, by D7, B -* A £ a. But since R^abc, B ->• A I a. ad s8. Suppose R^abc, and define y = (D:(PB)(B -»• D e a & B e c*)}. Then where a, b c £ K^, y e K^ and R^ac*y. Suppose next R^y*de, where y* • {A: ~A i y} and, for a reductive argument, A e d and A i b*. Then ~A e b. Let a be non-degenerate; since D8 implies BIO this is permissible. Then some E i e; so A -*• E i y*, and ~(A E) £ y. Therefore some wff B, B ~(A -*• E) £ a and B £ c*. Hence by D8, ~A -»• ~B e a; and also ~B I c. Since R^abc and ~A £ b, ~B e c however, which is impossible. Hence there is a 7 £ K^ with the requisite properties: the remainder is established by applying lemma 4.4. _ ad drl. Let x be the class of theorems of L. Then O^x. Suppose A -»• B e a and A e x, for a £ F^. Then as h L A , |- jA B B by BR1. Hence B £ a. In short (Px e (OjX & I^axa). Finally apply lemma 4.4 or Zorn's lemma. ad dr2, dr3. The rules yield A v ~A and B -»• A v ~A respectively. Theorem 4.10 (I) Where U is a set of wff not L-derivable from regular set S of wff, there is a [canonical] L-model which verifies every member of S and falsifies every member of U. (IA) Let L be any one of the extensions of system C considered. Where U is a set of wff of L which is not L-derivable from regular set S of wff of L, then there is a reduced L-model (i.e. a valuation on an L m.s.) which verifies every member of S and falsifies every member of U. Proof. Let S 1 be the theory-closure of S, i.e. S 1 = {D: ( P B ^ . . . ^ £ S) &...& B^ -*• D},

and let U^ be the ideal-closure of U, i.e.

A U. ={C: (PA 1 ,...,A m £ U) 1|-T C i V-"V 1 1 m L 1 m Then i) S^ is an L-theory containing S; ii) U- which contains U is closed under disjunction; iii) S* is disjoint from U .

ad i).

Suppose D e S^ and j-jD -»- E.

Then for some wff B 1 »...,B n e S,

315$

4.6 THE BASIC PROOF OF STRONG COMPLETENESS:

THE 1NVUCT1VE

STEPS

f-T B, &...& B -»• D. Hence by R3 I-T B- &.. .& B E. So E e S. as ' L I n ' L I n 1 as required. Suppose next D e S^ and E e S^. Then for some B ^ , — B _ . , C lt ...,C^€ S, | - L B^ &...& B^

D and |~L C^ &...& Cfc

E.

Hence

|-T B, &...& B. & C, &...& C, D & E, whence D & E t S, . L 1 j 1 k i ad ii). The argument that when C, D e U ^ C v D e is dual to the argument just given for adjunction-closure of S. . ad iii) Suppose, on the contrary, that some wff A belongs to both S^ and Uj. Then for some wff B-,... ,B € S and C. ,... ,C e U» h T B i & . • B A and i n l m L l n |~tA ->. C, v...v c , whence I - T B, &...& B ->-. C, v... v C , and so S I~TU. ' L 1 m ' L I n l m L Since S^ is an L-theory disjoint from U, lemma 4.4 applies to provide a prime regular L-theory T which includes S and S^ but is disjoint from U^. In case (I) form the canonical L m.s. M with base T^ = T. By lemma 4.5, M is an L m.s. In case (IA) form the T-canonical L m.s. JJpAgain by lemma 4.5, M^, is a reduced L m.s. A canonical valuation v^, [v^] of the language of L in M^ [M^] is defined thus: (v)

for every sentential parameter and every a e K^ [a e K^],

v T (p,a) = 1

iff p e a

[v L (p,a) = 1

iff p e a ] .

It is shown by induction from basis (v), (III)

I(D,a) = 1

iff D e a, for every a e

where I is the valuation associated with v^ in

[a e KjJ and every wff D of L, [v^ in M^].

The inductive steps for &, v, ~ follow at once from the induction hypothesis and the prescription for I, as the case for ~ illustrates, ad I(~B,a) = 1 iff I(B,a*) ^ 1, by the interpretation rule. iff B t a*, by induction hypothesis, iff ~B e a, by definition of a*. ad The inductive step for -»• breaks into two cases, depending on whether reduced models occur or not. i).

a e K^.

If B -*• C e a then, if R^abc and B e b then C e c for every b

and c in K^, in virtue of the definition of R^. tion hypothesis and the interpretation rule for

Hence applying the inducif B

C e a then

I (B + C,a) = T. For the converse, suppose B ->• C i a. It has to be shown that for some b, c e B e b, C i c and R ^ b c ; for then the result, that I(B -*• C,a) ^ 1, follows using the induction hypothesis and the interpretation rule for -*. 1'irstly define b' = {D: |-^B -»• D}. Then b' e K^; for closure under adjunction fol3ows from A4, and closure under provable L-entailment from R3 or R4. Define c' = (C: X 1 " 0 ) ^ e a & D e b')}; then as in the case for q2 in lemma 4.5, c' e K^, and R^ab'c'. Further B e b f , but C i c'. For suppose otherwise, C e c ' . Then for some Bi, Bj + C e a and f- B ->• B^. Hence by R3, |— B^ -»- C B •*• C: thus B C e a, contradicting assumptions. Hence for some b', c' e K^, R^ab'c', B e b' and C I c'. The conditions for corollary 6 to lemma 4.4 are satisfied, and accordingly there are b, c e K^ such that R^abc, C i c, and, since b' £ b, B e b. Alternatively, Zorn's lemma may be applied directly to replace b* and c' by appropriate b and c in K^, as follows. Maximalize first on the 316$

4.6 FURTHER DETAILS IN PROVING

STRONG

COMPLETENESS:

T-ENTAILMENT ANV

NON-VEGENERACy

class of z such that R^ab'z and C / z to obtain a, c e K^ such that R^ab'c and C i c, and second on the class of y such that R^ayc and b' £ y, to obtain a, b e K^ such that R^abc and b e y .

The second case has already been

supplied in the alternative proof of lemnui 4.4.

As for the first, maximalise

on the class Y of L-theories z such that R^ab'z & C I z. vides an L-theory c such that R^ab'c & C I c.

Zorn's lemma pro-

To show c is prime, suppose,

on the contrary, A v B e c, A / c, B / c. Form A & c and B & cj^ as before these sets are L-theories which properly include c. But since R^ab'c & (c £ A & c) >. R^ab'CA & c ) , it must be the case, as c is maximal, that A & c 4 c and B & c I c; but as in lemma 4 this leads to a contradiction, ia) a e K^. The case (which is detailed in SE III) is similar to (i) except in the following respects:- b' is redefined as {D: B D e T}. Then b' is a T-L-theory. c', defined c' = {C:(PB)(B -»• C e a & B e b')} as before, is < T-L-theory. But showing that it is, that when A e c' and A C e T then C e c', requires B4. By assumptions D A e a and D e b ' , for some D; what is to be shown, since D e b ' , is that D -*• C e a. What is thus required is that D A D C e T, and given only A ->• C e T, B4 is essential to bridge the gap. In order to show, moreover, that C I c' it is essential to apply B3 as the argument attempting to derive a contradiction from the hypothesis that C e c' reveals. For some B^, B^ C e a and B -»• B^ e T, but in order to show, contrary to assumptions that B -*• C e a, we need to know that B^ •+• C -*•. B C e T, which B3 guarantees, but suffixing does not. Thus closure under L-entailment, as opposed to T-entailment, is essential in weaker L systems to complete the induction step for But closure under L-entailment does not ensure that for T (constructed as in lemma 4.4) R^Taa holds, because L is in general properly included in T. Hence in order to design a canonical model for L where the language of L includes negation, we are forced to replace T = by a set to which T belongs, but which also contains non-uniformly, for each a e K^, an element x such that R^xaa. Where JJj is non-degenerate further details are required in establishing hypothesis (III). The rules for connectives & and v, since they do not introduce new situations, cannot introduce degenerate situations; and the rule for ~ only introduces a* which is not degenerate where a is not. One half of the rule for -*• is unproblematic since the universal quantification can be restricted to exclude degenerate situations. For the other half we supposed, in case (ia), that B -*• C I a and found b, c e K^, such that B e b, C i c and R^abc. It has to be shown, where a is non-degenerate, that we can find non-degenerate b, c with the appropriate property. Suppose as before B -»• C I a and a is not degenerate, so that D e a for some wff D. There are two cases. First, |- B -*• C. If either BIO or Bll is a theorem, the main cases in point, then D -*•. B B (in case of Bll, since f-B •* B). Hence B -»• B e a. Thus as |- B B B + C, B - ^ C e a contradicting the assumption. (In short for any condition a, a B •*• C e a). Second, then ~f-B -»• C. Define b' = {B': f-B B'}, and c' = { C : (PB')(B' •* C'e a & B' e b'). Then b' is non-degenerate since B e b' and C I b', and c' is not-degenerate since B e e ' and C i c', the latter being proved as before. Finally in applying lemma 4.4

4.6 COROLLARIES

OF STRONG COMPLETENESS,

SUCH AS COMPACTNESS

or Zorn's lemma we maximalise on non-degenerate situations.1 the proof of (III). Since for A e S, A e T, by (III) I(A,TL) = 1;

hence ^

Further since for B e U, B ^ T, by (III), I(B,Tt) ^ 1;

This completes verifies A.

hence M

falsifies B.

Corollaries 1 (Completeness)• If A is L-valid, or reduced~L-valid in the case of extensions of system C, then A is a theorem of L. Proof. Suppose A is a non-theorem of L. Set U = {A} and S = L and apply theorem 4.10. 2 (Implicational adequacy) . |- A

B iff A L-implies B.

Proof.

Combine lemma 4.2 (5) with the completeness result. 3.Every absolutely consistent L-theory (i.e. L-theory to which some wff does not belong), and hence every non-degenerate L-theory, has a (non-degenerate) L-model. Furthermore, where system L is weakly relevant, every L-theory has an L-model. Proof. Let S be an absolutely consistent L-theory and let U = {D} where D is a wff, guaranteed by absolute consistency, not in S and so not L-derivable from S. Then theorem 4.10 supplies a L-model for S, i.e. an L-model which verifies all wff of S. Now suppose L is weakly relevant; let S be any L-theory, and let D represent some sentential parameter not in the vocabulary of L. Then D is not derivable in L u {D} from S; for suppose otherwise sj-D. Then for some finite subset A^, ..,, A^ of S, &...& A^ D, contradicting weak relevance since D does not occur in S. Hence S is an absolutely consistent theory in L u { D } . The first part of the proof now ensures an L-model for S. 4 (Compactness). If S is a set of wff of L such that every finite subset of S has a non-degenerate L-model (i.e. one such that the base theory T is non-degenerate) then S has a non-degenerate L-model. Proof. Let S' be an arbitrary finite subset of set S of wff. Since S' has a non-degerate L-model, S' is absolutely consistent; for every element of S' belongs to a base theory T but some wff of L does not belong to T. Hence S also is absolutely consistent, since a given arbitrary sentence is only L-derivable from S if it is L-derivable from a finite subset of S. Therefore by corollary 3, S has a non-degenerate L-model. Note that an L-model is never null; and it is universal, i.e. its base T is universal, iff it is trivial, i.e. absolutely inconsistent. 5-L m.s. L2 m.s., and L3 m.s. each validate exactly the same class of theorems, namely those of L. That completes the semantical analysis of the main class of affixing logicc under investigation. There are, of course, other ways of approaching the semantics given, for instance, through the theory of theories (as in SE I), and, connected therewith, there are other ways of construing the semantical theory given, with worlds interpreted syntactically (e.g. through collections of sentences) or epistemically or ontically. There are, furthermore, alternatives to the semantics; there is much that can be done with them and shown using them; but there are also limits to their applicability and usefulness. All these things will concern us, especially in the remainder of this chapter and in the next chapter. 1

The case for t, elaborated in chapter 5, is also modified. Suppose t e a where a is of course non-degenerate; then for some D, t -»• D. Hence x = {A: |t -*• A} is not degenerate; then extend, or maximalize on, non-degenerate x such that Px and x c a. The rule for ° is handled like the rule for

318$

4. 7 NEW MOVES MAVE IN REALISING

THE AMERICAN

PLAN AT THE HIGHER

VEGREE

§7. Alternative semantics for affixing systems: the American plan3 with classical-style negation and without stars3 completed. Just as there were at the first degree different - though not really rival - semantics (see 3.1 and 3.2), so there are at the higher degree alternative semantics, many of which can be obtained by ringing changes on the two-valued relational semantics given. Among the alternatives are non-normal semantics (given in chapter 10), operational semantics (studied in 11), algebraic semantics (in 12), Boolean-valued semantics (worked out for R, in BV), metavaluational semantics and American plan semantics (both of which were investigated at the first degree stage in 3.2). Here we show how the American plan can be extended to provide semantics for a range of relevant affixing systems.1 Given the Australian method just developed, with its two-valued relational semantics and starred negation, there are, in realising the American plan, these new moves (most of which having been seen can be seen to have been glimpsed from the Australian plan):i) Important features of the American plan at entailment stages are taken over, in particular &, v and but the determination of truth in a value l's belonging but also from value O's not

the first degree and first degree not only the evaluation rules for model and validity, not just from belonging (see 3.2).

ii) A second three-place relation S in addition to relation R is added to model structures to enable the evaluation of falsity in an implicational wff. In part, but only in part, S compensates for the sacrifice of the star operation. iii) Some worlds, especially the base worlds of model structures, are considered as being related as mates. This is the remaining American compensation for the loss of star. Move iii) is, perhaps, less important and can, in the case of weaker relevant logics, be avoided entirely. Although the American plan can be made good, it is substantially less satisfactory than the Australian plan, both in the way it forces the doubling up of conditions for less basic axioms, and in its accommodation of negation postulates, especially Contraposition (which is the Achilles' heel of the American approach). Moreover, working through the American plan repeatedly suggests the star operation and negation rule of the Australian plan, and indeed a reasonable treatment of Contraposition and Reductio appears to require some surrogate of star. So perhaps a major virtue of the American approach is that it provides yet another approach to and explanation of - what it was set up partly in opposition to, and was supposed to be getting away from - the star negation rule and its role in the Australian plan. Nonetheless it remains a substantial virtue of the American approach that it deflects, and helps refute, the repeated criticism that relevant logic semantics must make use of the star operation and cannot be outfitted with a classical negation. In elaborating the American approach we separate unreduced modellings for system B, and then for some extensions of B, from reduced modellings for C (which contains Contraposition)- and extensions of C. 1. Classical-style semantics for B and their adequacy. An American B model structure (B a.m.s.) is a structure M = where K is a set of worlds, 0 (comprising the regular worlds) and P are subsets of K, T and U are elements (the positive and negative base worlds) of K with T in 0 and U in P, x

The remainder of this section is largely a condensation of Routley 82b, in which omitted details will be found.

379$

4.7 CLASSICAL-STYLE

EVALUATIONS

UNVER THE AMERICAN

PLAN

and R and S are three-place relations on K, such that the following definitions apply and postulates hold, for all a, b, c, d in K: dl.

a < b =

f Dt

(Px) (x e 0 & Rxab)

|50.

if for some x in P, Sxab, then a < b

It follows since T e 0 and U e P that dpO. pi. p2.

if RTab then a < b, and a < a. if a < b and Rbcd then Racd.

dj50. if SUab then a < b. }ll. for some x in P, Sxaa. $2. if a < b and Sacd then Sbcd.

As before the model structures can be varied in several ways without affecting their adequacy; e.g. T and U can be identified 1 , and in fact both can be removed; 0 and P can be equated and in stronger systems removed; the ordering relation < can be made a partial order; and stronger monotonicity conditions (like p2 and jJ2) can be required, for each place in R and S, etc. A valuation x in M is a function which assigns a value from IIII = {{l}, {0}, (1,0), { }} to each sentential parameter p at each world a such that for a, b in K (1) (2)

if a < b and 1 e T(p,a) if a < b and 0 e x(p,a)

then 1 e T(p,b), and then 0 e x(p,b). 2

Conditions (1) and (2) are the basic hereditariness conditions. Like most else under the American plan, in contrast with the Australian plan, they come in double form. Function T is extended inductively from initial wff (sentential parameters) to all wff by the following rules, which hold generally, for all wff A and B and every a, b in K: ~ rules:

1 e x(~A, a) iff 0 e x(A, a);

0 e t(~A, a) iff 1 e x(A, a);

& rules:

1 e x(a & B, a) iff 1 e x(A, a) and 1 e x(B, a); 0 e x(A, a) or 0 e x(B, a);

v rules:

1 e x(A v B, a) iff 1 e x(A, a) or 1 e x(B, a); 0 e x(A V B, a) iff 0 e x(A, a) and 0 e x(B, a).

0 e x(A & B, a) iff

Where, as classically and modally, there are just two values, 1 and 0, these are exactly the classical rules. And even where there are, as under the American plan, four values composed from 1 and 0, these & V ~ rules look exactly like world relativised classical evaluations. For the most part it is assumed that v is defined: this involves no loss in generality. rules:

1 e x(A -»• B, a) iff, for every b and c in K, if Rabc and 1 e x(A, b) then 1 e x(B, c); 0 e x(A -»• B, a) iff, for some b and c in K, Sabc and 0 e x(B, b) and 0 i x(A, c).

x

But the assumption that T = U is effectively a normality assumption which it is preferable to avoid. Though normality does hold for the systems considered (see 5), it can then be proved using the semantics, rather than assumed in showing their adequacy.

2

AS in 3.1, values 1 and 0 can be unproblematically equated with {1} and {0} respectively.

320$

4.7 VOUBLE TRUTH, VOUBLE

VALIDITY,

AND THE DOUBLE

IMPLICATION

LEMMA

Lemma 4.5. Hereditariness Lemma. Where T is a valuation in an a.m.s. M, then for every wff A and every a, b in K. (It) (2+)

if a < b and 1 e X(A, a) then 1 e T(A, b), and if a < b and 0 e X(A, a) then 0 e X(A, b.

Proof is by joint induction from basis (1) and (2). for (It) there are these cases:

In the inductive argument

ad &. Suppose a < b and 1 e x(B & C, a). Then 1 e t(B, a) and 1 e x(C, a), whence by induction hypothesis 1 € x(B, b) and 1 e x(C, b). So 1 e x(B & C, b). ad Suppose a < b and 1 e T(~B, a). Then 0 e x(B, a), whence by induction hypothesis from (2t), 0 e x(B, b). Therefore, by the ~rule, 1 e x(~B, b). ad -*-. Suppose a < b and 1 e T(B C, a). In order to show 1 e T(B -»• C, b) , suppose for c and d in K that Rbcd and 1 e T(B, C). It suffices to show 1 e T(C, d). Since a < b and Rbcd, by p2, Racd. Hence as 1 e T(B -* C, a) when 1 € T(B, C), 1 E T(C, d). In the argument for (2t) there are three similar cases; ad Suppose a < b and 0 e x(~B, a). and 0 E T(~B, b). ad etc.

two are given.

Then 1 e x(B, a), so by (It) 1 e t(B, b),

Suppose a < b and 0 e T(B -*• C, a).

Then for some c and d in K, Sacd and

It suffices to show for 0 e r(B -*• C, b) that Sbcd and (the same) etc. and Sacd, so by {(2 Sbcd. The wff A is 0 I T(A, therein, (and the (doubly)

But a < b

semantical notions defined are based on double, or increased truth.1 A doubly true (2-true) on T in a.m.s. M just in case 1 e x(A, T) and U). Wff A is (doubly) valid in M iff A is 2-true on all valuations and otherwise invalid in M and M provides a countermodel structure to A a.m.s. and such a valuation concerned, a countermodel to A). A is valid iff A is (doubly) valid in all a.m.s.

Interconnected with 2-truth and double validity are double implication relations. With respect to a given a.m.s., wff A doubly (or 2-) implies B on T iff, for every a in K, if 1 e x(A, a) then 1 e x(B, a) and if 0 e x(B, a) then 0 e x(A, a). A 2-implies B iff, for every valuation x in every a.m.s., A 2-implies B on x. Lemma 4.6. Double Implication Lemma. (1) Where x is a valuation on M, if A 2-implies B on x then A -*• B is 2-true on x. (2)

A 2-implies B iff A + B is doubly valid.

Proof. As to (1), given A 2-implies B in x it suffices to show (i) 1 e x(A -»• B, T) and (ii) 0 I x(A -*• B, U). For (i) suppose RTab and 1 e x(A, a), for arbitrary a and b in K. It suffices to show 1 e x(B, b). Since RTab, by dpO, a < b; hence by hereditariness 1 e x(A, B). But A 2-implies B in x, whence 1 e x(A, b). For (ii) suppose SUab and 0 e x(B, a) for arbitrary a and b in K; to show 0 e x(A, b). By df50, a < b, hence by hereditariness 0 e x(B, b). But then by 2-implication, 0 e x(A, b). x

The notion of double truth was introduced, though in a very different role, in ancient Indian thought.

321$

4.7 PROOF OF THE SOUNDNESS

THEOREM WITH DOUBLE

VALIDITY

As regards (2), one half (LH to RH) follows from (1) simply by universal quantification. For the converse, suppose A -*• B is doubly valid. Then for arbitrary M = < T , U, 0, P, K, R, S > and T in M, 1 e T(A B, T) and 0 / T(A •*• B, U) . Suppose further (i) for arbitrary a E K, 1 E T(A, a). It suffices for one half of 2-implication to show 1 E T(B, a). Since by pi, a < a, for some d e 0, Rdaa. Form a new a.m.s. M' = < d , U, 0, P, K, R, S > , i.e. an m.s. differing from M only at first base; and define an indentical valuation x' in M', i.e. for a in K, x'(p, a) = T(p, a) for every sentential parameter p. Then x = x' by induction; and M' is indeed an a.m.s.1 As A B is valid, it is 2-true in M f , so 1 £ x'(A •*• B, d), i.e. 1 £ x(A -*• B, d). But Rdaa, hence as 1 e x(A, a), 1 £ x(B, a) as required. Suppose next (ii) for arbitrary a e K, 0 £ x(B, a). It remains to show 0 e x(A, a). The argument is like that of (i) but uses jil in place of pi. By jil, for some e in P Seas. Form a new a.m.s. M" with e as second base in place of U (alternatively, the mated bases T, U can be replaced together by the pair d, e), and determine an identical valuation of x on M". Since A •*• B is valid, it is 2-true in M", so 0 t- x(A -*• B, c). But Seaa, so as 0 £ x(B, a>, 0 e x(A, a). Theorem 4.11.

Soundness theorem.

Every theorem of B is doubly valid.

Proof is by the usual inductive argument, for which it is enough to show that the rules preserve validity and the axioms are valid. (I)

The rules of B preserve validity.

ad Rl. Suppose first for arbitrary x in M, 1 e x(A, T) and show 1 e x(B, T). But, as A -*• B is valid, A 2-implies B, whence the result. Suppose further 0 I x(A, U) and show 0 i x(B, U). 0 e x(B, U) then 0 e x(A, U), whence the result. ad R2.

But again A 2-implies B, so if

The result is immediate from the valuation rules for &.

ad R3. Suppose B -»• C -»•. A C is not valid. Then there is a countermodel x in M. There are two alternatives to be dealt with: either (i) 1 i x(B -*• C A •* C, T) or (ii) 0 e x(B -*• C A ->• C, U). Suppose first (i). Then for some a, b e K, RTab and 1 e x(B C, a) but 1 I x(A -*- C, b). By dpO, a < b, so 1 i x(A C, a) by hereditariness. Hence for some c, d e K, Racd and 1 e x(A, c) but 1 I x(C, d). But 1 e x(B + C, a), so 1 ^ x(B, c). Therefore A does not 2-imply B, and A B is not valid. SuDpose next (ii). Then for some b, c e K, SUbc and 0 £ x(A -»• C, b) but 0 I x(B -»• C, c). By d|50, b < c, so 0 € x(A C, c) by hereditariness. Hence for some d, e e K, Scde and 0 e x(C, d) but 0 i x(A, c). But 0 k x(B C, c) so as 0 e x(C, d), 0 e x(B, e). Therefore A does not 2-imply B and again A -*• B is not valid. Hence since it is not valid under either alternative, it is not valid. ad R4.

Similar to R3.

ad R5. Given that A -»• B is valid, and hence A 2-implies B, in showing that ~B -»• ~A is valid, by showing that ~B 2-implies ~A, there are two cases, namely i) if 1 e x(~B, a) then 1 e x(~A, a), i.e. if 0 e x(B, a) then 0 e x(A, a), and ii) if 0 e x(~A, a) then 0 £ x(~B, a), i.e. if 1 £ x(A, a) then 1 e x(B, a). But these cases are guaranteed by the fact that A 2-implies B. 1

For none of the modelling conditions involve T and U essentially; other elements of 0 and P will do as well. But ensuring this has complicated formulation of the conditions, e.g. iSO in place of d|50 and hence involvement of class P.

322$

4.7 PRELIMINARIES

TO COMPLETENESS

THROUGH A CANONICAL

MODELLING

Verification of R5 reveals the point of doubling up: the whole complication of doubling is required to yield Contraposition principles in a natural way under the American plan. (II)

The axioms of B are valid.

Three examples illustrate the method completely.

ad A2. To show that A & B 2-implies B it is enough to show that (i) if 1 E x(A & B, a) then 1 E X(A, a) and (ii) if 0 E x(A, a) then 0 E T(A & B, a). But these connections are immediate from the evaluation rule for &. ad A4. It is enough to show (i) if 1 e x(A B &. A -»• C, a) then 1 e t(A -»-. B & C, a) and (ii) if 0 E T(A B & C, a) then 0 e T (A B &. A C, a). As to (i), on the hypothesis, by & and ->• rules, for every b and c in K, if Rabc and 1 e T(A, b) then 1 e X(B, c) and also 1 E T(C, C), whence 1 e x(B & C, c); so 1 e T (A -»-. B & C, a). As to (ii), on the hypothesis, for some b and c in K, Sabc and 0 e T(B & C, b) and 0 I T(A, C). So either 0 e x(B, b) or 0 e T(C, b). If 0 E T(B, b) then 0 e X(A ->- B, a); while if 0 € x(C, b) then 0 E x(A + C, a). So in either case, 0 e x(A B &. A C, a). ad A9. If 1 e x ( — A , a) then 0 e x(~A, a) and 1 e x(A, a). Similarly 0 e x ( — A , a) iff 0 e x(A, a). Hence — A 2-implies A, and also conversely. The completeness argument uses canonical model structures whose elements are characterised, as in §6, in terms of theories and prime theories. Only new details are given. For a, b, c in K^, S^abc iff whenever ~B e b and "A I c then ~(A -*• B) e a for all A and B. of wff B such that ~B I x; rably a prime L-theory.

Where x is a prime theory in 0^, let y be the class

then P L is the class of all such y.

P

is

demonst-

L

The canonical a.m.s. for L on bases T, U is the structure < T , U, 0^, P^, K^, R^, S L > . It remains to determine T and U appropriately in 0 L and P^ respectively. A useful but inessential auxiliary semantical notion, which enables previous work to be taken over, is that of the opposite of a prime theory. Where a e K, a*, the opposite of a, is defined as {B : ~B I a}. Then a* e K. In these terms, P^ = {x* : x e 0^} and, as it will turn out, U = T*. So it remains to determine T, which is done in terms of the classes of wff to be verified and refuted. Theorem 4.12. Strong Completeness Theorem. Where set W is not L-derivable from regular set S of wff, there is a (canonical) valuation x in a (canonical) B a.m.s. which 2-verifies (i.e. makes 2-true) every member of S and 2-falsifies every member of W (i.e. for B in W, 1 I x(B, T) and 0 e x(B, U) where T, U are the base elements of M^). Proof.

Let Sj be the theory-closure of S, i.e.

S, = {D: (PB,, ..., B e S) k B. & ... S B -*• D> and let W. be the ideal1 1 n L 1 n i closure of W, i.e. W = {C: (PA,, ..., A e U) (- T C -»-. A. v ... v A }. Then, as 1

1

TO

1

ID

in theorem 4.10, S^ is an L-theory containing S, W^ which contains W is closed under disjunction, and S^ is disjoint from W^.

Under these conditions, there is

a prime regular B-theory which includes S^ but is disjoint from W^.

Let U = T*,

and take the canonical a.m.s. Mg for B on T, U so defined. (I)

M_ is a B a.m.s.

Since it is immediate that T e 0 T and U e P , it remains

323$

4.7 MAIN VETAILS

OF THE STRONG

COMPLETENESS

ARGUMENT

UNVER THE AMERICAN PLAN

to establish the modelling conditions. But pi and p2 are proved as in §6, and earlier results can also be used to establish propositions concerning S^ given the following connection RS.

S L abc iff R L a*c*b*,

which serves well as a guiding relation in linking Australian and American plans. ad JO.

Suppose for some x e P^, S^xab, i.e. for some x* e 0^, R^x*b*a*.

Then by

dl and its canonical equivalent - since a ^ b iff a c b - b* c a*, whence a c b. ad f>l. Since by pi, a* s a*, for some x* e 0^, R^x*a*a*. x e P L , S^xaa. ad j>2. Suppose a c b and S L acd. i.e. S^bcd, as required.

That is, for some

Then R L a*d*c* and b* c a*, so by p2, R^b*d*c*,

Almost needless to say, these parasitic ways of establishing modelling conditions, which route through earlier results (in §6) and make much use of the dastardly star, can be avoided, and conditions proved in a pure star-free way. Define a (canonical) valuation x in Mg as follows: 1 e x(p, a) iff p e a and 0 e T(p, a) iff ~p e a, for every a in K^ and every sentential parameter p. With this induction base it is shown (II)

(1°) (2°)

1 e T(A, a) iff A e a, and 0 e T(A, a) iff ~A e a, for every wff A and every a e K^.

There are familiar induction steps for each connective. ad &. 1 i.e. iff 0 e T(C, theory. ad 1 TTe. iff

e T(B & C, a) iff 1 e T(B, a) and 1 e T(C, a), i.e. iff B e a and C e a B & C e a, since a is a theory; 0 e T(B & C, a) iff 0 e T(B, C) or a), i.e. iff ~B e a or ~C e a, i.e. iff ~(B & C) e a, since a is a prime e x(~B, a) iff 0 e x(B, a), i.e. iff ~B e a; 0 e x(~B, a) iff 1 e x(B, B e a, i.e. iff ~~B e a, using Double Negation.

ad ->-. What has to be shown for (1°) is B C e a iff 1 e x(B C, a), i.e. iff (b, c e K L ) if R L abc and 1 e x(B, b) then 1 e x(C, c), i.e. iff (b, c e 1^) (if R^abc and B e b then C e c). But this is established in §6. For (2°) what is required is ~(B C) e a iff 0 e x(B -»• C, a), i.e. iff (Pb, c e 1^) (Sjabc and 0 e X(C, b) and 0 I x(B, c)), i.e. iff (Pb, c e 1^)(R^a*c*b* and ~C e b and ~B t c), i.e. iff (Pb*, c* e K^)(R^a*c*b and B e e * and C I b*), i.e. iff (Pd, e e K L )(R L a*de and B e d

and C e e).

However ~(B -*• C) e a iff B

_by the previous cases iff, for some b, c e K^, R^ade and B e d

C I a*, i.e.,

and C I e.

(III) Since A e S, A e T, by (II), 1 e x(A, T). But 0 I x(A, U) iff ~A e d, i.e iff ~A i T*, i.e. A e T, i.e. if 1 e x(A, T). Hence A is 2-true on x. Since further for B e W, B i T, \ i x(B, T). Hence B is not 2-true on x, and in fact x in M_ 2-falsifies every member of W. ~B

Corollaries 1.

Completeness.

Every doubly valid wff of B is a theorem.

Proof. Let A be a non-theorem of B. Let W = {A} and S = B, and apply the theorem, since A is not B-derivable from B. Then A is 2-falsified, and hence not doubly valid. Other corollaries such as compactness follow as in §6. 324$

4.7 HOW TO ACCOMMODATE

POSITIVE

EXTENSIONS

OV 8 UNDER THE AMERICAN

PLAN

2. Some positive extensions of B and the problem with negative extensions. Without much doubt, virtually any extension of B by positive axiom schemes that can be accommodated under the Australian plan can be handled by the American plan upon mirroring the required semantical postulate involving R by one involving S. Such positive schemes are those involving connectives -»-, & or V (and perhaps others such as fusion, *, and fission, V), but not negation or its derivatives such as material implication. So long as there is no "crossing" or "mixing" of Is and Os, mirroring of R conditions by S conditions is straightforward; but when negation enters "crossing" occurs and the straightforward method is lost, as we shall see. Here only a few sample positive extensions of B will be given, by axiom schemes like those required in reaching systems like R, T and E; but any of the positive postulates listed earlier (in §3) would be similarly accommodated. B3.

A

B -»-. B

C -*•. A -*• C

B5.

A

(A

B6.

A

BR1.

A -t (A -»• B) -*•B

B)

A -*• B

A -*• B ->• B

q3. 43. q5. 45. q6. 46. drl. drl.

If R abed then R b(ac)d; If Sazb & Szcd then (Px)(Saxd and Sbcx). 2 If Rabc then R abbe; If Sabc then (Px)(Saxc and Sxbc). If Rabc then Rbac; If Sabc then Scba. Raxa for some x e 0; Saax for some x e 0.

For every postulate there are two cases, a 1 case and a 0 case. The 1 case is the same, both for soundness and completeness, as under the Australian method (of earlier sections), while the 0 case is an S dual of the 1 case. Some examples of details for adequacy will make this clear. ad B3. What has to be shown is 1) If 1 C T(A -*• B, a) then 1 E T(B •*• C -*•. A C, a), and 2) If 0 e t(B C A •*• C, a) then 0 e t(A B, a). The first 1) is but a transcription of what happens under the Australian method: for the transcription write 1 e T(D -»• E, a) for I(D -»• E, a) = 1. For 2) suppose 0 e t(B C -*-. A -»• C, a) but 0 I x(A B, a). Then for some z and b in K, Sazb and 0 e X(A -»• C, z) and 0 t X(B -»• C, b). Similarly for some c and d Szcd and 0 e x(C, c) and 0 I x(A, d). Now by 43, for some x Saxd and Sbcx. Since Sbcx, 0 e x(B, x), but then as Saxd, 0 e x(A, d), which is impossible. ad q3 and 43. Proof of q3 is as under the Australian method (see §6), and 43 is a variant on q3 after translation. For 43 translates to: Ra*b*z* and Rz*d*c* implies (Px)(Ra*d*x* and Rb*x*c*), which is a rewrite of q3 (interchange c and d and star all elements). ad B5. In the second case to show 2) If 0 e x(A •*• B, a) then 0 e x(A A -»• B, a), suppose otherwise. Then for some b and c, Sabc and 0 e x(B, b) and 0 I x(A, c). By 45, Saxc and Sxbc. Since Saxc and 0 t x(A A •*• B, a) if 0 e x(A B, x) then 0 e x(A, c); so 0 i x(A B, x). Hence as Sxbc and 0 e x(B, b), 0 e x(A, c) which is impossible. ad 45. The condition translates to: which is a starred rewrite of q5.

if Ra*c*b* then (Px)(Ra*c*x* and Rx*c*b*),

A strong adequacy theorem now follows, as before, for these positive extensions of B. The easy duplicating of modelling conditions that worked for positive extensions of B breaks down when negation enters. Consider, in particular, what is required to 2-verify Contraposition, the critical scheme; that i) wherever

325$

4. 7 VALIDATING

CONTRAPOSITION

BY THE METHOD Of STAR

IMITATION

1 e x(A B, a) then 1 e t(~B ~A, a) and ii) wherever 0 e t(~B ~A, a) then 0 e T(A B, a). As to i), suppose then 1 e R(A B, a) and for arbitrary b and c in K, Rabc and 1 e x(~B, b), and try to show 1 e T(~A, c), i.e. 0 e x(A, C). Then 0 6 T(B, b), and using the remaining data, if 1 € T(A, b) then 1 e t(B, c). There is no way, however, in the semantical framework so far developed that a conditional concerning l's memberships and a premiss regarding O's membership will yield a conclusion concerning O's membership: Os and Is do not cross or interact. The same impasse confronts us as regards ii). There is no direct way through modelling requirements, e.g. on R and S, that Contraposition can be nailed down. There are however several ways of adjusting modellings so that Contraposition is verified. Apart from the brute force method (discussed and dismissed in Routley 82b) there are, for instance, the following methods. Method 2. Star imitiation through mates. The effect of a star operation can be obtained without altering the negation evaluation rule to one involving star. That is, the negation rule is not tampered with, but modellings are enriched^ It is assumed that worlds come in mated pairs, that every world has a mate a (which maybe a itself: hermaphrodites are not excluded). More exactly, a further relation M is added to model structures (and so to models); it is required i M firstly that M be an operation, and the image of a under M is written a , and M M M M M M secondly that Rb d c iff Sbcd and Sb d c iff Rbcd, for every b, c, d in K. Finally, the valuation function T is assumed to conform generally to the conditions 1 e T(p, b M ) iff 0 I T(p, b),

(3)

and

0 e x(p, b M ) iff 1 I T(p, b), M for every b (and so b ) in K and every sentential parameter p. (4)

1 e r(A, b M ) iff 0 I x(A, b),

and

(4+) 0 e x(A, b M ) iff 1 i x(A, b), every wff and every b in K.

for

Lemma 4.7.

(3+)

Proof is by induction from basis (3) and (4). All the induction steps are elementary cases, those for ->• using the modelling conditions imposed. Proofs of soundness are not affected by the further requirements associated with M, since what is valid in all models is valid in all M restricted ones, M nothing depending on the generality of the models. For completeness, set b = b* for every b e K^ . Then all modelling conditions associated with M are satisfied. Thus M-restricted models are adequate, and in terms of these it is a straightforward matter to model various negation postulates. Again only two sample postulates are tabulated, for reasons that will become evident (if not already so): Axiom scheme A

B

~B -»• ~A

Corresponding modelling conditions I if Rabc then H a c ¥ if Sabc then S a c V 1

J

Weaker assumptions would suffice, e.g. full operationality is probably not required. On how weak the requirements can be see below.

326$

4.7 VINDICATION

OF THE STAR NEGATION

Axiom scheme A -+ ~A

RULE FROM THE AMERICAN

SEMANTICS

Corresponding modelling conditions

(

~A

Raa^a Saa^a

Proofs of completeness are much as in §6. following sorts of details:ad Contraposition.

Continuing the argument in i) above, it follows from

0 e T(B, b) that 1 I T(B, b M ). M

As to soundness there are the

Since Rabc, R a A * 1 , hence as 1 e T(A

M

B), if

M

1 e T(A, C ) then 1 e T(B, b ). So 1 i T(A, c ), whence 0 e T(A, C) as required. As to ii), assuming 0 e x(~B ~A, a) for some b, c, Sabc and 0 e £(~ A » b ) a n d m 0 i T(~B, C), whence 1 e T(A, b) and 1 i T(B, C) and so 0 e x(B, C ) and 0 i T(A, b ). Hence as S a c V . 0 e T(A -»• B, a). ad Reductio.

For i) assuming 1 e T(A -*• ~A, a) it has to be shown 1 E T(~A, a),

i.e. 0 e x(A, a). But as Raa M a, either 1 it t(A, a M ) or 1 e t(~A, a). If M 1 I T(A, a ) then 0 E T(A, a). So either alternative yields the required result. As to ii), suppose 0 e T(~A, a) but 0 I T(A -*• ~A, a). Then as Saa a, either 0 i T(A, a H ) or 0 i x(~A, a). But if 0 ^ T(A, a M ) then 1 e T(A, a) so 0 i T(~A, a). Hence both options contradict 0 e T(~A, a). In fact were it simply required that M be an involution, which it might as well be, then both arguments for ii) reduce to those for i) using the lemma. Indeed S conditions can disappear and therewith the need for S. And of course the whole semantical theory can be considerably simplified using the star rules for negation given in the following 1 e x(~A, b) iff 1 i x(A, b M ) ;

Lemma 4.8.

0 € x(~A, b) iff 0 i x(A, b M ) . Among other things then, the semantics provides an indirect vindication of the star rules from a more classical viewpoint. However under an improved version of method 2 stronger relevant logics can be given American-style semantics without so blatantly imitating star (see below). Method 3. ->- rule doubling. Much as we complicated the notion of semantic implication to 2-implication to ensure Rule Contraposition, so we can strengthen systemic implication, through the evaluation rules for -»•, in a parallel way, to validate Contraposition. Such a strengthening of the rules is readily suggested by the argument given in i) above; for what is required to close the argument for Contraposition is that 1 e x(A -*• B, a) also guarantees that wherever 0 e x(B, b) then 0 e x(A, c) provided Rabc. The amended

amended 2 -*• rules

rules which emerge are these:

1 e x(A •*• B, a) iff, for every b and c in K, if Rabc then if 1 e x(A, b) then 1 € x(B, c) and if 0 e x(B, b) then 0 e x(A, c) 0 e x(A -*• B, a) iff, for some b and c in K, Sabc and either 0 e x(B, b) and 0 I x(A, c) or else 1 e x(A, b) and 1 I x(B, a).

Lemma 4.9.

i) ii)

1 e x(~B •*• ~A, a) iff 1 e x(A B, a) 0 e x(~B -*• ~A, a) iff 0 e x(A •*• B, a).

327$

4.7 AMENDING

THE AMERICAN

PLAN FOR REDUCED

MODELLINGS

Proof is by the amended -»• rules. Since the amended rules thus force Contraposition, adoption of the amended rules will not lead - assuming the rules work - to supersession of earlier labour which catered primarily for systems without Contrapositon. But finding rules which might succeed is one thing, showing that the rules do indeed work is quite another, and more laborious, task. 3. Reduced classical-style semantics for system C, and extensions and complications thereof. An American C model structure (C a.m.s.) is a structrue M = < T , K, R, S > where K is a set of worlds, T belongs to K and R and S are threeplace relations on K, such that the following definitions apply and postulates hold, for all a, b, c, d, e in K: dl.

a < b -

RTab

d5.

S 2 a(bc)d « D f (Px)(Saxd & Sbcx)

d7.

S/Ra(bc)d ->Df (Px) (Saxd & Rbcx)

pi.

a < a

d4.

S 2 a[bc]d = D f (Px)(Saxd & Sxbc)

d6.

R/Sa[bc]d = D f (Px)(Raxd & Sxbc) , etc.

p2. if a < b and Rbcd then Racd j2. [For Bl] ql. Raaa 41* [For B3] q3. If R 2 abcd then R 2 b(ac)d (j3. q4. If R 2 abcd then R 2 a(bc)d «|4. qm3. If R/Sa[bc]d then qm4. S/Rd[ab]c qm5. Iff S/Rabcd then qm6. R/Sb(ac)d

if a < b and Sacd then Sbcd Saaa. if S 2 a[bc]d then S 2 d(ab)c if S 2 a[bc]d then S 2 a(bc)d If R/Sa[bc]d then R/Sa(db)c Iff S/Rabcd then S/Ra(bc)d

A valuation T in M is defined as before. Function T is extended to all wff by the ~ and & rules previously given and the amended -*• rules. Lemma 4.10. Hereditariness Lemma. involves some new cases:

The formulation is as before, but the proof

New -*• cases. Assuming 1 E T(B, c) it can be shown as before that 1 E T(C, d). Now assume 0 e x(C, c); to show 0 e T(B, d). As before by p2, Racd, so as 1 E T(B C, a) when 0 E X(C, c), 0 e T(B, d). The generalised argument given in the •*• step for (2t) already suffices. Truth and validity can now be defined in the usual way, that is (as on the Australian plan) doubling is not required. A wff A is true on T in M just in case 1 e T(A, T); A is valid in M iff it is true in all valuations T in M; and A is valid iff it is valid in all C a.m.s. While 2-implication is still required, it now emerges naturally from the amended •*• rules. Lemma 4.11. Implication Lemma for Reduced Modellings. (1) A 2-implies B in T iff A B is true on x (2) A 2-implies B iff A •*• B is valid (double validity is not required). Proof. (2) follows from (1) by quantificational logic. As to (1), suppose A -*• B is true in x , i.e. 1 e x(A B, T). Then as RTaa, for every a e K, when 1 e x(A, a), 1 e x(B, a), and when 0 e x(B, a), 0 e x(A, a), i.e. A 2-implies B on x. The converse requires 2-implication. Otherwise there is no guarantee of the 0 clause in the -*• rule. Suppose A 1-implies B on x . To show 1 e x(A -»• B, T), suppose further RTab for arbitrary a and b in K. It has to be shown that (i) when 1 € x(A, a) then 1 e x(B, b) and (ii) when 0 e x(B, a) then 0 e x(A, b). 111$

4. 7 SOUNDNESS

ANV COMPLETENESS

THEOREMS

FOR SYSTEM C UNDER THE AMERICAN

PLAN

Since RTab , a < b. Then both (i) and (ii) follow by hereditariness combined with 2-implication. Theorem 4.13.

Soundness Theorem.

Every theorem of C is valid.

Proof enlarges on the corresponding result for B, and postulates drawn from B are treated precisely as before: so it remains to validate Bl, B3 and D4 ? . ad Bl. In virtue of the Implication Lemma it suffices to show 1) if 1 e T(A &. A B, a) then 1 e x(B, a), and 2) if 0 e T(B, a) then 0 e x(A &. A -*• B, a). As to 1), 1 e t(A, a) and 1 e x(A -*• B, a), so as Raaa, 1 e x(B, a). As to 2), suppose 0 i x(A &. A B, a). Then 0 I x(A, a) and 0 i x(A -*- B, a). Since Saaa, if 0 e x(B, a) then 0 £ t(A, a), so 0 / x(B, a). 1) follows by Contraposition. ad B3. What is to be shown is 1) if 1 e x(A -+ B, a) then 1 e x(B •*• C A -*• C, a) and 2) if 0 e x(B -»• C -»•. A -»• C, a) then 0 e x(A -»• B, a). The argument is case-ridden: there are 8 eventual subcases. These cases will simply be illustrated. ad 2). c in K either or

Suppose 0 for which 2.1) 0 € 2.2) 1 e

e x(B •*• C -*•• A -*• C, a) but 0 i x(A -»• B, a). Sabc x(A B, b) and 0 i x(B + C, c) x(B C, b) and 1 / x(A C, c).

Then for some b and

ad 2.2). Then for some d and c in K such that Rede either 2.21) 1 e x(A, d) and 1 I x(C, e) or 2.22) 0 e x(C, d) and 0 t x(A, e). But Sabc & Rede imply that for some x, Sadx and Rbxe and also imply, for some other x, Rbdx & Saxe. Hence, for 2.21), since 0 t x(A -»• B, a), as Sadx, 1 e x (B, x). Since further 1 6 x(B -»• C, b), as Rbxe, 1 e x(C, e), which is impossible. Similarly, for 2.22), as Rbdx, 0 e x(B, x). But Saxe, so 0 e x(A, c), which is again impossible. ad D4. The argument is as in the previous section from amended -*• rules. conditions are required.

No

The completeness argument differs from that given for system B in that T-Ltheories replace L-theories. i^, R^,, S T , etc., are now defined in terms of T-L-theories, where T is some (prime) regular theory. Where T is a prime regular theory the T canonical L a.m.s. is the structure M^ = < T, K^,, R^, S T > where as before prime restrictions are taken to obtain unbarred predicates. In order to exploit previous work the operator * is again introduced as an inessential auxiliary, with a* - {B : ~B I a) for a e It follows that (RS) S^abc iff R T a*c*b*, i.e. iff, in virtue of Contraposition, R T a*bc. These interconnections greatly reduce the task of verifying modelling conditions, especially those for B3. Theorem 4.14. Strong Completeness Theorem. Where W is not C-derivable from regular set S, there is a canonical valuation x in a canonical C a.m.s. M which verifies every member of S and falsifies every member of W. Proof is like that for the corresponding theorem for system B, with differences as remarked. A prime regular C theory T is obtained as before, but now a T canonical C a.m.s. = < T , K^,, R ^ S T > on base T is formed.

329$

4.7 EXTENSIONS

(I)

IJj, is a C a.m.s.

OF C, BOTH STRAIGHTFORWARD

ANV

PROBLEMATIC

Much is proved as in theorem 4.10 of §6.

ql, q2 and q4 may be established precisely as there. showing that T e K^. ad 41'

By (RS) this amounts to R T a*a*a* which follows from ql.

ad 43.

Expanding this is:

if S^azd & S^zbc then for some x, S^dxc & S T abx, i.e.

by (RS) if R T a*d*z* & R T z*bc then for some x, R^fcxc & R ^ b x . ad qm3.

In particular,

Note that Bl is used in

i.e. if R^azc & S T zde then, for some x, R,j,adx & S^cxe.

But this is q3. That is by (RS)

if R T ac*z*& R T z*de then for some x, R ^ d x & R T c*xe, i.e. q3 relettered.

The

other conditions for B3 are similarly verified. A canonical valuation T is defined as in the corresponding theorem for B, and (II) is established in much the same way. But note that as in theorem 4.10 of §6, both B3 and B4 are required in the induction step for ad amended -»- rules.

By theorem 4.10, B -*• C e a iff (b, c e K^) if R^abc and

B e b then C e c, i.e. by Contraposition iff (b, c e K^,) if R^abc then if B e b then C e show B ~(B S T abc in

c and if ~C e c then ~B e b.

By induction hypothesis, this suffices to

C c a iff 1 e r(B -»• C, a). •*• C) e a iff B E C t a*, i.e. iff the negation of the above but with place of R T a*bc using (RS), i.e. iff 0 e T(B -»• C, a).

Corollaries such as completeness, compactness and implicational adequacy follow as before. Some extensions of C are moderately straightforward, though again caseridden, for instance Absorption:-

A -*• (A -*• B) -*. A

B

q5 set

q6.

If Rabc then R 2 abbc

46.

If Sabc then S2a[bc]c

qm7.

If Rabc then R/Sa [bc]c

qm8.

If Sabc then S/Rabbc

The really disconcerting new phenomenon is that with remaining axioms required for R and T problems emerge which beset not merely negative axioms like Reductio, D3, but also positive axioms such as EAssertion, B6. Much as previously with Contraposition, a l-to-0 cross-over is called for, which however the rules do not sanction. Consider to illustrate, the putative validation of B6, the only scheme outstanding in an American semantics for system R. It has to be shown, for one case, that if 1 e t(A, a) then 1 e t(A B ->• B, a). Suppose otherwise; so both 1 e x(A, a) and for some b and c in K for which Rabc either 1.1) 1 e t(A -»B, b) and 1 t t(B, c) or 1.2) 0 e t(B, b) and 0 I t(A -»• B, c). Consider 1.2). Since 0 E T(A -*- B, c) if Sabc then if 0 e T(B, a) then 0 e T(A, b) and if 1 € T(A, a) then 1 E T(B, b). But without some crossing from belonging to 0 to not belonging to 1 or vice versa, no trouble with the remaining assumptions 1 E T(A, a) and 0 E X(B, b) can emerge. Thus C a.m.s. strongly counter Ackermann fallacies such as B6. With Reductio there is a similar "problem".1 ^ o r in the light of 3.9 these outcomes can be seen not as problems but as desirable features. Perhaps the American approach has more things going for it than we earlier admitted. 330$

4.7 RESOLVING

DIFFICULTIES

THROUGH

(SUB)THEORIES

OF COMPLETE

ANV POSSIBLE

WORLDS

There are several ways of varying attractiveness, of trying to readmit principles like B6 and D3. These methods include Brute force. The method works of course, after its own fashion: it succeeds so well it could be used to model the higher degree of virtually any relevant logic. Appeal to possible and complete worlds. Further amendment of the rules, allowing for cross-over by way of an additional three-place relation on worlds. Mateship, an improved version of star imitations. The third of these methods has not been seriously investigated, but the two ideas involved are worth mentioning. One is that a relation, Uabc, say, is introduced that stands in for Rabc*, such as Sabc does for Ra*bc; the other is that amended -*• rules have further "mixed" clauses that incorporate 1 to 0 cross-over and vice versa. The second of the methods, though ultimately unsuccessful, merits investigation, especially because it is entangled with the important matter of recovery of possible world semantics (without presupposed possibility assumptions) from more general world semantics (a matter taken up again in later parts of the book, e.g. 5.3 and chapter 13 ff.). The class K of worlds contains a subclass C of complete worlds (i.e. worlds where at least one of A and ~A hold, for each wff A) and another subclass H of possible worlds (i.e. worlds where at most one of A and ~A hold, for each wff A). It also contains, as model structures may or may not indicate, the intersection N of these worlds; that is N = C fl H is the class of modal worlds, complete possible worlds, to which possible world semantics typically restricts worlds. These further classes of worlds can be added separately or in combination to C a.m.s. But for the further classes of worlds to accomplish the requisite work, e.g. facilitating crossing-over, modelling and valuation conditions have also to be imposed, in a way now illustrated. A C-enlarged C a.m.s. is a structure M = < T , C, K, R, S > where all is as before except that further C c K and cl.

If Rxbc then Sxbc,for x in C.

A valuation x in M is also defined as before except that it is further required c2.

For x e C, if 0 I x(p, x) then 1 e x(p, x), for every sentential parameter p.

Lemma 4.12. Completeness Cross-over Lemma. 0 I x(A, x) then 1 e x(A, x).

For every wff A and every x e C, if

Proof is by straightforward induction. The adequacy theorem for C is unaffected by change to enlarged models. Soundness is as before. For completeness, define C = {a : a* c a}. Then if x e C, x* c x, so if R^xbc, by p2 effectively, R T x*bc, i.e. S T xbc, confirming cl. For c2, 0 i x(A, x) implies ~A i x, which implies A e x*, which implies A e x, as x e C, which implies 1 € x(A, x). This enables simple, but fundamental, extensions of C like the following to be accommodated: D2.

A v ~A

s2(T) 114$

T e C

4.7 CONDITIONS

TOR WORLD CONSISTENCY

ANV WORLD COMPLETENESS,

AND THEIR

PROBLEMS

For soundness, suppose otherwise 1 i x(A v ~A, T) for some C-enlarged m.s. Then 1 I T(A, T) and 1 i x(~A, T), i.e. 0 I x(A, T). But since by s2(T), I £ C, by the lemma, 1 e t(A, T) also, which is impossible. For completeness it is enough to show T* c T. But for every A, because A V ~A £ T, A e T or ~A e T, i.e. A i T* or A E T, as required. World completeness does not however suffice for the validation of Reductio and EAssertion, for which the much more demanding matter of world consistency is also involved. As regards modelling, consistency is the dual of completeness. An H-enlarged C a.m.s. is a structure M = < T , N, K, R, S > where further H c K and hi. h2.

For x £ H, if Sxbc then Rxbc; For x £ H, if 0 £ x(p, x), then 1 £ x(p,x) for every sentential parameter p.

Lemma 4.13. Consistency Cross-over Lemma. 0 £ x(A, x) then 1 t x(A, x).

For every wff and every x e N, if

Again the adequacy theorem for C is unaffected. For the complete possible worlds of modal logic, C and H are simply combined (i.e. the only ifs of hi, h2 and the lemma are replaced by iffs). But life, particularly attempts to prove the hard parts of the adequacy theorem, is made easier by introducing completeness and possibility separately, not jointly. A CH-enlarged C a.m.s. is a structure M = < T , C, H, K, R, S > where C c K and H c K and all of cl, c2, hi and h2 hold. Adequacy is again unaffected, and putative modelling conditions for the outstanding schemes of R and T can now be supplied. B6.

A

A

D3.

(A

~A)

B

B

~A

q6 set

If Rabc then Rbac; If Sabc then Scba; If Rabc then, for some x e H, Scbx and a < x; If Sabc then, for some x e C, Rbxc and x < a.

S3 set

For some x e C, Raxa and x < a; For some x e H, Saax and a < x.

Proof of soundness is straightforward if tedious. Proof of completeness is however far from straightforward, and appears to be ruled out entirely as regards conditions for B6.1 Accordingly, the remaining investigation has focussed on the closely related fourth method. Under the fourth, mateship, method, worlds are mated, and cross-over occurs between mated worlds. A C a.m.s. with mateship relation M is a structure M = < T , K, R, S, M > which adds to a C a.m.s. a two-place relation M in K such that

1

Consider the final condition for B6. In virtue of the first two conditions which are unproblematic the supposition Sabc can be replaced by Rba*c. Given this supposition what has to be shown is that a* can be extended to a complete theory x* (not in its own a difficult feat) but without enlarging the class of consequences in c. Unfortunately this seems to be impossible. For suppose p v ~p C £ b but p V ~p i a*; then extending a* to a complete theory does enlarge c.

332$

4.7 HOW CROSS-OVER

ml. m2.

WORKS: DETAILS

OF THE MATESHIP

METHOD

If Mab then Mba, and If Mab then Racd iff Sbcd.

The valuation function x is constrained by the condition m3.

If Mab then 0 e x(p, a) iff 1 i x(p, b), for every sentential parameter p. Conditions ml-m3 are imposed precisely in order to enable proof of the following cross-over result. Lemma 4.14. Corollary.

Mated Cross-Over Lemma.

If Mab, then 0 e x(A, a) iff 1 I x(A, b).

If Mab, then 1 e x(A, a) iff 0 i x(A, b).

The corollary follows by symmetry of M, and proof of the lemma is by induction from the given basis. Adequacy results for system C and extensions already dealt with transpose unproblematically. For completeness, define Mab as a - b*. Then ml-m3 are immediate. Sought extensions such as B6 and D3 are modelled as follows: B6.

A -*•• A

D3.

A

~A

B -»• B

~A

q6 set

S3 set

If If If If

Rabc Sabc Rabc Sabc

then then then then

Rbac; Scba; for some x, Mxa and Scbx; for some x, Max and Rbxc.

f For some x, Max and Raxa; t For some x, Mxa and Saax.

The details required for adequacy go as follows: ad B6. It is enough to show 1) if 1 e t(A, a) then 1 € x(A B -*• B, a) and 2) if 0 e x(A B -»• B, a) then 0 e x(A, a). For 1) suppose 1 e x(A, a) but 1 T t(A •*• B B, a). Then for some b and c in K such that Rabc either 1.1) 1 e t(A -»• B, b) and 1 I x(B, c) or 1.2) 0 € x(B, b) and 0 I x(A B, c). Option 1.1) is readily ruled out by the first modelling condition which permutes a and b. For 1.2) apply the third condition. Since Rabc, for some x, Scbx and Mxa. As 0 d x(A B, c) and 0 e x(B, b), 0 e x(A, x). Hence as Mxc, by Mated Cross-over 1 i x(A, a) which is impossible. Under case 2) there are two more rather similar cases. ad q6 set. The first condition follows directly using B6, as under the Australian theory (see §6). Suppose, for the second, Sabc. Then Ra*c*b*, so by the first condition Rc*a*b*, whence Scba. For the third, suppose Rabc. Then permuting, Rbac, Rbc*a*, Rc*ba*. So setting x = a*, Rc*bx, i.e. Mxa and Scbx. The fourth is similarly established. ad D3. It is enough to show 1) if 1 e x(A ~A, a) then 1 € x(~A, a) and 2) if 0 e x(~A, a) then 0 e x(A -*• ~A, a). As to 1) suppose otherwise 1 e x(A •*• ~A, a) but 1 i x(~A, a), i.e. 0 t x(A, a). By the first modelling condition for some x Max and Raxa. Hence if 1 e x(A, x) then 1 e x(~A, a), whence 1 i x(A, x). So by the cross-over lemma 0 e x(A, a), which is impossible. For 2) suppose 0 e x(~A, a), i.e. 1 e x(A, a), but 0 i x(A -*• ~A, a). By the second modelling condition, for some x, Mxa and Saax. So as 0 e x(~A, a), 0 e x(A, x). Therefore by the cross-over lemma, 1 I x(A, a), which is impossible. ad s3 set. The Australian modelling condition for D3 is Raa*a, the adequacy of which is already established. Set x = a*. Then Max and Raxa. Similarly Ra*x*a*, so Saax. 333$

4.7 THE VERY LJMJTEV SUCCESS

OF THE AMERICAN

PLAN

Theorem 4.15 Strong Adequacy Theorem for stronger relevant logics such as R and T under the American plan with mateship. Statement and other details of proof are like those for C. The theorem similarly follows for E and similar systems upon complicating the structures by class 0 and conditions thereon for BR1. Add to mated m.s. a class 0 of (regular) worlds included in K and containing T. Then BR1 is modelled as follows:BR1.

A -d- A

B -> B For some x in 0 and some z, Mxz and Saaz.

Corollaries such as compactness follow in the usual way. The mateship relation, though it suggests the star operation, does not enable the definition of star, since neither functionality condition is guaranteed. Nor, for a similar reason, does it permit contraction of model structures by definition of S. The addition of a functionality requirement, m4.

If Max and Mbc then a = b,

does allow definitional elimination of S thus: Sabc (Px)(Rxbc & Mxa). But condition m4 is now required to establish m2. Suppose for m2, Mab. If further Racd then Sbcd, by quantificational logic alone. For the converse suppose further that for some z Rzcd and Mzb. Since Mab and Mzb, a = z, so Racd. (Strictly an ordering relation a < z would suffice, but symmetry of M converts this to what is effectively an identity relation.) The penalty for the natural and obvious elimination of relation S is the definitional restoration of star. Since M is symmetric it supplies an operation, * say, on worlds, defined through the connection b = a* iff Mab. But then the star rule follows and values can be deflated from four to two. Lemma 4.15

Star Rule Lemma.

1 e x(~A, a) iff 1 i t(A, a*).

Proof. 1 e T(~A, a) iff 0 e T(A, a), i.e. by Mated Cross-Over, iff 1 E T(A, b) where Mab, i.e. iff 1 I X(A, a*). The American plan is, in the end, when duly enriched with the Australian mateship component, little more than a very circuitous and cumbersome way of avoiding the much more intuitive Australian plan. §5. The iicreducibility of models for sublogics of C3 the addition of Disjunctive Rules in unreduced modellings and their role in reducing modellings. It might be thought that the procedure adopted of carrying more complex unreduced modellings throughout the semantical investigation is unnecessary. It is not: we show next that (unlike the situation with positive logics, studied in SE III, where a primeness theorem conveniently holds) the class of L m.s. cannot in general be restricted to reduced L m.s. Though the argument is given only for the system G it extends to a whole class of systems which lack forms of Exported Syllogism (i.e., B3 or B4). Theorem 4.16.

1. 2. A -*• C) v D, is not 3. 1

p q =>. q -»• r p •>• r is not a theorem of G. The rule of Disjunctive Suffixing, (A B) v D -o (B admissible in G. Disjunctive Suffixing is however admissible in G+. 1

C

Results like 2 and 3 also hold for Disjunctive Prefixing, and in fact for other disjunctive rules. 334$

4.8 PROOF THAT DISJUNCTIVE

SUFFIXING

IS NOT ADMISSIBLE

IN SVSTEM G

4. Models for G cannot be restricted to reduced models; for that A is valid in every reduced G m.s. does not imply that A is a theorem of G. Proof 1. It is shown that 6 = (q -+ r p -»• r) v ~(p q) is not G-valid by constructing a G countermodel, i.e. a G m3s M with valuation v in M is designed which falsifies 0 . ~ The Gm3s M = < T , 0, K, R,

* > is defined as follows:

K - {T, T*, a, a*, b, b*, c, d, e},

0 = {T, e};

R holds just in the following cases: RTab, Rbcd, Racd, RT*ab, and RTxx for every x e K; < holds in just the following cases: T* < T, a < b, b* < a*, x < x for x e K; and operation * is such that (T*)* = T, (a*)* = a, (b*)* = b, i.e. (y*)* = y* for {T, a, b}, and x = x* for {c, d, e}. A function v in M is defined thus: v(p, a) = 0 = v(q, a), v(p, c) = 1, v(q, c) = 0, v(r, d) = 0; otherwise v may be determined arbitrarily, so let us say that otherwise v(s, x) = 0 for x e K and sentential parameter s. M and v may be represented pictorially thus:

t b* l t i la*

(i) (ii) (iii)

M is a G m3s. v is a valuation in M. v in JJ falsifies 3.

Proof of (i) consists of straightforward case by case verification of pl-p5, p8 and p9. For x e 0, x* < x is given. It is worth noting that M falsifies the conditions q3 and q4 for Exported Syllogism. (ii) is immediate since v(s, x) never has value 1 when x < y holds. Proof of (iii) assumes a classical metalogic. Since Rbcd & I(p, c) = 1 ^ I(r, d), I(p r, b) = 0. To show I(q -»• r, a) = 1 it is enough to show that there is no case where Raxy & I(q, x) = 1 for x, y e K. There is only one case where Raxy holds, viz, Racd, and then I(q, c) = 0. Hence since RTab I(p r -»•. r -*• q, T) = 0. Further as there is only one R-lead from T*, viz. RT*ab and v(p, a) = 0, I(p q, T*) = 1 holds vacuously. Since T* < T, I(p q, T) = 1. Finally then 1(8, T) = 0. 2. p -*• q v ~(p q) is a theorem of G, but, by 1, 6 is not as it would be were the rule admissible. 3. a) Suppose (A B) V D is a theorem of G+. But G+ is prime since it is metacomplete, by Meyer 73a. Hence j- A -»• B or f- D. In either case however f- (B -»• C -*•. A •* C) v D. b) in 4 below.

Alternatively, 3 may be proved by a semantical argument like that

4. Suppose on the contrary, for every wff A, A is a theorem of G iff A is valid in every reduced G m.s. Then the rule of Disjunctive Suffixing is 335$

4.8 MODELLING

LOST DISJUNCTIVE

RULES IN UNREDUCED

STRUCTURES

admissible in G, contradicting 2. It remains to derive the rule. (Ine argument is essentially that veryifying suffixing in G+ in SE III.) Since only reduced G m.s. are considered all biconditionals in lemma 3.2 hold. Suppose for arbitrary reduced G m2s M = < T , K, R > and arbitrary v and associated I in M, I (A -> B V. D, T) = 1. It suffices to show that I(B + C A C) V. D, T) = 1. Since I (A ->- B v. D, T) = 1 either I(D, T) = 1 or I (A + B, T) = I. Suppose first I(D, T) = 1; then, by the rule for v, I (y v D, T) = 1 where y abbreviates B C A C. Suppose on the other hand I (A -*• B, T) = 1; then for every d e K, if I(A, d) = 1 then I(B, d) = 1. To show I(y, T) = 1, and hence I(y V. D, T) = 1, it suffices by lemma 3.2 to assume for arbitrary b that I(B C, b) = 1 and to show I(A C, b) = 1. For the last suppose Rbde and I(A, d) = 1. But then I(B, d) = 1, and so as I(B C, b) = 1 and Rbde, I(C, e) = 1. Hence I (A C, b) = 1. What Disjunctive Rules do, among other things, is to validate material implication principles, which may not however hold in weaker systems. Disjunctive Suffixing, for example, validates a material form of suffixing, namely A ^ B =>. B C A C, in G as follows: (- (A B) V ~(A B), whence by Disjunctive Suffixing, (- (B C A C) V ~(A -»• B). Nothing however stops the modelling of disjunctive rules such as Disjunctive Suffixing in unreduced structure, and there is sometimes much point in doing so. And what holds for Disjunctive Suffixing holds more generally, for other disjunctive rules. In unreduced modellings disjunctive rules paralleling systemic rules such as C V A, C v (A + B) -ft. C V B and E V (A B), E V (C D) -t> E V (B -»• C -»-. A D) are lost from logics not containing corresponding forms of the rules. Such disjunctive rules have, however, proved important in the investigation of dialectical set theory.1 It is fortunate then that such rules can be retained in unreduced modellings, by adoption, in the standard way, or corresponding modelling conditions. We tabluate disjunctive rules and corresponding semantic postulates in the usual way:- 2 jrl. a e 0 > jr2. a e 0 & (Rbxy jr3. a e 0 & jr4. a e 0 > jr5. a € 0 >

JR1. C V A, C V (A B) -e> C V B JR2. E V (A B), E V (C D) E V (B C A D) JR3. C V (A + ~B) -o C v (B -*• ~A) JRA. C V A -o C V ~(A ~A) JR5. C V (~A •*• A) -o. C V A

Raaa R 2 abcd > (Px)(Racx & (Py) & Rayd)) Rabc > Rac*b* Ra*aa* Raa*a

Such rules typically replace corresponding nondisjunctive forms; in particular, JR2 renders Affixing redundant, JR3 makes the matching Rule of Contraposition redundant, and JR5 replaces Rule Reductio or LEM. Soundness is proved for these JL systems in the usual way, that is, it is shown that each rule preserves validity in the appropriate semantics, making use of its corresponding semantic postulate. ad JR1. Assume C v A and C V (A -*• B) are valid. Now for some arbitrarily chosen valuation in some m.s. let I(C, a) = 0 for d e 0. Then, by the assumption of validity, I (A, d) = 1 and I (A ->• B, d) = 1; and hence by Rdbc and

l

2

See Brady's work in PL.

The modellings for JR1-JR5 are due to R.T. Brady; they were originally incorporated in his 'A semantics for some quantified relevant logics', typescript, 1980.

336$

4.8 ROUTINE SOUNDNESS

I(A, b) = 1, I(B, c) = I(C v B, d) = 1.

1.

ARGUMENTS,

AND NEW TWISTS

IN

COMPLETENESS

But by jrl, Rddd, and so I(B, d) = 1.

Thus

ad JR2. Assume E v (A B) and E v (C -*• D) valid. Now, let I(E, T) » 0, with T an arbitrary selected base element in an arbitrary m.s. Then, using the validity assumption, I (A -»• B, T) » 1 and I(C -»• D, T) = 1. Hence whenever Rtcx and I(A, c) = 1, I(B, x) = 1, and whenever RTyd and I(C, y) - 1, I(D, d) = 1. Suppose RTbz and I(B -*• C, b) = 1. Then for every x, y in K for which Rbxy and I(B, x) = 1, I(C, y) • 1. Suppose further Rzcd and I(A, c) = 1. Then by jr2, for some x^ e K, RTcx^ and, for some y^ e K, Rbx^y^ and RTy^d. It follows in turn that I(B, x^) « 1, I (C, I (A -»- D, z) = 1, I(B + C

yp

=. 1, and I(D, d) = 1.

Hence reassembling,

A + D, I) • 1 and I(E v (B + C

A -»• D), T) = 1.

ad JR3. Assume C v (A -*• ~B) valid. Let I(C, T) = 0. Then I(A -»• ~B, T) = 1. So whenever RTde and I(A, d) = 1, I(~B, e) - 1. Suppose RTbc and I(B, b) = 1. By jr3, RTc*b*. Hence, if I(A, c*) » 1, I(~B, b*) = 1. So as I(B, b) = 1, I (~A, c) - 1 and hence reassembling I(B -»• ~A, T) = 1. But then I(C v (B -*• ~A), T) = 1. ad JR4. Assume C V A valid. Let I(C, T) = 0. Then I(A, T) - 1. By jr4, RT*TT*. Hence RT*TT* and I(A, T) « 1 and I(~A, T*) = 0. Therefore for some b, c € K, RT*bc, I (A, b) - 1 and I(~A, c) = 0, and so I (A -*- ~A, T*) - 0. Hence I(~(A -*» ~A), T) = 1, and I(C v ~(A ~A), T) = 1. ad JR5. Assume C v (~A + A) valid. Let I(C, T) = 0. Then (~A -»• A, T) = 1, and so whenever RTbc and I(~A, b) = 1, I(A, c) = 1. By jr5, RTT*T; and hence if I(~A, T*) = 1, I(A, T) - 1, that is, if I(A, T) = 0, I(A, T) = 1. So in any case, I(A, T) = 1. Hence I(C v A, T) - 1. Completeness requires some new twists, primarily complication of the definition of regularity to ensure closure under the Disjunctive Rules of the system, and derivatively adjustment of the Extension and Priming Lemmas. An L-theory a is (fully) regular iff all theorems of L are in a and a conforms to whichever of the following closure conditions match Disjunctive Rules of L: (i) (ii) (iii) (iv) (v)

C v A e a & C v ( A - » - B ) e a > C V B £ a . EV(A-»-B)ea&EV(C->D)€a>Ev(B^C^. C V (A ~B) e a > C V (B -*• ~A) e a. C v A e a > C v ~(A -»• ~A) e a. C V (~A -*• A) e a > C V A e a .

A->D)ea.

In the completeness argument that follows we focus on the case where all closure conditions (i)-(v) are satisfied; but essentially the same argument succeeds, with appropriate rewriting, where only a subset of the closure conditions is satisfied. A JL-theory is an L-theory which satisfies the closure conditions (i.e. (i)-(v) in the case focussed upon). B is JL-derivable from A, written B, iff B is derivable from A by successive applications not only, as usual, of Adjunction and Provable Implication (i.e. A -*• B, where A B is provable) but also of JR1-JR5 (more generally, of as many of JR1-JR5 as hold). A set T of wff is JL-derivable from a set S of wff, written S T , iff, for some A,, ..., A 1' * m

e S and some B., ..., B 1* ' n.

T, A. & ... & A k , B. V ... v B . * 1 m'JLl n

pair of sets of wff < S , T > is JL-maximal iff (1)

T is not JL-derivable from S, and

337$

A

LEMMA TO ACCOMMODATE

4.8 A NEW EXTENSION

(2)

S U T = set of all formulae of JL. A -»- A and hence A f-^ A.

Lemma 4.16.

DISJUNCTIVE

RULES

Note that S U T =

A, since

Iff < S , T > is JL-maximal, S is a prime JL-theory.

Proof is like the corresponding lemma of 4.6 except for the new closure conditions, for which the following case is typical:Suppose C V A € S, C V (A ->- B) e S but C V B i S. Then C V B e T. & (C v (A -»• B)) |-JL C v B, S T, contradicting (1).

Since (C V A)

Lemma 4.17 (Extension Lemma). Where T is not JL-derivable from S, then there is a JL-maximal pair <S', T' > such that S c s ' and T c T;. Hence S' is a prime JL-theory. Proof proceeds as in 4.6, except that L-derivability is replaced by JL-derivability. To complete the proof it is enough to show the following: where B" = B & B', C" = C V C', B' & A. k_ C' and B k T C V A., each of B" & A. ' ' l 'JL 'JL x l k_ C", B" k C" V A. and B" k C" follow. JL

ad

J 1J

B" & A. k 1

J LI

J LT

B" k

J LI

C".

J LI

Since k

J LI

C v C', C1

since k T C' ad

1

C" v A.. 1

J LI

1

B" k T C".

B" f-JL "

v

B

( "

A^.

J LI

1

J LI

B, and since k

J LI

Then by transitivity, B" k

J LI

J LI

&

1

1

JL

1

1

c

1

B & B' •* B, B" k

C" v A..

(C" V B") & (C" V A.).

J LI

(JG)

J LI

1

Then, by transitivity B" & A. k T C".

By the previous step, B" k

J LI

B" k

J LI

(B & B') & A. -> B' & A., B" & A. k T B & A., and

C".

Since k

1

(C v C') v A., C v A. k ad

k

Since k

VJ LI

C" v A.. 1

C v A -*• 1

C" v A.. 1

Hence as B" k

JL

C" v B",

(C" V B") & (C" V A.) + C" V (B" & A.), 1

1

Suppose it can be shown that,

where A [-JL B, C v A

C v B, for all A, B and C.

Then applying (JG) to the first step, C" v (B & A ± ) f-JL C" v C", whence B" C" V C".

JL

Since C" V C" f-JL C", B" H J L C".

ad (JG). Proof is by induction on the derivation of B from A. It is shown that C can be disjoined at the front of each step of the derivation.1 There are the following cases according as the named rule is applied:ad

Provable implication.

the same rule C v A |ad Adjunction. C~V (A & B). ad JR1.

Since then f-^ A -»• B,

C V A

C V B.

Since k

(C V A) & (C V B)

C V (A & B), C V A, C V B

JL

(D V C) V A, D V (C V A) k

JL

JL

Similarly, D V (C V (A + B)) f-JL (D V C) V (A + B). D V (C V (A ->- B)) |-JL (D V C) V B.

f-JL D V x

D V (C V B).

k JL

Since k T D V (C V A)

(D V C) V B

C V B, whence by

Since ^

(D V C) V A.

Hence, D V (C V A),

(D V C) V B

D V (C V B),

Hence, as required, D V (C V A) , D V (C V (A V B))

(C V B).

The method is a fruitful one, and has been applied to advantage in relevant logic, e.g. by Meyer and Dunn 69. 338$

4.8 SHOWING

ad JR2-JR5.

JL MODEL STRUCTURE

THAT THE CANONICAL

IS NOT

FRAUDULENT

Similar to JR1.

Lemma 4.18 (Priming Lemma). Where T is a JL-theory and U is a set of wff disjoint from T which is closed under distinct disjunction (i.e. for distinct A and B, where A e U and B e U, A V B £ U), then there is a JL-theory T' such that T c x*, T 1 is disjoint from U and T' is prime. Note that if T is a regular JL-theory then T* is a regular JL-theory. Proof. Xhe proof is as in 4.6 except that L-theories are replaced by JL-theories and L-derivability is replaced by JL-derivability. However, it is necessary to ensure that, since T is a JL-theory, if A,, .... A e T and A, & ... & A I- , J 1 ' m 1 m 'JL Bj v —

v B n then

v ... v B n e T.

But for each application of a disjunctive

rule, membership in T is preserved by the rule. Priming Corollaries, now stated in terms of JL, follow as in 4.6. Lemma 4.19.

The canonical JL m.s. is a JL m.s.

Proof is like the corresponding result in 4.6. The only old details that require checking are those concerning 0 and 0 J L > and they are routinely confirmed. It remains to verify jrl-jr5. ad jrl. Let a e 0 . Suppose A •*• B e a and A e a. Then B v (A -* B) e a and B v A e a. Since a e 0 , by JR1, B v B e a and B e a. Thus R aaa. J LI

ad jr2.

JL

Let a € 0 T T and R 2 .abcd, i.e. for some z R JL

JL

x' = {A: (PB) (B -*• A e a & B e c)}.

JL

abz and R

JL

zed,

Let

As before x' e K J L and R JL acx'.

Let

As before, y' e K J L and R J L b x * y \

y' = {A: (PB) (B - ^ A e b & B e x ' ) } .

As to

Rjjay'd, suppose A -*• B e a and A e y'. Then, for some C, C A e b and C e x'. Also, for some D, D C e a and D e C. Hence (C A •*. D -»• B) V (A -»• B) e a and (C A -»•. D + B) v (D -»• C) e a. Then, since a e 0 J L > by JR2, (C ->• A D -»• B) V (C -»• A

D

B) e a and C + A + . D

Since R^^zcd, B e d . ad jr3.

Suppose a e 0 J L and R ^ a b c .

and so ~B ->• ~A e a. Hence R JL ac*b*. ad jr4.

Let a e 0

ad jr5.

B e z.

Let A -*• B e a.

Then A -»• — B

Since a e 0 J L > by JR3, (~B

e a and

~A) V (~B + ~A) e a,

Since R n a b c , ~B e b > ~A e c, and hence A e c* > B e b*.

.

Suppose A ^ B e a* and A e a.

Then ~(A -*• B) I a.

By JR4

~A), and so A & ~B -*> ~(A & ~B •*• ~(A & ~B)) , whence A & ~B -»~(A ->• B) . if A & ~B e a, ~(A -»• B) e a, and hence A & ~B i a. Accordingly

A 1 a or ~B i a.

~A

JL

Since R J L abz, D

Then priming corollaries extend x' and y' to x, y e Kj^.

(~B -> ~A) V (A •* — B ) e a.

A -t» ~(A Since a e

B e a.

Since A e a,

Let a e Oj^*

A -k A, and A

and hence ~A v B e A.

~B i a and B e a*.

Suppose A -»• B e a and A e a*. B

~A V B.

Hence Rj^a*aa*. Then ~A i a.

By JR5,

Since a e 0 J L , when A -»• B e a, ~A V B e a,

Since a is prime and ~A £ a, then B e a.

Hence R

aa*a.

Theorem 4.17 (Strong completeness). Where U is a set of wff not JL-derivable from a set S of wff containing all theorems of JL, there is a (canonical) JLmodel which verifies every member of S and falsifies every member of U.

339$

4.8 REDUCED MODELLINGS

FOR CERTAIN SUBSYSTEMS

OF C; AND SUPERSTRONG

RELEVANT

LOGICS

Proof is like the corresponding theorem in 4.6, but requires some additional details, e.g. in showing that given sets are JL-theories. The standard corollaries, given in 4.6 follow as before. The strategy applied in the completeness argument for JL-systems, of closing regular theories under further rules, enables the blockages encountered at the end of 4.6 in obtaining reduced modellings using systems weaker than basic system C to be got past. System C is only basic with respect to a particular, though important, method of establishing completeness. Vary the method appropriately and subsystems of C can be fitted out with reduced modellings. A basic system relative to the method used for JL systems is the system JBB obtained by adding JR1-JR3 to B (thereby rendering Affixing and Rule Contraposition redundant). JBB modellings can be reduced, 0 can be contracted to {T}. Soundness is of course automatic. In completeness, closure order JR1-JR3 closes the gaps principles of C beyond B were required to plug under the original method (as a matter of inspection of completeness proofs, JRl compensates for Bl, JR2 for B3 and B4 and JR3 for D4). §9. Semantical analysis of various principles of superstrong relevant logics, and present limits of the semantical methods. Although modelling conditions have been given for a large array of logics extending B, only irrelevance inducing principles extending R have in fact been modelled. That is, no relevant extensions of R, or superstrong relevant logics as we might call them, have been modelled, though the principles involved play an important part in the argument of 3.6 against Anderson and Belnap's choice of systems.1 It is time to effect some rectification. There are, however, serious limits to the extent of rectification that can be made within the semantical framework so far developed, and many open problems remain. One class of extensions of system R has succumbed to semantical analysis, Minglish principles relevantly strengthening R. That the principles to be considered preserve weak relevance when adjoined to R has already been proved (see the discussion in 3.6 under type 6). Correspondences between the principles and their modelling conditions are tabulated in the usual way (for M5 reduced modellings are used):Scheme

Modelling Condition cMl.

Ml.

A & ~A -*•. A -*- A

M2. M3.

A A

~A 4 B + A V B ~A & B A V (A - B)

cM2. cM3.

M4.

A

(~A -*• A) V (A & ~A -»• B)

cM4.

M5. M6.

A V ~(A ~A & B

M7.

A & B

M8.

A & ~A & B

1

B) V (~A . A -»• B) (~A -*• A) V(A + B) (~A •*• A) V (A + B) A

B

cM5. cM6. cM7. cM8.

Rabc >. a < b* V V a < a* Rabc >. a < b* v Rabc & Rede >. a V b < d* V b < e Rabc & Rade >. a v a < e v b < b* Rabc >. b < T V a Rabe & Rade >. a V b < d* V b < e Rabe & Rade >. a V a < e V b < d * Rabc >. a < b* v

a < c V b < c a < c V b < c < b* V a < c v d < c < b* V a < d* V b < d* v b < e < b * V RT*bc < b* V a < c < c v a < d* V b < e a < c v a < a*.

There are two reasons for this; first, that the earlier investigations (in §§3-6) were undertaken before much thought was given to superstrong relevant logics, and secondly that several of the principles leading to superstrong systems proved recalcitrant to semantical analysis. 340$

4.9 ADEQUACY

OF MODELLINGS

FOR MINGLISH

PRINCIPLES

RELEVANTLY

EXTENDING

R

Proofs of adequacy are presented in the now familiar way, i.e. every scheme is validated using its modelling condition, and each modelling condition is shown to hold in a canonical modelling given its scheme, and, sometimes, those of system R. Since many of the proofs are similar, some details are omitted. 1 ad Ml. Suppose I(A & ~A, a) = 1, Rabc, and I(A, b) = 1. Then I(A, a) = 1 and I(~A, a) = 1, and hence I(A, a*) = 0 . By the modelling condition cMl, a < b* or a < c o r b < c o r a < a*. But a i* b*, since if a < b* then b < a* and I(A, a*) = 1. a £ a*, since if a < a* then I(A, c) = 1. If b < c, then I(A, c) = 1. Hence, I(A, c) = 1 and I(A A, a) - 1. ad cMl. Assume Then for all A, and further A e B, C and D, B e ~D c a. Since

Rabc for prime RMl theories a, B, A - * B £ a & A e b > B e c . a. It suffices to show A e c, a, B I b*, C e b , C ^ c , D e a ((A & D) V (~B & C)) & ~(A

b and c, where RM1 is RM+M1. Suppose a t b*, b t c and a t a*, whence a c c. Now for some wff and D I a*. Hence ~B e b and & D) & ~<~B & C) (A & D) V

(~B & C) (A & D) V (B & C), and a is an RMl-theory, ((A & D) V (~B & C)) & ~(A & D) & ~(~B & C) e a > (A & D) v (B & C) (A & D) V (~B & C) e a. Since A e a and D e a , A & D e a and (A & D) v (~B & C) e a. Since ~D e a, ~(A & D) e a. Since B e a, ~(~B & C) £ a. Hence, ((A & D) V (~B & C)) & ~(A & D) & ~(~B & C) e a and (A & D) v (~B & C) + (A & D) v (~B & C) e a. Since Rabc, (A & D) v (~B & C) c b > (A & D) v (~B & C) e c. Since ~B e b and C e b, ~B & C e b and (A & D) v (~B & C) e b. Hence (A & D) v (~B & C) e c. Since C i c, ~B & C I c. By primeness of c, A & C e c , whence A e c. ad M2. Suppose I(A, a) « 1, and let Rabc and I(~A & B, - 1, I (B, b) - 1, I (A, b*) - 0 and, by cM2, a < b* or a I (A, a) = 1 and I (A, b*) = 0, a £ b and then a < c or b I(A, c) - 1 and I(A v B, c) - 1. If b < c, I(B, c) - 1 Hence I(A v B, c) » 1 and I(A & B + A v B, a) - 1.

b) = 1. Then I(~A, b) < c or b < c. Since < c. If a < c, and I(A v B, c) = 1.

ad cM2. Suppose Rabc for appropriate prime theories a, b and c. Let a t b* and a £ c. Let A £ b. Then for some wff B and C, B € a, B i b*, C £ a, C i~c. Also ~B £ b. Since by M2 B & C -*-. ~(B & C) & A -*• (B & C) V A is a theorem, and since B & C £ a, ~(B & C) & A •*• (B & C) v A £ a. As Rabc, ~(B & C) & A £ b > (B & C) v A £ c. But ~(B & C) £ b, because ~B £ b, by Simplification. So, as A £ b also, (B & C) v A e c, i.e. as c is prime, either B & C f c o r A e c . But B & C / c since C d c, again using Simplification. Hence A £ c. ad cM4. Make use of non-degenerate theories, i.e. suppose Rabc and Rade for appropriate prime non-degenerate theories. Let a £ b*, a t d*, b t b*, b ^ d* and b e. Let A £ a. Then for some wff B, C, D,~E and F, B £ a, B I b*, C £ a,~C i d*, D £ b, D I b*, E £ b, E I d*, F £ b, F I e. Hence ~B £ b, ~C £ d, ~D e b and ~E e d. In addition, since c is non-degenerate, let G / c. Now unpack the theorem (A & B & C) V (D & E & F) -»• (~(A & B & C) & ~(D & E & F) -*- (A & B & C) V (D & E & F)) V (((A & B & C) V (D & E & F)) & ~(A & B & C) & ~(D & E & F) •*• G). ad M6. Suppose I(~A & B, a) - 1, but I(~A -»• A, a) = 0. Let I(A, b) • Rabc; to show I(B, c) • 1. By the data I(~A, a) - 1, I(B, a) - 1 and I(~A, d) • 1 and I(A, e) « 0, for some d, e e K. For this d and e, by ional semantic postulate, a < b* or a < c or b < c* or b < e. Let a < Then b < a* and I(A, a*) • 1. Since I(~A, a) = 1, I(A, a*) - 0, which

x

1 and Rade and the additb*. is imposs-

Full details for cases which are abbreviated or omitted are given in R.T. Brady, 'Further relevant extensions of R'. All the modelling results for these Minglish extensions are due to Brady.

3 41

4.9 COROLLARIES

FOR THE MODELLINGS

OF RELEVANT

MINGLISH

EXTENSIONS

OF R

ible. Let b < d*. Then I(A, d*) = 1, and, since I(~A, d) = 1, I(A, d*) = 0, which is impossible. Let b < e. Then I(A, e) = 1, which is impossible. Finally let a < c. Then I(B, c) = 1. ad cM6. Suppose Rabc and Rade, for appropriate prime theories a, b, c, d and e. Let a b*, b d*, b t e and A e a; to show A e c. For some wff B, C and D, Bea, B^b*, Ceb, C ^ d * , D e b and D i e. Also, ~B e d and ~C e d. Since ~(~B & C & D) & A ^ (~(~B & C & D) ~B & C & D) V (~B & C & D A) is a theorem, and a is a prime theory, ~(~B & C & D ) & A e a > (~B & C & D) -* ~B & C & D e a o r ~ B & C & D " > A e a. Since B ~(~B & C & D) is a theorem and B e a, ~(~B & C & D) € a. Since A € a, ~(~B & C & D) & A e a, and hence ~(~B & C & D) ->~B&C&Deaor~B&C&D->Aea. Suppose ~(~B & C & D ) ^ ~ B & C & D e a . Then ~(~B & C & D) e d, ~B & C & D e e. Since it is provable that ~C -»• ~(~B & C & D) and ~C e d, ~(~B & C & D) e d. Hence, ~B & C & D e e. Since demonstrably ~B & C & D D, D e e, which is impossible. Hence, ~ B & C & D ^ - A e a. Then as Rabc, l N , B & C & D e b > A e c . Since ~B e b, C € b and D e b, ~B & C & D e b, and so A e c. Hence a c c, as required. The semantics given both raise some questions and yield results. For example, the modelling condition for M2 is similar to the additional postulate, Rabc > a < c or b < c, that yields the system RM. This raises the questions of whether R+M2, like RM, is decidable, and of whether it has semantics with strong simplifying features, as has RM (cf. SEI, p.222), where the 3-place relation R can be replaced by a 2-place relation. The similarity of the modelling conditions also at once suggests various systemic inclusion relations (not merely that R+ML c RM), some of which are recorded in the following result: Theorem 4.18. (1) Ml is a theorem of systems obtained by adding to R any of M2, M4, M6, M7, M8. (2) R + Ml is properly contained in R + M4, R + M6, R + M8. Proof of (1) consists in deriving cMl using the modelling conditions for the additional schemes. In the case of M2 and M8 this is immediate, by Addition. ad M4. By substitution in cM4, when Rabc either a ^ b* or a ^ c or b ^ b* or b < d* or b < c. So as in system R Rabc > Rbac, when Rabc either a < b* or a < c or b < c or a < c*. ad M6 and M7.

Similar.

As to (2) the matrix models CM and BM (of 3.6) show that containment is proper. The success achieved with Minglish extensions of R has unfortunately not been repeated with other extensions of considerably more interest, especially extensions of R, E and T by qualified factorisation principles such as Strong Distrubution. Indeed with several superstrong principles the modelling techniques so far deployed appear to fail: details of completeness arguments break down rather conspicuously. Thus the semantical methods have apparent, but not very sharply determined, limits, and leave several open problems, many of them Don't Cares, but some of them nasty. The limits of the semantical methods employed for affixing systems are not confined then to the failure, thus far encountered (in the hands perhaps of inept researchers), to resolve the matter of the decidability of stronger relevant systems such as T, E and R. Another serious, and little understood, limitation surfaces in the shape of several formulae - none of them fortunately relevantly irresistible for the main notions, such as deducibility, under investigation - which resist semantical analysis within the framework so far developed

342$

4.9 TWO

FAMILIES

OF WFF WHICH APPEAR TO RESIST MATHEMATICAL

ANALYSIS

(they will all fit, of course, into the general framework of the universal semantics, arrived at in chapter 9). Some of these formulae can be accommodated by minor elaboration of the semantical framework; the formula scheme B C -*•. A A analysed later (in chapter 7) provides an example. There are others which cannot - hard empirical investigations seem to show - be so easily accommodated, though the reasons why they cannot remain opaque. (Our suspicion is that an adequate theory of translation of these recalcitrant formulae into semantical conditions would reveal that in many cases the resulting conditions include Skolem functions in an uneliminable way: for a beginning on such a translation procedure for relevant logics, see Morgan in DRL.) Among such formulae are the following (in most cases we cite a condition which is sufficient for soundness on the right of the formulae), arranged in approximate family groups:1. Connexive principles: The basic principle is Aristotle; ~(A -*• ~A). Of course the addition of this connexive wff would render even system B simply inconsistent, since (~B C & C ~(C & ~C), but simple inconsistency is no bar to semantical modelling in the case of weaker affixing systems, and may be of considerable dialectical interest (cf. Mortensen in PL). However in the case of stronger affixing systems - R at least - such simple inconsistency induces triviality (the proof is given in ABE, pp.461-462). Thus the system R + Aristotle reduces to the trivial system, TRIV, which has no semantical modelling within the affixing framework (TRIV can of course be modelled within the wider universal semantics: see Routley and Meyer 76). The proof is as follows: Since TRIV includes classical logic S, any affixing modelling for it can be reduced to a single world classical model (using the condition T < a: see qll, §3). But a modelling for S together with a contradictory pair is impossible. For let B and ~B be the contradictory pair. By qll, T < T*, and, by S2, T* < T; hence T = T* effectively. Hence I(B, T) = 1 = I(~B, T) + I(B, T*) = I(B, T), i.e. I(B, T) ^ I(B, T), which is impossible. More complex connexive theses such as Boethius, A -*• ~B -*•. ~(A -»• B), similarly appear to resist first-order modelling in B models. (But as to how to escape this impasse, see chapter 13). 2.

Peirce-type principles: (i)

Peirce:

((A -> B) -*• A)

A

(Px) (Raxa &

(y) (Rxyz > y < a)).

For stronger systems such as R and its extensions the modelling condition for Peirce's law is however unproblematic, since the addition of the principle leads to classical two-valued logic. Indeed Peirce's law reduces even the pure implication part of R, R , to classical implicational logic, as the following argument 1 shows: (A ->. B + A) A -> A (A°B-»-A)-»-A-»-A C->A->A->A->. C A A ° B -»• A A B -»• A

by Peirce (putting B -»- A for B) using fusion, replacement by R^ by Modus Ponens o-exportation.

But A ->-. B A and Peirce, together with principles of R^, provide a full axiomatization of the classical implicational calculus (see Prior 62, p.302). The argument of course makes use of the fact that fusion, conservatively extends R.. ^ r o m R.K. Meyer, 'Peirced clean through', typescript 1975, from which the general argument concerning Lukasiewicz-style weakenings of Peirce is also drawn.

343$

4.9 INTRACTABLE

PRINCIPLES

OF PEIRCE ANV MINGLISH

TYPES

Not only Peirce but all its Lukasiewicz-style weakenings also collapse R^ into S , and so R into S. Rose showed that the characteristic axiom of N 1 N+l Lukasiewicz N + l logic is, essentially, A -»• B -*• A A where A = A and A A^ ° A (from the Lukasiewicz viewpoint the N4"*1 order forms of Contraction and Peirce are two sides of the same coin). By precisely the same argument as above, we can derive A N ° B -»• A from N-Peirce, whence in R^, by Contraction, A ° B -*• A and Peirce follow. It will become evident that several of the Peirce type principles below will likewise collapse R to classical logic (and perhaps E to modal and thence to classical logic). Thus each of (iii), (iv) and (ix) below reduce R to S. For instance, (vi) yields by rewriting (A -»• B) -*• B -*-. (B •*• A) which together with principles of R provides Meredith's axiomatisatlon (with no unconnected variables) of classical implicational logic (see Prior 62, p.302). (ii)

Dilemma:

(A -»• B) -> C (A + B) B Racd > (Px)(Raxd & (y, z)(Rxyz > Rcyd))1

(iii)

C -»•. ((A -+ B) + A) A Racd > (Px)(Rcxd & (y, z)(Rxyz > y < d))

(iv)

(A + B A) C A Racd > (Px)(Raxd & (y, z)(Rxyz > y < d))

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi) 3. (i) (ii)

((A B C) •*• B) A •*• B Racd > (Px)(Raxd & (y, z)(Rxyz > Rycd)) Roll: (A -> B C) (C A -»•. A) Racd > (Px)(R2c(ax)d & (y, z)(Rxyz >. y < d) ((A -»• B) •*• C) B f. B -*• D -*•. A -*• D R 2 acef > (Px, w)(Raxw & Rcwf & (y, z)(Rxyz > Ryew)) (A •*• C -»-. D) B C A •*• B D R 2 acef > (Px)(Raxf & (y, z)(Rxyz > R 2 c(ey)z)) (A -»• C -f. C) -*•. B + C •+. A •* B C R 2 acef > (Px)(Raxf & (y, z)(Rxyz > R2c(ey)f > R 2 c(ey)z)) (A + B) C B ->- C Racd > (Px)(Raxd & (y, z)(Rxyz >. c < z)) (A

B -»-. B) -»-. A -*• B

As for Peirce.

Further Minglish and Suppression principles. Minginterm: A + C-*. B C A ->. B Rabc & Rede & Refg > Rade & Rbdc EMingle:

A-»-B->. A -»• B

A -»• B

These include

C Rabc & Rede > Rade & Rbde

For want of anywhere better to locate it we also include the following rather different suppression principle in this group (it is a thesis of R and also, like EMingle, of S3): :

For these Peirce type principles and of most the soundness requirements we are entirely indebted to M.K. Rennie. The principles and their names are drawn largely from Prior 62. 344$

4.9 FAMILIES

(iii) 4.

Albert:

OF FACTORISATION

((A

PRINCIPLES

B) -*• B) -*• A -*•. A

WHICH RESIST SEMANTICAL

A

MODELLING

Rabc > (Px) (Raxc & (Rxde > Rdbe))

Factor and summation principles: (i)

(ii)

Factor:

A ->• B -»•. C & A ->- C & B.

Summation:

A -»• B -»-. C v A +

CvB.

Rabc > (Px, y) (b < x < y < c & Raxy)

In the framework of system B + (D4) these principles are each deductively equivalent to the paradoxical scheme B -»• C -»-. A •*• A, and they will accordingly be treated in chapter 7. In a more general framework, without affixing rules or their equivalents, however, Factor and Summation appear not to lead to paradox. A modelling theory for this more general setting is given in chapter 13, but a neat theory for factorisation principles has not been found yet. Precisely the same set of points apply in the case of (iii)

Inclusion:

A -»• B

A & B

A.

In systems which contain B2 and D4, Factor and Summation lead to two classes of qualified factorisation principles, of much independent interest: 5.

Strong Replacement principles: (i)

AC-replacement:

(A -*• B) & (B & C -»- D) A & C -»- D Rabc > (Px)(Rabx & Raxc & b < x).

(ii)

DC-replacement:

(A -»• B V C) & (C D) A -»• B V D Rabc > (Px)(Rabx & Raxc & x < c).

DC-replacement was (erroneously) validated by the initial operational semantics for the positive part of R, R+, and is a theorem of the stronger resulting positive Urquhart system, UR+, where it i£ readily validated operationally. AC-replacement follows from DC by Contraposition in the full system UR, and is (indirectly) validated operationally there. These principles appear, then, to be more naturally handled within strong operational semantics (see chapter 11), or in semantics which vary conjunction and disjunction rules (on which see chapter 13). Closely connected with the replacement principles (see 3.6) are 6.

Cut principles, including: (i)

Strong Distribution:

(A & B -»-. C) & (A B V C) A -»• C Rabc > (Px, y)(Rabx & Rayc & x < y & x < c & b < y ) .

The failure of completeness for this condition is perhaps the most exasperating failure of the semantical methods at the sentential level to date. In a way typical of several of the outstanding postulates, the completeness argument breaks down at two crucial points, firstly in finding x and y satisfying all the requisite conditions, and secondly in priming both x and y since usual corollaries to the priming lemma do not apply. (ii)

Mints:

(B ->-. A -»• C) & (A B V C) A C Rabc > (Px, y)(Rabx & Raxy & Rybc & x < c).

In the framework of systems like R containing Commutation and Ass, where Mints is provable, the principle can, of course, be modelled, e.g. by the condition Rabc > (Py)(Racy & Rybc) (i.e. identifying x and c in the more comprehensive

345$

4.9 NONIMPLICATIONAL

condition).

AND NEGATION

PRINCIPLES

WHICH ESCAPE THE SEMANTICAL

For completeness set y^ = {D: (PC)(C - ^ D e a & C e c ) } .

For Ry^bc suppose A -»• B e y and A e b, and show B e e .

NET

Then Racy^.

For some C, C -»- (A

B)

e a and C e c. By Comm, A (C -»• B) e a, so as Rabc, C -»• B e c, whence by Ass, Bee. Then maximize y, in Ry^bc. 1 7. Anderson nonimplication principles: The key principle (which will subsequently be both generalised and specialised) is (i)

Anderson:

A / B

C -*• D.

Among special cases are

(ii) A / B A ->• B, which is closely related to the Lewis axiom (B9) distinguishing strict from material implication, and (iii)

A + A

A -f A.

These principles are dealt with in the expanded framework of chapter 10. 8. Finally there are miscellaneous principles involving negation that have so far escaped our semantical net (this may simply be due to a shortfall of energy). Examples are: (i)

A & B C (A & ~B C) -*• (~A & B C) * ^ RabjCj & Rab 2 c 2 > (Px, y) (Raxy & (bj < x v x < b,,) & (b 2 < x V x < bj) & (y < c 1 v y < c 2 ) *

(ii)

~A -»• A

B -»- A

Rabc > (Px) (Rax x & x < c).

This principle, like many of the foregoing principles, is of course readily accommodated in the less discriminating modal logic framework (where (ii) is nothing but the modal paradox DA -*•. B -*• A). Not so productive of irrelevance are the next two principles: (iii)

(iv)

~A -»• A ~(A ^ ~A) (Pw, x, y, z) (Rayz & Ra*wx & y * < w & x < y & z < w & z < A ~(A ~A), 2 or equivalently 1 A & B ->~(A + ~B) J

x*)

for a e 0 Ka aa , tor a e u

In fact the modelling condition is just jr4 (of §8), which was proved adequate for the disjunctive strengthening JR4 of rule (iv). Some of the problems so far open can be solved by enlarging the semantical apparatus thus far introduced, and making expansions of the semantics is often a good initial way in which to try to resolve recalcitrant modelling problems. 3

1

2

Mints, which served to separate E from system NR, can be further analysed semantically upon splitting relation R into a pair of relations (i.e. syntactically construing the connective as =>, which is analyzed as • ->-): see the semantics for systems such as NR in chapter 10.

According to Brady who has studied (iii) and (iv), they 'do not seem to be capturable in the Routley-Meyer sentential semantics'.

3

This is true also as regards the vexed question of stronger quantified relevant logics.

346$

^ K M T&YWKNTI LOGICS ASSEMBLED AND

CONVENSEV

By enlarging the semantical apparatus and varying the semantics we are able to obtain modellings (still however leaving much to be desired) for Aristotle's principle ~(A -»• ~A), we are able to get to grips with Anderson's questions concerning nonimplicat ions, and so on (see especially chapter 10). The enlargement of the semantical apparatus is in any case essential to handle functors and constants, many of them of considerable logical importance, which have been set aside. In fact much of the remainder of the book is a matter of expanding the semantics to accommodate these and more logical apparatus. We begin on this expansion at once by considering the analysis of functors and constants that have had a significant role in the brief technical evolution of relevant logics, but which have so far been excluded. (Go to chapter 5.) §20. Synopsis of certain frequently referred to systems and their model structures. The systems and structures presented are not included for their importance - though all have some importance - but for facility of reference, since they are mentioned with some frequency in the text. The basic system B and its model structure have been given sufficient separate attention (on p.287ff. and p. 298, respectively). So also, until simpler structures emerge, has system G, the classical foundation (including all tautologies and admitting Y)> and the D systems, such as DK and DL (of p.289) which will figure more prominently when the matter of deep relevant logics and deducibility is further investigated (in 15). But system C, a basic reduced-modelling system, has not. System C has as postulates just the rules R1 and R2, of Modus Ponens and Adjunction, the Distributive Lattice Schemes (A1-A8 of B) and these schemes: B2. A4.

A & (A -»- B) — A ->• A

B

B3. D4.

A B C -»• A C -*» B A -*• ~B -»•. B -»• ~A.

A C m.s. is a structure < T , K, R, * > , with a < b

RTab, satisfying pl-p3 as

2

for B, i.e. a < a, R Tabc > Rabc, and a** = a, and further Raaa, R 2 abcd > Rb(ac)d, and Rabc > Rac*b*. The big-three systems, R, E and T, have already been presented in 3.5 (p.64), and their model structures can now be pieced together therefrom (using 4.3). But more complex model structures admit of considerable simplication. Consider, in particular, that for system R, so pieced together by adding B m.s. postulates q3, q5, q6 and s4. It simplifies to the archetypal R m.s. < T , K, R, * > satisfying just the following conditions: a < a, i.e. RTaa R 2 abcd > R 2 acbd Rabc > Rac*b*

Raaa R 2 Tabc > Rabc a** = a

The archetypal R m.s. may be connected with the original R m.s. by the following steps: first, B5, the scheme corresponding to q5, is equivalent in the context of other R postulates to D3, or more to the point, to Bl; so Raaa, ql can replace q5. The interchangeability of ql and s3 semantically is shown thus, starting from s3: Raa*a, Ra*aa, Ra*a*a*. Given q5, substituting and using RTaa, Raaa, i.e. ql. It suffices, to complete these connections, to derive B5 from D3 (the syntactical route here is simpler). Prefixing D3, (~B -»-. A •*• ~A) -*•. B -»• ~A, whence, commuting and contraposing, (A A B) •*•. A -»• B, i.e. B5. Secondly, q6 may be derived in the archetypal R m.s., and correspondingly used to derive the condition that yields it. One direction goes thus: by substitution in the third condition RTbx & Rzcd > (Py)(RTcy & Rybd), i.e. b < z & Rzcd > Rcbd, applying the fourth (monotonicity) condition; so identifying b with z, q6. Conversely, q6 applied to q3 yields the third condition. Other details are trivial. A relational semantics for system E is set out in Appendix 1. 34 7

5.1 THE AWJTJON

OF CONSTANTS

SUCH AS t ANV A? PROS ANV CONS

CHAPTER 5 FURTHER INVESTIGATION OF RELEVANT AFFIXING SYSTEMS AND THEIR PARTS

In this chapter we assemble a somewhat motley collection of results, topics, and problems left over from or suggested by the preceding chapter. We investigate the logic of constant t and the theory of modality it yields. We consider reformulations of the affixing systems using alternative connectives, and we look at the addition of new connectives to the systems, in particular the connective ° of intensional conjunction or fusion. We also examine the matter of fewer connectives rather than more, and study the parts of B and its extensions. We apply the relational semantics provided for B and its extensions in the previous chapter to prove such desirable features of the systems as normality, Hallden reasonableness, separability, and decidability. In order to establish such features, where they can be established, it is often valuable to vary the models used, and it is important, both here and elsewhere, to be aware of the limitations of the main models presented. The matter of variations and limitations sets the scene for several of the later chapters, where some of the limitations are overcome. (As to present open problems, see further DRL.) §2. The conservative addition of constants, the logic of t3 and inadequate definitions of routine modalities. A sentential constant that has played a central role in the semantical and algebraic analyses of exportative relevant logics is the constant t (or, equivalently, the absurdity constant A of Ackermann 56, since t ** ~A and A ** ~t). Constant t can play a similar, and equally important, role in the case of weaker affixing logics, as the algebraic analyses of these logics (in chapter 12) should make evident. There are various reasons for not introducing constants such as t and connectives such as ° at the bottom of relevant logics, as integral components of the logics rather than additions. One reason is that to be observed in the case both of t and of the additions may cause collapse, or undesirable results, in the case of sought extensions and applications of the systems (consider Et + C8 below). Another reason is that the additions may introduce assumptions or principles at odds with the intended intuitive interpretation of the system and motivation for it. Such is, for example, one of the reasons for objecting to the attempt to add classical negation to relevant logic. The same can and has been said about the inclusion of constants like t in relevant logics; thus it seemed to Robinson 57 in reviewing Ackermann 561 'that the introduction of such a symbol [as A ] is not quite in keeping with the general spirit of the paper1. However so long as it can be shown that a constant connective can be added conservatively to a system there can be little objection to its addition for technical purposes. Since constant t can be added conservatively to relevant affixing systems of interest, there can, it seems, be little objection to its use, for example, in the algebraic analyses of these logics. Harder questions may be asked, however, about the use of t to define modalities (as we shall see later in the section). The proposal is that necessity, for example, be defined in terms of deducibility from necessary truths, which gets written down as DA t A. For the definition to carry the intended interpretation the connective must be read as an entailment, and t must be taken as somehow representing all necessary truths. Ackermann's A is the same as f (elsewhere in the text and in BV): it should not be confused with F which, like the intuitionistic A, implies everything. A fuller discussion of constants, but only in the framework of system R, may now be found in Meyer 79. 34 8

5.1 DEFINITIONS

OF MODALITIES

IN TERMS OF IMPLICATIONS

FROM

VARIOUS

G1VENS

In principle there may seem to be no problems about meeting either of these requirements. The arrow of relevant logic can always be construed as entailment, though it may represent better or worse attempts at capturing the intuitive notion, and in the absence of further conditions constant t can be construed as we like, and so in the intended way. Thus the definition may seem, under the intended interpretation, almost trivially correct and - or because circular. But the definition is really far from assumptionless, and presupposes, and yields, the entailments (t -»• A) -*• DA and DA -*. t -> A. In any case the definition is logically important, and appears at first sight to furnish a pretty treatment of modalities within the framework of relevant logics weaker than system E where direct definitions of • which assign necessity some of the correct properties are impossible. That is to say, in E the connective • may be defined (in ways that are open however to substantial criticism) as follows: •A (A -* A) -»• A, i.e. in effect t can be represented by identities such as A -> A; but in weaker systems, notably T and subsystems, such an internal definition of the absolute modality • is not possible if • is to have even some of the expected properties of logical necessity (see ABE, p.49, for a special case of the more general point). The proposal that absolute modalities be defined in terms of constants added to relevant systems was made by the founding father, Ackermann 56, who defined OA in the A -extension II" of II* as ~A /I; but this proposal was criticised by Anderson and Belnap (59, p.109) on the Ockhamist grounds that the point of extending II1 to II" to define modalities is lost given that the modalities can already be characterised, as they showed, in IT * itself. If however entailment is not captured by II' (or equivalently E) but only in terms of a weaker system, this criticism loses its force; and Ackermann's proposed definitions merit reconsideration - though not necessarily all of the conditions Ackermann imposed on A . In particular Ackermann's rule (£)

A

B, (A -*• B) & C -»- A - * C •*• A (effectively A -*• B, D(A -*• B a. C)

DC)

- which is of course admissible in E - is evidently a suppression principle of an undesirable kind which permits the deletion of provable entailments in given contexts. The proposal that absolute modalities be defined in terms of entailment or relative modalities (according to Lewis 'the colloquially more frequent' ones) did not originate with Ackermann. It appeared much earlier in Lewis's work (e.g. in Lewis and Langford 32, p.161) with the germ of the idea, that what is necessary is what follows from the given, already discernable in Lewis 12. In Lewis constant t (or in Lewis's symbols Q) represents the given, or what is logically given (or, for epistemic modalities, known); and what is possible is what is consistent with the given, i.e. O A A q t, an equivalence which follows from DA **. t -»• A. Lewis, especially in his earlier work, did not differentiate logical and epistemic modalities sharply enough: the reduction of absolute modalities calls for different "givens" in each case what is given logically in one case, i.e. t, and what is given epistemically, or is known, i.e. V , in the other case - and there are problems in the epistemic case that do not arise in the logical case, in particular transmission problems. Thus it is unsatisfactory to try to characterise KA, it is known that A, as V A, for this leads to the incorrect result that what is deducible from what is known is also known. A better strategy is to construe the epistemic modalities in the way von Wright does (in 51), the analogue of • being taken as V, it is verified that; and then defining VA as V-*• A. Even this course is not without its problems, as known difficulties from the logic of

349

5.7 EXTENSIONS

OF Bt, THE DISTINCTIVE

t POSTULATES,

AND THE

t-RULES

confirmation transfer to verification (see Hempel 45). We shall allow however for epistemic and other interpretations of the modality sign and corresponding reinterpretations of t, and accordingly will not rush directly to S5-style postulates for •. It seems evident, for example, that the characteristic S5 scheme fails for verification - ~VA V~VA has to be rejected - whereas the defects of ~0A -*• O Q A , if any, are rather less obvious. We have already characterised a formal language SLt which contains the constant (or zero-place connective) t. The basic affixing logic Bt has as language SLt and the same axiom schemes and rules as system B. As well as extensions of B we consider, at the same time, extensions of B with further connectives, e.g. of B°, by the addition of t. Among the extensions of Bt (Bt°, etc.) of interest are those which add certain selections of the foilowing schemes:Cl.

t

; equivalently

C2.

(t

C3.

t ->-. A

A) -*- A

C4.

A & (A ->• B) & t

C5.

(A

A B

B) & t -»-. B ->• C

C6.

t -»•. A v ~A

CR7.

A -t> t

C8.

A

A

C

A

t -»• A

Further schemes, e.g. the characteristic S4 and S5 postulates, will be introduced shortly. Some of the systems in the literature that result using t postulates are these:BE: B + {CI, C3, Dl}. BE is the basic logic (called B) of AL II. Et: BE + {B3, B5, C2, D3, D4}, i.e. T + {C2, C3>. Et~is a conservative extension of Anderson-Belnap system E by constant t; it is studied in ABE and SE IV. The modality-collapsing principle C8 is of passing interest because it can be adopted in place of Commutation or Assertion in axiomatising Rt, i.e. Rt = Et + C8. Thus too Et + C8 is not a conservative extension of E. In outline the argument that Et + C8 yields Commutation is as follows:Et permits Restricted Commutation. By C8 and C2, A t -»• A, which, using replacement of coentailments, turns Restricted Commutation into full Commutation. Unless Rt is to make a large completeness claim, that whatever is true is provable, t has to be reinterpreted, e.g. as the class of truths, not merely necessary truths. The distinctive t postulates - which tie t precisely to the regular situations - are not C2 and C8 but their rule analogues CR1 and CR7, i.e. t, as the (logically) given, distinctively conforms to the two way rule: A** t -*• A. Frequently this rule can be replaced by the postulates CI and C3. Lemma 5.1. In any extension L or Bt which may be formulated with just Modus Ponens and Adjunction as primitive rules and all its axiom schemes as implications (what we call a smooth formulation), t is distinctive, i.e. Cl and C3 hold, iff the following two way rule is derivable: A*-* t

A

t-rule

Thus t is distinctive in exportative systems such as T, E and R iff the t-rule

350$

5 . 7 SEMANTICS F O R CONSTANT t

obtains. Moreover in these systems where A -»• (B C) A B -»•. A C - or B1 - is derivable, CI, i.e. t, need not be assumed in order to establish the equivalence of C3 with CR7. Proof. It is evident using t -»• t, a substitution case of A ->• A, and Modus Ponens that CI is equivalent to CR1. Again as A -*• A is a thesis, CR7 at once yields C3. For the converse connection let A be a theorem of L. It is shown by induction on the proof of A, using a smooth formulation of L, that t -»• A is also a theorem. There are these cases: i. C ->• D is an axiom scheme of L. Then by prefixing, |- C -»• C C D, L

and so [-

t -»• (C -»• C)

t -»• (C -> D), whence by C3, t -*• (C -»• D) is a theorem

of L. ii. C results from predecessors in the proof sequence for A by Modus Ponens, say from B and B -*• C. Then by induction hypothesis, t -»• B and t -*• (B -> C) are theorems. Hence by CI, B -*• C, and so by Syllogism t -»• C. Alternatively by the implication distribution rule t -*• B -*•. t -*• C, whence again t -*• C. iii. C results by Adjunction, say C is D & E. By induction hypothesis t D and t -*• E are theorems. Hence, by Composition, so is t ->• C. Where t is distinctive, an Lt model structure simply amounts to an L m.s. But in the more general case - where t is in effect an arbitrary constant an Lt m.s. enlarges an m.s. by adding a further one-place attribute (or set) to the structure. We present two equivalent ways, both of which have occurred in the literature, in which this may be done. A Bt m.s. is a structure M = <0, K, R, P, *> where <0, K, R, *> is a B m.s. and P is a property on K. P can be construed, without too much distortion, as a possibility predicate. As well as pl-p4, the following semantical postulate is adopted: p5.

a < b & b < c > a < c .

Postulate p5 illustrates yet another variation that can be made on basic model structures. For p5, and indeed a postulate requiring that < is a partial order, can be added to the requirements on a B m.s. without affecting the class of theorems of logic B, as the completeness proof for B readily shows. Thus a Bt m.s. is really no more than a B m.s. which takes care of constant t. Where t is not simply a passenger constant, but pulls its weight by at least satisfying t-distinguishing schemes CI and CR7, properties 0 and P can in fact be equated (as in AL II); but the reduction of 0 to singleton T is then blocked in the case of some extensions of system C, most notably system E. The interpretation function I is extended to cope with t thus: vi.

I(t,a) = 1 iff, for some b in K, b < a and Pb.

Where t is distinctive, the rule can be simplified to the important rule: vi.a.

I(t,a) = 1

iff a e 0,

since 0 = P and a < a and since b < a & Ob > 0a can be required without affecting proof details. Rule vi.a confirms the intended interpretation of t as representing the class of theorems; for it shows that t marks out the regular worlds, i.e. those where the theorems hold. The modelling conditions for C schemes are as follows, with ri corresponding to Ci:

134$

5.1 MODELLING INITIAL

rl.

AND ALTERNATIVE CONDITIONS FOR POSTULATES IN t

Px, for x in 0, i.e. Ox > Px [or, in reduced m.s., PO]

r2.

(Px)(Px & Raxa)

r3.

Px & Rxab > a < b

r4.

x < a & Px > Raaa

r5.

(Px)(x < a & Px) & R 2 abcd >. R 2 b(ac)d

r6.

Px & x < a > a* < a

r7.

Pb & b < a > Oa

r8.

Rabc & d < b & Pd >. a < c [where t is distinctive, Rabc & Ob >. a < c]

It follows from rl and r7, that Oa iff (Pb)(b < a & Pb). Thus the conditions for the distinctive t requirement suffice for the coincidence of evaluation rules vi and vi.a, as claimed. P can accordingly be eliminated where t is distinctive. The reduction of P to 0 when t is distinctive suggests that alternative rules may succeed even when t is not distinctive, and such proves to be the case. In other words, the evaluation rule for connective t can be always condensed to: vi'.

I(t,a) = 1

iff Sa

alternative t rule.

To apply this rule we replace P in Bt m.s. by property S for which we require p5'.

a < b & Sa >. Sb, for a, b e K:

p5 can be chopped.

If S is used in place of P, supplementary r-postulates have to be amended respectively to:rl'.

Sx, for x in 0 [or equivalently in reduced m.s., SO].

r2'.

(Px)(Sx & Raxa)

r3'.

Sa & Rabc > b < c

r4'.

Sa > Raa

r5*.

Sa & R 2 abcd > Rb(ac)d ;

and so on.

Call an L m.s. where S replaces P a varied L m.s.; in varied L m.s. t is evaluated according to the alternative t rule. In cases of main interest, where t is distinctive, S will coincide with 0. But the procedure is worth sketching because it illustrates the way in which constants other than t can be treated semantically. In establishing adequacy of the semantical analysis, we first elaborate the soundness argument of the previous chapter. To begin there is a new case to consider in lemma 4.1. ad t. A is the constant t. Suppose a < b and I(t,a) = 1. Then for some c, Pc and c < a. But then by p5, c < b; whence I(t,b) = 1. There are also several new cases in the soundness theorem 4.9. ad CI. Since by rl, Px for arbitrary x in 0 and x < x, I(t,x) = 1 for x in 0 by rule vi. Hence I(t,T) = 1, as required. ad C2. Suppose I(t -»• A,a) = 1; to show I (A,a) = 1. Then for every b and c such that Rabc and I(t,b) = 1, I(A,c) = 1. By r2, for some x, Raxa and Px. Thus as x < x, I(t,x) = 1, whence I(A,a) = 1. ad C3. Suppose I(t,a) = 1; to show that I (A -*• A,a) = 1, suppose for arbitrary b and c, Rabc and I(A,b) = 1. Since I(t,a) = 1, for some d, d < a and Pd. By p2 then, Rdbc, so by r2, b < c. Hence, as I(A,b) = 1, I(A,c) = 1.

352$

5.7 DETAILS OF THE COMPLETENESS ARGUMENT FOR STEPS INVOLVING t

ad C4. Apply r4. ad C5. Apply r5. ad C6. Suppose I(t,a) = 1. Then, for some b, Pb and b < a; so by r6, a* < a. Hence, as in the case of D2, I(A v ~A,a) = 1. ad CR7. Suppose that t A is not valid. Then for some situation a in some model, I(t,a) = 1 ^ I(A,a). Since for some b, b < a and Pb, by r7, Oa. Consider a new model with base a. Since I(A,a) ^ 1, A is not valid, thereby verifying CR7. ad C8. Suppose I(A,a) = 1, Rabc and I(t,b) = 1; to show I(A,c) = 1. As I(t,b) = 1, for some d, d < b and Pd; so by r8 a < c, whence I(A,c) = 1. To establish completeness of Lt systems the completeness proof for L affixing systems is likewise elaborated. The requisite new canonical notions are defined as follows:Firstly, P^a iff whenever t A e L, A e a; i.e. if OA e L then A e a. Now define the wide Lt m.s. M as the structure

K^, R^, P^> with base

T^ = L, and the canonical Lt m.s. Mj^ as the structure

K^, R^, P^, *> where,

as usual, unbarred relations are the restrictions of barred relations to prime theories. Secondly, P T a iff (A) (t A e T > A e a). The T-wide Lt m.s. M is defined as the structure and the T-canonical Lt m.s.

as the structure
R^,, P T , *> .

The inductive step for constant t in the completeness argument goes thus:ad t. To show t e a iff (Px e K^,) (P^x & x £ a), for a e K^, suppose first P^x & x £ a. suppose t e a .

Since P^x and t -*• t e 0^, t e x; Let x = {A: t -*• A e 0 T ).

hence t e a .

Conversely

Then by maximalizing on x in the

now familiar way (set out in SE IV), there is an x' e K^, such that Px' and x' £ a. Alternatively lemma 4.4 may be applied to avoid use of Zorn's lemma. The case where a e K^ is a routine variation. ad rl.

Suppose x e 0^.

Then, as j— t, t e x.

Hence whenever t -*• A e T,

A e x, that is P^,x, as required. ad r2.

Let x = {A: t

A e 0 T >.

Then by definition of P, Px, and also x e K.

Suppose to show Raxa, B -*• C e a and B e x . Then t B e T whence B -*• C -»-. t -»• C_e T, so t_-> C e a. But t -*• C -»• C e T, by C2, so C e a. Hence for x in K, Px and Raxa. Now maximalize (as in SE IV) to replace x by a similar element y in K. ad r3. Suppose Px, Rxab and A e a. Since Px, if t -*• B e T then B e x , for every B. Thus by C3, A -*• A e x; and so A e b, as required. ad r4. Suppose x £ a, P T x , A -*• B e a, A e a. Since t -* t e 0 T and P T x , t e x, and so t e a. ad r5.

Hence by C4, B e a as required.

As in the r4 case given x £ a and P^,x, t e a .

Thus R^aaa. The remainder of the

verification is like q3, except that, since t e a, C5 can be used instead of B3. ad r6. Suppose P^b and b £ a. Then t - » - A e O ^ > A € a , for every wff A. Since t -»-. A v ~A» A v ~A e a; ad r7.

whence a* < a.

Suppose b < a and P T b and let A be a theorem.

theorem so t

A e T.

Since

By CR7, t -*• A is a

A e b and as b £ a, A e a.

353$

Hence a is

5.1

COROLLARIES OF ADEQUACY FOR Lt SYSTEMS: CONSERVATIVE EXTENSION RESULTS

regular, i.e. O^a. ad r8. Suppose Rabc, d < b, Pd, and A e a; to show A e c. t -> A e a and by d < b and Pd, t e b, whence by Rabc, A e c.

By C8,

The following adequacy theorem now follows using the parallel result from chapter 4: Theorem 5.1: Strong adequacy theorem for Lt systems. The statement is just as for L systems. Corollary 1. The same result holds where varied Lt m.s. replace Lt m.s. (and so S modelling conditions replace P conditions). In particular then, wff A of Lt is a theorem iff it is true in every varied Lt m.s. Proof. Soundness is a trivial variation of theorem 4.1. For completeness we build on theorem 4.10. Let S T a iff (Pb e 1^) (P L a & b £ a), for a e Then, as in (III) I(t,a) = 1

iff S L a iff t e a .

Further the postulates on

ST follow directly using results on P T . L u Corollary 2. Where Lt is an extension of Bt with t distinctive, then 0 and S may be equated in varied Lt m.s. and 0 and P in Lt m.s. without affecting Lt theoremhood. Proof. Soundness is trivial, since m.s. with the elements equated are m.s. For completeness we apply corollary 1 and show O^a iff S^a. Suppose O^a. Since all theorems are in a, so by distinctiveness is t; P^a as well.

Suppose S L a.

A, every theorem is in a; A e a);

Since t e a so O^a.

and since

whence S^a, and A for every theorem

a

Suppose finally P^ > i.e. (A)(|-^t -*• A >.

then t e a, so O^a.

Corollary 3. Lt is a conservative extension of L for many of the systems considered; in particular, where t satisfies no further postulates or just the postulates for distinctiveness. (Otherwise it has to be shown that the modelling conditions for further semantical postulates corresponding to t postulates do not enlarge the semantical postulates for L.) Proof. Since Lt contains L one part is trivial. Suppose, for the hard part, A is a wff of L which is a theorem of Lt. Then A is Lt-valid and so, as A contains no occurrences of t and there are no distinctive t semantical postulates, is L-valid. Hence A is a theorem of L. In case t is distinctive the argument depends on the simplification of the evaluation rule to vi.a and the consequent elimination of rl and r7. This corollary is not nearly as restrictive as may at first sight appear. For it suffices for algebraic investigations. Moreover in the case of systems E and R, t's distinctiveness is sufficient to characterise t without further postulates. Corollary 4. Et and Rt may be reaxiomatised as simply adding the rule for t's distinctiveness to E and R respectively. Hence by corollary 3, Et is a conservative extension of E and Rt of R. Proof. In the case of Et it suffices to show that (t + A ) + A is a theorem of E + {CRl, CR7}. A semantical argument establishes that it is, for the modelling condition r2 for (t A) -*• A is exactly condition drl for E after P and 0 have been equated, as corollary 2 permits. (Of course if E with t in the vocabulary is formulated with rule DR1 as primitive, (t A) -*• A is immediate from t.) To obtain A -*• . t -*• A for Rt apply commutation to t A -*• A. This happy result does not extend to the weaker systems when t is exploited to define modalities. Even the addition of DA -*• A, i.e. of C2, to

354$

5.7 PROBLEMS WITH THE CHARACTERISATION OF MODALITIES THROUGH t ; AND EtS

system Tt (or indeed to certain exportative sub-systems of T) collapses Tt into Et. The semantical argument of corollary 4 already shows this, but a syntactical argument delivers DR1 directly, as follows: A - f r t + A - ^ A - ^ A - * - . t ->- A-fr A A -*• A, by C2. Although non-exportative systems with C2 need not reduce to Et [for sublogics of S2 do not exceed S2 through the addition of the S2 theorem ((p-* p) A) A)], the adoption of C2 does lead to undesirable suppression principles at odds with the case of chapter 3 for non-exportative systems as those really representing entailment. C2 itself involves suppression of t in an implicative position. This is not the only way in which use of t to define modality involves relevance-restricted paradoxes of implication. Another minor dilemma which emerges can be put as follows:- either the set of necessary truths t represents is complete - unlikely, and in conflict with canonical modellings where t comes down to the theorems of the logic and also with known procedures for eliminating t in terms of conjunctions of cases of the law of identity - or it is less than complete - in which case the adequacy of the definition seems to rest on restricted paradoxes of the form that necessary truths are entailed by variable-sharing instances of the law of identity. Even if the t-reduction of modalities were satisfactory - it is not - it would not help with the wider project of marking out relevant implication in general and defining entailment as necessary relevant implication! For from this perspective logical necessity is required to characterise entailment, so entailment cannot be used, non-circularly, to characterise it. Because of this we subsequently (in chapters 8 and 10)introduce necessity as an independent primitive, not characterisable in terms of t. Properly axiomatised the addition of • to system T does not collapse T, or its theory of necessary implication, to E. Nonetheless the characterisation of modalities through t and entailment is worth pursuing just a little further. One reason is that it makes relatively straightforward, what has sometimes seemed a problem, a semantical analysis of an S5 version of Et. For the modelling conditions that follow it is assumed that t is distinctive (or that varied Lt m.s. are used, in which case S supplants 0):Axiom scheme C9.

t •*• A

CIO. ~(t

Modelling condition t

A) -*-. t

(t

A)

~(t -»• A)

R 2 abde & Ob & Od > (Px) (Raxe & Ox) Rabc & Ra*de & Ob & Od > (Px)(Rc*xe & Ox)

In a similar way modelling conditions may be designed for other modal postulates1,, e.g. for relevant versions of the characteristic postulates of modal systems such as S4.2 and S4.3. Scheme C9 is derivable in Et, though not in weaker systems; but CIO is not derivable. The system Et5 of S5 Entailment adds CIO to Et. The given, t, is not the only constant of importance that has figured in relevant logic, nor is its use in trying to characterise necessity the only important role that has been assigned to t. Other constants of note are the constants f, F and T, which, together with t, play a valuable part in characterising common types of enthymematic reasoning within the framework of sentential relevant logic, and which enable straightforward formulations of "relevant intuitionism" and of "relevant orthologic" (all topics taken up in chapter 6). 1

Also of course for other principles, though again there are limitations. For example the qualified suppression principle (cf. 3.5), (A -»-. A & t -»• B) A -»• B, has the modelling condition Rabc > (Py, z)(Raby & Ryzc & Ox & b < x). For completeness, tentatively define y = {D: (PC) (C D e a & C e b)} and z = {E: (F) (E F e y > F e c)}. The latter is the problem. 355$

5.2 INTENSIONAL DISJUNCTION,

AND LEWIS'S

EARLY EXCURSIONS INTO RELEVANT LOGIC

§2. Alternative and extra connectives, including intensional intensional conjunction, fusion, consistency s and independence; Anderson and Belnap's "case" against disjunctive Syllogism.

disjunctions and

Recently logicians - with some notable and other less notable exceptions - have tended to become extremely parochial as regards the connectives they are prepared to consider in their logics and as to what they are prepared to admit is intelligible. Like herd animals they tend to follow one another in the choice of the same routine set of extensional connectives {&, v, 3, =} with some of these elements perhaps defined in terms of others. If they admit intensional connectives at all, it will be almost as conformist, with • or Q taken routinely as primitive. This wasn't always so. In the early days of modal logic before the entrenchment of classical logic, logicians were much more adventurous and less inhibited. We are thinking of investigators such as McColl, Lewis and members of the Harvard school such as Scheffler and Nelson. Lewis presented his first alternatives to the theory of implication of Principia Mathematica using the connective V of intensional disjunction, and many of his subsequent investigations were based on the relation of consistency. The early C.I. Lewis can indeed be seen as a precursor of relevant logic. His initial attempts at formalising implication in fact resulted in relevant logics - in systems closely allied to system R - and he was, to begin with, singularly uncomfortable when his subsequent attempts at formalisation led to the paradoxes of strict implication (see his confession in 30). In the first theories of strict implication, those of Lewis 12 and 14, strict implication was a relevant implication. In both papers Lewis separates extensional (non-exclusive) disjunction from intensional disjunction, the 'disjunction of the dilemma' (14, p.241) the truth of which 'is independent of the truth of either member considered separately' (12, p.524). It is intensional disjunction, not the extensional one, that represents the disjunction of ordinary discourse. 1 Whereas material implication is defined in terms of extensional disjunction, strict implication or implication proper is to be defined, analogously, in terms of intensional disjunction, A -*• B ~A V B where V symbolises such disjunction or fission. Thus the logical behaviour of implication - whether extensional, modal, or relevant - turns on that of (intensional) disjunction. According to Lewis (12, p.530) a 'few simple changes' in PM suffice for the axiomatisation of intensional disjunction. The basic idea was to keep the sentential postulates of PM, except for the first postulate, the principle of addition, as postulates for intensional disjunction, and to add further postulates for extensional connectives. What Lewis specifically rejected was the scheme A A V B, that is, none other than the paradox A ->-. ~A B. In his first attempt at a logic of intensional disjunction and implication Lewis added conjunction as a further primitive, in his second, 1914, attempt extensional disjunction was taken as primitive and the PM postulates for extensional disjunction in essence adopted. The system Lewis actually presented in 1912 was a rather deficient one which did not contain several of the principles, e.g. for extensional connectives, that he cites as correct elsewhere in the paper. Among the primitives of the fuller Lewis system which contains these correct principles - L"? as we shall call it - are these x

The view that everyday disjunction is tied to conditionality is now commonplace: cf. Strawson 52, p.91: in all, or almost all, the cases where we are prepared to say something of the form p or q, are also prepared to say something of the form if not-p, then q ... But, even if correct, this does not on its own settle the question as to whether everyday disjunction is sometimes extensional.

356$

5.2

LEWIS'S EARLY ATTEMPTS TO FORMULATE THE LOGIC OF STRICT IMPLICATION

connectives: &, V (or v may substitute for &) . Defined are implication, in terms of V, and extensional disjunction, by the usual De Morgan connection. The postulates of the fuller system rewritten in schematic form, with the presented system displayed on the left and with implicational translations where appropriate written on the right, are as follows:A V A

A

A -* ~A

A V B +. B V A A V

(B

V C)

B -* C

A, A

~A -»• ~B -»•. B B V (A V C)

A V B +. A V C

A & B

~A

A

(B

B + C

+ C) A ^ B

—A B +.

A

+ C

A + C

A B

A, ~A v B

B

~P

B

A V B ->. A V B

A -»• B -»-. ~(A & ~B)

A v B ^ B v A

A & B

B & A

A -*• A — A -v A The reason for the final axiom is in order to obtain, what Lewis concedes, all forms of Contraposition and of De Morgan principles; and the Contraposition principles in turn yield Double Negation. It is evident, in the light of recent investigations, that L12 is a subsystem of R, which contains the full implication-negation fragment of R (see the formulation of R-> in ABE, pp. 142-3), but which is incomplete in its conjunctive part. L12 lacks from R, what Lewis would clearly have accepted and did accept in his 1914 system, Composition, Distribution and Adjunction. It is amusing to reflect now that Lewis's first attempt at formulating a good implication to replace material-implication yielded a relevant logic and one very close to system R. Lewis's second attempt at formulating the logic of strict implication, in 1914, which combined the intensional disjunction principles of L12 with the extensional disjunction principles of PM, was not so successful. The initial formulation of the 1914 system contained, as postulate S9, a version of the principle of Antilogism, A V (B v C) -*• B V (A v C), which (as Parry was to show) collapsed the system into classical logic. Lewis had, however, some reservations about S9 - among other things it led, he argued, to (such paradoxes as) p -»• q p & ~q -»• r and to p -»• ~(q & ~q) - and accordingly he suggested replacing S9 by M6, i.e. A v (B v C) B v ( A v e ) . However the modified system, which Lewis never really endorsed, is in equal trouble owing to the inclusion of the principle of Summation, B C A v B A v C, which yields in the context of the remaining postulates A -»-. B A, and so again classical logic (for requisite details see chapter 17, where summation and factor principles are studied). Lewis was later to realise that the inclusion of A V (B V C) B V ( A V C ) i.e. of Commutation, as a correct implication principle was wrong. Interestingly the deletion of this scheme from the modified 1914 system would have yielded the system I (of SI, also studied in chapter 17) which contains as its first degree part the system FD of first degree entailment (2.11). In short, all the ingredients for a paradox-free syntactical analysis of entailment already appear in Lewis; but the approaches were never followed up correctly, and were soon to be dismissed.

357$

5.2 THE WEAKNESS

OF ANDERSON

AND BELNAP'S

CRITIQUE

OF LEWIS AND

STRICT-IMPLICATION

By 1914 Lewis had already begun to leave the road to enlightenment, having rediscovered in 1913 the modal definition of strict implication known to medieval logicians. The modal reduction was presented in 1914, and later, by way of the equation A V B ~ Q ~ ( A v B) 1 , an equation Lewis had earlier, and correctly, rejected (12, p.526) along with the equation of everyday disjunction with extensional disjunction. The interrelation of extensional and intentional disjunctions, along with the matter of principles correct for each disjunction determinate, has been an important issue in the dispute between stricters and those who like their entailment without paradoxes, Anderson and Belnap especially. The central part of Anderson and Belnap's critique (in 62 and ABE, p.164) of Lewis's "independent argument" for the paradox A & ~A -»• B, which (as ordinarily presented) turns on the issue of disjunction, may be summarised as follows (after Curley 72, p.199):Lewis claims firstly that A entails B means that B is deducible from A 'by some valid mode of inference' and secondly that there is a valid mode of inference from A & ~A to B. We accept the first point but deny the second. Lewis's independent argument is not valid because it uses Disjunctive Syllogism. This principle is, of course, valid in one sense, namely that of strict implication. But if Lewis is presupposing such a criterion of validity, then he is begging the question; for that criterion would commit us directly to the paradox without any need for the argument. Should Lewis attempt to escape the charge of questionbegging he encounters, however, a trilemma concerning the interpretation of disjunction; either (i) he interprets disjunction extensionally throughout the argument, in which case Disjunctive Syllogism is invalid, or (ii) he interprets disjunction intensionally throughout, in which case Addition (in the form A A V B) is invalid, or (iii) he commits a fallacy of ambiguity, treating the disjunction of 'A or B' both extensionally (in deriving it from A) and intensionally (in going on to use it in Disjunctive Syllogism). Lewis and most stricters would certainly want to avoid alternatives (ii) and (iii): A -*• A V B does not hold when intensional disjunction is construed strictly, and the "hard" argument would definitely lose its strength if it relied on an ambiguity in disjunction. The weak point in the Anderson and Belnap critique has always appeared, to strict critics, to lie in alternative (i). And Anderson and Belnap have scarcely presented a case likely to persuade reluctant stricters, or even swinging voters with relevant sympathies, that Disjunctive Syllogism, formulated with extensional disjunction, is defective, or that inferences using it are invalid. Indeed they have perhaps somewhat confused the issue by making it appear that inferences using it are in a sense correct, if not valid, through the emphasis placed on proofs of the admissibility of the rule y of Material Detachment for entailment logics, that where A & (~A v B) is provable so is B: 'the disjunctive syllogism, so construed, is an admissible rule' (ABE, p.299). The central point however is that their case against Disjunctive Syllogism, and consequently against the hard Lewis argument, does not stand up to much examination. 1

The modal mistake for A V B <*• ~(~A « ~B). The modal mistake was also incompatible with Lewis's proposal (14, p.246) for eliminating the paradoxes of strict implication.

34 7

5.2 FAILURE OF THEIR CRITICISM

OF DISJUNCTIVE

SYLLOGISM

AS A RELEVANCE

FALLACY

Their main criticism is that A & (~A v B) B 'commits a fallacy of relevance' - when v is construed extensionally; for otherwise 'the real flaw in Lewis's argument is not a fallacy of relevance but rather a fallacy of ambiguity' (ABE, p.165). It is apparent however that Disjunctive Syllogism does not commit a fallacy of relevance in any obvious way, since B is relevant to, and shares variables with, A & (~A v B) ; for B is a component of A & (~A v B). That is, Disjunctive Syllogism satisfies the 'necessary condition for the validity of an inference from A to B', 'that A be relevant to B' (ABE, p.17). So wherein lies the fallacy of relevance? Anderson and Belnap can hardly argue validly: the conclusion of the hard Lewis argument commits a fallacy of relevance; the premisses other than Disjunctive Syllogism do not; so Disjunctive Syllogism does. For it is not just that such an elimination argument is too like Disjunctive Syllogism itself for comfort: it is also that responsibility for the emergence of a fallacy cannot be distributed back on one premiss in this simple way. There are, as Anderson and Belnap's formal analyses of relevance (e.g. by way of subscripted proofs) help reveal, two sorts of relevance, principle or implicational relevance which Disjunctive Syllogism prima facie exhibits but A & ~A -*• B does not, and systemic relevance, which the system E has in not containing any implicational irrelevances, but which no systems of strict implication have. But a principle on its own cannot be accused of systemic irrelevance (unless a principle be identified with the system consisting of its own singleton, and even then the system consisting of Disjunctive Syllogism alone is not irrelevant). Since Disjunctive Syllogism appears, to lead to neither sort of irrelevance, we are obliged to reject Anderson and Belnap's main criticism: while it is a good criticism of classical and strict systems it is no criticism of Disjunctive Syllogism or a cogent reason for singling it out for special criticism. There are things seriously wrong with Disjunctive Syllogism (see e.g. 2.9), but such irrelevance on its own is not one of them. (And derivatively there are ways in which Disjunctive Syllogism may be seen as a fallacy of relevance: see Routley 82a and DS). Evidently, however, any system containing an implication (and thus Modus Ponens and Rule Syllogism) and extensional conjunction and disjunction and also Disjunctive Syllogism exhibits systemic irrelevance? True: any system with the apparatus for Lewis's argument exhibits such irrelevance through the paradoxes. But that does not on its own put the heat on Disjunctive Syllogism; for relevant systems may be designed upon deleting practically any one of the principles deployed in the Lewis argument, e.g. there are relevant connexive systems with Disjunctive Syllogism (cf. chapters 1 and 2). Some further case against Disjunctive Syllogism is called for; and the case cannot be that the principle involves extensional disjunction. For firstly what is in question is whether the principle in this form is valid; and secondly Disjunctive Syllogism and the paradox argument can be reformulated so that disjunction nowhere figures: simply replace ~A v B by ~(A & ~B). It is somewhat harder to argue that an intensional conjunction has been confused with an extensional one (though it can be done); more important, the matter of disjunction is not of the essence. Anderson and Belnap 62 facetiously suggest that no-one except a logician making jokes has ever accepted "or used extensional Disjunctive Syllogism.1 Curley, taking this charge seriously, assembles historical evidence (72, p.200) which indicates that the charge is false. Furthermore Disjunctive Syllogism and its mates are substantially used in mathematics, not merely in such corrupted areas as modern set theory and recursion theory, but even, as investigation of relevant arithmetic have begun to make clear, in elementary arithmetic. 1

Lewis may have encouraged this view with his repeated claim that Disjunctive Syllogism is scarcely ever used, and is indeed a rather useless logical (footnote continued on next page) 359$

5.2

THOUGH

THEIR CASE MAY FAIL, THEY HAVE V1SL0VGEV

DISJUNCTIVE

SYLLOGISM

The remainder of Anderson and Belnap's case against Disjunctive Syllogism is, at bottom, that the principle is not a thesis of certain preferred systems, in particular of the system of tautological entailments and of subscripted formulations of E which are supposed to answer to the notion of proof from or using hypotheses. While this is true, it does not offer a case for rejecting Disjunctive Syllogism without much further ado; and, moreover, the appeal to E and subsystems begs the question in much the way that an appeal by Lewis to the strict definition of validity would in defence of Disjunctive Syllogism. The availability of a subscripting formulation of E, in which track is kept of premisses used, shows very little; for the negation and disjunction rules Anderson and Belnap offer (in contrast to the implication and conjunction rules) are pretty ad hoc, and moreover are easily varied to fit a range of other systems - including strict systems (see e.g. 7). Building on the semantics of the system of tautological entailments, there is a case to be made against Disjunctive Syllogism; but unfortunately Anderson and Belnap have not made it, even in their epic work on entailment. Anderson and Belnap have however cut the ground from under easy acceptance of Disjunctive Syllogism and the Lewis paradox arguments, and they have gone a long way in providing really viable options to systems of strict implication. In doing so they have, incidentally, confirmed Lewis's early vision of rival systems of implication and largely undermined his mature views as to the unassailability of strict implication and its paradoxes. In 1914 (p.247) Lewis was not writing merely of the systems that subsequently came to be known as systems of strict implication: But even if, in the end, we prefer the false simplicity of material implication, this much, at least, remains clear; the present calculus of propositions is only one among a number of such systems, each of which may be self-consistent and a possible choice as an applied logic. Yet later Lewis was to so overstate his case as to claim (30, p.38): There was no way to unexpected theorems accepted laws as to any formal logic at

avoid the principles stated by these without giving up so many generally leave it dubious that we could have all.

And similarly (see Bennett 69, p.236): There is no way of escaping from the paradoxes except by forgetting to ask yourself some question of logic, or repudiating the plainly indicated answer. Just how wrong Lewis was, Anderson and Belnap have demonstrated; and underwriting some of their claims is one of the burdens of this text. The stumbling block for Lewis, as for Moore, Lewy, and many others seeking a satisfactory account of entailment, lay in the principles of Disjunctive Syllogism and (Rule) Antilogism, which Lewis briefly contemplated abandoning (14, p.246), 1 (footnote 1 continued from previous page) principle. In ordinary discourse, we seldom have occasion to make an inference which has this form ... (Lewis and Langford 32, p.243). Lewis's claim is certainly mistaken, Lewis and Langford themselves make heavy use of Disjunctive Syllogism, as a paradigmatic entailment principle, in their argument that deducibility comes down to the converse of strict implication. Furthermore Disjunctive Syllogism (or an equivalent such as Rule Antilogism) figures crucially, and it seems essentially, in various metalogical results concerning modal logics, e.g. in the proof of completeness of quantified S2; and it also plays a critical role in applications of logic in mathematics.

360$

5.2 LOGIC AND SEMANTICS

FOR FISSION

AND

CONSISTENCY

but saw no way around. The blockage lay, then, to put It in semantical terms, in the unwarranted modal restriction to the possible, in the failure to see the role and importance of impossible situations in the assessment of deducibility (cf. 2.9). We turn back to the investigation of relevant accounts of intensional disjunction. In fact there are at least two non-modal notions with some claim to represent intensional disjunction, distinct notions which systems like R conflate, but genuinely entailmental systems distinguish; namely the relation V already characterised, which is interdefinable with entailment (or, more generally, implication) and consistency, and a relation W which corresponds, in De Morgan fashion, to the relation ° of fusion or intensional disjunction, and is defined: A W B = D f ~(~A • ~B). Since connectives V and o are interdef inable with -»•, relevant affixing systems with negation can be recharacterised with either intensional disjunction or consistency (or inconsistency) as alternative primitive in place of entailment. Most simply this can be done by replacing -*• by V (or ^ ), defining and retaining the postulates already given for B or extensions. But it is evident that more elegant axiomatisations can be obtained, e.g. the lazy procedure suggested would give for R5 the primitive rule: ~A V ~B -o ~B V ~A, which would, in a more elegant formulation, give way at least to: A V B-> B V A. The semantical rules for V and o are precisely those that result from their definitions in terms of ->-, namely I (A V B,a) = 1 iff for every b and c in K such that Rabc, either I(A,b*) = 1 or I(B,c) = 1; and I(A 0 B),a) = 1 iff for some b and c in K for which Ra*bc I(A,b) = 1 = I(B,c*). These rules simplify in the expected fashion for suppressive logics. Thus in the modal case where Rabb stands in for the two-place accessibility relation Rab and a = a* for every a e K, the rule for V reduces to that for D(A v B) and the rule for 9 to that for 0(A & B), i.e. the modal mistakes ensue. And in the case of system R, because Commutation ensures that Ra*bc iff Rba*c, Ra*bc iff Rabc* since Rba*c iff Rbc*a; thus in R the rule for O (wrongly) reduces to that for fusion ° (to be given shortly). The semantical analysis of systemic consistency facilitates comparison of the relevant theory of consistency with Lewis and Langford*s strict theory (32, p.l53ff). For the semantics furnishes relevant countermodels to such strict theses as the S2 Consistency Postulate, A $ B A « A (32, theorem 19.11), and so shows that this postulate is not a theorem of properly relevant logics. The Consistency Postulate is of course - in consistency as distinct from modal formulations - a paradox, amounting to none other than A -*• ~A -*-. A -*• B. A more detailed comparison of relevant with strict consistency principles is worthwhile; we make a modest beginning on this task by indicating the extent to which the strict consistency principles Lewis and Langford 32 deduce hold for relevant affixing logics (however we omit their T-principles, 17.7 ff). We set out in each case, in left to right order, the Lewis and Langford theorem label, the scheme in question prefixed by an acceptance (|—) or rejection (—|) sign, and the relevant principles beyond those of B and definition of
h A & B

17.,12

b

17.,13

A

A

0

equivalent to D3.

B

B ** ~(A 0 ~B)

h

~(A 0 ~A)

17.,2

h

A 0 B H- B o A

17.,3

h

A

equivalent to D4.

B & A 0 C -*•. B 0 C

B3, B5, D4 361$

5.2 A VETAJLEV

COMPARISON

OF RELEVANT WITH STRICT

17.31

|- A

17.32

|-A^C&B-»-D&AOB^.

B & ~(B « C) -»-. ~(A 0 C)

CONSISTENCY

B2

C O D

Proof can avoid Lewis and Langford's use of B -»• D -*. ~D -*• ~B, B -> D -»-. C ~D . C B D C ~D A ~B, A->-C&B->D->-. Principles used are B3, B4, B5, D4. Also,

Antilogism, as follows: ~B, A C •>. C ~B -»•. A + ~B, A -»• C & A O B + CoD. without B5, f- A + C & B + D-*-.

A V B ••> C 0 I). 17.33

i— A - C & B ^ D & ~(C O D)

17.4

-j (A & B) O C

(B & C) O A;

~(A

-| (A & B) 0 C

(A & C) « B;

O B)

B2, D4

etc.

These rejected principles are derivationally equivalent to Antilogism. 17.5

|

(A O C) & ~(B 0 ~C)

~(A 0 B)

B2, D4

17.51 I- (A -»• ~C) & (B -> C) - ~(A O B ) B2, D4 These principles and several that follow illustrate well, what Lewis and Langford remark upon (32. p.156), that 'many of the laws governing the relation of consistency follow from the Principle of Syllogism'; a little more exactly many depend upon the availability of Conjunctive Syllogism (B2) and of Contraposition (D4). 17.52

f- (A

B) & (A -»• ~B) -»• ~(A 0 A)

17.53

(- (A -*• B) & (A 0 A) -*• (B 0 B)

B2, D4 B3, B4, B5, D4, BR1

Proof is, in outline, as follows: B -*• ~B -*•. A •*• B ->. A ~B, B •*• ~B A B A B -*-. A -*• ~A, since A -»• ~B A B ->. A ~A, B -»• ~B -»-. A -*• B -»-. A ~A, A B B -»• ~B ->•. A -*• ~A, A->-B->. A O A + B O B . In principles like 17.53, 17.3 and 17.32, the principles of exportative relevant systems, notably E and T, enable the proof of results which depend for their proof in S2 on Antilogism. 17.54

|- (A -»• B) & ~(B O B ) + ~(A 0 A)

As for 17.53

17.55

|- A 0 A & ~(B 0 B)

As for 17.53

17.56 Since

~(A -»• B)

A + B & A O A -»-. A 0 B

&

Qh

A o A, |-

&

B3, D4, B5

OA & (A -*• B) -*•. A 0 B.

This gives the standard

relevant and modal qualification on Boethius, A -*• B -»-. A • B. Langford remark of 17.56 (p.157):

Lewis and

a self-consistent proposition is consistent with all its implicat ions. The principle A B A 0 B, which might be expected to hold, does not, in fact hold without exceptions. There are propositions which are not consistent with any other proposition; yet such propositions have implications. Hence if A -»• B holds, it requires the further condition A 0 A to assure that A is consistent with B. Needless to say, this is not the relevant reason for qualifying Boethius; for the claim that there are propositions that are not consistent with any other propositions, is but a variation on a paradox of implication. Relevantly there are no propositions (apart from artificial constants such as F) which 1 are not consistent with some other propositions. For consider any wff A This is not satisfactorily formulable systemically at the sentential level, but it should be a theorem of a complete extended propositional logic that (p)(Pq) q o p.

362$

5.2 A/KEY DIFFERENCE: LOGICAL

IMPOSSIBILITIES

WHICH ARE

SELF-CONSISTENT

and let B be any other wff weakly irrelevant to A. Then as A does not entail ~B, A is consistent with B. The relevant reason for qualifying Boethius is, in part, that there are counterexamples to the unqualified form. For let A be p & ~p and B he p v ~ p ; then A * B, but also ~(A $B). 17..57

B

(A > B) & ~(A 0 B) ->. ~(A 0 A)

B2, D4

17.,58

H

(A 0 A) & ~(A 0 B) -> ~(A -+ B)

B3, D4, B5

17. 59

H

(A 0 A) & (A

17.,591

h A Q A ->- ~(A •*• B & A -»• ~B)

17.,592

h A 0 A -»•. A O B v .

17.,6

h A -»•. A

17..61

h ~(A Q A) -*• ~A

B)

~(A

~B)

17.56 B2, D4 (cf. 17.57) B2, B4

A 0~B

OA

D3 (cf. 17, D3

H <JA ** A v A, but |- OA ">•. A O A, with 0 A = D f ~0~A. Proof of the relevantly correct half goes as follows: A -»• ~A ~A, D(A -> ~A) O A , A ->• ~A ~ O A , A -»•. A OA. The proof adds to B«, D3, Necessity Distribution and the S4-type principle A -*• B •>. D(A ->• B). The rejection of A o A -*• OA is one of the crucial separation points between relevant logics and strict implicational systems. The strict argument for A 0 A QA depends, of course, on the characterisation of strict implication, and runs as follows:-»• ~0(A & A), & A) A ~A, ~QA -*•. A -f ~A, ~(A -tf ~A) ->• <)A. By this (invalid) argument stricters rule out statements which are self-consistent but logically impossible, incorrectly assimilating all such statements to A & ~A. Examples of logical impossibilities which are self-consistent are not hard to come by once the paradoxes of implication have been seen through, indeed most non-implicational impossibilities are of this sort. Consider, to take a very simple case, ~(A -»• A). Since •(A -*• A), A), i.e. ~(A -»• A) is logically impossible. But ~(A -»• A) is self-consistent, since it is not the case that ~(A-»- A) -*•. A -*• A. 1 18..1

18.12

^

-OA

18.13

H

Q~A " ~A 0 ~A

~(A

18.14

-|

DA

18.2

-|

A -*• B ** ~(A & ~B 0. A & ~B)

o A)

~(~A O ~A);

QA «». ~A

A

The strict proof of this principle depends on repeated applications of Antilogism. 18.3

-f

A 4 B

18.35

H

^(A & B & C)

A & B

0. A & B A & B & C

-\ A & B & C

0. A & B & C; A & B & C

«*•. A & 6 9 C;etc.

This set of rejections completes the comparison of relevant consistency principles with those of Lewis system SI. The essential differences between the accounts derive from the definition of strict implication in terms of a one-place modal connective and. one upshot of this, Antilogism. 1

The apparent truth of ~(A -*• A) A. (A -»• A) raises Anderson's problem (in 63) about the system in representation of the fact that ~(C D) does not imply A B, for any wff A, B, C and D. Is it enough to have _] ~(C -»• D) -+. A B, or should a semantically complete relevant logic include the thesis |- ~(C -*• D) A. A -*• B? We have answered that the rejection is enough, but the semantical conditions for such theses as (C A D)A. A -*• B and its special cases such a s A ^ A f A + A merit further investigation (for which see chapter 10). 363$

5.2 THE UNACCEPTABILITY

OF LEWIS'S

CONSISTENCY

POSTULATE

System S2 results from SI by the addition of the postulate 19.01 & B) OA (Possibility Simplification) or what is strictly equivalent 19.1, ~(A
•: h A 0 (B & C)

A 0 B

D4

A $ C

D4

h

A 0 (B & C)

19.,43

H

A 0 (B & C) -v. B 0 C

19..44

D4

h

A 0 (B & C)

(A 0 B) & (A 0 C)

19.,45

H

A 0 (B & C)

(A 0 B) & (A 0 C) & (B 0 C)

19..451

H

A <> (B & C)

(A 0 A) & (B 6 B) & (C 0 C) & (A Q B)

(B 0 C) Thus the analogy Lewis and Langford observe (p.170) between & and 0 is substantially weakened in relevant systems. 19.7

-| A « A

A « B v, A • ~B;

-) (A © B) v (A O

A $ A.

The acceptable half of 19.7 is given by 17.592. According to Lewis and Langford (p.173) 19.7 is a 'very important' expansion principle. Granted it is a principle analogous to the extensional expansion principle that plays a critical part in Lewis's independent argument for the positive paradoxes, namely A A & B v A & ~B: it is also equally obnoxious. 19.72

-\ ~ 0 A

~(A O B) & ~(A 0 ~B) ;

~ <>A -»• ~(A • B) & ~(A 0 ~B) .

So, on this note of rejection, ends our comparison of Lewis and Langford's sentential consistency principles with corresponding relevant principles. Lewis and Langford consider along with consistency, at the outset at any rate, a further connective, that of independence (e.g. 32, p.122). There are in fact two familiar independence relations worthy of systemic investigation, both of them commonly relegated to metasystemic investigations these days because sufficiently discriminating logics to mirror their features have not won proper investigation. The relations are characterised thus: B is independent of A iff A does not imply B; and A and B are mutually independent iff A does not imply B and B does not imply A. As usual we define these relations generally for implication as a determinable: independence relations for entailment determinates then emerge as special cases. Relevant affixing systems yield many of the expected properties of these independence relations, but not all of them. For example, the semantical methods of chapter 5 (and parallel matrix methods) will show that in system B Double Negation, — A -*• A, is independent of Identity in the standard metasystemic sense of postulate independence (a revealing discussion of this notion can be found in Hughes and Londey, pp. 131 ff.); but it is a theorem of B that A ->• A ~~A -*• A, and also conversely. That is, Double Negation, so represented, is not inde-

3 64

5.2 FURTHER

INTENSIONAL

CONNECTIVES:

INDEPENDENCE,

AND

FUSION

pendent of Identity. There is undoubtedly something strange about the latter result. Our explanation will be elaborated later (in chapter 14, where the logic of independence is further studied): the explanation is, in brief, that the familiar notion of logical independence is one linked with, and a discrimination of, the truly rigorous entailmental relation corresponding to propositional identity. For this entailment, replacement of coentailments is not admissible, so A A -»-. ~~A •*• A fails. The strangeness of some affixing results concerning independence is, of course, minor indeed compared with the strict implicational theory, according to which no propositions are independent of impossible ones, and no necessary proposition is independent of any other proposition whether necessary or not. These results, even if familiar, are scandalous: it is surely the case that irrelevant propositions are mutually independent. A statemental relation - with little in the way of historical precedents, except insofar as it is confused with the consistency relation, as happens in system R and its extensions - which has come into its own with the development of relevant logics is that of fusion or intensional conjunction, symbolised Fusion has proved important in obtaining sequence formulations and algebraic formulations of relevant logics, and in linking relevant logics with the theory of combinators (see 12). Fusion could, of course, have played a similar role in the theory of strict implication, especially the investigation of the entailmentally important S2 and S3 systems, as we will show in passing; but the historical emphasis on the reduction of two-place modal connectives to one-place ones has obscured such roles.) The logical behaviour of fusion is completely determined, at the sentential level, by an importation-exportation rule R6.

(A ° B) + C H A + . B

C

(Residuation,or °-Portation)

That this rule enables the derivation of all the other expected properties of fusion in main systems of relevant logic will become clear from its semantical analysis and its algebraic treatment (in 12). For example, that <> is commutative in R, but not in E and sublogics, is a consequence of the fact that Rule Commutation holds in R but not in E.1 Proof of A ° B •*• B ° A in R° goes as follows: Since B ° A -*• B ° A, B •*•. A •*•. B ° A by R6. Thus, by Rule Commutation, A -*•• B -*•. B o A, whence A « B + B » A by R6 again. Syntactically the importance of fusion, especially from the point of view of Gentzenisations of relevant logics, is that it permits the bunching of appropriately nested antecedents. The (bad) case for taking fusion as a kind of conjunction is that it conforms to some of the most undesirable properties of classical conjunction, most conspicuously Rule Exportation. Because of the failure of Rule Exportation, fusion does not admit, either relevantly or modally, of simplification, i.e. A ° B -*• A is rejected, and B ° A + A is rejected in S3 and its sublogics, though not in the overpowered S4. Thus an essential feature of (normal) conjunction is lacking. Logics L° and Lt°, with languages that contain ° as a further primitive connective, result by adding to the postulates of L and Lt respectively the Rule R6 or Residuation. An L° m.s. is an L m.s. for which the following postulate - an optional extra for L m.s. - holds:p6.

a < d & Rbca >. Rdcd

The semantical evaluation rule for fusion is the following: 1

System R is opened to further damaging counterexamples through its (derivable) fusion principles. For example, according to R, where A B -»• C, C follows from the mere compatability of A and B. This would afford, e.g., a (fallacious) proof of the axiom of choice from the compatability of the axioms of set theory with the continuum hypothesis (as W. Harper remarked). 34 7

5 . 2 THE LOGIC AND SEMANTICS

vii.

OF

FUSION

I(A ° B, a) = 1 iff, for some b, c e K, Rbca and I(A,b) = 1 and I(B,c) = 1.

Truth, validity and so on for L° systems are defined in the established fashion. o o Theorem 5.2. Strong adequacy theorem for L and Lt systems. The statement is the same as for the L and Lt systems trom which the fusion systems dervie, and combines soundness and strong completeness statements. Proof simply elaborates that already given for L affixing systems. To show soundness it suffices, given previous results, to prove the inductive step for ° in the hereditariness lemma 3.1 and to prove that R6 preserves validity. For the first, suppose a < d and I(B ° C,a) = 1. Then for some b and c, Rbca and I(B,b) = 1 = I(C,c). By p6 the same holds with Rbcd replacing Rbca, so I(B ° C,d) = 1. For the second it is enough to show that if D is the premiss and E the conclusion of an application of R6 then if I(D,T) = 1 then I(E,T) = 1. Suppose first I (A 0 B C,T) = 1 and I(A,a) = 1, for arbitrary a e K. To show, as required, that I(B -*• C ,a) = 1 suppose Rabc and I(B,b) = 1. Since I (A ° B -*• C,T) = 1, if I (A ° B,c) = 1 then I(C,c) = 1. But since Rabc and I(A,a) = 1 and I(B,b) = 1, I(A ° B,c) = 1. Hence I(C,c) = 1. For the converse, suppose I(A -»-. B •*• C,T) = 1 and I(A ° B,a) = 1. Hence for some b, c, e K, Rbca and I(A,b) = 1 and I(B,c) = 1. Thus I(B C,b) = 1 and I(C,a) = 1 as required. To establish strong completeness it suffices to add to the and (IIA) the inductive step for i.e. I(B ° C,a) = 1 iff B 0 sider the (IIA) case. Suppose first, I(A ° B,a) = 1, i.e. for R^bca and I(A,b) = 1 = I(B,c), i.e. by the induction hypothesis

proof of (II) C e a. Conb, c e K , A e b ana

B e e and, for every A', B', A* -+ B' e b & A* e c => B' e a. Since A ° B -> A ° B, P A B ->-. A ° B, by R3. Hence B A ° B € b, and so A ° B e a, as required. For the converse suppose A ° B e a, and let b* = {C: A ^ C} and c' = {C: |- B -»• C>. Then b* and c' are T-theories; further A e b', B e c' and R b'c'a*. As to the last, suppose D E e b' and Dec'. Then |- A -*• (D E) and |- B -*- D; so [- A ° D e E, and |- D -* A ° D ->. B A 0 D, whence A B A ° D, and so |- A ° B A " D. Thus |- A ° B E; s o E e a . As before, apply Zorn's lemma or 4.4: maximize on b' to get a theory b such that R^bc'a and A e b, and then maximize on c' to get a theory c such that R^bca and B e e . Therefore, by the induction hypothesis, I (A ° B,a) = 1. Corollary. For each of the L (and Lt) affixing systems considered, L° is a conservative extension of L. Proof of the hard part depends on the fact that there are no special modelling conditions just corresponding to fusion postulates. Suppose A is a wff of L and a theorem of L°. Then, by soundness, A is L°-valid, and hence, since A contains no occurrences of and given the fact adduced, A is L-valid. Thus A is a theorem of LAlthough fusion can be added conservatively to L sentential systems, and is a useful connective to introduce for algebraic and Gentzenisation exercises, there are some grounds for doubt about fusion and for thinking that it should not figure in the ultimate larger scheme of things (i.e. it is, like classical negation, a connective only good within a limited class of deductive situations) . In particular, Dunn has adduced an excellent reason1 for not admitting fusion, as characterised through R6, within dialectical set theory (for example, as presented in system DST of UL), namely that it trivialises the J

The argument was subsequently discovered independently by Slaney.

366$

5 . Z FUSION

INTERFERES

WITH SIMPLE

RETENTION

OF LOGICO-SEMANTICAL

PARADOXES

theory. Consider the (set) theory obtained by adding a comprehension principle and quantifier principles to B° + B2 (in fact weak - > — s u b s y s t e m s of B° which delete rules of affixing will suffice for the argument). The theory is trivial, as the following variant of the Curry-Moh Shaw Kwei paradox argument shows:(Pw) (x) (x e w **. x e x 2 w e w

2

p,

w e w -*. w e w

2

•*• p,

w e w

w e w -+. w e w

p), by Comprehension, where A 2

A ° A.

applying recognised quantif icational steps1 by Simplification

w e w2,

by R6 from Identity

w e w -»-. (we w -»• w e w 2 ) & (w e w 2 -»• p) , by Composition w e w -»-. w e w

p, by B2, i.e. Conjunctive Syllogism, and its rule derivative,

w e w 2 -*• p, applying R6 w e w/ \

Modus Ponens steps from previous premisses.

Fusion is accordingly a relation to which we give only qualified endorsement. Though not nearly as classical as classical negation it is too classical, in its Portation behaviour, for relevant dialectical purposes.

§3. Semantically closed relevant systems, dialectical resolution systemic semantical antinomies3 inconsistency and incompleteness.

of the

Even within sentential relevant logics a sufficient approximation to semantically closed systems can be achieved to enable the formulation of various semantical paradoxes. Provided that we can locate some wff, p^ say, which says of itself that it is not true and can represent this and the Tarski T-scheme within the system, it is unnecessary to expand the language SL to include names of its expressions and predicates concerning these, such as 'is true'. Let us add to system L the connective T, which reads 'it is true that' and satisfies the T scheme TS.

TA «*• A;

most simply at the sentential stage, T can be introduced by the definition TA • y A, with T really conforming to an oversimple vanishing theory of truth. (Alternatively, and a little more sophisticatedly, quasi-quotes or better a quotation function qu, like that of GR, may be introduced along with the predicate Tr which satisfies the scheme Tr £u(p) **• p. But though this makes the truth theory and its semantics more interesting, nothing essentially new for the purposes of our discussion is added, since TS and other wanted schemes can be recovered by abridging Tr £u to T). Falsity may be defined, in one of the standard ways, thus: FA = f T~A. Then FA ~TA, since T~A ~TA. Relevant affixing systems already contain, through the coimplication connective a moderately good approximation to a statemental identity (for the virtues and shortcomings of this explication, see 14 and Routley 77). Now the statement the simple liar paradox yields, p^, is one that is identical with the statement of its own non-truth, i.e. the paradox is explicated formally through the equation LP. 1

" ~TP;l .

These steps are not however beyond all suspicion, as Makinson has pointed out.

367$

5.3 ELEMENTARY

EXPRESSION

OF SEMANTICAL

ANTINOMIES

WITHIN THE LOGIC

Empirical considerations (are generally taken to) show that such an equation results from assertions such as 'This very statement is not true', since the statement in question is just 'This very statement is not true'. TS and LP lead at once, by Rule Syllogism, to p^ ** and so, by Simplification and Rule Counterexample, to p^ &

(and also Tp^ & Fp^) .

The dialectical resolution of this and similar semantical antinomies is thus simply the naive one, that the argument is valid, and that true contradictions can be established. Contrast the classical positions according to which not both TS and LP can be expressed in sentential logic (at least without trivalising it), a claim which comes down, since the definitional introduction of T can hardly be ruled out, to the inexpressibility classically of a statement p^ satisfying LP. (Let no classical logician complain then, without reflection upon the variety of notions classically inexpressible, about the fact that classical negation, being a classically-restricted notion, cannot be defined over all the deductive situations of relevant logic.) Because p^ ordinarily is expressible, e.g. in natural languages, and ought, it certainly seems, to be expressible, classical-style logics have a paradox to cope with. That is one of the penalties of accepting the paradoxes of implication, one of the "harmless" outcomes. The semantical antinomies create a problem not just for classical-style logics, however, but for any theory that is based on a logic which is not paraconsistent; and the more sophisticated semantical antinomies cause difficulties, through semantical analogues of Curry-Moh Shaw Kwei paradoxes, for overpowered paraconsistent logics (as we have seen already in the case of the logical paradoxes). Semantical analyses for dialectical relevant logics containing constants like p^ which are both true and false, or more radically coentail their own negations, are easily devised (and may be found in 6.5, along with a discussion of the relations of systemic truth and falsity with meta-truth and meta-falsity of the classical semantics given). The inclusion of inconsistent and paradoxical statements within relevant logics raises several questions about the representation of inconsistency, and correlatively of incompleteness, within the relevant logic framework. As already observed (in chapter 2) both inconsistency and incompleteness can be incorporated within the semantics of relevant logics through incomplete and inconsistent situations, in terms of which we can say that some statement A is inconsistent [incomplete] in a given situation when both A and ~A [neither A nor ~A] hold at a. And the semantics can do this either in a two-valued fashion, or more explicitly in a four-valued way with values specifically representing inconsistency and incompleteness. The question is bound to arise as to whether these notions can be satisfactorily incorporated in the syntax of relevant logic. The introduction of an inconsistency connective P with the expected properties is a straightforward matter. PA reads 'A is inconsistent' or perhaps because inconsistency in the actual situation may be considered paradoxical, 'A is paradoxical'. P is semantically determined by the following connection: I(PA,a) = 1 iff I(A,a) = 1 = I(~A,a), i.e. iff I(A,a) = 1 and I(A,a*) ^ 1. It is easy to show that P considered as a new one-place connective is adequately characterised through the following axiom scheme: PA

A & ~A.

368$

5.3

REPRESENTATION

OF AN INCONSISTENCY

FUNCTOR,

AND PARADOXICALNESS

AS A VALUE

The hereditariness lemma for P uses p4, the verification of the P axiom is immediate, and the remainder of the soundness argument is as before. The only new detail in the completeness argument is the inductive step showing that in the canonical model I(PA,a) = 1 iff PA e a, which follows using the axiom scheme and the induction hypothesis. Alternatively then, P may be satisfactorily introduced within the context of relevant logic by the definition PA =T Df A & ~A. Where LP is a further postulate Pp^ is of course a theorem; but this fact does not trivialise any relevant logic worth its salt (i.e. which is at least paraconsistent). In contrast, if L is a logic which is not paraconsistent (e.g. classical logic or a modal logic), L can contain no paradoxical statements. In terms of the connectives T, F and P a kind of three-valued logic can be built up within relevant logic, indeed exactly the kind of three-valued logic to which direct investigations of the paradoxes in terms of 3-valued logic have led and, given weak conditions of adequacy, should lead (cf. Asenjo 66, Priest 79). Since in B definitionally enlarged |- T~A ** FA, |- F~A «* TA and }- PA ** P~A, the following negation matrix which sums up the relations is forthcoming:-

T

F

P

T

F P

F

F

T

F

P

F

P

T

P

P

T

T

Similarly the conjunction and disjunction matrices shown are derivable within system B, where the results entered in the body of the table are those implied given the determination shown at the edges, e.g. in the & matrix the pair (T, F) gives F because j- TA & FB ->- F (A & B) , (P, P) gives P because |- PA & PB -*• P(A & B), etc. The procedure does not lead, however, to a complete matrix for implication, but, even in stronger systems, at most to a partial table, e.g. while -*(T, F) is determined as F, -»-(F, T) and -*(T, T) both have to be left open. Thus the logic is only three-valued in its extensional, & - v — , part, so to speak. The matrices derived are those of Lukasiewicz's system X^; but the intended interpretation of values is different, and this leads to a difference in the class of designated values. Whereas Lukasiewicz's third value was intended to represent (future) indeterminacy, the third value now is interpreted in terms of inconsistency rather than incompleteness and so becomes designated since paradoxical statements are true. This change in interpretations and designated values in fact takes us from the matrices of 1L to those of the system RM„, already encountered. 1

Strictly, the matrices are those for the extensional (& v ~ part) of RM^ which also has matrices for -»• and t, and derivatively = defined:

A => B

A & t -*• B.

The & ~ ^ matrices are those of the Asenjo-Tamburino logic of paradox (in 75). The extensional RM^ matrices derived in fact provide the full matrices of Priest's logic of paradox,where the matrix for follows using the definition: A + B = -A V B. These matrices are the same as Kleene's weaker matrices except Df in designated values, and are among the C-systems studied in GR.

369$

5.3 PARADOXICALNESS,

HEGELIAN

CONTRADICTORINESS,

ANV INCOMPLETENESS

AGAIN

There are good reasons for supposing that inconsistency, syntactically symbolised by P, does not represent full paradoxicalness. There are many inconsistencies which do not have the pull of paradoxes like the liar, which there is little or no inclination to say are true. What then distinguishes paradoxes from other contradictions? It is tempting to suppose that the property of coimplying their own negations is the relevant distinction that A is fully or liar paradoxical, PA, iff A ** ~A.1 The semantics of IP is readily determined from this definitional equivalence. IP is suitably stronger than P; for while PA -»- PA (given Reductio: otherwise the implication weakens to a rule connection), the converse does not hold, except under classical extremism, flP provides a considerably better representation of paradoxicalness than P. And other notions which look like candiates to represent paradoxicalness reduce to P. One such example is Hegelian contradictoriness. A is a Hegelian contradiction, HC(A), iff A = A =. A = ~A. Representing identity in affixing systems by content coinclusion, HC(A) = f A ** A **. A ** ~A. This reduces, however, in exportative systems simply to liar paradoxicalness, IPA, i.e. A ** ~A. For HC(A) IPA in B, as A ** A, and the converse IPA -*• HC(A) is proved as follows:A ~A -»-. A A A -> ~A, A ~A -»-. A A ->-. ~A -* A, A ** ~A ->. (A -»- A ->. A -»• ~A) & (A -*• A ~A •> A), whence by T3.7, A ** ~A A ** A A ** ~A. Similarly from A ~A ->•. A -*• ~A A A, A ** ~A ~A A A ->• A, A ~A -»-. A +* ~A A ** A follows. A similar argument shows that in non-exportative systems: HC(A) -fc {PA; so Hegelian contradictoriness adds but little to the study of liar paradoxicalness. Whereas inconsistency has an appealing representation within the syntax developed, incompleteness does not. Let YA symbolise 'A is incomplete'. Y is semantically characterised through the following evaluation rule: I(YA,a) = 1 iff I(A,a) + 1 and I(~A,a) 4 1, i.e. iff I(A,a*) = 1 4 I(A,a); for A is incomplete in a iff neither A nor ~A hold in (or belong to) situation a. Thus a is incomplete at a iff A is inconsistent at a*: Incompleteness is appropriately correlative to inconsistency because * is involutory. Despite this correlativity, Y is one of a cluster of more classical notions which is not syntactically expressible within the framework of the relevant logics thus far developed. Were classical negation, symbolised by overlining, and conforming to the evaluation rule: I(A,a) = 1 iff I(A,a) ^ 1, syntactically available it would be a simple matter to express Y; for YA ** A & -A, so Y could be introduced definitionally. And then familiar 4-valued matrices with Y as a value could be derived in place of the 3-valued matrices above. But, for excellent reasons, classical negation is not generally available. In case the fact that Y is not representable is taken as a black mark against relevant systems, it is worth observing that incompleteness is not classically representable either. A larger syntactical apparatus combining classical and other elements is required, and accordingly will be developed shortly. %4. Classical-type connectives and arypto-relevant logics; so-called classical (C) and superclassical (K) relevant logics. After the tirade issuing from Pittsburgh against classical negation, it came as something of a surprise that classical negation (i.e. a negation satisfying the classical semantical rule) could be conservatively added to relevant systems, even to stronger systems such as R, without disturbing relevant insights of the original systems so extended. Moreover the conservative extensions enabled some interesting semantical simpl1

Priest has an argument (see PL) that such a characterisation fails to include certain logical paradoxes satisfactorily. Given that Priest is right, the question of distinguishing fully paradoxical assertions is open.

370$

5.4 THE DISTURBING

EFFECTS

OF ADDING

CLASSICAL

NEGATION

TO RELEVANT

LOGICS

ifications to be made and pretty results established (see, e.g., CRI and CRII). It came as even more of a surprise, as something of a shock, that these "classical-relevant" systems could be made even more classical as regards negation, into superclassical systems, still without collapsing back to classical or modal logics. It is these systems which are not really relevant, in that they are not weakly relevant, that we now investigate. The introduction of classical negation, say, into the syntax is (like the introduction of incompleteness) not without difficulties. It is one thing to simply add to the syntax and have it conform to the postulates which we know independently lead to the classical rule, I(A, a) = 1 iff I(A, a) i* 1. This much has been accomplished, in main features, already (in chapter 4). It is another and more vexatious thing to require that conform to the classical rule from the outset. For this cannot be accomplished without collapsing the hereditariness requirement. For consider the inductive step for classical negation in the lemma establishing hereditariness. What is to be shown is that whenever a < b and I(A, a) = 1 then I(A, b) = 1, i.e. whenever a < b, then for every A, when I(A, b) = 1 then I(A, a) = 1, i.e. in effect whenever a < b then b c a, and canonically, when a <= b then a = b. Thus the semantical addition of classical negation requires a < b iff a = b, collapsing <. Thus adding classical negation to relevant logics resembles adding classical negation to intuitionistic logic, where hereditariness is similarly disturbed. However with relevant logic the disturbance is not calamitous: it can be made without collapse of relevant logics even to modal logics. There is already something of a plethora of classical relevant systems in the literature and, as a result, some confusion. Initial investigations concerned the system R, but most of the results obtained can, as so often, be generalised to other systems. The first classical relevant systems, the CL systems as they are called (extending the label CR) were obtained in effect by adding to positive relevant logic L (which may contain such connectives as ° and V) a classical negation , i.e. a negation conforming (as it proved) to the classical rule. That is, in the notation to be adopted, CL is L+ . It soon emerged however that could be added to full relevant logics, without collapsing positive insights, and the resulting L systems could be arrived at in two theoremwise equivalent ways, either by simply adding to L or by adding another one-place connective * to the syntax of CL, and defining . Hence the equation L~ = CL* (and of R~ with CR*, as in BV).

ive

System L~ results from L by the addition of the primitive one-place connectsubject to the following postulates

CI.

A

A

CRI.

A & B ->- C -t> A & C

+ B

(Rule Antilogism)

Irrelevance, in such forms as C & C -*• B, is immediate from CRI. In fact, CL results in a precisely similar way from L + , i.e. L shorn of its negation postulates; and also results from L£, i.e. the implication-conjunction part of L provided V-Composition is added, where classical v is defined A V B

(A & B). 1

Results for these systems, selected primarily with a view to showing completeness, include the following: CT1.

J

A & A -»• B

The latter formulation is used in QRI, p.317 ff. classical v amounts to v. 371$

It soon turns out that

5.4 THEOREMS

OF CLASSICAL RELEVANT

LOGIC

USEV

IN ESTABLISHING

COMPLETENESS

Proof. Apply CRl to A & B -»- A, then use Cl. Proof, like all those that follow, makes use of the positive, or implication-conjunction part of L, i.e. effectively B+ theorems. Where elementary these theorems will simply be applied without citing further proof. CT2.

+

A

Proof. CRT1. CT3.

A;

A ** A

Apply CRl to A & A

A.

A&B-J-C-frA&C-^B. A & A & B

Proof.

B;

By CT2 and CRl.

A & (A v B)

For the first, apply CRTl to A & B -> A & B.

analogous rule t o A & B - > A & B , ition. CRT 2.

A

Proof. CRT3. MT1.

B.

B A

B B

giving A & 5 & B

For the second, apply an B;

then use the v defin-

A

B & A

B.

A -> B -b. B -»• A.

Hence by CRl, A

B -t> B & B + A.

By CRT2 and CT2.

(Replacement of coimplication) .

A

B

C(A) -> C(B).

Proof, for each system, is by induction over connectives in C(A). the induction step for CT4.

CRT3 provides

A v B **• A v B (for systems that contain V).

Proof.

A

A & B, B + A & B ;

hence by V-Composition, A V B

A V B + A and A V B -»• B, by &-Composition, A V B CT2, A V B ->• A V B.

A & B,

A V B.

Since

whence by CRT3 and

By virtue of CT4 and MTl, it is unnecessary to distinguish V from V, so V need figure no more. Also CT3 thus gives A & (A v B) B. CT5.

B

CT6.

A V A.

By CT5.

CT7.

A & B

A V B.

By V definition, CT4, MTl, T2.

CT8.

A v B

A & B.

Similar to CT7.

CRT4.

A V A.

A & B

C -fr A

Proof outline. A

•*•

By CT1, CRT2, CT4.

A & B

B V C C

A V B

C

B & C

B & (A V B) -ft, B & C

A

B V c.

CRT5.

A & B ->•. C V D -A A 4 D

Proof outline.

CRT6.

A & D

A I B

+

B V C

+ C V D - » A 4 C V D - > B - > A I C I D

+ B,A->BVDH>

A

B.

Proof as in 3.2.

372$

+ B-TA5,D+.

BVC.

5.4

STRONG

ADEQUACY

RESULTS

FOR CLASSICAL RELEVANT

SYSTEMS

EXTENDING

C

Consider now reduced modellings for CL systems: unreduced modellings introduce minor complications best dealt with separately. A basic system here is CC, i.e. the classical negation extension of C+(or . A CC m.s. M is a structure < T , K, R > with T in K and R a 3-place relation such that, generally cqO. q3.

RTab iff a = b R 2 abcd > R 2 b(ac)d

ql. q4.

Raaa R 2 abcd > R 2 a(bc)d. x

What this adds to the positive m.s. is the C-condition, RTab > a = b; and more generally CL m.s. and L m.s. (which may include *) result from L m.s. by the C-condition. For example, an R m.s. adds the C-condition to a R m.s. (as given in 4.10). But the C-condition also enables some simplification, specifically removal of monotonicity conditions as redundant, e.g., p.4, R Tabc > Rabc, is trivially derivable. Theorem 5.3.(Strong adequacy of CL and L systems extending system C by postulates like those modelled in chapter 4 except for those whose modelling involve 0 or P). The statement is as before, mutatis mutandis. Proof. Soundness is simplified to the complexity level of modal logic; more exactly, the soundness lemmas of 4.5 trivialise with the trivialisation of hereditariness. Generally, wherever I (A B, T) ^ 1, then for some a, for which RTaa, I(A, a) = 1 4 I(B, b) - from which point verification of schemes can proceed often as before (in 4.6). For when I (A ->• B, T) ^ 1 then RTab A whence a = b and so RTaa. Thus to verify Cl, it is enough to show that when I(A, a) = 1, i.e. I(A, a) = 1, then I(A, a) = 1. For CRl, suppose that in some m.s., based on T, since RTaa that I(A, a) = 1 = I(C Z a) f I(B, a), i.e. I(A, c) = 1 = I(B, a) i I(C, a). But then I(A & B ->• C, T) i 1. Remaining postulates of C are verified as before. Completeness requires (where there is no specific machinery for isolating null and universal theories) the use of non-null theories, or nondegenerate theories. 2 Here it is required that T-CL theories (similarly hereafter L~ theories) are nondegenerate. The two-fold point is shown in these facts:Fact 1. Every prime (n.d.) T-CL-theory is consistent and ~ complete, i.e. A e a iff A d a for every wff A. As to consistency, suppose otherwise A e a and A e a. Then A & A e a, whence by CT1, a is universal, contradicting nondegeneracy. For completeness, since a is non-null, B e a for some B; and so by CT5, A v A e a, whence by primeness A e a or A e a. Fact 1 together with an induction hypothesis yields the required canonical interpretation condition, I(A, a) = 1 iff A e a. For I(A, a) = 1 iff I(A, a) ^ 1, i.e. iff A I a, i.e. iff A e a.

*A C+ m.s. has to compensate for the absence of s4 by the addition of q4. In the case of a C m.s., it suffices to add to a C m.s., as set out in 4.10, the C-condition, RTab > a = b. 2

Where classical negation is involved the net effect is much the same: in CRI non-null theories were used, but a cut-down to theories which were n.d. was made in lemma 5, p.60; in QRI n.d. theories were used and similar work came in showing that newly defined theories were appropriately n.d. But where classical negation, which symmetrises nullness and universality, or completeness and consistency, is not present there may well be point in separating nullness and universality: see some of the modelling conditions of §5, where specific machinery for isolating degenerate theories is introduced.

373$

5.4 ADJUSTMENTS

FOR SYSTEMS

REQUIRING

UNREDUCED

MODELLINGS

OR THE LIKE

Fact 2. Where a and b are prime (n.d.) T-CL-theories, if a c b then a = b, i.e. the theories are maximal in an important sense. For suppose a c b but for some A, A e b but A i a. Since A e a, by completeness, A e b, contradicting consistency. Fact 2 guarantees the C-condition, since RTab iff a c b. Otherwise the completeness argument is as before (in 4.6), though the nondegeneracy of newly introduced or extended theories has to be confirmed at each relevant point. As T-CL-theories are used, there is no difficulty in proving RTaa. But for weaker systems, both this postulate and the other half of cqO have to be adjusted. Also both to accommodate constant t and for unreduced modellings, and for modelling systems like E, the property 0, with 0T, has to be included in m.s. Then for weaker systems and unreduced modellings cqO is replaced by cqa. cqb.

Rxaa, for some x in 0, i.e. a < a in the usual unreduced sense; For x in 0, iff Rxab then a = b, the O-C-condition.

and

These conditions of course condense to (Px e 0)Rxab iff a = b. The interpretation rule for t is again simply, I(t, a) = 1 iff Oa, the basic postulates for t being as before the two-way rule, A
New decqa is pi

But then by Fact 2,

Classical negation enables the easy modelling of some principles, and correspondingly the axiomatisation of some conditions, that escaped the semantical net before, for example

J

It is suggested in QRI, p.330, that the condition T £ K is problematic. It is not with unreduced modellings, that T is closed under provable CL-implication is almost immediate from the extension lemma. Appeal to system CAQ was unnecessary at this point. (Note that there are significant defects in QRI, pointed in an errata the editors promised to publish with that paper, but which did not appear. These defects do not materially effect nonquantificational results.)

374$

5.4 SYNTACTICAL

A & (A A B

AHV SEMANTICAL

FEATURES OF THE STAR

B) => B, or equivalent ly A V B

CONNECTIVE

Raaa, for a e 0, or in reduced m.s. RTTT.

Soundness is straightforward, and completeness makes use of the following useful minilemma: For x e 0, where f- A => B and A e x, then B e x . Since x is regular A => B e x, so A & (A V B) e x, whence by CT2 and MTl, B e x . To show Raaa for x e 0, suppose A -»• B e a and A e a, and show B e a. But since (A -»• B) & A e a by the minilemma B e a. The star connective, *, which is a syntactical image of the semantical star Operation, is a one-place connective which can be added to any of the systems considered. The basic postulates on *, to be added in particular to CB and extensions, are these: Dl. D3.

A* & B* -»• (A & B)* A** ** A

D2. RD1.

(A V B)* A* V B* A •*• B - A* B*

These are essentially the postulates required for establishing semantical adequacy (see QRI, p.322 ff.). Semantics for starred classical relevant logics add an operation * to CL m.s.; * is a 1-place operation on K such that a** » a for a e K. The interpretation rule for connective * is naturally: I(A*, a) = 1 iff I(A, a*) = 1. Negation extensions of B ative examples are:

are matched by star extensions of CB*.

Axiom scheme

Modelling condition

A v A* A •*• B B* A* (A -*• A*) ->• A*

x = x*, for x e 0 Rabc > Rac*b* Raa*a.

Illustr-

The systems introduced are interconnected through the following definitions, or the coimplications they yield:- For example, in CL*, negation may be defined thus, A = D f A * ; while in L , syntactical star may be defined, A • ^ ~A (provided L includes or, since negation order is immaterial, ~A. Given these definitional interconnections, the L and CL* systems are theoremwise equivalent, i.e. L = CL*. Proof of equivalence follows simply from adequacy results, granted one further adequacy result, sought independently, namely Theorem 5.5.(Strong adequacy for starred systems).

The statement is as before.

Proof. For soundness, it is enough to verify new postulates (and this is done in outline in QRI, p.333); one example: ad D2. I((A v B)*, a) = 1 iff I(A v B, a*) = 1 iff I(A, a*) = 1 or I(B, a*) = 1 iff I(A*, a) - 1 or I(B*, a) = 1 iff I(A* v B*, a) = 1. For completeness, define a* = (A: A* e a}. Then * is well defined, the modelling conditions are satisfied, and I(A*, a) = 1 iff A* e a. Details are given in QRI, p.333. Theorem 5.6. (Equivalence). A is a theorem of L translation) A is a theorem of CL*. Proof.

iff (under definitional

In CL*, I(~A, a) = 1 iff I(A*, a) = 1 iff I(A*, a) * 1 iff 1(A, a*) t 1, as

375$

5.4 THE LIMITED ADMISSIBILITY

OF CONNECTIVES

WHICH INDUCE IRRELEVANCE,

ETC.

required; and in L~, I(~A*, a) = 1 iff I(~A, a) = 1 iff I(~A, a) ^ 1 iff I(A, a*) = 1, again as required. A much more syntactical proof is given in QRI, p.334. Connective * may be simply added to B or an extension thereof. Then classical negation is definable, A • ^ ~A* (or = (~A)*), and so L* = CL*. For, to add to the previous theorem, I(A, a) = 1 iff I(A*, a*) ? 1 iff I(A, a) f 1. addition of *, like , accordingly induces irrelevance.

The

What is the position with respect to additional connectives, especially irrelevance-producing connectives such as and *, that can be added to relevant logics? Plainly there are various different stances that can be taken, from the tough relevant position that use of such connectives is never to be countenanced to the irrelevant (or irreverent) position that their use is always permissible. Here an intermediate position is adopted: that it is (of course, just on liberal principles) perfectly alright to investigate the logical behaviour of such connectives, for instance in theories that don't purport to be true, and it is permissible to include them as technical adjuncts, even in (what purports to be) the true theory, provided their addition affords only a conservative extension of the underlying theory. In most crucial cases, in particular set theory, they will not afford merely conservative extensions, but will damagingly extend the theory. But what is wrong with introduction of such classical connectives where the extensions thereby made are not conservative, is not confined to evident damage effected in such extreme cases as theory trivialisation. Such connectives, since axiomatically constrained, narrow for many purposes inadmissibly - the class of intrepretation of the theories under investigation: they impose assumptions that may not hold or that are undesirable.1 Even the admission of irrelevance-generating connectives, like classical negation, in conservative extensions of the theory, involves of course modification of requirements such as those of weak relevance, e.g. to certain classes of connectives and constants (something connective t already required, since t p •*• p). In a similar way, as in original significance set theories, abstraction axioms are qualified to holding for (certain) relevant connectives. Some pretty things, on the far-classical side of the relevant enterprise, can be done using classical negation, in combination with (positive) relevant logics and the usually admitted constants t and f. One striking example is the new classically-based axiomatics for relevant logics. Although these have been provided so far only for systems tied to R, they could without much doubt be accomplished for a range of other relevant logics (how will be indicated).2 As some of us do not have much time for classical negation, because of its numerous bad traits, we shall be brief in details as to the tricks that can be accomplished. Consider first a system MCRt (the system was called R+ t in Meyer 74, but ^hese points are much elaborated elsewhere, e.g. UL and EMJB. The matter of additional connectives, especially those that lack natural language interpretations, is discussed in detail in SL, chapter 5. 2

The work for R is carried out in Meyer 74 (this article, I, is the only member of the series). What follows in the text makes use of Abraham 77, which is based on Meyer 74.

376$

5.4 CLASSICALLV-BASEV

AXIOMATISATIONS

OF CLASSICAL

RELEVANT

LOGICS

this system is not to be confused with the system R+ together with the constant t, hence the relabelling). MCRt has the same language as CRt, and classical connectives => and — are defined in the standard way in terms of set { , &, v}. The postulates of MCRt are these: MCI. A, where A is a truth-functional taugology in &, v, ently any set of axiom schemes for classical logic S. MC2.

A«B°C°D=>.

MC3.

t ® A

MR1.

(i.e. y).

2

=>, =;

or equival-

A « A o B o D o C

A A, A ^ B

B

MR2.

A ° B : > C - » A : > . B -*• C

In designing like systems corresponding to other relevant logics (e.g. in designing MCCt), the main work would go into replacing MC2 (which concentrates the fusion schemes of R) bv fusion postulates of other systems. The main result linking R+ with MCRt (first proved in Meyer 74) is Theorem 5.7. (Classical axiomatisation of R+). Where A and B are wff of R+, then A is a theorem of R+ iff t => A is a theorem of this new-fangled MCRt, and, furthermore A -*• B is a theorem of R+ iff A => B is a theorem of MCRt. This "classical axiomatisation" theorem may be established directly, given present machinery, as a corollary of the following adequacy theorem: Theorem 5.8. (Adequacy for MCRt with respect to CRt models). of MCRt, A is a theorem of MCRt iff A is CRt-valid.

For every wff A

Proof of soundness is by the usual inductive argument that the axioms are valid and the rules preserve validity, and is presented in detail in Abraham 77, pp. 36-7. The completeness argument is also given in Abraham* pp.37-44, but unfortunately appeals back to the classical axiomatisation theorem in proof of one of the lemmas needed (Lemma 14, p.41) Given lows thus: CRt-valid, Further, A CRt-valid,

an independent proof of theorem 5.8, an extension of theorem 5.7 folLet A and B be wff of CRt. Then A is a theorem of CRt iff A is i.e. iff t => A is CRt-valid, i.e. iff t 3 A is a theorem of MCRt. -> B is a theorem of CRt iff A -*• B is CRt-valid, i.e. iff A B is i.e. if A ^ B is a theorem of MCRt.

One of the extensions of CRt, by axiom t -*•. t -*• t, with corresponding modelling condition RabT iff a = b = T, is worth separate mention because the world structures in Meyer 74 are required to satisfy this further condition. All the results can be reworked with m.s. which satisfy the extra modelling conditions. These tricks performed with classical negation extensions of L+ can be elaborated for full L systems. But again, they have only been worked out in the case of system R so far. Just as MCRt was connected with R+, so the system MGRf (called R in Meyer 74) can be connected with R. System MCRf results from MCRt by the addition of DNL, A = — A , where A = d J A f. Constant f may either be taken as primitive and t defined as f, or else f defined as t. extends to the following theorems. Theorem 5.9. (Adequacy for MCRf with respect to CR* models). MCRf, A is a theorem of MCRf iff A is CR*-valid.

377$

The analogy

For every wff of

5.4 SUPERCLASS!CAL

RELEVANT,

KL, SYSTEMS

COLLAPSE

THE STAR OPERATION

ENTIRELY

Theorem 5.10. (Classical axiomatisation of R). Where A and B are wff of R, then A is a theorem of R iff t => A is a theorem of MCRf, and furthermore, A -> B is a theorem of R iff A o B is a theorem of MCRf. Still more astonishing than the classical systems are the superclassical "relevant" systems, the K systems. KL = CL + Contraposition in , D4, i.e., A -*• B -*•. B ->• A. D4 is not a theorem of CR or CR* as the addition of classical negation which has the following matrix

-

-3

-2

-1

-0

+0

+1

+2

+3

+3

+1

+2

+0

-0

-2

-1

-3

to the Belnap MQ matrices soon reveals.

For take A « B = -2, for instance;

the

assignment falsifies D4. The modelling condition for D4, in the CL setting is d4. Rabc > Racb; that is, the D4 requirement, d4, with the stars removed. And this, more generally, is what, as we shall see, the systems do, remove the stars. But, first, the adequacy of the modelling condition: For soundenss, given I(A B, a) = 1 it has to be_shown, for arbitrary b and c in K assuming Rabc and I(B, b) = 1, that I(A, c) = 1, i.e. I(A, c) i 1. But by d4 when I(A, c) = 1 then I(B, b) = 1; so as I(B, b) 4 1, I(A, c) = 1 as required. For completeness, assume Rabc and A -*• B e a and A e c, for arbitrary a and c (in K^ say, to take the_unreduced case)? to show B e b. Then, as by D4, B A e a, when B e b then A e c, i.e. when A e c, B e b as required. Accordingly, adequacy of KL follows from that for CL (which may or may not include connectives *, etc.). KL is so far simply a straightforward extension of CL. But now consider systems L with (simple) reduced modellings, i.e. extensions of system C, C+, etc., or more generally systems with Contraposition, D4. Then a KL m.s. is precisely a CL* m.s. for which x = x* for every x e K. For example, a KR m.s. is just a CR* m.s. with x = x* always, as the following table comparing their modelling conditions shows: RTab iff a = b Raaa R 2 abcd > R 2 acbd Rabc > Racb

RTab iff a > b Raaa R 2 abcd > R 2 acbd Rabc > Rac*b*

In short, such KL systems collapse the * operation, which is often viewed as the divide between relevant and classical frameworks (but, as we have seen several times over, it is not). It is for this sort of reason, in particular, that it has been thought that the K systems would reduce at least to modal systems, and considered remarkable that they do not. They do not collapse; for Strong Distribution, (p & q -*• r) & (p -»• q v r) •*. p •*• r, is not a theorem of KR, as the following countermodel shows.1 Consider a 3-world KR m.s. where K = {T, a, b} and R holds of the following triples and all permutations of them: TTT, Taa, Tbb, aaa, bbb, aab. Then M = < T , K, R > is a KR m.s. Define a valuation v on M so that v(p, a) = v(q, b) = v(r, T) = v(r, a) = 1 and v(p, T) = v(p, b) = v(q, T) = v(q, a) = v(r, b) = 0. Then I(p & q r &. p - > q v r , a) = l but I(p -»- r, a) = 0, falsifying Strong Distribution. Since Strong Distribution is a theorem of modal system S2, and therefore all its extensions, KR does not reduce to a modal system. 1

This countermodel is due to A. Abraham, who independently, discovered the K systems and several of their features. 161$

5.4 STRANGE

FEATURES OF SYSTEM

KR, THOUGH

IT IS

CRYPTO-RELEVANT

Corollaries are that neither Antilogism, A & B -»• C A & C B, nor Factor, A -»• B -»-. A & C + B & C, are theorems of KR. In fact both would collapse KR beyond S2, to S. But let us consider fairly direct syntactical arguments. Firstly, if Antilogism were a theorem Strong Distribution could be derived. By Antilogism, together with resources of KR, A & B C coimplies B & C A, and B C v A coimplies A -*• (C & B) (by this sequence:

B + C&A, C&A->B, C & B

A).

Hence

(A & B C) & (B -»•. C v A) B & C -»• C & B. But B & C -* C & B coimplies B -*• C, whence Antilogism, by the following sequence: B & (C & B).->- C, B & C -»• C, C & C •*• B, B C. Secondly, Factor would deliver Antilogism as follows: A & B + C-*-. C + A & B , A & B -»- C A&C-"A&A&B. But by DSyll, A & A & B -»• B, whence Antilogism by Prefixing. Incidentally, these arguments help reveal, once again, that while Disjunctive Syllogism may be representative of modal logic at the first degree, it no longer is at higher degrees: that role is assumed by the stronger principle of Antilogism, which depends on the combination of factorisation features along with Disjunctive Syllogism. The next shock - perhaps these things are no longer shocking, or never were - is that KR is a crypto-relevant logic, that is, its positive part is relevant, as is that of CR. That KR is a crypto-relevant logic may be shown using what has come to be called the Church lattice (and matrices) diagrammed thus (with T and t designated), and with associated implication matrix: T

t

f

F

Matrices for & and v are given ^ by lattice meet and join respect*T T F F F ively. To establish weak rele*t T t f F vance of KR+, let A B be a wff F f T F t F of KR+ such that A and B share no F T T T T sentential variables. Assign matrix value T to every variable of A and value t to every variable of B. Then, by induction, A takes value T and B takes value t, and so A B takes value F, and A -»• B is not a theorem of KR+. KR+ does not reduce however to R+. The scheme A ° A v. A -»• B is a theorem of KR+ expanded by ° but not of corresponding R+; and this scheme may be expanded to remove fusion, for instance, to the system-separating wff (A A + B) -*• B V. A -*• B. The modelling condition RxxT, corresponding to wff A ° A v. A B, is indicative of the logical troubles with (and evil of) systems like KR; the condition in effect assures us - what is false - that if p holds anywhere then p is true, i.e. holds at T.1

§5. Nondegeneracy more explicitly redone3 and the truth-preservation

s

null and universal worlds} classical negation connective. Rather than imposing the require-

ment that null or degenerate situations are excluded in order to satisfy classical principles, an alternative is to build into the semantical theory conditions to the effect that worlds are occupied, not trivial, and so on. There are two ways to accomplish the latter:' either to introduce properties of worlds, such as the occupation predicate C (of chapter 7); or to introduce as elements of x

More is known about KR, for example that there are infinitely many non-equiv2 3 alent wff, p, p , p , ..., resulting from a single variable p by fusion (where p n + ^ = p n °p and p* = p). Further results concerning KR are assembled in 'Super-Boolean relevant logics', to appear.

379$

5.5 MODELLINGS

WITH THE NULL AND UNIVERSAL

WORLDS

EXPLICITLY

INTRODUCED

world-set K null and universal elements, e and u. That is, under the second course which will now be pursued, e, the null (or empty) world, is a member of K where nothing, no wff, holds; and u, the universal (or trivial) world, is a member of K where everything holds. An enlarged L model structure (L.e.m.s.) accommodating e and u, is simply a structure which adds to an L m.s. the elements e and u in K; specifically (to build on extensions of B m.s.) it is a structure < T , 0, K, R, *, e > where e € K and further the following definitions and conditions are satisfied for every a, b e K: du.

u = D f e*

epl.

Reue

ep2.

Ruab >. a « e v b = u

ep5.

e j* u. 1

An (e) valuation v in such a model M is one which satisfies also the following further valuation conditions:E-condition: U-condition:

v(p, e) = 0, for every sentential parameter p; v(p, u) = 1, for every sentential parameter p.

These conditions force ep5.

Otherwise semantical details are as before (in 4.4).

The remaining modelling conditions are deployed in the following lemma. Lemma 5.2 (Degeneracy lemma).

For every wff A, I(A, e) = 0 and I(A, u) = 1.

Proof is as usual by induction from the given basis. ad I(~B, e) = 0 iff I(B, e*) = I(B, u) = 1, which holds by induction hypothesis. Similarly I(~B, u) = 1 since I(B, e) ^ 1. ad &.

Almost immediate from the &-rule and the induction hypothesis.

ad -»-. I (A B, e) = 0, since by epl Reue. I (A •*• B, u) = 1, since by ep2, for every b and c in K, when Rube b = e or c = u. In the completeness argument, it is enough to establish the further modelling conditions, Define e , e in the canonical model, as the null set A. Then since, byJ definition of u, A e u iff A e e *, i.e. ~A I A u is the class of ' c c c wff.

Then epl and ep5 are immediate.

a ^ A; A

to show b = u £ .

As for ep2, suppose R^u c ab and further

But by hypotheses, some A e a and whenever, for any B,

B e u c , i.e. always, B e b, i.e. b = u c >

With adequacy established, wff previously requiring nondegenerate canonical modelling can be given more explicit semantical postulates, e.g. as follows: A B B A -*. B A A B v ~B

a ^ e & Rabc > b < c b ^ e & Rabc > a < c a ^ e > a* < a

*As an optional extra it can be required that e i 0. This further condition, not required here, is exploited in chapter 12 in the design of matrices for >-systems,

3B0

5.5 REMODELLING

CLASSICAL

NEGATION

(AND PRINCIPLES)

THROUGH

THE ENLARGED

SEMANTICS

Connectives whose semantical analysis calls for nondegenerate situations, at least in canonical modellings, can also be profitably recast within the enlarged modelling framework. Such is the case both for classical negation, already analysed, and for the truth-preservation connective, >, yet to be introduced into the object language.1 The addition of classical negation proceeds in much the same way as before, but a different axiomatisation will be used. L results from L by the addition of these postulates: CA1. CAR1.

B -*•. A v A C V (A B) -» C V (B ->- A)

CA2.

A & A -»• B

An L~ e.m.s. is an L e.m.s. satisfying only the conditions explicitly presented before such that also ep4. ep5\

When a 4 e and b 4 u and a < b then a • b, and e i 0

Thus, to put things together, a B e.m.s. is a structure such that du and epl-ep5' hold where as usual a < b (Px e 0) Rxab and also a < a. By virtue of the collapse of < to = through ep4, former monotonic requirements become redundant. Similarly the need for an hereditariness requirement on valuations disappears. To each additional axiom scheme or rule that is added to B (or B+ or B& ) to form the system L , the corresponding semantical postulate (as given in %. 3) is added. Everything else is as before except that there is a (new) rule for the evaluation of , namely, for every c e K, I(A, a) = 1 iff I(A, a) 4 1 where a is n.d., i.e. a 4 e and a 4 u; and, otherwise, I(A, e) 4 1 and I(A, u) = 1. The proof of the adequacy of the enlarged semantics for classical negation systems parallels proof of adequacy of similar semantics for the truth-perservation connective >. Since > proves definable in terms of A > B A v B, this is not altogether surprising. But that L e.m.s. are exactly the same as L e.m.s., for corresponding extensions cf B, is not altogether expected. Since accordingly, proofs of adequacy of L and L** systems parallel one another, duplication is avoided by considering them together. Before this is done the truth-preservation connective deserves a better introduction. It is often claimed that 'if ... then' is sometimes used in a purely truthpreserving way, so that 'if A then B' holds at (or "is true in") the set-up or world a under consideration iff when A holds in a then B holds in a, without involving any set-up or world other than a. Such a connective, called not altogether satisfactorily "the truth-perservation connective", is characterised, that is, through the semantical rule I (A > B, a) - I iff I (A, a) = 1 > I(B, a) = 1. The systemic relation is so chosen because it matches, as the semantical rule reveals, the purely truth-preserving connection much used in the metalanguage. Plainly such an extensional connective as >, evaluated in an entirely intraworldly way, is very different from an implication or conditional which is properly intensional. While its narrow extensionality severely delimits any inter-

1

The material which follows is directly based upon R. Brady's 'Truth-preservation in the Routley-Meyer semantics'; and what has preceded in this section is substantially influenced by this work of Brady.

381$

5.5 AXIOMATISATION OF THE THEORY OF "TRUTH-PRESERVATION"

est > may have as an implication, its investigation is nonetheless of much technological interest. In classically-based systems such as modal logics, the truth-preservation connective is just the material-implication However, in the relevant semantics 3 does not in general represent the truth-preservation connective, and the two connectives ^ and > coincide only for worlds a such that a = a*. Generally, in worlds semantics, there are four possible relationships between worlds a and a*, namely a = a*, a* < a, a < a*, and (a* £ a and a £ a*). However, on the evident semantics for >, it soon emerges that a 5 b > a = b for non-trivial worlds a and b, and so two cases, a* < a and a < a*, disappear. The collapse of the ordering relation £ could be anticipated since > can be directly defined through classical negation, but ~ leads to the collapse of The underlying reason is again that the inductive step for > in the hereditariness lemma fails unless the ordering £ is symmetric. For suppose to show when a s b and I(B > C, c) = 1 that I(B > C, b) = 1. Then suppose further I(B, b) = 1, and try to show I(C, b) = 1. To make use of the relation I(B, a) = 1 > I(C, c) = 1 and complete the argument, using the inductive hypothesis it is essential that a 5 b guarantees b < a. Thus there are essentially two restricted logics of truth-preservation with just the connectives &, and >, corresponding to the two cases, a = a* and (a* £ a and a £ a*). If a = a*, then as before > reduces to material-implication =>, and the connectives & and > are represented by the classical 2-valued matrices. If however a* ^ a and a £ a*, then & and > are represented by a set of 4-valued matrices. Eventually (in 12.8) this matrix logic will be axiomatized, and most of the systems with the connective set &, >} now to be considered will be shown to be conservative extensions of this 4-valued logic. To consider the logical behaviour of > in isolation from outright classical connectives such as , > is added on its own to the logics of chapter 4. Where L is any such system, i/* is the system resulting by adding the two-place connective >. Truth-preservation > is not already definable in relevant logics, since its addition quickly induces irrelevance, even in B'* (as axiom F3 shows). results from L by the addition of the following postulates: Fl. F3.

A & (A > B) -*-B C A V (A > B)

F2. FR1.

B ->-. A > B D V (A -»• B) -*• D V. B > C

L

A > C

A L > e.m.s. is an L e.m.s. Indeed the only way the semantical analysis of L differs from that of L lies in the evaluation rule for > (in place of ), which in e enlarged m.s. is as follows, for every a e K: I (A > B, a) = 1 iff I (A, a) = 1 > I(B, c) = 1, for a 4- e; I (A > B, e) 4 1.

and, otherwise,

Adequacy arguments for I/* and L systems elaborate on those using e enlarged L m.s. But it is not entirely a matter of further details, since the notion of regularity is given a new twist in the completeness proof. We shall however shorten the story by building on previous results where we can. The degeneracy lemma (5. 2 ) is stated and proved as above, but there are new cases for > and for ": ad >. I(A > B, e) = 0 is given in the rule for >. I(A > B, u) = 1 iff I(A > B, e*) = 1, i.e. since by ep5 e* 4 e iff I(A, e*) = 1 > I(B, e*) = 1. But the latter is, by induction hypothesis, always true.

165$

5.5 PROOFS OF SOUNDNESS

ad

.

FOR L~ AND L> WITH RESPECT

TO ENLARGED

L

M.S.

Immediate from the interpretation rule.

The preliminaries to soundness (lemmas 4.1 and 4.2) are proved exactly as in chapter 4, once it is established that (2)

when a < b and I(A, a) = 1 then I(A, b) = 1.

Suppose a < b. If a jt e and b ^ u then, by ep4, a = b whence (2) is immediate. If a = e or b = u then, by the degeneracy lemma, either I(A, a) ^ 1 or I(A, b) = 1, whence (2) (degenerately) follows. Theorem 5.11 (Soundness Theorem). Where A is a theorem of L~ or of L > than A is appropriately L-valid, i.e. in the requisite e enlarged L m.s. Proof.

It is enough to validate new axiom schemes and rules:-

ad Fl. Suppose I(A & (A > B), a) = 1. Then, I(A, a) = 1 and I(A > B, a) = 1. Hence, I(A, a) = 1 > I(B, a) = 1, and so I(B, a) = 1. ad F2. Suppose I(B, a) « 1. Then, by degeneracy lemma, a j e. I(A, a) = 1 then materially I(B, a) =1, and so I(A > B, a) = 1.

Hence, when

ad F3. Suppose I(C, a) = 1. Then, by degeneracy lemma, a ^ e. Let I(A, a) = 0; then, I(A, a) = 1 > I(B, a) = 1, and so I(A > B, a) = 1. Hence, in any case, I (A v (A > B), a) = 1. ad FR1. Suppose I(D V (A •* B), T) = 1. Then I(D, T) = 1 or I (A B, T) = 1. Let I(D, T) = 0, whence I(A B, T) = 1. Then, for all x, y € K, when RTxy and I(A, x) = 1 then I(B, y) = 1. Suppose further RTab and I(B > C, a) = 1. Then a < b, a ^ e and, I(B, a) = 1 > I(C, a) = 1. By (2), I(B > C, a) = 1 > I(B > C, b) = 1, and hence b ^ e. Suppose next I(A, b) » 1. If b = u, then I(C, b) = 1. If alternatively b ^ u, then by ep4, a = b, and hence RTba and I(B, a) = 1. Thus I(C, a) - 1, and so I(C, b) = 1. In either case then I(C, b) = 1, and hence I(A > C, b) = 1. (A > C), T) =1, and so I (D V ((B > C) (A > C)), T) - 1.

Thus I((B > C) •*

ad CA1. Suppose I(B, a) = 1. Then a ^ e. If a = u, then I(A v A, a) = 1, by the degeneracy lemma. If alternatively a ^ u, then, I(A, a) = 0 > I(A, a) = 1, and hence I(A, a) = 1 or I(A, a) = 1. So, I(A V A) = 1. Thus either way, I(A V A, a) = 1. ad CA2. Suppose I(A & A, a) » 1. Then I(A, a) « 1, I(A, a) » 1 and a 4 e. If a ^ u, then I (A, a) = 0 and I (A, a) = 1, wliYfcfi'is impossible. Hence a = u^ and so I(B, a) = 1. ad CAR1. Suppose I(C v (A -»• B), T) = 1. Then I(C, T) = 1 or I (A -»• B, T) = 1. Suppose I(C, T) 4 1, and hence I (A B, T) = 1. To show I(B -»• A, T) = 1, suppose RTab and I(B, a) = 1 and show I(A, a) = 1. On these assumptions a ^ e and a < b. If b = u, then I(A, b) = 1. So assume b ^ u. Then, by ep4, a = b, and hence RTba. So I(A, b) = 1 > I(B, a) = 1. Also a 4 u, and hence I(B, a) » 0. Thus I(A, b) = 0. Also b 4 e, and hence I(A, b) - 1. Since then I(B + A, T) - 1, I(C v ( S + A ) , T) • 1. The main adaption of the completeness argument of chapter 4 derives from an amended account of regularity and of the base^heory of the canonical model. The idea is to have all the regular theories of L and of L satisfy, respectively, the closure conditions:

383$

5.5 VARYING THEORY

>closure: closure:

CLOSURE

CONDITIONS

whenever D V (A •*• B) e a whenever D V (A -»• B) e a

FOR THE COMPLETENESS

ARGUMENT

then D V (B > C A > C) e a; then D V (B A ) e a.

An i/'-theory is an RL > -theory and is >-regular if, in addition to the standard requirements, a satisfies >-closure. Similarly, an L -theory is an RL -theory and is -regular if, in addition to the standard requirements, a satisfies "-closure.

In the canonical

e.m.s. < T , 0 , IC , R_ , *, A > , T is an approp-

>

>

riately chosen RL -theory and 0 L is the class of >-regular L -theories; wise things are as before.

Similarly, in canonical L~ e.m.s.

other-

The priming lemmas

extend straightforwardly (as in 4.8) to RL 5 - and RL~-theories. New details in showing that the canonical e.m.s. are e.m.s. go mainly into establishing ep4, which calls for these preliminaries: > condition: and every a e condition: a

where a # A, A > B e a iff A I a or B e a, for every wff A and B

where a 4 A and a 4 u , A e a iff A £ a, for every wff A and every

* *L"

Proofs.

For the > condition, assume a ^ A for a e Kj^-

Firstly, let A > B e a

and A e a. Then A & (A > B) e a, so by Fl, B e a. Hence, when A > B e a either A £ a or B e a. For the converse, let A £ a. Since a ^ A, let C e a, for some wff C. By F3, A v (A > B) e a, so A e a or A > B e a, and hence A > B e a. Alternatively let B e a; then by F2, A > B e a. Hence, when either A £ a or B e a, A > B e a. For the ~ condition, assume a ^ A and a f6 u £ for a e K^-.

Suppose, firstly,

A e a and also A e a. Then A & A e a, so by CA2, B e a whatever B. Hence a = u c contradicting assumptions. Thus when A e a, A £ a. For the converse, suppose Ala. Since a ^ A, for some B, B e a, so by CA1, A v A e a; whence A e a or A e a, and so A e a. ad ep4.

The proof depends on which condition holds.

> case:

Suppose a e 0^, b ^ A, c j u^ and R^abc,

standard fashion, b c c.

where a, b, c e K^.

Let C £ c, since c ^ u £ .

Then, in

In order to show R^abc,

suppose A -*• B e a and A e c, and show B e b. Then, in turn ((B > C) -»• (A > C)) V (A B) e a (since (- A -*• B V A), ((B > C) -»• (A > C)) V ((B > C) -> (A > C)) e a (since a is >-regular), and ((B > C) -> (A > C)) e a (since j- A V A -*• A). Since R^abc, B > C e b > A > C e c, and hence A > C ^ c > B > C ^ b . Thus by the > condition, c = A or (A e c and C £ c) > b = A or (B e b and C I c). But A e c and C £ c both hold, b ^ A holds by assumption, and hence B e b. Therefore c. whence c <= b, and so b = c. case: b c c.

Suppose a e 0 L , b / A, c ^ u £ and R^abc, where a, b, c e K^. To show R^acb, suppose A -*• B e a and A e c.

(A B) e a (since |- A B V A), (B and B -»• A e a (since |- A v A -*• A). and so A £ c > B £ b.

Then

Then, in turn, (B

A) v

A) v (B ->_A) e a (since a is "-regular), Therefore, B e b > A e c, since R^abc;

By the ~ condition, for every wff A, for all a e K^,

A £ a iff a = A or (A e a and a ^ u ). Therefore, c = A or (A e c and c ^ u c ) 3S4

5.5 REAXIOMATISATION OF THE "TRUTH-PRESERVATION"

THEORY

> b = A or (B e b and b 4 u ). Since A e c, c 4 u , and b ^ A c c Therefore R^acb, and so c c b, whence b • c.

B e b follows.

Theorem 5.12 (Completeness Theorem). The formulation adapts the standard statement; so in particular, where A is valid in requisite e enlarged L m.s., A is a theorem (of L > or L , as the case may be). Proof is like proofs of corresponding theorems in chapter 4. The new induction steps for > and in proving that for the canonical interpretation I, I(A, c) * 1 iff A e c, use the > condition and condition respectively. When reduced modellings are used, as they may be for stronger systems, some simplifications result, along the lines illustrated in chapter 4. As usual 0 = {T}, SO 0 can be elided from enlarged L m.s. and a < b reduces to RTab. More striking however is a simplification that can be made of the axiomatisation of > itself. A single two-way rule can replace all of F1-F3 and FR1. The rule is a double disjunctive strengthening of the extensional portation rule A & B

C ++

A

B > C

(I)*.

which at once invites comparison with the fusion rule A ° B -*• C * * A -*•. B

C

(II).

Rules (I) and (II) are of course direct intensional substitutes for the following disastrous portation rule of classical and intuitionist logic: A & B

+ Co-fA-*-. B + C

(III).

Thus as fusion accomplishes "conjunction" of nested implications in (II), so truth-preservation achieves exportation from conjoined antecedents in its material analogue (I). Scheme (I) is readily validated using the semantical analysis of L .

Conversely, (I) enables the derivation of F1-F2 as follows:

ad Fl.

(A > B)

ad F2.

B & A

(A > B), so by (I), (A > B) & A -»• B, whence Fl. B, whence F2 by (I).

But to arrive at F3 and FR1 it appears necessary to disjunctively strengthen (I), first to A & B ad F3.

C V D <M>. A

Since C & A

CV

(B > D)

A v B, by (I'), C

(I'). A v (A > B), i.e. F3.

ad (I). Since B & A ->-. (A > B) V B, by (I'), B -*•. (A > B) v (A > B), so B -»-. A > B, i.e. F2. By F2 and V-Composition, B v (A > B) A > B, whence B v (A > B) «*•. A > B. This enables the derivation of (I) using (I*), thus: A & B + D A & B -* D V D b e c a u s e of this property, > lends itself naturally to deduction theorem roles.

385$

5.5 THE ADEQUACY

A

OF THE RULE

FOR L>

(R>) AS SOLE POSTULATE

D V (B > D) 4-p- A -*. B > D. To recover FRl, (I) is further, and finally, strengthened to E V (A & B ->. C V D) <*+> E V (A

C V (B > D))

(R>).

Though (R>) is adequate, as is now shown, much of the original appeal of (I) has been lost in the progressive disjoining. Theorem 5.13.

(R>) can serve adequately, as sole postulate for >, in L > .

Proof. (1) (R>) enables the derivation of F1-F3 and FRl. immediately derivable, as the following sketch reveals:-

Firstly, (I') is

LHS(I') -*> LHS(I') v LHS(I') -«• RHS(I') v RHS(I') RHS(I'). F1-F3 follow as above. Secondly, E V (A & B -*• C) H E V (A able as follows: E v (A & B -*• C) E v (A & B -»• C V C) «** E V (A . C v (B- > C)) E V (A B > C)

But then (I) and B > C) ia deriv, ,

by (R>) using F2.

Thirdly, then, FRl can be derived thus: D V (A -*• B) D V ((B > C) & A + C) -o D V (B > C -»-. A > C). The last step follows by the two-way rule just derived. As for the first it can be established for example, as follows: Let F abbreviate (B > C). Then D V (A B) (D V (A B)) & '(F + F) D V (A B) & (F + F)) D V ((B > C) & A B & (B > C)) D V (B > C) & A C using F2 (and the same stragegy over again). (2)

(R>) preserves validity, and hence is admissible.

LHS to RHS: If I(E, T) = 1 then I(E v G, T) = 1 for appropriate G, so suppose I(E, T) 4 1, and accordingly, given the premiss is true in the m.s., I(A & B -*. C v D, T) = 1. Then whatever x and y, I(A & B, x) = 1 and RTxy > I(C v D, y) = 1. Suppose, further, for the conclusion, RTab and I(A, a) = 1. Then a < b and a 4 e. Finally it can be assumed that I(C, b) 4 1, since when I(C, b) = 1 the conclusion is straightaway true. There are two subcases: i) b 4 u. By ep4, a = b. Hence when I(B, b) = 1, I(B, a) = 1, so I(A & B, a) = 1, so I(C v D, b) = 1. Hence I(D, b) = 1. Thus as b 4 e, I(B > D, b) = 1, whence I(C v (B > D), b) = 1. ii) b = u. Then by a lemma, I(B v D, b) = 1 whence again I(C V (B > D), b) = 1. Hence in either case, I(A -* C V (B > D), T) = 1, as required. RHS to LHS: It can be supposed that I(E, T) 4 1 (since the contrary case is immediate), so I(A C V (B > D), T) = 1. Thus whatever x and y in K, when I(A, x) = 1, I(C V (B > D), y) = 1. Suppose for the conclusion RTab and I(A & B, a) = 1. Then a < b and a 4 e. Hence too I(A, a) = 1 and I(C v (B > D), b) = 1. Again there are two subcases: i) b 4 u. By ep4, a = b. Then I(B > D, a) = 1 whence, since I(B, a) = 1, I(D, a) - 1, and I(D, b) = 1, so I(C v D, b ) = 1. ii)

b = u.

Then by a lemma, I(C > D, b) = 1.

3 86

5.6 NORMALISING

M.S., I.E. AVJOJNJNG

Hence in either case I ( A & B - * C V D ,

A CONSISTENT

BASE

WORLV

T ) = l , completing the argument.

The question of whether the extensions resulting from the addition of > are conservative remains open. Closing the question would involve showing that the addition of semantical postulate ep4 does not enable any new wff, with only connectives from &, ->•}, to be validated.

§6.

What have been called normal model structures1:

y and its limits.

The

models considered thus far have been unusual from a modal viewpoint in that not even the base world T (representing the factual world) or the regular worlds of 0, have been assumed to be consistent. For logics like G where the law of excluded middle LEM is a theorem, it has of course been assumed that the regular worlds are complete, that is, that x* < x for x e 0; but not that they are consistent. It is time to make amends: we will show that for an extensive class of consistent logics the semantical modelling can be based on consistent worlds. This has an important corollary: that for such logics the rule y of Material Detachment (A, A ^ B -*> B) is admissible. That is, for such consistent logics the addition of rule y leads to no new theorems. An L m.s. M is normal iff it satisfies the postulate pO.

T = T*.

Frequently structures which are not normal can be expanded to structures which are normal and which perform the requisite semantical work. Such normalisation is achieved, where it can be, by introducing additional elements which are assigned the right properties to ensure that the structure is normal and retains other defining features (those of the given m.s.). Specifically, for the project at hand, which is to show that normal model structures suffice for validity, we proceed as follows:Where T' is a situation not in K, the normalisation of L m.s. M = < 0 , K, R, [P], * > ajt T' is the structure M' « < 0 ' , K', R', [P',] * ' > , where K' = K U (T'}; 0 f - 0 U {T'>; R' is like R on elements of K, R'T'T'T', and where just one of a, b e K', the other being in K, R'T'ab iff R'Tab, R'aT'b iff R'aTb, and R'abT' iff R'abT*; 0' is like 0 on elements of K and O'T'; similarly where P figures in M, P* is like P on elements of K and P'T'; *' is like * on elements of K and T'*' = T'. Lemma 5.3. (i) R' is well-defined, because except for R'T'T'T' relation R' reduces to relation R. For, again except for the case R'T'T'T', relation R' holds among elements of K' iff R' holds among elements of K' upon replacing occurrences of T' in either of the first two argument places by T and occurrences of T' in the final place by T*. Similarly (ii) a <' b if a < b for a, b E K. Otherwise where a is T' it is replaced by T and where b is T' it is replaced by T*. Proof of (i) takes the following elementary route:- Since R'T'T'T' it is enough to consider cases where the new element T' occurs at most in 2 places of R'. If T' occurs in just the first place, then R'T'ad iff R'Tad iff RTad. for a, d e K. If T' occurs in the first and second places, then R'T'T'a iff R'T'Ta iff R'TTa iff RTTa. If T' occurs in the first and third places, then R'T'aT iff R'TaT* iff RTaT*. And so on. As to (ii), for a, b e K, a <' b iff (Px e 0') *In the light of subsequent paraconsistent insights, to call such structures abnormal m.s.

387$

it might have been better

5.6 NORMALISATION

WORKS

FOR

CONVENTIONALLY

NORMAL

SYSTEMS

EXTENDING

G

R'xab, i.e. iff (Px e 0)(R'xab V R'T'ab) i.e. iff (Px e 0) Rxab, i.e. iff a < b. The shuffles are similar where a is T' or b is T'. (The latter 'or' is exclusive: R'T'T'T' is not defined componentwise. 1 ) Corollary.

T* < T* < T.

Proof. T* < T iff (Px e 0') R'xT'T, i.e. iff (Px e 0) RxTT, i.e. T < T, by the lemma. There are two cases. Either an appropriate x in 0' is also in 0, in which case R'xT'T iff R'xTT, i.e. iff RxTT, or else R'T'T'T* which reduces by two stages to RTTT. But T < T so T' < T. The other inequality will follow from this by the next lemma, or can be shown directly along the following lines: T* < T* iff (Px e 0') R'xT*T*, i.e. iff (Px € 0) RxT*T*, i.e. T* < T*. Lemma 5.4. Given M - < T , 0, K, R, TP,] * > is a fully monotonic L m.s. (such that T* < T), its normalisation M' = < T ' , 0', K', R \ T P ' J *'> at T* (with T* jf K) is also an L m.s. 2 , where L is any conventionally normal system extending G. (i) System G is conventionally normal. So (ii) is virtually any system extending G by any combination of postulates listed in 4.1 whose modelling conditions (in 4.3) are not too complex: specifically they are conjunctions of R and 0 statements or else are implications whose antecedents involve only a single R statement. Examples under (ii) are extensions of G by any combination of Bl, B2, B5, B6, BIO, Bll, D2, D3, D4, D5. So (iii) are many extensions of G which include as well postulates of 4.1 whose implicational modelling conditions involve two R statements in the antecedent, provided the extensions enable the derivation of the Disjunctive Rules JRl and JR4 of 4.8. Examples under (iii) are extensions of G+{JRl, JR4} by B3 or B4. So also (iv) is G+{JR1, JR4}. Thus all the systems singled out for notice in are conventionally normal except for system N, i.e. contrast, is in. But the notion of conventionally open-ended; and we should be prepared to admit any ing the lemma as conventionally normal.

4.1, and many other systems 3 , T-W. System N + JRl, by normal is deliberately left system demonstrably satisfy-

In sum, all the main relevant logics so far modelled, except the L-W systems (for L = T, E and R), are conventionally normal. And even in exceptional cases of these sorts such normality can be restored by the addition of Disjunctive Rules JRl, C V A, C V (A B) - C V B , and JR C V A - C V ~(A ~A). Proof is a matter of the detailed verification of semantic postulates for M', since M' is otherwise well-defined, and is normal. The key to verification of 1

A componentwise definition would erroneously induce T < T*.

2

An L m.s. is fully monotonic if R is monotonically decreasing under < in its first two places and monotonically increasing in the third place. Since (by theorem 4.10) L m.s. can always be replaced by fully monotonic L m.s. without changing the class of theorems of L there is no loss of generality in assuming fully monotonic L m.s.

3

Including some important systems that have come into prominence since 4.1 was completed. There are also however some significant exceptions, e.g. the orthosystem OR (i.e. R less Distribution) studied in chapter 6, for which rule y fails.

388$

5.6 NORMALITY

VETAILS

FOR MODELLING

CONDITIONS

FOR G AND SOME

EXTENSIONS

the postulates is supplied by the preceding lemma. Monotonicity making use of the inequality T* < T is however also important; for this commonly permits occurrences of T* in the final place of relation R to be replaced by T. Observe that full monotonicity of M ensures needed transitivity of By p6, a < d & Rbca > Rbcd, a < d & (Px e~0) Rxca > (Px e 0) Rxcd, i.e. a < d & c < a > c < d . Consider first modelling conditions for system Glad pi'.

Where a e K, part (ii) of the preceding lemma shows a <' a.

Otherwise

the requirement R'T'T'T' shows T' <' T'.

t

t

ad p2'. Where a, d e K, if a <' d & R'dbc then a < d & Rdbc (where c is c or T*) so by p2, Rabc+, and so R'abc. Here, as in subsequent steps, the preceding lemma has been repeatedly applied, unremarked at the points in question. Where a = d = T' R'abc coincides with R'dbc and p2' is a tautology. Where a = T' and d e K, if a <* d & R'dbe then T < d & Rabc^, so by p2, RTbc', whence R'abc. Where a € K and d - T', if a <' d & R'dbc, then a < T* & RTbc'''. Since T* < T, a < T, so by p2 again Rabc^, whence R'abc. ad p3'.

Immediate from definition of *'.

ad p4'. Where a, b e K, p4' follows at once from p4'. Where a = b = T', p4' is a tautology. Where a = T' and b e K, if a <' b then T < b, so b* < T*, by p4. But this just is b*' < a*', i.e. b* <' T'. Where a e K and b = T', if a <' b then a < T*, so T < a* by p4 and p3. But this just is b*' < * a *'ad p6' and p7' (required for full monotonicity). Similar to p2'. For example, in the case of p6', where a = d = T', p6' is a tautology; otherwise p6' follows from p6, in one case using the relation T* < T. ad s2'. Where x e 0, s2' follows from s2; otherwise, where x = T' s2' follows from the equation T'*' = T', which establishes normality of M'. Since J3' is normal the lemma is proved for system G. Establishing the result for extensions of G is a matter of verifying modelling conditions corresponding to additional postulates of the extensions. ad ql', and similarly s3' and s5'. For a e K the primed result follows from the unprimed, ql' from ql. Otherwise it is guaranteed by R'T'T'T'. The pattern is similar for any modelling condition that is a simple conjunction of (unnegated) R and/or 0 statements. ad q2', and similarly for q5', s4', (making use of T' = T'*') and many other one premiss conditions. Generally q2' follows from q2, except where i) c in the antecedent R'abc is T' or ii) all of a, b, c coincide with T'. In case i), where it is assumed that not both a and b are T', R'abc yields RabT* (with the dashes on a or b removed, in case either is T'). Then q2 implies that for some x RaxT* & Rabx, whence R'axT & R'abx, as required for q2'. In case ii), take x as T'. ad q3', and similarly q4 and many other two premiss conditions. Where all of a, b, z, c, d in q3', i.e. R'abz & R'zcd > (Px)(R'bxd & Racx), are in K, q3' follows from q3. So it is also in all other cases except the following problematic cases i) z or d is T 1 , and ii) all elements in one or both antecedents coincide with T'. As to 1), suppose first z - T'. Then R'abT' & R'Tcd yields RabT* & RTcd"f (with d"f either T* or d, as the case may be), which since T* < T yields RabT & RTcd , which gives by q3 again (Px) (Rbxd"1" & Racx) whence (Px) (R'bxd &

389$

5.6 PROOF OF THE ADEQUACY

OF NORMAL

M.S.

FOR CONVENTIONALLY

NORMAL

SYSTEMS

R'acx). The further case where d = T' is also implicitly dealt with. As to ii), suppose all of a, b, c, d (at least) coincide with T'. Then the consequent of q3' follows from R'T'T'T', taking x as T'. Otherwise suppose a = b = z = T'. So assume R'T'T'T' & RTcd+. To apply q3 replace R'T'T'T' by RTTT which jrl provides. 1 Then for some x (in 0) RTxd+ & RTcx, whence R'T'xd & R'T'cx as required. Suppose finally, z = c = d = T'. Assume R'abT' & R'T'T'T'. This time replace R'T'T'T' by RT*TT*, which jr4 yields 2 and apply q3 to obtain for some x in 0, RbxT* & RaTx, whence R'bxT' & R'aT'x, as required. ad jrl and jr4.

Trivial.

Theorem 5.14. Where L is G or any conventionally normal extension of G (e.g. as given by the preceding lemma), A is a theorem of L iff A is valid in every normal L m.s. Proof. If A is a theorem of L, it is valid in all L m.s., and so is valid in all normal L m.s. This establishes soundness, without need to consider base shift. Suppose for the converse that A is not a theorem of L. Then by theorem 4.10 there is a fully monotonic (canonical) L m.s. M = < T , O, K. T, [P], * > and a valuation v and interpretation I on M for which I(A, T) 4 1. Moreover since A v ~ A is a theorem of G, and T includes all theorems, T* < T. Hence by the lemma, its normalisation M' = < T ' , 0', K', R', [P',1 * > is also an L m.s. Define a new valuation v' in M as follows for every sentential parameter p: v'(p, a) = v(p, a) for a e K;

v'(p, T*) = v(p, T).

Then (I) v' is a valuation in M'. Since v' is otherwise well-defined it is enough to show that v' conforms to the hereditariness requirement, i.e. generally for a, b e K' (1)

a < b & v'(p, a) = 1 > v'(p, b) - 1.

Where a, b e K the condition carries over from that for v. Where a = b = T' the condition reduces to a tautology. Where a = T' and b e K the condition reduces to T' < b & v(p, T) = 1 > v(p, b) = 1, which is guaranteed by that for v since T' < b iff (Px' e 0') R'x'T'b iff (Px e 0) RxTb (the same whether x' is T 1 or not) iff T < b, Where a e K and b = T' the condition reduces to a < T' & v(p, a) = 1 > v(p, T) = 1. But if a < T' then a < T* so, as T* < T, a < T, whence the reduced condition follows from that for v. (II) Where I' is the interpretation given by v', for every a, b e K' and every wff A, (2)

a < b & I' (A, a) = 1 > I' (A, b) = 1.

Proof of (II) is exactly as for lemma 4.1. It uses (1) as the induction base and the fact that M' is an L m.s. and so supplies requisite conditions. (III)

Where I' is as above for every a e K and every wff B, I'(B, a) = I(B, a).

Naturally ql [Assertion] if available will do as well. case q2 if available will also serve. 2

Condition s3 [Reductio] if available will do as well.

390$

But note that in this

5.6 PROOF OF THE ADMISSIBILITY

OF y ANV INCLUSION

OF CLASSICAL

LOGIC

Proof of (III) is by induction on connectives in B from the given basis for v'. The induction steps for & and v are immediate; and so is that for ~ since for a e K, a*' = a*. ad Suppose I(B -»• C, a) 4 1. Then for some b, c e K, Rabc and I(B, b) = 1 4 I(C, c). So for some b, c e K', R'abc and by induction hypothesis, I'(B, b) = 1 4 I'(C, c). Hence I'(B -*• C, a) 4 1. For the converse suppose I'(B -»• C, a) 4 1. Then for some b, c e K', R'abc and I'(B, b) = 1 4 I'(C, c). Where b, c e K all primes can be dropped, applying the induction hypothesis and lemma, whence I(B C, a) 4 1. So let b or c be T'. The case where b • c = T* illustrates the procedure for each of the 3 cases. Since by a corollary T* < T* < T by (II) I'(B, T) = 1 and I'(C, T*) 4 1. Hence by induction hypothesis I(B, T) - I f * I(C, T*). But R'aT'T', so by a lemma RaTT*. Therefore, particularising, I(B f C, a) 4 1. Now since I(A, T) 4 1, by (III) I* (A, T) 4 1. Hence as T' < T by a corollary, I'(A, T') 4 1, by (II). Therefore I' on M' falsifies A, which shows that A is invalid in a normal L m.s. This theorem breaks down for sublogics of G such as B. For A v ~A is valid in every normal B m.s. though it is not a theorem of B. The theorem only succeeds of course for certain extensions of G for which modellings are available : elsewhere (e.g. in chapter 12) we will encounter systems with modellings extending G for which y fails, and so for which normal modelling cannot be obtained. Corollaries system L.

1.

Rule y, of Material Detachment, is admissible for each such

2.

Classical sentential logic S is a sublogic of each such L.

Proofs. 1. Suppose A and ~A v B are theorems of L. To show as required that B is also a theorem of L it suffices to show that it is valid in all normal L models. Let S be any such model with interpretation I. Then I(A, T) = 1 = I(~A v B, T) by the initial assumption. Hence I(~A, T) » 1 or I(B, T) = 1 , and so I(A, T*) = 1 or I(B, T) » 1. But as the model is normal, T « T*, so distributing in I(A, T) = 1, 1 = I(A, T) 4 1 or I(B, T) = 1. As the first alternative is impossible, I(B, T) = l 2 , i.e. B is valid in S. 2. As semantical arguments show 3 , every classical tautology (in & — V — ) is a theorem of G, and so of every extension L or G. Thus given, by 1 that y is admissible in L, S is fully included in L. Alternatively take any axiomatisation of S with y as sole rule. It is straightforward to show semantically that each such axiom is a theorem of G and so of L. 1

The algebraic proof of y in ABE is accordingly more general inasmuch as it establishes y for non-normal systems, such as system I (of SI) beyond the scope of the semantics se far elaborated. But the semantical analysis of I given in chapter 7 may be applied, much as above, to show that y holds for I. However the algebraic proof also works for systems (e.g. some from 4.9) which have thus far escaped the semantical net entirely.

2

This is the notorious extrasystemic use of y in establishing y. The matter is discussed at length elsewhere: for references and some discussion see Routley 82b.

3

0r as the alternative argument in this proof shows.

391$

5.7 HALLVEN

REASONABLENESS,

PROVUCT M.S. ANV R-CLOSEV

M.S.

§?. Hallden reasonableness and normal characteristic matrices. A logic L with sentential language SL (or SL°, etc.) is reasonable in the sense of Hallden - or Hallden reasonable for short (after Hallden 51) - iff, whenever A v B is a theorem of L where wff A and B share no sentential variable, either A is a theorem of L or B is a theorem of L. Being Hallden reasonable is (as argued in 1.6) a requirement of adequacy in an acceptable sentential logic - a requirement which leading modal explications of entailment such as S2 and S3 notoriously fail to satisfy (see Hallden 51). We generalise the argument (in onable. The product M" of L m.s. M * * > is the structure~ for a e K and a' Oa and 0 ' a \ R" < a , a ' > < b , b ' > < a, a* > *" is < a*, a'*'> .

SE II, p.70ff.) that R and NR are Hallden reas= < T , 0, K, R, * > and M' = < T ' , 0', K', R', R", *" > , where K" is the set KXK* of all ordere K, T" is the pair < T , T * > , 0" < a , a ' > iff < c , c ' > iff both Rabc and R'a'b'c', and

Lemma 5.5. For any L m.s. (without P), elaborating a B m.s., whose semantical postulates can be put in the form of a conjunction of conditions on R, or R and 0, or a particular quantification thereof, or in the form of a conjunction of conditions on R (and 0 and *) implying a conjunction of conditions on R (and 0 and *) or a particular quantification thereof, the product of L m.s. is also an L m.s., that is to say, the product preserves structure. Thus ql-ql5 are included as structure preserving, but ql7, ql8 and q20 are not; and sl-s5 are also included. In short, most of the extensions of B m.s. introduced in 4.3 are structure preserving. Proof is by straightforward verification of cases and of compounding principles. Some examples from the semantical postulates for B m.s. show how the procedure goes. ad pi". By pi and pi', Ox & Rxaa & 0'x' & R'x'a'a' for some x and x*. Hence 0 " < x , x ' > & R " < x , x ' > < a , a' > < a, a ' > for some < x , x ' > , giving pi". This illustrates the second compounding principle. ad p2". Similar, upon combining p2 in the form Ox & Rxad & Rdbc > Rabc with p2' in similar form. This illustrates the third and main compounding principle. A L m.s. is R-closed iff for every a e K, for some b, c e K, Rabc. The condition is a 3-place version of the corresponding modal notion (i.e. (Pb)Rab) used for modelling deontic logics, but its relevant status is very different. Most relevant logics of relevant interest so far modelled can be equipped with R-closed m.s., as the following lemma shows:Lemma 5.6. for which i) ii)

Where L is any of the extensions of B Cor B°] modelled (including B)

ql or s3 holds, i.e. either Assertion or Reductio is a theorem of L, or non-degenerate canonical modelling is not required,

then A is a theorem of L iff A is true in all R-closed L m.s. Proof. i) is immediate, by quantificational particularisation. As for ii) if A is a theorem then it is true in all L m.s. and so in all R-closed ones. For the converse, suppose A is not a theorem of L, and consider the canonical L model M which falsifies A. The null and universal theories, e and u, are prime ~c L-theories, which by ii) are not excluded from M (e.g. in order to push through ~c classical negation or reduction features). But R^aeu. Hence M c is R-closed, so

392$

--- 5.9 THE ARGUMENT

TOHALLVEN

REASONABLENESS

A is not true in an R-closed L m.s.

FOR MAIN RELEVANT

LOGICS

•-

Theorem 5.15. Where L [L°] is any one of the extensions of B [B°] introduced with R-closed m.s. for which the product preserves structure (e.g. for which the preceding lemma holds), L is Hallden reasonable. Proof. Let A and B be wff for L [or L°, as the case may be] which share no sentential variable. Suppose neither is a theorem of L. Then there are valuations v in L m.s. M and v' in M' which falsify A and B respectively, by a completeness theorem. Then there is a valuation v" in the product M" of M and M' which falsifies A v B, thus establishing Hallden reasonableness. Define v" as follows, for < a , a ' > e K": v"(p, < a , a ' > ) = v(p, a) for every sentential parameter p in A; v"(q, < a , a ' > ) = v' (q, a') for every such q in B; and otherwise v" is determined arbitrarily so long as it meets valuation conditions. Then v" is a valuation in M" since v is in M and v' in H'. Now let D^ be any wff all of whose sentential variables occur in A, and similarly D., in B. is (I) ^

Then

I"0>A» < a , a * > ) = I(D A , a) and I"(D D , < a , a' > ) = I' (D , a*), for every < a , a ' > B fl

£ K".

Proof is by induction from a given basis. The inductive steps for &, v, and ~ are elementary, as the last will indicate. ad *

(first case).

I"(~C A , < a , a ' > ) = 1 iff I"(C A , < a , a ' > *") * 1, i.e. iff

I"(C A , < a * , a'*® > ) * 1, i.e. iff I(C A , a*) ^ 1 by induction hypothesis, i.e. iff I(~C A , a) = 1. ad -»•. D. is B — A

C for some wff B and C whose variables are restricted to A.

Suppose I"(B C, < a , a* > ) ^ 1. Then for some < b , b ' > and < c , c ' > in K", R " < a , a ' X b , b' > < c, c ' > , I(B, < b , b ' > ) = 1 and I(C, < c , c » > ) t 1. By definition of R", Rabc and by induction hypothesis I(B, b) = 1 and I(C, c) t 1, whence I(B -*• C, a) ^ 1. For the converse suppose I(B -»• C, a) £ 1, whence for some.b and c in K, Rabc and I(B, b) = 1 and I(C, c) ^ 1. By the given modelling condition, R'a'b'c', for some b'atid c' in K', whence R " < a , a ' > < b , b * > < c , c ' > , and by-induction hypothesis I"(B, < b , b ' > ) = 1 and I(C, < c , c ' > ) i 1, whence I"(B - C, < a , a ' > ) * 1. ad Similar to the previous case. (The details are as given in SE II, theorem 5, except that R-closure replaces R-reflexivity.) Finally, since I(A, T) - I'(B, T') ^ 1, by (I) I"(A V B, < T , T * > ) ^ 1, establishing that M" indeed falsifies A V B.

Hallden reasonableness of a system is a necessary condition for its having a normal characteristic matrix. A matrix S = < M , D, &', v', , > for a sentential system L with connective set {&, v, ->•} is normal iff for every p, 0 e H , P &' a e D iff p e D and 0 £ D, p V* a e D iff P £ D or O £ D and ~'p £ D iff P t D. Now suppose L has a normal characteristic matrix, and assume to show Hallden reasonableness that for some A and B which share no variables ~ k A and ~ k B . It suffices to show that ~ k A v B, i.e. to find an assignment w L L L in # such that w(A V B) I D, i.e. by normality, for which w(A) £ D and w(B) i D.

393$

5.7 THE CONNECTION

WITH NORMAL

CHARACTERISTIC

MATRICES

FOR MAIN SYSTEMS

The remaining argument resembles that in the preceding theorem. By the assumptions there are assignments, by v and v' of elements of M to variables of A and B such that v(A) i D and v'(B) I D. Finally fix a new assignment v" that agrees with v on variables of A and, as is possible, with v' on variables of B. v" is indeed an assignment and does what w is required to do. Evidently Hallden reasonableness uses only some of the conditions that having a normal characteristic matrix supplies. A simple set of properties that uses them all is assembled in the following result Theorem 5.16. Where L is a sentential logic whose connective set includes {&» v, , L possesses a normal characteristic matrix iff i) ii) iii) iv)

L y L L

is simply consistent, i.e. not both A and ~A are provable for some wff A, is an admissible rule of L, includes all classical tautologies, and is Hallden reasonable.

Proof. Suppose L has such a matrix. Then iv) has just been proved, ii) follows as in §6, iii) derives from the fact that truth tabular conditions are guaranteed by normality, and i) from the fact that not both A and ~A can be jointly designated given negation normality. The converse 'proceeds by constructing a normal characteristic matrix by Lindenbaum's method' (Kripke 65, p.207). Combining previous theorems, it follows that main relevant logics have normal characteristic matrices. Theorem 5.17. Where L is any one of the extensions of G with R-closed m.s. for which the product preserves structure and for which the structure is conventionally normal, L has a normal characteristic matrix. Corollary. G, DK, DL, E, R, T and many other relevant logics have normal characteristic matrices. Since Lewis system S3 is not Hallden reasonable, though it has R-closed m.s. for which the product preserves structure, the argument suggests (but does not prove) that system S3 is not analysable within the semantical framework so far developed. As we shall see however (in 7), a simple liberalisation of the framework facilitates the treatment of systems like S3.

§fl. Parts, initial separation results, and modelling simplifications and variations. Among the many ways in which world semantical methods can be fruitfully applied is in establishing that certain systems are parts of other systems, that, for instance, the pure entailment part of E (the system E^ of ABE) contains all the theorems just in entailment (in connective -+) of the full system E. That it does is proved below (see Appendix 2). Results or classes of results of this sort which segregate a system into parts in terms of its operations (connectives) are commonly called separation results. They show that certain parts of a logic (those concerning connectives only) are properly separable from the larger logic, in that crucial properties (e.g. well-formation, provability) can be confined just to local matters of the parts concerned. In particular proofs of a theorem of any one of the parts do not have to go beyond the resources of the part and track through the larger logic. To show how easy semantical methods can make otherwise often difficult separation results, consider the mainly semantical proof that R+ is the positive part of R, that, in other jargon, R is a conservative extension of R+. One half

394$

5.8 SEMANTICAL

DERIVATION

OF SEPARATION RESULTS

FOR RELEVANT

LOGICS

of the result is of course immediate; for as R+ is a subsystem of R, every theorem of R+ is a theorem of R. For the much harder converse, suppose A is a wff of R+ but not a theorem of R. Then as A is not R-valid, there is an interpretation I extending v on an R m.s. M = < T , K, R, * > for which I(A, T) 4 1. But, as a little argument shows, this also furnishes an interpretation I (= I) also extending v on an R+ m.s. M = < T , K, R > for which I (A, T) 4 1, whence A is not R+-valid, and so not a theorem of R+. Since M simply omits * and its conditions, the only connectives of R+ are used in establishing modelling conditions on the canonical structure M , it is trivial that M is an R+ m.s. It remains to prove that I~(A, T) • 1. The crucial feature of easier separation arguments emerges at this stage: that the (semantical) rules for each connective are independent of those for other connectives. Thus there is no obstacle to showing by induction that, for every wff B of R+, for every a e K, I~(B, a) - I(B, a). The induction basis is given, and requisite induction steps for immediate. Since I(A, T) 4 1, I~(A, T) 4 1•1

& and v are

The argument generalises to many other cases. For the semantical methods applied, in proving completeness, depend essentially only on the presence of connectives -*• and &, and the semantical rules remain duly independent. Hence the adequacy proofs given suffice also to prove, in the way illustrated with R and R+, the following separation result for systems modelled:Theorem 5.18. Where L' is any part of logic L° which contains only one of the following sets of connectives explicitly &;

&, o;

V;

&, V, o;

&, v, ~

then (i) wff A is a theorem of L' iff A is L'-valid, where L'-validity differs from L<>-validity only in deleting interpretational clauses for connectives of L° which do not occur in L'; (ii) where A is a wff of L', A is provable in L' iff A is provable in L°, that is where A is derivable in L°, A is derivable in L' from those axiom schemes and rules of L° which contain only connectives of L'. The connective t may also be generally eliminated from any system L' listed in the theorem which contains t, provided only that L' contains compensating t-free axioms. If, as with weaker systems, the only postulates containing t are CI and C3 then t may be eliminated without any compensating postulates, a fact which may be proved either from the semantical analysis above or syntactically (as in ABE, or SE IV: see Appendix I). With the deletion of negation or of negation and disjunction, various important simplifications can be made in the modellings. 2 For instance, in the case of the positive, i.e. L+, systems not only can * be deleted from corresponding m.s. but 0 may be replaced by T, i.e. the m.s. can be reduced, and P may be definitionally eliminated, through the connection Pa iff T < a; so lr

The more complex argument for the result, where fusion is integrally involved, is given in Lemma 14 of SE I(pp.224-6).

2

The simplifications are explained again, in different detail, for the case of system E in SE IV: see Appendix 1.

395$

5 . 8 SIMPLIFICATIONS

ANV VARIATIONS

UPON THE SEMANTICS

FOR PARTS OF

SYSTEMS

result the positive model structures of SE III.1 When, further, disjunction as well as negation is deleted from logic L, relation Rabc between elements of K may be analysed into relation a ® b < c where ® is an operation closed on K and < is as before an order; and then conditions on relation R may be replaced by more algebraically amenable conditions on operations ® (or °).2 In the reverse direction this operational semantics may be expanded from L c to the whole of logic L by suitable including the model structures both a non-prime and prime theories, that is, K as well as K. Such operational semantics are elaborated and applied in chapter 11. There are alternative model structures, built on the operational semantics r for L & , for positive relevant logics. These structures can be straightforwardly extended to model classical relevant logics. For example, a Meyer R+ m.s. M is a structure < 1 , K, + > where the base world 1 (so relabelled because it behaves in main respects like an identity) belongs to K, ° is an operation on K satisfying conditions for a De Morgan monoid 3 and + is a further operation on K. The interpretation rule for & is the normal one, that for -*• is the standard operational rule, namely I (A -*• B, a) = 1 iff for every b e K, if I(A, b) - 1 then I(B, a ° b) = 1, but that_for v is the following new rule: I(A v B, a) = 1 iff for some b, c e K for which a = b + c, I(A, b) = 1 = I(B, c).u

When however subsystem L', lacks connective &, the techniques for proving separation of theorem 5.18 no longer apply: the methods we have applied in proving completeness and the syntactical elimination of t both break down. To capture separation results in the absence of &, we may either resort, as in SE I and AL II, to algebraic methods, or else modify the completeness proof, as in SE IV and Appendix 2. With pure implicational systems, where L' contains only connective -*-, separation methods split according to the strength of the system. Theorem 5.19. Where L is either basic system B or a simple extension of it such as B + Bl, and L is the part of L which contains only the connective -+, then where A is a wff of L (i) A is provable in iff A is provable in L, and (ii) A is a theorem of L iff A is L-valid and iff A is L r -valid. -y Proof. Where A is a wff of B^ it is shown in AL I that A is a theorem of or B+ iff A is of the form C C. Hence, since B is a conservative extension of B+, ^ h e simplification depends on Meyer's result that, for main positive relevant logics, A v B is a theorem of L+ iff either A is a theorem of L+ or B is a theorem of L+; for some details and references see ABE, p.378ff. 2

For these conditions in semantical and algebraic form see, respectively, chapters 11 and 12. The relevant part of chapter 12 builds on AL I and incorporates AL II.

3

That is, the structure is that of a partially ordered (under <), square-decreasing (i.e. a ° a < a) commutative monoid, i.e. 0 is associative, 1 is an identity w.r.t. and ° is monotonic (specifically if a < b then a ° c < b ° c).

''This semantics for R+ is due to Meyer. The rule for disjunction is an interesting special form of that subsequently adopted for connexive logics: see Routley 78. 396$

5.8 MORE PIECEMEAL

SEPARATION

RESULTS

CONCERNING

PURE IMPLICATIONAL

PARTS

A is a theorem of B^ iff A is a theorem of B. Furthermore, then, A is a theorem of iff A is B-valid, and so iff A is B & -valid. A similar argument works in the case of B + Bl and for what we shall call simple extensions of B. This result evidently leaves a certain amount to be desired. For it does not connect B^ theoremhood with B^ validity, (but this can easily be done as in theorem 5.21 below), and, worse, it leaves simple extensions of B only circularly specified, and inadequately instantiated. There is, moreover, a class of systems in the vicinity of DK and DL - systems apparently like simple extensions - whose implicational parts remain to be determined with rigour. For affixing systems with contraction the situation is mare satisfactory (and arguments can be duly semantical). Theorem 5.20. Where L is any suitable extension of B (with the same language) which includes Contraction, i.e. B5, and L^ is the part of L which contains only the connective then (i)

A is a theorem of L^ iff A is L^-valid, and

(ii)

where A is a wff of

A is provable in

iff A is provable in L.

Proof. When L includes Contraction a modification of the main completeness argument can be applied. The modification which involves new definitions of L theory, T-L^-theory and I^abc, in particular, is presented in detail in the Appendices (§10 of Appendix 2). Use of Contraction occurs essentially in the inductive step for ->-. A suitable extension of B + B5 is any for which the requisite semantical postulates can be established among the new definitions. These include E and R; hence the Corollary.

The result holds for T, E and R.

The theorem does not extend to contractionless (W-less) extensions of B such as L-W or R-W or N (i.e. T-W). However that the full and positive contractionless systems are conservative extensions of their pure implicational parts, and that these pure parts have the expected semantical analysis may be shown, somewhat indirectly, by algebraic methods. 1 The techniques relied upon in the preceding theorem can be adapted to show that for any system L for which Commutation, i.e. B7, also holds, the theorem also succeeds for the implication-negation part of L. Theorem 5.21. Where L is any suitable extension of B which includes Contraction and Commutation, and L~ is the part of L which contains only the connectives -»• and then (i)

A is a theorem of L~ iff A is L^-valid, and

(ii)

L is a conservative extension of L~.

Proof Is as in the preceding theorem. Corollary.

J

The theorem holds for system R;

thus R is a conservative extension

See 12.5 and 12.6. For use of these methods in showing that R is wellaxiomatised, see Meyer 72a.

397$

5 . S THE SIMPLE STRUCTURE

or R~.

OF THE IMPLICATION-NEGATION

PART OF SYSTEM

B

Hence also, by previous results, R is well-axiomatised.1

As these techniques break down in the absence of Commutation, resort must again be had to other more piecemeal or indirect methods. For logics like basic system B where the implication-negation part has a very simple structure, it is not difficult to resolve the remaining important separation detail. Let T be the class of wff of B~ defined recursively as follows: (i) if A is of the form C •*• C, then A e T; and (ii) if A e F and B results from A by substituting — C for a well-formed part C of A, then B e T. Further, if B results from a wff D of B~ by zero or more applications of rule (ii), substituting ~~C for C or C for — C in B, we call B a transform of D and write B alternatively as t(D). Theorem 5.22. Where A is a wff of B~, i.e. a wff of B containing only connectives -•• and A is a theorem of B~ or of B iff A e F, i.e. iff A is a transform of an instance of the principle of identity C -*• C. Proof. Since B B, C -»• — C and — C C are theorems of B~, one half of the result follows by the principle of substitutivity of coimplicants, DR2 of (4.1, where proof for B~ is a special case of that given there). The converse exploits a specialised semantical modelling for B~. Define an m.s. M = < T , K, R, * > as follows:- a e K iff a is a set of wff of B~ closed under applications of rule (ii), i.e. if A e a so do its transforms; T = T; Rabc iff, for a, b, c e K, whenever B -»- C e a and B e b then C e c; a* = {A: ~A i a} for a e K. Then (I) M is an m.s.; in fact a reduced m.s. The result follows readily from the facts that RTab iff a c b, for a, b e K, that * is an operation in K, and that T e K. Modelling conditions are then almost immediate. As usual v is so defined that v(p, a) = 1 iff p e a , for every parameter p and every a e K. Then v is a valuation in M. Proof by induction that (II) I(A, a) = 1 iff A e a, for a e K and every wff A of B~, follows the of theorem 4.10 (III). The only new details appear in the induction step where under the assumption that B C e a it has to be shown that b = {C' B -> C* } and c = {A: (PB) (B A e a & B e b) } both belong to K; but this consequence of the requirement that, for d e K, if A e d and B' ->• C' e d do t (A) and t(B') t(C').

lines for -»: [is a then so

Since only members of T are verified on M, it follows that A is a theorem of B~ or of B only if A e T. Corollaries. 1. 2 duced B—valid. -b

A is a theorem of B~ iff A is B~-valid, in fact iff A is re-

Soundness is by direct verification. Conversely, suppose A is not a theorem of B~. Then A I T, so the reduced m.s. M of the theorem above falsifies A. 2.

B is a conservative extension of B~.

3.

B~ is decidable.

Accordingly the partly algebraic procedures of SE I can be avoided in the case of R. 2

Thus it is the mix of positive connectives with negation that forces unreduced modellings. 181$

5.9 THE FUNDAMENTALS OF FILTRATION METHODS

FOR ESTABLISHING

DECIDABILITY

For the important range of systems between simple extensions of B and system R, resort to algebraic methods can once again often be used to show that implication-negation parts of the systems have the desired properties. Such is the strategy of Meyer-Routley 74, a generalisation of which fits seamlessly into chapter 12 (§6). It will then follow that main affixing relevant logics modelled are well-axiomatised. 1

§9. Decidability and undecidability. I. Filtration of relational ures. The decidability of various weaker relevant systems can be

model

struct-

established by filtration means, as is now shown. 2 What is called filtration method is used to establish the finite model property, that each non-theorem of a system is falsified in a finite model. By the completeness theorem there is, for each non-theorem A, some model which falsifies it. That model, which will in general be different for different A, is not finite. The filtration method, where it works, enables a new model, which is finite and which also falsifies A, to be defined in terms of that model. The fundamentals of the method are these:A finite class of wff associated with A, ^(A) say, is first determined: 3 •(A) will need to include at least all well formed parts of A if, as required, the semantical evaluation of A through its parts is to carry over to the new model. Secondly, an equivalence relation, dependent o n ^ i . e . on ^(A) for a given A), is defined on set of worlds K of the original model. is simply a = j, b iff (B e

, I(B, a) = I(B, b);

The basic definition

but variations on this defin-

ition are important in accommodating recalcitrant modelling conditions.

Relation

essentially enables features of the old model which do not affect A's evaluation to be discarded and indifferent wgrlds to be equated.

Thus a new finite

model can be defined in world set K = {a:

= {b e K: a

a e K} where

b}.

then, new relations and operations are defined in K, commonly pointwise, and a valuation v in K is defined in terms of the restriction of v to It remains, finally, to show that the new model has the requisite properties. Given it does v in K will duly falsify A, establishing the finite model property. Decidability follows therefrom in the way explained in the corollary to theorem 5.23 below. A class of ^ of wff of L is appropriately closed iff it is closed under subformulae and contains ~B whenever it contains unnegated wff B (and B when it contains negated wff ~B).14 Thus for each wff of L there is an appropriately closed finite class of wff. Where ^ is appropriately closed define, for a, b e K, a = ^b iff (B e . I(B, a) = I(B, b). Then a iff a* = ^ b * , i.e. (B e

I(B, a*) = I(B, b*).

Relative to

define equivalence classes,

*In such claims a certain asymmetry between & and v - already evident in axiomatisations as regards Adjunction, V-Composition and Distribution - is assumed. It is taken that v is added to •+-& logics, not vice versa, and not v to pure implicational systems. 2

The method varies that of SI;' for application of filtration methods for proving decidability, see also Segerberg 68 and (especially) 71, p.63ff.

3

More generally, a class ^ ( A )

can be associated with class of wff A .

''Basic filtration methods require closure of ^ under subformulae only, i.e. where A is a subformula of B and B e ^ then A e Finiteness would not be lost by simply closing under truth-functional composition as well as subformulae, as will be done later.

399$

5.9 FIRST FILTRATION DEFINITIONS ANV PROOFS OF VESWEV

a,., = {b e K:

a

b}.

PROPERTIES

In terms of these new elements, which comprise a class

K = la: a e K), define a new model M in K. Then T is given and a* = a* but there are significant options for how suitably invariant definitions of remaining elements of A are framed. An initial set of definitions goes as follows: Oa

iff

a < b Rabc

I(B

(B e iff

iff

( B e ¥) (B, C e

B, a) = 1; (I(B, a) = 1 > I(B, b) = 1); (B ->• C eflf& I(B

C, a) = 1 = I(B, b) >. I(C, c) = 1).

(Where constant t occurs in the language and satisfies schemes CI and C2, Oa can be more satisfactorily defined as I(t, a) = 1, with t assigned to ) Define a (first) filtration M of Lm3s M through represented M/^, as the structure < T , 0, K, R, * > . Define a putative valuation v in M as follows: for every sentential parameter p, and every a e K, v(p, a) = 1 iff v(p, a) = 1 and p e Lemma 5.7.

(i) (ii) (iii) (iv)

Oa > Oa Rabc > Rabc Where M is a B m.s., then M is also, Where v is a valuation in M, v is in M.

Proof. As to (i), suppose for B e I(B B, a) 4 1. Then Rabc, I(B, b) = 1 and I(B, c) = 0. Then since Oa, b < c, whence I(B, b) = 1 4 1. ad (ii). Suppose for B C e Rabc, I(C, c) = 1, as required. ad (iii).

I(B

C, a) - 1 and I(B, b) = 1.

Then, given

Since T e K iff T e K, and T e 0, by (i), f e 0.

0 and R are relations on K which are independent of the choice of elements of their relata. ^Similarly * is an^extensional operation. For a* e K since a^* e K, and where a =^b, a* = a* = = b*. It remains to establish semantical postulates. Since < is an inclusion, pi and p5 are immediate. ad p^. Suppose, to establish Rabc, B -> C e I(B -»• C, a) = 1 and I(B, b) = 1 for arbitrary wff B, C. It has to be shown that I(C, c) = 1. Since B -+ C e KB C,^a)^= 1 and, by hypothesis a ^ I(B C, d) = 1. Since also by hypothesis, Rabc and I(B, b) = 1, I(C, c) = 1 as required. ad p3.

(a*)* = (a*)*

= a.

ad p4. Suppose a < b and B e * and I(B, b*) = 1, i.e. I(~B, b) 4 1. What is to be shown is that I(B, = 1, i.e. I(~B, a) 4 1: in short if I(~B, a) = 1 then I(~B, b) = 1. But as a < b, and B e ^ implies ~B e if I(~B, a) = 1 then I(~B, b) = 1.

ad p8. ^Suppose, in order to show b < c, B e ^ and I(B, 1 for arbitrary B. Since Oa and B e I(B B, a) = 1, and further since Rabc and I(B, b) = 1, I(B, c) = 1, as required.

400$

5.9 RESTRICTED

INTERPRETATIONS,

& PROOF THAT B HAS THE FINITE

ad p9. By p9, for some d € K, Od and Rdaa. A A AAAA Od and Rdaa. ad (iv).

MODEL

PROPERTY

Hence, for d e K, by (i) and (ii),

v is an appropriate function in

* ^ which respects

For v(p, a) = 1 & a ^ b >. v(p, a) = 1 & p e * & (B e *) (I(B, a) = 1 > I(B, b ) = 1) > p e * & v(p, b) - 1 > v(p, fi) = 1. Lemma 5.8. For ever^ wff A e ^ I(A, a) = I(A, a), where I is the interpretation function, in M which extends v to all wff (in Proof is by induction, from the already established basis, v(p, a) = v(p, a) for each sentential parameter p e ad &. I(A & B, a) = 1 iff I(A, a) = 1 = I(B, by hypothesis, i.e. iff I(A & B, a) = 1. ad v.

2),

i.e. iff I(A, a) - 1 = I(B, a),

Similar.

ad I(~A, a) = 1 iff I(A, a*) = 0, i.e. iff I(A, a*) = 0, i.e. iff I(A, a*) = 0, by induction hypothesis, i.e. iff I(A & B, a) = 1. ad -*•. Suppose, to show I (A •*• B, a) = 1 > I(A -*• B, a) = 1, that I(A -»• B, a) = 1, Rabc and I(A, b) = 1, for arbitrary b, c e K. It has to be shown that I(C, By induction hypothesis I(A, S ) = 1 and by (ii) of the previous lemma, Rabc. Thus, as I(A •*• B, a) = 1, I(C, c) = 1 whence by induction hypothesis again I(C, c) = 1. For the converse suppose I(A B, a) 4 1. Then for some b, c e K, Rabc, I(A, 6) = 1 and I(B, c) 4 1. Thus, by induction hypothesis, I(A, b) = 1 4 I(B, c), and by definition of R, since A -»• B e ^ by hypothesis, I (A -*• B, a) 4 1. Theorem 5.23. Basic system B has the finite model property, i.e. for each nontheorem A of B there is a finite model which falsifies A. 1 Proof. Let A be a non-theorem of B. By completeness there is a valuation v in a m.s. M such that I(A, T) 4 1 where I is the extension of v and T the base of M. Let • be the appropriate closure^of A. By the preceding lemmas I(A, T) 4 1, i.e. there is a valuation v in m.s. M = M / * which falsifies A. Since * is finite, M/^ is finite; in fact where ¥ has n elements, K has no more than 2 n elements. Corollary.

B is decidable.

Proof. Since B is finitely axiomatisable 2 and has the finite model property, we can specify enumerations

of all derivations of theorems of B, and

J

2

A model, and similarly an m.s., is finite iff the class of worlds K is finite.

Strictly B has more rules than are usually admitted in (too narrow) accounts of finite axiomatisability. These additional rules make no difference to arguments like that which follows, but other features of finite axiomatisability are important to it.

401$

5.9 DECIDABILITY ESTABLISHED AND EXTENDED TO EXTENSIONS OF B

of all finite B models.1 Then there is an effective procedure which, for each wff A of B, yields the answer YES or NO to the question whether A is a theorem of B; for example (following Segerberg 71, p.35) as follows:START. Go to STEP 0. STEP n (for n > 0). If n is even, test whether D

n/

is a derivation for

2 A. If it- is go to STOP 1; otherwise go to STEP n + l . If n is odd test whether M, .N is a B-model. If it is and it falsifies A, go to STOP 2; ~(n-1) .

'2

otherwise go to STEP n+l. STOP 1. STOP 2.

Stop! Stop!

The answer is YES. The answer is NO.

To extend the theorem and corollary to systems beyond the basic system B it suffices to show for each given system L that, where M is an L m.s., so is M; and this is a matter of verifying that the further semantical postulates of L beyond those of B hold in M. But, to put it mildly, by no means every semantical postulate considered in chapters 4 and 5 will succumb to this treatment (as the undecidability of Q+ will demonstrate, and as the present undecidabilicy of rhe decidability of E, R and T indicates). We consider a few postulates of inte.vv.st that do succucsb:-

j. i, and so Raa*a. --r so simple, and require complication - tall- of 6 and g

of defin-

1 r nc t; I (3 B, a) » 1 = K B v ~B, a), and reestabt ic - awc-s nc,. ra ts (i) and (iii) of the first lemma. As to (i), suc-Dose. t.o :ake the alternative case, I (B, a) = 0 = I (~B, a) for B e Since Ua, by 3.- a* a: tnus n B , &«) = 1 implies I (B, a) = 1 contradicting I(B, a) = 0. Now co verify Ox >. x* £ x, suppose Ox, B e * and I(B, x*) = 1. It suffices to show I(B, a) = 1. Since Ox and B e * , I(B v ~B, x) = 1, so I(B, x) = 1 or else I(~B, x) = 1, i.e. I(B, x*) = 0. But the latter is impossible, so K B , x) -1.

ad s4. Enlarge * as follows:- Whenever B C e * a contraposed form equivalent to ~C ~B 6 that is to say the form coincides with ~B modulo double negation which removes doubly iterated negations. Evidently the enlargement * + is finite where * is finite. Write the form equivalent to ~C ~B as |~C ->- ~B | , that is j~C! j~B j . Now suppose, to establish s4, Rafic, B -> C £ * +. I(B ->- C, a) '= 1 and I(B, c*) = 1. It has to be shown that I(C, b*) = 1. Then Specification of such enumerations is standard in logic textbooks; for the first see almost any text that present Godel's incompleteness theorem in due detail (e.g. Mendelson 64), while for the second see texts on model theory. In each case minor adaptions are required to accommodate B.

185$

5.9 PROBLEMS WITH EXTENDING VECWABIL1TV,

AND SECOND FILTRATION DEFINITIONS

|~C| + |~B| , and I(|~c| + |~B|, a) = 1 because I(~C ~B, a) = 1; so I(|~C|, b) = 1 > I(|~B|, C) = 1, by Rafic. Hence I(~C, b) = 1 > I(~B, c) = 1; thus I(B, c*) = 1 > I(C, b*) = 1, whence I(C, b*) = 1. In some cases it is necessary to make use of restricted modellings or of features of the canonical modelling. Since the invalidation procedure used in establishing the finite model property always begins, in effect, with the canonical falsifying model, features of this model can be legitimately assumed, as in the next cases. ad qlO. As the canonical model is nondegenerate, for some wff D, I(D, a) = 1, whence by BIO, I(B -*• B, a) = 1. Redefine < as follows: a £ b iff (B e * , B + B el 1 ).

I(B, a) = 1 > I(B, b) - 1,

and rework the basic argument accordingly. To show, as required given Rabc that b ^ c , suppose B e ^ , B B e * and I(B, b) = 1, and show I(B, c) = 1. But as ftafic, under these conditions I(B, c) = 1. Theorem 5.24. For each extension L of B or B+ considered (i.e. for extensions by selections of the postulates dealt with, both by first and subsequent filtration methods), L h a s the finite model property, and is decidable. A recurring problem that confronts attempts to include other important semantical postulates within the scope of the theorem just stated is the matter " ng rtew occurrences of wff, implication wff especially, to The en irises sharply and sometimes irresolubly (cf. below), where new equivali?«es have to be introduced, as with q3 and q4; but also occurs in -J tuations, as with q7. What has to be shown there is that given Rabc, Xcai: i e. wherever B -> C e ^ and I(B C, b) = 1 = I(B, a) then I(C, c) = I, cy E7, I ((B C) C, a) = 1, the desired result would then follow using '3cv were (B C.) C in But there is no obvious way of ensuring that it is, of sufficient generality, which keeps ^ duly finite. This problem can however be surmounted by switching to a different definition of filtration, straightforwardly adapted from modal logical applications. In oiace of the initial set of definitions, a second set goes (in basic form) as follows: SaSc iff, for some a e a , b f c b , c e c , Rabc; and Oa iff, for some a e a, Oa. With second filtrations as so far defined some previously excluded verifications are made easy; two examples: ad qT. ^Given Rabc for some a e a, b e b and c e c, Rabc. whence Rbac. ad q^.

Then by q7, Rbac,

Similar, using the fact that x e K iff x e K.

Then a (second) filtration M of L m.s. M through M / ^ is the structure < $ , 6, Y 7 T 7 * > ~ ; v in M is defined as before. (By the previous argument it suffices to filtrate the canonical model, to consider canonical filtrations. The restriction is important in trying to verify p2.) The previous lemmas transfer largely intact, though some new argument is required. (i) and (ii) of the^ first lemma are immediate. A A AFor (iv) and parts of (iii), it is /\ important that < A XA AAAA A yields an inclusion. But a < b iff (Px)(Ox & Rxab), so given a < b, for some Xj e x, a^ e a and b^ e b, Ox^ & Rx^a^b^, whence a^ < b^. Hence by lemma 4.1 for every wff B, I(B, a.) = 1 > I(B, b.) = 1.

403$

But a 1 e a, so for every wff B e

5.9

SHORTCOMINGS

OF SECOND

FILTRATIONS

AND AN INTRODUCTION

I(B, a^) = I(B, a), and similarly for b;

TO THIRD

FILTRATIONS

and therefore for every B e

I(B, a)

= 1 > I(B, b) = 1. Of the basic modelling conditions, Pi, p5 and follow directly from definitions given: only p2 remains problematic. As to suppose that a ^ 3 and ftSfic. It has to be shown that &af>c, i.e. for some a^ e a, b^ e 6 and c^ e c, Ra^b^c^.

But by assumption for some a^ £ a, bj e 6 and c^ e c, Ox & A

A

Rxajd^, i.e. a^ < d^, and Rd^^c^, for some x e x and d^, d 2 e d.

The latter is

the snag (rather as with q3 and q4): if d 1 and d„ were the same the desired verification of p2 would follow by p2 itself. But d^ and d 2 are only ^-equivalant, i.e. d^

d^;

whereas the argument requires that for every B

merely every B - * - C e ^ , B that for a l =j, a, bj

C £ d^ > B

C e d2«

b, c 1 = j, c and dj^

C, not

For canonically it is given

d = j, d 2 , (D) (D e a^ > D e d^) and

(B, Cf) (B -*• C e d 2 & B e b j >. C e c^), while what has to be shown, for the same given data, is (B, C)(B

C e a^ & B e b^ >. C e c^).

This difficulty1 would

however be surmounted if the (canonical) model used in filtrating were duly restricted to There are various ways of imposing such a restriction, different from the method of first filtrations. Simple adornment of the second method AAAA restricts relations and operations to H'. Thus on the finalised second method Rabc is redefined as: Pa^ e a, b^ 6 b, c^ e c R^a^b^Cj, where R ^ b c iff (B -»• C e (I(B C, a) = I(B, b) > I(C, c) = 1), i.e. canonically (B C e ¥) (B -»• C e a & B e b >. C r c ) . The problem with is then overcome (since for B -»• C £ B - > C £ d ^ > B - > - C £ d 2 ) , and lemmas 5.7 and 5.8 go through for various nonexportative logics. The approach through restriction also suggests however another more flexible method, namely just restrict each world or theory (or a suitable closure thereof) to A third set of filtration definitions, those for third filtrations, takes off most directly from canonical modellings. In simplest form a = a fl 'J'; more complex forms consider some suitable closure, under e.g. deducibility (and priming), of a fl for each a. That is, in the simple form one effectively modifies the equivalence relation by distributing B e ^ into the consequent, to obtain:

a

b iff (B)(I(B, a) = 1 & B e

I(B, b) = 1 & B e *) .

Then,

as the definition suggests, one takes a = {C: I(C, a) = I & C £ , so that once again a = £ iff a = b. Then set a* = a*", Rafic iff RJ (a fl (b fl *) (c fl *), Oa iff 0 L (a 0 *), to give a third filtration of canonical m.s. ^

through

But

to arrive at interesting further results in this way appears to require taking deductive closures (at least) of a fl ^ and this leads to complications more easily handled through the alternative operational semantics for relevant logic (developed in 11). For instance, it is sometimes important that where A is in a filtrated element a' and k A -*• B then so is B; however resolution by taking the

^abbay's claim to have proved that system B+ and various consequence relations have the finite model property (73, p.351) appears to encounter just this difficulty. On the other hand, an argument using unadorned second filtrations will presumably succeed for weaker consequence relations and, more interestingly, for classical relevant systems where the ordering requirement associated with p2 can be removed. So such second filtrations - which often work nicely for modal logics - are not entirely beside the point in relevant investigations.

404$

5.9 OTHER DECIDABILITY

METHODS,

LIMITATIONS,

AND UNDECIDABILITY

RESULTS

deductive closure c(a') of a' may destroy primeness, thereby calling for models such as the operational semantics supply, which include unprimed as well as prime theories.1 Decidability theorems for relevant logics, reached by applying filtration methods to (relational) semantical modellings for the logics, can be obtained for several more systems than results so far assembled may indicate. But it is, by and large, easier to establish these results using the operational semantics for relevant logics than the relational semantics so far explained - and promoted, because conceptually superior. Since the operational semantics prove intertranslatable with the relational semantics, in a sense neither yields anything that the other does not, in particular relational decidability is recoverable in principle from operational decidability. That does not however undermine claims as to conceptual inferiority or ease of filtration, for such properties are simply more "opaque" than the identity relation intertranslatability determines. Indeed filtration methods themselves appear quite sensitive to form; ease of applications of the methods certainly is. Because the operational approach is easier, and ties more directly with other methods for showing decidability, a further major round of decidability results is postponed until later (chapter

11).

At the same time alternative decision procedures - through other systematisations .the operational semantics naturally lead to, especially Gentzen formulations of relevant logics - will be introduced. But despite these easier or alternative methods, indeed despite the variety of methods for establishing decidability now known 2 (and much labour expended trying to apply them), the decidability of the original "relevance logics" (especially of the main systems E, R and T of ABE) long resisted all methods. The reason is the obvious one increasingly conjectured as new assaults on decidability failed, and related logics such as Q+ and recently KR proved undecidable - that these "Pittsburgh systems" are in fact undecidable. The current situation as regards decidability status is indicated in the following diagram:

1

The essentials of one filtration method of this type may be found in Fine 74.

2

Both the methods for proving decidability and those for establishing undecidability (usually consisting of translation into known undecidable problems in algebra or geometry) are pretty piecemeal and extraordinarily sensitive to variation. The whole area seems to be going in the (ill-behaved) direction of that of solutions to differential equations. On Gentzen and algebraic methods for establishing decidability of relevant logics (e.g. R and E^) and their relatives (e.g. RM) see ABE, and references cited therein (e.g. Meyer's thesis which gives details of Gentzen methods for OR). For other decidable systems cited in the diagram see not only chapter 11 but also the ANU Logic Group publications (listed in the front of this book) and Giambrone 82.

34 7

5.9 THE PROBLEM

OF THE UNDECIVABILITY

Decidable By filtration methods

B, G, DK, DL, ...

By Gentzen methods

(R-W)+, (T-W)+ OR, E~, R ^

By other methods e.g. algebraic

R£, R~, RM

OF THE ORIGINAL

Unresolved

T^, T~, T£ (E-W)+, E-W

RELEVANCE

LOGICS

Undecidable Q+ (and conservative extensions thereof 1 ) KR E, R, T; E+, R+, T+ (and many extensions thereof)

The situation as regards relevant logics is accordingly much more interesting than that with any other much developed class of sentential logics (modal logics provide an obvious comparison). For not only is there an expected class of decidable logics, indeed a considerable variety of such logics, and also some so far unresolved related systems, but there is a significant class of undecidable logics. Moreover the undecidable logics include not only artificial systems specially contrived to show undecidability as with undecidable modal logics, but as well as rather natural relevant systems like Q+ (a neighbour of R+) and the independently motivated but curious crypto-relevant logic KR, the original relevant logics of Ackermann and of Anderson and Belnap. 2 While the undecidability of E and R undoubtedly increases their interest as heterodox logics, it appears to diminish still further their (tenuous) claims to capture, respectively, the logics of entailment and relevant implication at the sentential level. For what sentential principles (in &, v, ~) are entailmental theses should, it has hitherto (correctly) seemed, be effectively determinable. The quest for the sometimes elusive features of entailment (and its relatives) will be further pursued in part II. There the range of systems answering to the genuine features of entailment will be further narrowed, to a restricted class of D (deducibility) systems, among which a (minimal) choice will tentatively be made. The leading D systems are decidable (as will be explicitly shown in 11.14), and exhibit, along with other prized properties, the important feature of depth relevance (see 15.6) which the original relevance logics lack. In the course of the venture, the logic of relevant implication - of which entailment is the necessitation (cf. 10.1) - will be marked out, again tentatively: clearly though it is about as far removed from system R as "the" logic of entailment is from E and NR. Many other things of fundamental philosophical import will be encountered on the way. For example, the closely related issue of the features and behaviour of conditionals will be seen close up - a duly relevant theory of conditionals will then be discerned, which is appropriately different from the shoddy American wares that dominate present systems engineering - and features of propositional inclusion, often confused with entailment, and of its biconditional, propositional identity, will emerge as the further logical landscape unfolds.

x

The extensions include full systems * \th negation. The further agrument is simply that if L' is a conservative extension of L and L is undecidable then L' is undecidable.

2

0n the undecidability of Q+ see Meyer and Routley 73, on KR see Urquhart 82, and as to E, R and T and their positive parts, see Urquhart 83. Undecidability results are further discussed in Chapter 11.

406$

AJ. 0

WHAT IS MEW IN THE SEMANTICS FOR E OVER THAT FOR OTHER KFLEVANT

LOGICS

APPENDIX 1 THE SEMANTICS OF ENTAILMENT - IV: Es H' AND II" The relation of entailment, which holds when and only when one proposition follows logically from another, is central both in logic and philosophy. Enlarging on Ackermann's claim (in [1]) that his system II' of rigorous implication formally captured the relation, commonly equated with entailment, of inclusion of logical content, Anderson and Belnap have proposed their system E of entailment, an adaption of Ackerman's II', as a formal analysis of the relation of logical consequence at the propositional level (see [2]). Whether or not E is the, or a, correct analysis of entailment at the propositional level, it is, we believe, far superior to such rival analyses as strict implication. Yet it is said to be a scandal that philosophers have not generally been prepared to buy one or other of Lewis's systems of strict implication or Tarski's metalogical version of firstdegree strict implication. But it is no scandal: the glaring inadequacies of strict implication as an account of entailment are well-known, and it is also known that E avoids these inadequacies (see [2]). The scandal is that logicians should still be prepared to accept, as an account of entailment, an account with such unsatisfactory properties as strict implication (or its metalogical variant) merely because it is entrenched, when better motivated analyses with more satisfactory properties are now available; and that logicians have not been more active in trying to elaborate adequate theories of what is, after all, a fundamental relation of their discipline, namely logical consequence.1 The main object of this paper is to further Anderson and Belnap's cause by giving a semantics for the system E, and at the same time for Ackermann's systems H' and II". The semantical analysis is then applied to solve important formal problems for these systems, and to relate these systems to others. To provide semantics for E, II' and II" we extend the situational analyses of relevant logics developed in [15] and [16]. As before we consider a class of situations or set-ups K, which may include as well as the actual situation 0 (on normal modellings) both possible "worlds" and also incomplete and inconsistent situations (see [19]). The chief innovation in the semantical analysis of E we offer, over and above that given for logic R in [15], is the introduction of a new property P of situations. A situation a in K has property P iff each proposition which is necessarily true holds in a, i.e. a is a possible "world". P is required to cope with E in view of the way E introduces necessity as part of the logic of entailment, most obviously through the postulates NA -> A and NA & NB ->- N(A & B) and through the (derived) rule of necessitation. The point of property P, from the logical engineering standpoint, is to get round the problems raised by the+fact that it is impossible to obtain, in the way it was for the positive system E (analysed in [17]), a prime regular theory T which validates the rule of necessitation^ The class of theorems, in terms of which T could be defined in the case of E , fails to be prime in the case of E; for A v B may be a theorem of E when neither A nor B is (e.g. p v p). Admittedly, the presence of P, and its apparent uneliminability, makes the semantics of E more cumbersome and less attractive than that of some of its relevant rivals such as R and T. We have designed the paper so that the bulk of it and all the more important results, some proof details excepted, may be read without diversion beyond the paper to scattered papers on the semantics of entailment: much of the background material is, fortunately,,now assembled in [2]. J

With apologies to Geoffrey Hunter who made this Kantian point, with respect to if-then, in the preamble to [21].

34 8

A7.7

SYSTEM

E AW

E (OF "ENTAILMENT")

ANV THEIR

INTERCONNECTIONS

1. Syntax. The sentential language SL is a triple <S, 0, F> where S is a denumerable set of sentential paramenters, 0 is the set whose members are the unary connective - and the binary connectives &, v, and F is the set of formulae built up as usual from elements in S and the connectives in 0. We use p, q, r, etc. for sentential parameters in S and A, B, C etc. for arbitrary wff of F. The further binary connectives ° and -*• are defined: Dl. A ° B = D F (A -> B) D2.

A

B =

(A

B) & (B

A)

The sentential language SLt adds to SL the sentential constant t; and in the case of SLt F is the set of formulae built from sentential parameters and t by means of connectives. Church's conventions ([5], p.75) for the use of dots in place of brackets are adopted, and the binary connectives are ranked &, v, in order of increasing scope; otherwise ambiguities are resolved by association to the left. The logic E has language SLt and the following axiom schemes and rules: AO.

t

A + A

A2.

A -»• B

A4.

A & B + A

A6.

(A

A8.

A -> A v B

B -*• C

A + C

B) & (A •*• C)

Al.

(t •+ A)

A3.

(A

A5.

A & B + B

A

A -»• B)

A + B & C A9. A v B

B

A v B

A10.

(A •*• C) & (B

C)

All.

A & (B v C)

A & B v C

C

A12.

A -> A -»• A

A14.

A -»• A

Rl.

From A

R2.

From A and B, ito infer A & B (Adjunction)

A13.

A

-*•

B

B

A

B and A, to infer B (Modus ponens)

The rule of Necessitation NR: From A to infer t -»• A, is a derived rule of E. Proof. Where D are a complete list of the elements from S in C it follows by induction on the length of C that: (D -*• D ) & ...& (D, D, ) -»•. C -»• C is a theorem of E (see [3]). Hence since t (D D^ ) &... (D, -*• D^) by repeated applications of AO and A6, t C -*• C is a theorem of E for every wff C. Thus for each axiom scheme B of E, t -*• B. Since it is provable in E that t -»• (A -*• B) -»-. t A t B and (t A) & (t -*• B) -+•• t -»• A & B, the rule follows by induction over the proof of A. The logic E of Anderson and Belnap has language SL, and has the axiom schemes and rules of E except that A1 and AO are replaced respectively by Al 1 .

A -»• A -»• B

A7.

NA & NB +

B N(A & B)

where NA = ,, A -»• A A. dr Part of the point of the t-reformulation of E is to dispose of the intially troublesome axiom A7. We can justify concentration on E through the following known result. Theorem 1. E is a conservative extension of E. Proof (i) Every theorem of E is a theorem of E. ad Al 1 . It follows from Al by prefixing and suffixing that

408$

A7.J

E AND ITS PARTS

ARE CONSERVATIVE EXTENSIONS

OF E AND ITS

PARTS

(t + B) ad A7. Thus it t + A + t A

B +. A A -»• B •*• B, whence the result by Al. (t A) & - A A -*• A -*• A. (ii) If A is a wff of E and a theorem of E then A is a theorem of E. Proof is like [3], lemma 2. Let A ,...,A be the rproof of A = A in E, and let 1 n n = B^.-.B^ be all the sentential parameters in A ...A^. For this proof of A, define B, ) &...&(B -»• B ) and A. * as the result of replacing t3 by t Y r 6 l i m m i throughout A^. Then t* as (B

A,',...,A ' l n is a proof in E of A ' = A, since A n

is t-free.

n

The proof that the new sequence is a proof in E is by induction.

th At the i

stage A^' is obtained exactly as A^ is except in the case of "axioms Al and A14. If A^ is an instance of Al then A^' is proved using Al 1 ;

and if A^ is an

instance of A14 then A^' is derived as in the proof of rule NR above. The parts of E and E have as axioms and rules just those respectively, which include none but the connectives of their parts of E include the primitives t and -*• and all parts of E The parts of independent interest are the implicational parts

axioms of E and E, language. All the connective -»-. E^ (with postulates

AO - A3, Rl) and Ej (Al', A2, A3, Rl), the implication-negation parts E^ (Ej + A12 - A14) and =E—, the implication-conjunction parts E f (ET + A4 - A6) and E r l & I =& (ET + A 4 - A7), and the positive parts ET^ (E. + A8 - All) and E+ (Er + A8 - All) —1 is = =& with connective set {-»•, &, v}. "Theorem — " — —2. (i) E. a is a conservative extension of =Ecxr (ii) ET1" is a conservative extension of E + . Proof is like that of the previous theorem. The style of proof does not extend however to fragments without conjunction: the desired results may however be obtained either by Gentzen methods or by algebraic methods or by semantic means (as below). The logic IT' of Ackermann [1] has language SL and the schemes and rules:A + A (1) (2) A -»• B •*•. B •*•C -+. A C A -*• B -»-. C •*A• (3) C B (A -*•. A B) A -»• B (4) A & B -*• A (5) A & B + B (6) (A B) & (A -> C) -»•. A -»• B & (7) A -»• A v B (8) B -»• A v B (9) (A •*• C) & (B -»• C) -»•. A v B (10) A & (B V C) •*• B v (A & C) (11) A -»• B B -»• A (12) A & B A + B (13) A -*• A (14) £ A (15) From A and A + B to infer B (a) From A and B to infer A & B ($)

409$

A1.1 ACKERMANN'S

(y) (6)

SYSTEMS

n' ANV n" ANV THEIR INTERRELATIONS

WITH n ANV E

From A and A v B to infer B (material detachment) From A -*•. B C and B to infer A C

The logic H" has the^ language SLt and the schemes and rules of Ackermann's system II", where A = t; that is to say, to II' the following axioms and rules are adjoined; (16) (17) (O

A + A + AA & A -»• A From A B and (A -»• B) & C -»• A to infer C

A.

(Strictly Ackermann's system II" takes A as primitive in place of t, but the constants t and A are trivially interdefinable in the presence of negation). We shall replace II" by a deductively equivalent system II with a simpler rule structure. H has the axiom schemes of IT * together with the axiom t, or equivalently the axiom schemes of E, and together with rules (a) (3) and (y) the rule (6')

From A to infer t -»• A.

Thus II is deductively equivalent to E (formulated with the primitive rule of necessitation (6')) plus (y). Lemma 1. A is a theorem of II" iff A is a theorem of II. Proof (i) Every theorem of II is a theorem of 11". For both t and AO follow from (16) by negation principles, and (5') follows from (6) using AO as first premiss. (ii) Every theorem of II" is a theorem of II. Schemes (16) and (17) follow by negation principles, ad (6). t + B B + C-*. t C B C -»-. C -*•. A -*• (B C) A -»• C, by theorems of II. Hence since from B we can infer t -*• B, from B and A (B ->• C) we can infer A -* C. ad (£). From A B, (A B) & C A to infer A B, t -»• A -»• B v C to infer A ->- B, A + B v C to infer C, by (y) to infer t -»• C, by (6') to infer C A. Theorem 3. II is a conservative extension of II'. Hence too II" is a conservative extension of IT'. Proof is like that of the previous theorems. Full details are supplied in the original proof of Anderson and Belnap [3]. 2. Semantics. In the metalogic, in which we formulate and develop the semantics, we combine with English standard notation of classical quantification and relation theory formulated with connective set {&, v, D, =} and quantifiers V and 3, with (x)B (Vx)B. The discerning reader will observe however that we apply nothing like the full logical strength of classical quantification logic and relation logic; indeed we are willing to guarantee the reader, rightly uncertain about the consistency of the classical edifice, and so worried by our apparent uses of material detachment (y), that the metalogical argument can be carried out, with some adjustment, within the framework of a more adequate logic. Though we conform with the classical theory to the extent that we use standard notation of quantification logic, we do not intend the notation to have the standard referential interpretation. Anyone Platonistic or Democritean enough to swallow or buy incomplete and inconsistent situations, along with "possible worlds" as among what exists, as "part of their ontology", can of course take over the standard interpretation. But those of us who think that the semantics of entailment takes us beyond the possible, that in the semantics we are often talking about and calculating with situations that don't and can't exist, will opt for an alternative non-referential interpretation of the quantifiers, e. g. for some neutral or truthvalued or substitution interpretation.

410$

E MODEL STRUCTURES,

AJ.2

VALUATIONS,

INTERPRETATIONS,

TRUTH AND

E-VALIDITY

An E model structure (E m.s.) is a structure M = <0, K, R, P, *> where 0 e K, R is a ternary and P a unary relation on K, and * is a unary operation on K such that the following definitions apply and postulates hold for all, a, b, c, d, e K: dl.

a < b -

d2.

R z abcd 2

ROab (3x)(Rabx & Rxcd)

d3.

R a(bc)d -

pi. p2. p3. p4. p5. p6. p7. p8. p9.

a < a R 2 a(0b)c o Rabc R 2 abcd => R 2 b(ac)d Rabc => R 2 abbc (3x)(Px & Raxa) (3x)(Px & Rxab) =>. a < b a** = a Rabc = Rac*b* Raa*a

(3x)(Rbcx & Raxd)

For an arbitrary E m.s. M it follows for every a, b, c, d, e K: Tl. T2. T3. T4. T5. T6. T7. T8.

b < c & Racd ^ Rabd a < b & b < c ^ a < c R 2 0abc :3 Rabc a < b & Rbcd ^ Racd a < b ^ b* < a* 0* <0 c < d & Rabc ^ Rabd Raaa

(by P 2 ) (from Tl) (by p3, P 2 ) (from T3) (from p8) (from p4) (from Tl, T5, p8) (by p4, T4).

The modelling requirements may be strengthened in certain ways without Affecting the resulting class of valid wff, as the completeness proof will reveal, a4g. it can be required that the ordering < is a partial ordering and that P0. Further 0 may be deleted from E m.s. and defined as £x(x € K) or £xPx. A valuation v of SLt in E m.s. M is a function which assigns a truth-value from set II = {T, F} to each sentential parameter p in S for each set-up a e K subject to the restriction that it respects ordering relation <, i.e. that for every p e S and every a, b e K, (1)

if a < b and v(p, a) = T then v(p, b) = T.

ia short, v is a function from (S - {t})xK to II satisfying (1) The interpretation I associated with valuation v in M is a function in ~ II satisfying the following conditions for every sentential parameter p e S, every A, B e F and every a e K: FXK

i. ii. iii. iv. v. vi.

I(p, a) = v(p, a); I(A & B, a) = T iff I(A, a) = T and I(B, a) = T; I (A v B, a) - T iff I
A wff A is true on a valuation v, or on the associated interpretation I, at an element a e K, just in case I(A, a) = T; otherwise A is false on v, or I, at a. Wff A is verified on v, or on the associated I, just in case I(A, 0) « T; and otherwise A is falsified on v, or on I. A is E-valid in E m.s. M = <0, K, 411$

AJ.3

THE STANDARD

ROUTE TO SOUNDNESS

FOR E ANV E

R, P, *> just in case A is verified on all valuations therein; and otherwise is invalid in M. A is E-valid just in case A is valid in all E m.s.; otherwise A is E-invalid. 3. Semantic entailment, verification and soundness. Where v is a valuation of SL in M, A entails B on v (on I) or wff A v-entails B, just in case for every a e K, if A is true on v (on I) at a then B is true on v (on I) at a. A entails B In M just in case A v-entails B on every valuation in M; and A E-entails B iff A entails B in every E m.s. In order to show that entailment connects ippropriatsly with verification, i.e. that A E-entails B iff A -»• B is E-valid, w prove first that the monotonicity condition (1) extends to all vff. Lemma 2. Where I is an interpretation of SL in 1 for ever/ wff A of SL and every a, b e K, (2)

if a < b and I (A, a) = T then I (A, b) = T

Proof is by induction on the number of connectives in A fiom the given inductive basis (1). The inductive step for uses T3, and that for t T2. Lemma 3. Where I is an interpretation of SL in M for every wff A, B of SL (3)

A entails B on I iff A + B is verified on I;

(4)

A entails B in M iff A

(5)

A E-entails B iff A

B is valid in M; B is E-valid.

Proof is like theorem 1 of [15]. Theorem 4. If A is a theorem of E then A is E-valid Proof simply varies lemmata 4 and 5 of [15]. Only the axiom schemes A1 and AO concerning t are new. ad Al. Suppose I(t A, a) = T for arbitrary a e K. Since by p5 and pi for some element d e K, Rada and Pd and d < d, Rada and I(t, d) = T; whence I(A, a) = T. Since then t -*• A entails A on I, (t A) -»• A is verified on arbitrary I. ad AO. Suppose for arbitrary a, b e K, I(t, a) « T, I(A, b) = T and Rabc. Then for some d e K, Pd and d < a. Then for that d, by T4 Pd and Rdbc; hence by p6, b < c. Thus by lemma 2, I(A, c) = T. Reassembling then, if I(t, a) = T then I (A -*• A, a) = T for every a e K, i.e. t entails A + A on I whence by lemma 3, t ->•. A -*• A is verified on arbitrary I in M. Corollary. If A is a theorem of E then A is E-valid. For every theorem of E is a theorem of E. It is perhaps worth showing how necessity-type postulates of E are vindicated by using conditions on property P. Consider the postulate Al' and suppose, for a reductio argument, that I (A -*• A B, a) = T but I(B, a) 4 T. Since, by p5, for some x Px and Raxa, if I(A -»• A, x) = T I(B, a) = T. Thus I(A A, x) = F; so for some b, c Rxbc and I(A, b) = T and I(A, c) = F. But, by p6, Px & Rxbc => b < c, whence I(A, c) = T - a contradiction. 4. Semantic completeness. A subset T of the set of wff F is an (intensional) E-theory iff T is closed under adjunction, i.e. wherever A e T and B e T then A & B e T, and provable E-entailment, i.e. wherever A e T and A B is a theorem of E B e T. Where T is an intensional E-theory, T is prime iff whenever A v B e T A e T o r B e T . T i s regular provided T contains all theorems of E, and T consistent provided T does not contain the negation of some theorem of E. normal iff T is regular, consistent and prime. Lemma 4. Where A is a non-theorem of E there is a prime regular E-theory T that A i T. Proof, by a Lindenbaum construction, is given in Meyer-Dunn [12].

412$

either is T is such

A1.4 COMPLETENESS

FOR E: CANONICAL

E M.S. ANV CANONICAL

VALUATIONS

Where T is any regular E-theory, an intensional T-theory a is any subset of F which is closed under adjunction and T-entailment, i.e. whenever A B e T and A e a then B e a. An intensional T-theory is an E-theory, since T is regular. Primeness, regularity, etc. are defined, by analogous definitions to those given above, for intensional T-theories. Theorem 5. If A is E-valid then A is a theorem of E. Proof. Suppose, in order to establish the contrapositive, that A is not a theorem of E. By lemma 4, there is a prime, regular E-theory T such that A i T. Define s a canonical E m.s. = T, K^, R ^ P T , *> with base T as follows:i.

K,j, is the class of prime intensional T theories. For every a, b, c, e K^,

ii. iii. iv.

I^abc iff, whenever A -*• B e a and A e b, B e c, i.e. (A, B) (A -*• B e a & A e b =>. B e e ) ; P T a iff (A) (t -*• A e T a. A e a); a* = {A: A 4 a}.

A canonical valuation v^, of SL in

is defined thus:-

For every sentential

parameter p e S, v.

v T (p, a) = T iff p e a, for a e K^.

After the usual pattern of such completeness proofs it is shown (i)-

is an E m.s.

(ii). I(D, a) = T iff D e a, for every a e K^ and every wff D e F, where I is the interpretation associated with v T in as A I T, I(A, 0 T ) 4 T by (II);

Completeness then follows;

for

and hence, applying (I), A is falsified in an

E m.s.; that is A_is not E-valid. Proof of (I). Let K be the class of intensional T-theories (not jus_t prime theories), and R and P corresponding relations (to R^, and P^,) over K. As in [15] and [17] postulates pi - p9 are established for R and P and then restricted to R.J, and P^,, using Zorn's lemma in non-immediate cases. As in [17] it follows that vi.

a < b iff a c b for every a, b e K, where _c is inclusion between theories and < is defined in terms of R^,.

0 T e K t , P T is a unary and R^, a ternary relation.

Next pi - p6 are verified

for R and P for elements of K. pi is immediate from (vi). ad p2. Suppose RObx and Raxc; by (vi) b £ x and by (ii) (A, B) (A •*• B e a & A e x ^ B e c)^ Suppose further A -»• B e a and A e b ^ Since b £ x, A e x, whence Bee. Hence Rabc as required; that is R 2 a(0b)c = Rabc. _ _ ad p3. Suppose Rabx and Rxcd. In order to show that for some y, Racy and Rbyd, let y be the set of all wff C such that for some wff B, B C e a and B e e , i.e. y = {C: (3B) (B -»• C e a & B e c)}. Then i.

y e K, since y is closed under adjunction and provable T-entailment Xfor details see [17], lemma 6). ii. Racy, immediately, iii. Rbyd. Suppose C D e b and C e y. Then, for some wff B, B ->_C e a and B e e . Since B -»_C e a, C + D + , B •*• D e a. Since Rabx, B -*• D e x; and since Rxcd^ D e d, as required, ad p4. Suppose Rabc. To show that for some y Raby and Rybe^, let y = {C: _(3B) (B-^Cea&Beb). Then as in the case of p3, y e K and Raby. To show Rybc, suppose C -»• D e y and C e b. Then for som£ B, B -*• (C -»• D) e a and B e b. Hence B & C -*• D e a and B & C e b, whence since Rabc, D e c, as required. 413$

A1.4 DETAILS

ad p5.

OF THE COMPLETENESS

Let x = (A: t

PROOF: MODELLING

A e 0 T >.

CONDITIONS

Then, by (iii), Px

AND THE INDUCTION

STEP

and, since x is closed

under adjunction and T-entailment, x e K. Suppose B -*• C e a and B e x . Since Bex, t B e T, whence B + C t ->- C e T. Hence t -»• C e a. But t •*• C -> C e T, so C e a. Thus,_by (ii), Raxa. Hence (3x) (P x & Raxa) for a e K. ad p6. Suppose Px and Rxab and B e_a. Since t -»-. B B e T, B B e x; hence by (ii) and Rxab B e b. Thus Px & Rxab implies a £ b, and so by (ii) a < b. The restriction to prime intensional T-theories is accomplished as in [15] and [17]. For postulates such as pi, p2 and p6 which can be put in prenex universal form, the restriction to elements of K^, is immediate. The restriction of postulates p3 and p4 is already dealt with in [17]. _ ad p5. The argument above shows that for a e K^, there is an x, at least in K, such that Px & Raxa. It is shown that there is an x' e K,j, with the same properties. Consider the set Y of all elements y of K such that x £ y and Raya. Then whenever Z is a subset of Y totally ordered by £, U Z £ Y; thus, by Zorn's_ lemma Y has a maximal element, x ' say. Since x £ x', Px' by definition of P, and Rax'a by construction. It remains to show that x' is prime, whence P T x' & R^ax' a follows;

but primeness follows just as in [17], lemma 6.

To complete (I) it is shown (a)

* is an operation on K^.

Details are the same as those given in [151,

lemma 13. (b)

p7-p9 are satisfied for a, b, c, e K T .

ad p7. A e a** iff A I a* iff A e a iff A e a. ad p8. Suppose (B 1 , C') (B* C' e a & B' e_b a. C' e c) and B C e a and _B e c*, i.e. B i c. By contraposition, C -*• B e a; hence by the first premiss. C i b, i.e. C e b* as required. _ _ ad p9. Suppose B C e a and B e a* Then B I a and_since B -»- C B v c is a theorem of E, B v C e a; that is since a is prime B e a or C e a. Since B i a, C e a. Hence R^,aa*a. Proof of (II) is by induction from basis (v). The inductive steps for connectives &, v, are as in [17], Theorem 2. ad~. I(B, a) = T iff I(B, a*) t T iff B e a*, by inductive hypothesis iff B e a ad t. Since I(t, a) = T iff (3x e K^) (P^x & x < a), it has to be shown that t e a

iff (3x e i^)(P T x & x £ a).

t - » - t e O ^ = > t e x , hence t e x; tea. then t

Let x = {A: t -»- A e 0 T ). B e 0T;

Suppose first, P T x & x £ a. and since x £ a, t e a .

Then x e K, Px, and x £ a.

but t e a so B e a by T-closure.

Since P T x,

Conversely suppose For suppose

Bex,

In short there is an x e K

such that Px & ROxa. Then exactly as in the cut-down of p5 to prime intensional T-theories we can find, by Zorn's lemma, an x' e K^, such that P^jc' & R ^ x ' a , whence the result. Corollary. Where A is a wff of E, if A is E-valid then A is a theorem of E. Hence for wff of E, A is E-valid iff A is a theorem of E. Proof follows from the soundness and completeness theorems using the fact that E is a conservative extension of g. In fact the completeness proof for E may be given without the detour through E, by defining P a as (A)(NA e T ^ A e a). It then follows, as a corollary, that

414$

AT.5 NORMALISATION,

THE ADMISSIBILITY OF

(Y), AND SYSTEM

EQUIVALENCE

E is a conservative extension of E. 5. JI and rule (y). An E m.s. M = <0, K, R, P, *> is normal iff it satisfies the postulate pO.

0* = 0

The normalisation of an E m.s. M = <0, K, R, P, *> at 0* is the structure M' = <0', K \ R', P', *'> where 0' I K; K =" K u {O 1 }; R 1 is like R on elements of K, R'0'0'0', and where a e K, b e K', R'O'ba iff R'Oba, R'abO' iff R'abO*, R'baO' iff R'baO*, R'bO'a iff R'bOa; P' is like P on elements of K, and P'O'; *' is like * on elements of K and 0'*' = 0'. Lemma 5. Where M = <0, K, R, P, *> is an E m.s. its normalisation M' = <0', K', R', P', *'> is a normal E m.s. Proof is a matter of detailed verification of postulates pl-p8 for M', since M' is otherwise properly defined, pi, p7 and p9 are almost immediate. p5 follows in the new case from P'O' & R'0'0'0'. The key to the case by case verification of the remaining postulates is the point that R' holds among elements of K' iff R' holds among members of K' upon replacing occurrences of 0' in the first two argument places by 0 and in the final place by 0*, except for the case RO'O'O'. The fact that 0* < 0 is important in dealing with replacement 0' in the final place of R'. Since 0'*' = 0 ' , M' is normal, Theorem 6. A is a theorem of E iff A is valid in all normal E m.s. Proof. If A is a theorem of E it is valid in all E m.s. including normal ones, thus establishing soundness. Conversely, if A is not a theorem of E it is invalid in some E m.s. M = <0, K, R, P, *> under valuation v and associated interpretation (I). Let M' = <0 T , K', R', P', *'> be the normalisation of M, and define a new valuation v' in M' as follows: v'(p,a) = v(p,a) for a e K, v'(p,0') = v(p,0); for every sentential parameter p. Then (I) v' is a valuation in M', i.e. it satisfies restriction (1). Proof is as in [15], Theorem 4. (ID. Where I' is the interpretation associated with v', for every a e K and every wff B, (a)

I' (B

a) = I(B, a) and

(b)

if I(B, 0) = F then I'(B, 0 f ) = F

Proof of (a) and (b) is by joint induction on the length of B; Proof is as in [15], Theorem 4 (except that the steps commuting the first two places of R are unnecessary in view of the new definition of R'). Since I(A, 0) = F, I'(A, 0') = F by (b). Hence A is invalid in the normal E m.s. M'. _ Corollaries 1. Ackermann's rule (y) holds for E, i.e. whenever A and A v B are theorems of E so also is B, and for E.

Proofs 1.

2. 3-5.

2.

A is a theorem of II iff A is a theorem of E.

3.

A is a theorem of II iff A is valid in all normal E m.s., i.e. A is normal E-valid.

4.

A is a theorem of II" iff A is normal E-valid.

5.

Where A is a wff of II', A is a theorem iff A is normal E-valid.

Applying the semantics and using the fact that E is of course a sublogic of II. are derived rules of E. By 2 and previous theorems.

415$

0=0*.

Conversely both necessitation and (y)

A1.6 DETERMINING

A SIMPLE

MODELLING

FOR THE FIRST DEGREE

OF E AND

IT

Thus normal E m.s. and indeed E m.s. are characteristic structures for IT * and 11", i.e. they verify all and only theorems: however those structures validate neither the rule of necessitation nor (6). In fact it is impossible to design the base 1 of E m.s. so that these rules are verified in T. Suppose that the nontheorem A v. _t -*• A, i.e. A ClA^ does not belong to T. Since all theorems of II belong to T, A v A e T. Hence A e T or A e T. But if A e T and necessitation is verified t -»• A e T, whence A v. t -*• A e T, contradicting assumptions. Though E and II have the same set of theorems, and the same t-free theorems as both E and H the systems are not equivalent under extension. Whereas E has the very real advantage of providing for the consistent and proper investigation of certain inconsistent systems, I I , by spreading even isolated contradictions about everywhere, does not. Thus E is a superior system to I I , and likewise E to I I 1 , not only because of its more elegant axiomatisation, but because of the larger class of absolutely consistent, though simply inconsistent, extensions it admits, and because its axioms can all be true and its rules preserve truth, i.e. they can all be validated in the actual situation T, without rendering non-theorems true. 6. The first degree of E and II 1 ^ A wff of g is first degree (f.d.) iff it contains no occurrences of •*• nested within -*•. It follows from the syntactical structure of f.d. wff that the relation R for f.d. semantics can be wholly replaced by the ordering relation since verification never requires moves more than one step from 0. Thus an initial f.d. m.s. M is a structure M = <0, K, *>, where 0 e K, < is a reflexive and transitive relation on K, and * is an operation on K such that, for a, b e K, 0 = 0*, a = a**, and if a < b then b* < a*. Valuations are defined as before, but the interpretation rule for can be simplified, since implicational wff are only evaluated at the base 0, to: I(B + C, 0) - T iff for every a, b e K if a < b and I(B, a) = T then I(C, b) = T. Lemma 6. Where A is an f.d. wff, A is a theorem of E iff A is valid in every initial f.d.m.s. Initial f.d.m.s. can be simplified; the ordering relation < can be eliminAn f.d.m.s. is a structure M = <0, K, *>, where 0 e K, 0 = 0* and a** = a. SXK A valuation is a function in II . Interpretation I associated with v in M is defined as before except for the clause for -»•: The new clause is this:I(A • B, 0) » T iff for every a e K, if I(A, a) = T then I(B, a) = T. ated.

Theorem 7. f.d.m.s.

Where A is an f.d. wff, A is a theorem of E iff A is valid in every

This result leads in turn to algebraic and other analyses of the f.d. of E: the details are supplied in [13] and [4]. 7. | is reasonable in the sense of Hallden. A logic with sentential language SL is reasonable in the sense of Hallden iff whenever A v B is a theorem of L where A and B are wff which share no element of S then either A is a theorem of L or B is a theorem of L. Fortunately E is reasonable in this sense. The product M" of E m.s. M = <0, K, R, P, is the structure <0", K", R", P", *"> where K" pairs for a € K and a' e K', 0" is the iff both Rabc and R'a'b'c', P" . Lemma 7.

The product of E m.s. is an E m.s. 416$

*> and M 1 - <0', K', R 1 , P*, *'> = K x K r is the set of all ordered pair <0, 0'>, R" iff Pa and P'a', and *" is

A7.7 PROOF THAT E AND IT ARE REASONABLE

IN THE SENSE

OF

HALLVTN

Theorem 8. E and II' are reasonable in the sense of Hallden. Proof is the same as [16], theorem 5. Where A and B are wff which share no element of S, suppose neither A nor B is a theorem of E. Then there are valuations v in E m.s. M and v' in M* which falsify A and B respectively. We define a valuation v" in the product M" of M and M' which falsifies A v B, as follows: for e K", v"(p, ) = v(p, a), for every sentential parameter p in A. v"(q, ) = v' (q, a*) for every q in B. v"(r, ) = v(s, a), for every other sentential parameter r, where s is the first parameter in A say. Since v" is a valuation in rf' it remains to show by induction, for each wff A# and B# all of whose sentential parameters occur respectively in A and B, that I"(A#, ) - I(A#, a) and I"(B#, ) - I*(B#, a'). Only the inductive step for -*• is marginally new. Case -*-. A# is C -*• D. Suppose I"(C -»• D, ) - F. Then for some , e K", R" and I"(C, ) « T and I"(D, ) = F. Thus by definition of R" and inductive hypothesis Rabc and I(C, b) • T and I(D, c) • F; so I(C D, a) - F. Conversely suppose I(C + D, a) • F. Then for some b, c e K, Rabc and I(C, b) * T and I(D, c) • F. By T8, R'a'a'a', so R" and by inductive hypothesis I"(C, ) - T and I"(D, ) - F; whence I"(C D, ) - F. Corollary. Logic E has a normal characteristic matrix. Proof applies a theorem of Kripke (stated without proof in [6]): E has the requisite properties. 8. Within E: E + . Where L is a sublogic of E [E] various clauses in the inductive definition of interpretation I may be rendered inapplicable, and thereby components of the model structure rendered otiose; most notably, * is otiose in modellings for systems which do not contain negation. A more striking reduction may however be made in the case of systems without negation, for P as well as * may be eliminated. Whereas an E4" m.s. is a structure <0, K, R, P> obtained from an E m.s. simply by deleting * and its associated conditions p7-p9, a reduced E m.s. is a structure <0, K, R> where 0 e K and relation R on K satisfies postulates pl-p4 and p5'. RaOa. The reduction is possible because in E + m.s. P may be defined as follows: Px • ^ 0 < x, without affecting the stock of valid wffs of E + . Then p6 is immediate by T3, and p5 follows using p5'. applying the definition,

It follows too,

I(t, a) - T iff 0 < a. Since P is not included in a reduced E m.s., the interpretational clause for constant t accordingly gets redefined in just this way. The semantical notions defined for E may be redefined for E + by replacing 'E' throughout by 'E+': in particular a wff A is [reduced] E + - valid iff A is valid in all [reduced] E* m.s. Where logic L is a part of E*" or g*", A is [reduced] L-valid iff A is a wff of L and is [reduced] E4*-valid. In [17] it was shown that A is a theorem of If*" iff A is reduced £*-valid. We connect our new results with this result. Theorem 9. (1) A is a theorem of g4" iff A is E^-valid (2) A v B is a theorem of E+ iff either A is a theorem of or B is a theorem of (3) A is a theorem of E + iff A is reduced E*-valid. Proofs (1) The proof is a special case of the corresponding result for E, upon dropping both negation and *.

417$

A1.8 THE POSITIVE

PARTS OF E ANV E ARE THE POSITIVE

(2) has already been proved in [7] and [8]. by semantical means on applying (1).

It may alternatively be proved

Since reduced E + m.s. are E + m.s. with Px

(3)

PARTS

0 < x, soundness is immediate.

For completeness amend the base 0^, of the canonical m.s. to the class of theorems of E + , It is immediate by the rules of E + that is prime. Because of this choice,

Hence

(a)

t - A e 0 T iff A e 0 T

(b)

P T x iff C>T c x

For suppose P T x and A e C>T; suppose t

A e 0T; (c)

by (a) t

is an E + theory, and by (2) it

A e C>T, so A e x.

For the converse

then as A e 0 T , A e x, that is P ^ .

B^aOa.

Since for some x, P ^ and R T axa, the result follows by (b) and monotonicity of R^,. + + Thus if A is a non-theorem of E there is a reduced E m.s. which falsifies A. Corollaries 1. E is a conservative extension of E + . 2. Where A is a wff of E + , A is a theorem of E + iff A is E + -valid and also iff A is reduced E + -valid. 3. E is a conservative extension of E + . Proofs. 1. Where A is a wff of E + , if A is a theorem of E then A is valid in every E m.s.; but then A is valid in every E + m.s., so A is a theorem of E*. 2.

Uses the fact that E + is a conservative extension of E + .

3. By 1 and 2 E is a conservative extension of E+, and by an earlier corollary where A is t-free A is a theorem of E iff A is a~theorem of E, whence the result. Algebraic proofs of these results are given in [103• 9.

Further within: E^ and operational semantics.

When we retreat to E & , (i.e. E&) bj

+

omitting disjunction from E , a further reduction, valuable from the prooftheoretical point of view, can be made: namely Rabc can be analysed as a + b < c, where < is an ordering relation on K and + is an operation on K. In fact the semantics can be made fully operational by eliminating < in favour of lattice operation +; then a < b iff a + b = b as usual. An operational E^ m.s. (op E^ m.s.) is a structure M = <0, K,

+> where

0 e K, < is a reflexive and transitive relation on K, and + is an operation on K such that for a, b, c, d, e K, i. ii. iii. iv.

0 + a = a, a + 0 < a, if a + b < d then b + (a + c) < d + c, (a + b) + b < a + b

SXK A valuation v in op E r m.s. M is a function in II which respects i.e. a ~ satisfies clause (1). Interpretation I, associated with v, is the extension of v to all wff of E r by clauses i, ii (both as before).

a iii' and iv'

I (A -*- B, a) = T iff, for every b, c e K, if a + b < c I(A, b) = T then I(B, c) - T. I(t, a) = T iff 0 < a. 418$

A7. 9 RECOVERING

iii"

OPERATIONAL

ER SEMANTICS:

THEIR

a

ADVANTAGES ANV

EXTENSIONS

In fact the rule iii' for may be further reduced to I (A -»• B, a) = T iff for every b e K if I (A, b) = T then I(B, a + b) - T.

and the condition iii correspondingly simplified to iii'' (b + a) + c < (a + b) + c Lemma 8. For every a, b e K and every wff A, if a < b and I(A, a) = T then I(A, b) = T. Truth, verification and validity are defined analogously to characterisations provided for E. Theorem 10. (1) A is a theorem of E. & iff A is E r&-valid (2)

A is a theorem of E. iff A is reduced E - v a l i d ot a A is a theorem of E. iff A is operationally E - v a l i d ot a Proofs (1) and (2) are established much as in the case of E + . As to (3), soundness is established along the lines laid down in the case of E. For completeness, suppose A is not a theorem of E c though a wff of E c . Let T be the (3)

a

a

v a canonical op E & m.s. M^ = intensional T theories;

£, +> as follows 0 T = T;

£ is set inclusion on K^,;

a + b = {C: (3A, B ) ( A e a & B e b & (A, B)(A -* B e a & A e b

Bee}

}-E A ->•. B

K^ is the class of

for a, b e K^

C) } .

Hence a + b —c

i.e. a + b £ c iff R T abc.

c

iff

Then much as for

the relational semantics it can be shown (I) (II)

M

is an op. E. a m.s.

I(B, a) = T iff B € a for every wff B and every a e

Corollaries

1. 2. 3.

E and E + are conservative extensions of E a. E is relationally and operationally complete. —a E, E, E + and E + are conservative extensions of E_. —

=

—&

By admitting both prime and non-prime theories (K and K of section 4) in the model structure the operational semantics may be extended to the whole of E and E at the expense of more complicated disjunction and negation rules. The operational semantics, though less elegant and less satisfactory for philosophical interpretational purposes, have considerable advantages for proof-theoretical purposes. For example the operational semantics for E. lead, as in [14], to —&

semantic tableau and Gentzen methods for E.. —&

10.

The pure implicational parts E^ and fij- Since conjunction with its pleasant

properties is not available in E^. and E^., some changes have to be wrought to adapt the style of completeness proof we have been using to these fragments.

We

say then that a set T of wff of E^ is an E^-theory iff T is closed under entailment, i.e. whenever A e T and A

B e T then B e T , and under provable E^-entail-

ment, i.e. whenever A e T and A

B is a theorem of E^ then B e T .

are defined similarly.

E^-theories

Where T is any regular E^-theory, a pure intensional

T-theory a is any set of wff of E^ which is closed under entailment and under T-entailment, i.e. whenever A T-theory is an E^-theory.

B e T and A e a then B e a.

419$

A pure intensional

A 1.10 ISOLATING

Theorem 11. (a)

THE PURE IMPLICATIONAL

FROM FULLER

SYSTEMS

A is a theorem of Ej iff (1)

(b)

FRAGMENTS

A

(2)

A

(3)

A

A is a theorem of E^. iff (1)

A is reduced E -valid

(2)

A

operationally E^-valid.

Proofs. The soundness halves of (a) (1) and (2) and (b) (1) are special cases of the soundness result for E. For the converse the completeness proof for E is modified as follows. Suppose A is not a theorem of Ej (or gj). Then there is a regular Ej-theory 0 T = T, the set of theorems of Ej (or Ej), such that A I T. Define a canonical Ej m.s. i.

= <0 T , K^,,

P^> with base 0 T as follows:-

is the class of pure intensional T-theories.

For every

a, b, c, d, e K.^,, ii.

IL,abc iff whenever A ,A ...A e b and A A -*•.... A -*• B e a then T l 2 n 1 2 n B e c; in short (a, B) (a B e a & a £ b =>. B e C), where a is a finite non-null sequence of wff, and a -»• B abbreviates A^ -»•. A^ -»• .. .An -»• B where a is the sequence A^, A ...An«

Thus where a + b

{B: (3a) (a •*• B e a & a £ b)}, I^abc iff a + b £ c (as in [9], except that '+' replaces '-').

The sequence A , ...,Anis used in place of

the single wff A to compensate for the fact that adjunction is not available; for A -*•. A -*• .. .A -»• B takes ther place of A &...A -»• B. ' l 2 n i n iii. iv.

P T is defined as before for (a) (1).

For (a) (2) and (b) (1), P T is

omitted. V T in M.J, is defined as before, i.e. so that V T (p, a) = T iff p e a .

It is shown (I) (II)

g^ is an Ej m.s. I(D, a) • T iff D e a, for every a e K,^ and every wff D of Ej.

Proof of (I) varies, and simplifies, since primeness does not have to be established, proof of the corresponding result for E. Details in the case of pl-p4, which are readily adapted from those given, are supplied in full in [9]. ad p5. P T ° T i s immediate; so too is I^aO.j.a, i.e. (a, B) (a •*• B e a & a £ C>T =>. B e a);

for as each element of a is an Ej theorem, (a -*• B) -»• B e 0^.

ad p6. As for E. Proof of (II) is like that for the corresponding result for E: details are supplied in [9]. In the operational cases, + is defined in the completeness proofs just as above. Corollaries 1. E, E , E. are conservative extensions of E T . & I 2. E, E , E,, E, E*, E, and E T are conservative extensions of E T . Proofs

We take illustrative cases. 1. Suppose A is a wff of E^ and a theorem of E.

Then since A

is valid in every E m.s. and a wff of Ej, A is valid in every 420$

A1.11/12

BOTH E ANV E ARE WELL-AX10MAT7SEV;

ANV FUSION AWEV

CONSERVATIVELY

E + m.s., i.e. A is E^-valid, whence A is a theorem of Ej. 2.

Suppose A is a wff of E^. and a theorem of E^.; in every reduced E^ m.s.

then A is valid

Hence A is reduced E^-valid, and so a

theorem of 11. Negation and well-axiomatisation. The strategies so far used to prove completeness all break down in the case of the implication-negation part Ey. That E is however a conservative extension of Ey may be proved by alternative methods, for example by algebraic means building on the embedding results of [10] and [22] using the algebraic formulation of E in [ ]. In the interest of completeness we record this result in the following theorem which answers all Anderson's questions, in [0], as to the proper parts of E (but for the details of the proof that E is indeed a conservative extension of Ey, the reader will have to await a forthcoming paper (see now [ANZ 3]). Theorem 12. Both E and E are well-axiomatised in the following sense : Where A is a theorem of E [E] and L is a part of E [E] with connectives ->-; -»-, -»-, &; &, v and postulates from among the postulates of E [E] containing just these connectives, then A is a theorem of L iff A is a wff of L. Proof combines previous theorems. We next extend this result to take in systems formulated using in addition the intensional connective 12. The addition of connective o. Where L is a sublogic of E TE] without connective L° is the corresponding logic with o added to the class 0 of connectives and subject to the rule R3

From A o B

C to infer A

B

C, and conversely.

(Residuation)

L° model structures are the same as L m.s. except that the following postulate, inessential to the class of theorems of L, is added: yiz. a < d & Rbca o. Rbcd. Likewise valuations and interpretations are defined as before except for the following new interpretational rule for o: I(A o B, a) = T iff, for some b, c e K, Rbca and I(A, b) = T and I(B, c) = T. Call the new interpretations o-interpretations. A is o-valid in L m.s. M iff A is verified on all o-interpretations therein; and A is L"-valid iff A is o-valid in all L m.s. These steps break down where L does not have connective &, but the argument may be varied exactly as in the proof of completeness given for Ej. That is to say, R^j, is redefined thus: R^,abc iff (a, B) (a -»• B e a & a c b 3. B e e ) . Then much as before if I(A o B, a) = T, A o B e a. For the converse suppose A o B e a, and let B » {C: A C e E^.} and c = {C: B

C e Ej}.

Then b and c are pure intensional T-theories with

Bee.

It remains to show R_bca. Suppose D D ....D -*• E e b and D ,...D e c. T l 2 n i n Then A •*•. D^ -»•.... D q -»• E and [-B + D^, |~B-»-D,...,[-B-»- D n > By prefixing and suffixing and contraction f- A -*•. B -»• E, whence, as before E e a; Rfbca. Corollary. Where L is as above, L° is a conservative extension of L.

421$

and so

A1.13 BEYOND E: MODELLING

OLD AND NEW EXTENSIONS

OF E AND E

The fact that both ° and t can be added conservatively to sublogics of E, including E, is important in rendering the algebraic treatment of these logics tractable (see [18]). 13. Beyond E: extensions of the semantics. We consider extensions of E and E by the following commonly adopted axiom schemes: Bl. B2. B3. B4. B5. B5*.

A A A A B A A B B -*•. A A & (A

B B A ->- A -> A v A, or, equivalently, v B) B

The following semantical postulates prove adequate: ql. q2. q3. q4. q5.

Rabc Rabc Rabc Rabc a* <

=> Rbac a ^ c v b < c => b < c => a < c a

Equivalent postulates in the case of q4 are the postulates 0 < a or 0 = a, and, in the case of q5, a < a* or a = a*. These systems result R: E + Bl; i.e. relevant logic R RM: E + Bl + B2; i.e. R-Mingle S4: E + B3; i.e. Lewis modal logic S4 S: E + B4 or E + Bl + B3; i.e. classical logic EDS: E + B5. EDS, a (proper) subsystem of S3, and so a proper subsystem of S4, is a new system of importance in the debate on entailment since it includes Disjunctive Syllogism B5' without conceding the higher degree of strict implication. An L m.s. is a model structure obtained from an E m.s. by adding semantical postulates corresponding to axioms of L over and above those of E. A wff A is L valid iff A is valid in every L m.s. Theorem 14. For each extension L of E or E by any selection of postulates from B1-B5, A is a theorem of L iff A is L-valid. Proof. For soundness there are these cases: ad Bl. Suppose I (A, a) = T & Rabc & I (A B, b) = T. Then since Rbac, I(B, c) = T as required. ad B2. Suppose I(A, a) = T & Rabc & I(A, b) = T. Then, since a < c or b < c, I(A, c) = T as required. ad B3. Suppose I(B, a) = T = I(A, b) and Rabc. Then, as b < c, I(A, c) = T, as required. ad B4. Suppose I(A, a) = T & Rabc & I(B, b) = T. Since a = 0 = c, I(A, c) = T, as required. ad B5. Immediate. For completeness there are these cases: ad ql. Suppose Rabc and A -*• B e b and A e a. Then A -*• B -»• B e a, so B e c. ad q2. Suppose Rabc, a ^ c, A e b for arbitrary A. Since a ^ c for some wff B, B e a and B I c. Since B e a, A v B e a; so by B2 A v B -»• A v B e a, whence A -*• A v B e a. Since Rabc and A e b, A v B e c. Hence by primeness A e c or Bee. But B i c so A e c. In short, Rabc & a £ c =>. b £ c, whence the postulate. In order to deal with postulates q3-q5 we rework the completeness proof to exclude from K^, degenerate situations, i.e. the null theory which includes no 422$

AT .13

TENSE-LOGICAL

EXTENSIONS

OF E, AND S4 SEMANTICS

AS A

BV-PROVUCT

wff and the universal theory which includes every wff: details of the reworking, which is straightforward, are supplied in [20]. ad q3. Suppose Rabc and A e b. Since a is non-degenerate some wff, B say, is in a. Then A -»- A e a, s o A e c . Hence Rabc =>. b _c c. ad q4, q5. Similar to q3. Modellings can also be provided for less commonly encountered axioms, e.g. the postulate 0 < a v 0 < a* suffices for the scheme A ->• A -*•. A -»• A considered in [0]. Moreover modellings can be furnished for a wealth of extensions of E corresponding to tense-logical extensions of S4. This is hardly surprising since the usual semantics for S4, based on a standard S4 m.s. <0, K, R> with a two-place relation R which is reflexive and transitive on K, can be recovered from the semantics we have provided for S4 upon defining Rab as Rabb. The evaluation rule for strict implication: I(B -JC, a) = T iff, for every b such that Rab, if I(B, b) = T then I(C, b) = T, is readily derived. For suppose I(B ->• C, a) = T in our semantics and suppose Rab and I(B, b) = T. Since Rabb, I(C, b) = T. Conversely suppose Rabc and I(B, b) = T and suppose I(B C, a) = T in the usual strict sense. By q4, b < c, so by monotonicity Rabb. Hence by the strict rule I(B, b) = T whence, since b < c, I(B, b) = T as required. The standard impoverished negation rule: I(B, a) = T iff I(B, a) 4 T, follows at once since B3 implies B5, whence a = a* for a e K. Define simpleminded (sm) validity in terms of evaluation by the standard rules. Theorem 15. A is a theorem of S4 iff A is sm valid in every standard S4 m.s. Proof. Soundness follows by direct verification of the S4 postulates. As to completeness suppose A is not a theorem of S4. Then there is an m.s. M = <0, K, R, P, *> and a valuation v on M which falsifies A. In fact a e K iff Pa, so K is a set of possible worlds. Define a new m.s. Mi = <0, K, R>. Since the standard rules follow as above,v on Mi sm falsifies A. To show Mi is an S4 m.s. it remains to show R is reflexive and transitive. Reflexivity is immediate. For transitivity suppose Rab and Rbc; since Rabb and Rbcc, by p2 for some z, Racz and Rbzc. Then by q3 z < c whence by monotonicity (T7) Racc, i.e. Rac. REFERENCES [0]

A.R. Anderson, 'Some open problems concerning the system E of entailment', Acta Philosophica Fennicafasc. 16 (1963), pp. 7-18.

[1]

W. Ackermann, 'Begriindung einer strengen Implikation', Journal of Symbolic Logic, vol. 21 (1956), pp.113-128.

[2]

A.R. Anderson and N.D. Belnap, Jnr., Entailment3 Princeton University Press, Princeton (1975).

[3]

A.R. Anderson and N.D. Belnap, 'Modalities in Ackermann's 'rigorous implication', Journal of Symbolic Logic vol. 24 (1959), pp.107-111.

[4]

N.D. Belnap, Jnr. 'Intensional models for first degree formulas' Journal of Symbolic Logic, vol. 32 (1967), pp.1-22.

[5]

A. Church, Introduction t'o Mathematical Logic3 volume 1, Princeton University Press, Princeton (1956).

[6]

S.A. Kripke, 'Semantical analysis of modal logic II. Non-normal modal propositional calculi' in The Theory of Models (edited Addison-HenkinTarski) North-Holland, Amsterdam (1965), pp.206-220.

[7]

R.K. Meyer, 'Metacompleteness', Notre Dame Journal of Formal Logic vol. 17 (1976), pp.501-516.

423$

AL.13 ORIGINAL

REFERENCES

FOR APPENDIX

1, WITH SOME

UPDATING

[8]

R.K. Meyer, 'On relevantly derivable disjunctions', Notre Dame Journal of Formal Logio3 13 (1972) 476-480; also in [2].

[9]

R.K. Meyer, 'The pure calculus of entailment is the pure calculus of entailment', forthcoming. (Also Appendix 2, this volume)

[10]

R.K. Meyer, 'On conserving positive logics', Notre Dame Journal of Formal Logic 14 (1973) 224-236; also in [2].

[11]

R.K. Meyer, 'E and S4', Notre Dame Journal of Formal Logic, vol. 11 (1970), pp.181-199.

[12]

R.K. Meyer and J.M. Dunn, 'E, R and (1969), pp.460-474; also in [2].

[13]

R. Routley, 'A semantical analysis of implicational system I and of the first degree of entailment' Mathematische Annalen, vol. 196 (1972), pp.58-84.

[14]

R. Routley, 'Analyses of implication and implication-conjunction parts of relevant logics', unpublished.

[15]

R. Routley and R.K. Meyer, 'The semantics of entailment I' in Truth, Syntax, Modality (edited H. Leblanc), North Holland, Amsterdam (1972).

[16]

R. Routley and R.K. Meyer, 'The semantics of entailment II', Journal of Philosophical Logic, vol. 1 (1972) pp.53-73.

[17]

R. Routley and R.K. Meyer, 'The semantics of entailment III', Journal of Philosophical Logic, vol. 1 (1972), pp. 192-208.

[18]

R.K. Meyer and R. Routley, 'Algebraic analysis of entailment I', Logique et Analyse, vol. 15 (1972), pp.407-428.

[19]

R. Routley and V. Routley, 'The semantics of first degree entailment' Nous, vol. 6 (1972), pp.335-359.

Journal of Symbolic Logic, vol. 34

[20]

R. Routley and R.K. Meyer, 'The semantics of entailment V',unpublished.

[21]

G. Hunter, Metalogic,

[22]

R.K. Meyer, 'Conservative extension in relevant logic' Studia Logica 31 (1972) 39-46.

Macmillan, London (1971).

UPDATING THE REFERENCES [14] and [20], which were never published, form the initial basis of this book. [9], also not published, now appears in Appendix 2 below. Volume 1 of T2], originally listed as appearing in 1972, finally appeared in 1975; volume 2 of [2] is apparently unlikely to appear for some years yet. [23]

R.K. Meyer and R. Routley, 'E is a conservative extension of E-, Philosophia 4 (1974) 223-49.

424$

A2. THE PROBLEM, ANV PRELIMINARY

VET AILS SETTING

E+ IN PLACE

APPENDIX 2 THE PURE CALCULUS OF ENTAILMENT IS THE PURE CALCULUS OF ENTAILMENT The system E T , the pure calculus of entailment, was Introduced and motivated at some length in [1] as a system capturing our intuitions about logical deducibility. Yet Ej is but a fragment of the Anderson-Belnap system E of entailment, and it has long been an open question (cf. [2]) whether or not E is a conservative extension of E^.; i.e., whether or not all formulas of E whose only connective is are theorems of E iff they are already theorems of E... If not, many good motivating remarks would have gone to waste; but in fact application of the methods developed in [3] and [4] suffices to settle the question affirmatively. Let E+ be the system determined by the negation-free axioms listed for E in [2] (with &, v primitive). We have immediately Lemma 1. E is a conservative extension of E+. Proved in [4]. An e+ m.s. is a structure K = < 0 , K, R > , where K is a set, 0 e K, and R is a ternary relation on K, subject to the following definitions and postulates, for all a, b, c, d in K. dl.

a < b = D f ROab

d2.

R 2 abcd = D f Ex(Rabx & Rxcd)

pi. p2. p3. p4. D5.

a < a R 2 0abc > Rabc R 2 abcd > Ex(Racx & Rbxd) Rabc > R 2 abbc RaOa

Let K be an e+. m.s. and let P be the set of sentential variables and S the set of formulas of Ej. A function v defined on P x K with values {1, 0} is a valuation of Ej in K provided that, for all p e P and a, b e K, (1)

v(p, a) = 1 and a < b > v ( p ,

b)=l.

The interpretation I associated with a valuation v of Ej in K is the unique extension of v to S x K determined recursively by (2)

I(A + B, a) « 1 iff for all x, y in K, if Raxy and I(A, x) = 1 then I(B, y) = 1.

A formula A is verified on interpretation I iff I(A, 0) = 1; A is verified on all interpretations in all e+. m.s.

A is E+-valid iff

Lemma 2. For all formulas A of E T , A is a theorem of E+ iff A is E4-valid. Proof in [3].1 We formulate E

axiomatically following Anderson, Belnap, and Wallace, with

1

This is not strictly sepaking accurate, since [3] has extra postulates and an extra primitive, the constant t. The extra postulates are redundant, since E+ was built up in [3] from weaker systems; that t is redundant for E+ is proved in [7].

4 25

A 2. SYSTEM E, AND A CANONICAL HOVEL STRUCTURE

FOR IT

the following schemes: Al. A2. A3.

(A + B) ->• ((B -*• C) -> (A - C)) (A -*• (A B)) -*• (A -*• B) ((A A) •*• B) -> B

Rl.

From A and A

B, infer B.

We use 'I' to refer to the set of theorems of E^. Tl. T2.

(B -»• C) -» ((A + B) -» (A + C)) (A (B -»• C)) - ((A B) (A

Included in I are

C))

An I-theory is any set T of formulas of Ej closed under entailment and modus ponens — B e T.

i.e. , if A••*• B e I and A e T then B e T, and if both A

B, A e T then

We henceforth use 'a', etc., as a metavariable ranging over finite nonempty sequences of formulas; reserving c for ordinary set-theoretic inclusion, where T is an I-theory a c x shall mean that the set comprising just the members of the sequence a is a subset of T. Where a is the sequence A^, ... A^, the A^ not necessarily being distinct, a -*• B shall abbreviate the formula ' A. -»•(...-•• (A -*• B) ...). 1 m

And we observe

(3) If T is an I-theory and a c T, then if either a -»• B is a provable entailment or belongs to T, then B e T.

ion

Henceforth let H be the set of all I-theories. on H by setting, for all a, b in H,

(4)

a -»• b = {B: Ea(a ^ - B e a & a c b ) } .

We define a binary operat-

We must show -*• well-defined on I-theories a, b by showing a •*• b closed under entailment and modus ponens. Suppose first that A -*• B e I, A e a b. By (4), for some a, a A e a and a c b. By (possibly repeated) application of Tl and Rl, (a -»• A) -»- (a B) e I, whence a -*• B e a and, by (4), B e a b. Suppose next that A + B, A e a - > b . TlTen"ttifere are a, a' c b such that both of a -*• A, a' -*• (A B) e a. Using C, C' respectively to abbreviate the latter two formulas, we us Al, Tl, and Rl to show that C -»• (C1 -»• (a* •*• (a •*• B))) e I; whence by (3), a' -*• (a B) e a; whence by (4), B e a -*• b, completing the proof that a -*• b is an I-theory. We now define a ternary relation R on the set of I-theories by setting, for all a, b, c in H, (5)

Rabc iff a + b c c.

The canonical i.m.s. H shall be the structure < 1 , H, R > , where I and H are as above and R is defined by (5). Lemma 3. The canonical i.m.s. H = < 1 , H, R > is and e+. m.s. Proof. It suffices to show that pl-p5 hold. We note first, for a e H, (6)

I -»• a c a.

Since for any member A e a, A -*• A e I, by (4) a c I -»• a; conversely, if A e I -»• a, for some a c a we have a A e I; by (3), A e a. So I a c a,

426$

Al.

PROOF THAT THE CANONICAL MOVEL STRUCTURE IS AN E+ STRUCTURE

completing the verification of (6). and with < defined by dl, (7)

It follows immediately, for ail a, b in H

a < b iff a s b.

For by dl, (4), (5), (6), a < b if Rlab iff I -> a c b, iff a c b. Our first conclusion from (7) is that pi holds - i.e., < is reflexive, on H. Our second is that p2 holds; for if R 2 0abc, by dl and d2 for some x in H, a < x and Rxbc; but obviously by (4) (8)

a c x implies a -*• b c x -»• b,

from which p2 follows by (5), (7), and obvious properties of set inclusion. p3 is a little more verification to show b choose a •* c as the x in some a <= a •* c, a -*• B e

complicated, though it obviously suffices for its (a -*• c) c (a •*• b) •*• c, by (5) and the fact that we may p3 such that Racx. Suppose then B e b -*• (a -»• c); for b. Let a be the sequence A^, ..., A n ; by definition

of c, each A^ e a -*• c, whence for each A^ there is a y^ <= c such that y^-*- A^ e a.

Let 8 be the sequence y^, ..., y .

Repeated application of Al, Tl, and

Rl, using (3), yields (a -»• B) -*• (8 B) e a, whence since a + B e b we have from (4) that 8 B e a b, whence since 8 is composed of members of c, B e (a -»• b) -»• c, which was to be shown to verify p3. Reasoning similarly, to verify p4 we must show (a -*• b) -»• b a subset of a -*• b. Suppose B e (a -*• b) -*• b. By (4), for some a c b, a -»- B e a b; by (4) again, for some a' c b, a' (a.-* B) e a, which is all it takes to show B c a -*• b, ending the verification of p4. To verify p5, it suffices by (5) to show (9)

a •*• I c a.

Suppose then B e a -*• I; there is then some 8 c I such that 8 B e a. Using A3 and transitivity, show (8-^B) -»• B e I, whence by closure of a under provable entailment, B e a. This ends verification of (9), and H is accordingly an e+. m.s., as promised in the lemma. The canonical valuation c is the valuation of Ej. in < 1 , H, R > such that for each p e P and a e H, c(p, a) = 1 iff p e a. (7) assures that c meets the condition (1) on a valuation. Letting K be the canonical interpretation associated with c, we have moreover Lemma 4.

Every theorem of E^, and only theorems of E^, are verified on the

canonical interpretation K just defined. Proof. We show, for all A e S', a e H, (10)

K(A, a) - 1 iff A e a,

whence the theoremhood of A (membership in I) amounts as promised exactly to verification of A on K (truth at I). Proof of (10) is bv induction, the base case being taken care of by definition of K. Suppose then that A is of the form B G, where on inductive hypothesis

3 64

A2. THE COMPLETENESS

ARGUMENT,

CONSERVATIVE

EXTENSION

RESULT,

ANV

COROLLARIES

(10) holds for all a in H, for B and C in particular. Then I(A, a) = 1 iff for all x, y in H, a x 5 y and B e x imply C e y, applying the inductive hypothesis and definitions. We must show that the long condition amounts simply to B C e a. On the one hand, if B -»• C e a and B e x then by (4) C e a -»• x, so that if B C belongs to a the condition is satisfied. Suppose on the other hand that B -»• C I a. Let [B] = (D: B D e l}. [B] is an I-theory; its closure under provable entailment is obvious; while if both B••*• C, B -»• (C -*• D) e I, by T2 and Rl B -»• D e I, showing [B] closed under modus ponens also and completing proof that [B] e H. Establishing that the long condition is not satisfied amounts now to showing that C i a [B]. Suppose the contrary, for reductio. Then there are B^, ..., such that B 1 -»• (B2 ... -»• ( B r -»• C) ...) e a, where each of the B^ is provably entailed by B. It easily follows from the replacement properties guaranteed by Al, Tl, Rl that 8 C e a, where 8 is the sequence consisting of n replicas of B. Repeated application of the contraction axiom A2 then yields B -*• C e a, contradicting the initial assumption, and ending the inductive proof of (10) and with it the proof of lemma 4. Theorem. The pure calculus of entailment is the pure calculus of entailment i.e., E is a conservative extension of Ej. Proof. Suppose A is a non-theorem of E... By lemma 4, A is invalid in the canonical i. m.s.; by lemma 3, A is not E+-valid; by lemma 2, A is not a theorem of E+; by lemma 1, A is not a theorem of E. So all formulas in the vocabulary of Ej are theorems of E only if they are already theorems of E^, whence, the converse being trivial, the theorem stands proved. Numerous other conservative extension results can be proved on application and adaption of the methods of this paper, importing results from [3], [4], and [5]. Accordingly one can show other relevant logics well-axiomatized in the sense that Routley and I showed R well-axiomatized in [6]; i.e., that on a natural choice of axioms the system determined by -»• axioms may be conservatively extended to that determined by & axioms; by •*•, & axioms, to -»•, &, v axioms, and thence to the full system; and similarly for other reasonable combinations.2 We choose as explicit corollaries, however, only the following. Corollary 1.

The system T^ determined by A1-A2, Rl, Tl, and A

A is the pure

calculus of ticket entailment - i.e., the Anderson-Belnap system T of [7] is a conservative extension of T^. Corollary 2.

The system R^ (Church's weak theory of implication) is the pure

calculus of relevant implication (the Anderson-Belnap R). Corollary 3. t -*• (A

The system E^ got by adding the sentential constant t and axioms

A) and (t -*• A) -*• A to E^ is a conservative extension of E^.

Proofs in each case mutatis mutandis. We note in particular that the special E-axiom A3 enters here only in the verification of p5 in lemma 3, irrelevant 2

That the particular illustrative sequence is conservative at each step holds for all E, R, T. In fact, adding V is no problem for the logics of [3], as is easily demonstrated using its methods; that for positive rigorous logics, including E+ and T+, adding -»• is no problem is a principle result of [4].

428$

A2. REFERENCES AW

ACKNOWLEDGEMENTS,

in [3] to the semantics of T+. prisingly, 3 is new.

FOR THIS APPENDIX

ONLY

2 is not new, having been proved in [8];

sur-

REFERENCES [1]

A.R. Anderson and N.D. Belnap, Jr., 'The pure calculus of entailment', Journal of Sympolic Logic 27, (1962), 19-52.

[2]

A.R. Anderson, 'Some open problems concerning the system E of entailment', Acta Philosophica Fennica 16 (1963), 7-18.

[3]

R. Routley and R.K. Meyer, 'The semantics of entailment III', Journal of Philosophical Logic 1 (1972), 192-208.

[4]

R.K. Meyer, 'On conserving positive logics', Notre Dame Journal of Formal Logic 14 (1973), 224-236.

[5]

A. Urquhart, 'Semantics for relevant logics', Journal of Symbolic Logic 37 (1972), 159-169.

[6]

R. Routley and R.K. Meyer, 'The semantics of entailment (I)', H. Leblanc, ed., Truth, Syntax, Modality3 North-Holland, Amsterdam, 1972, 199-243.

[7]

A.R. Anderson and N.J. Belnap, Jr., Entailment, Princeton, 1975. (This book, which will be the definitive work in the area, will contain substantial parts of other references cited.)

[8]

R.K. Meyer, 'Conservative extension in relevant implication', Studia Logica 31 (1973), 39-46.

ACKNOWLEDGEMENT: Without the contributions of Dunn, Routley, and Urquhart, this paper, which fulfills a boyhood dream, could not have been written. To conform with arguments in [3] and [6], (4) should have taken the form (4') a -»• b = (B: EA(A + B e a & A € b)}; old proofs that p4 held in analogues of lemma 3 broke down, however; Dunn suggested (4), so persistently over my objections that I worked out the details; he's right! But like results for implicational fragments of systems from [3] without A2 should use (4'). Dunn also suggested that it be required that I-theories be closed under modus ponens; any relevant logic in whose semantics Raaa holds generally (a consequence here of p4 and the basic pl-p2) is subject to this requirement. To Routley I owe not only the basic idea of a relational semantics for relevant logics, but also the specific suggestion that conservative extension results would fall out naturally. Similarly, the operational semantics suggested by (4) and (4') and implicit in the argument of lemma 3 I learned from Urquhart, who recorded his version in C5]; Routley had similar ideas. Others - e.g., Scott and Gabbay - were thinking along the same lines as Routley and Urquhart; ignorance of their work is my only excuse for not given them more credit. Similarly, without the problems of Anderson, Ackermann, and Belnap, particularly the former, I'd have been long since unemployed; alas, as Dunn has warned, the time has come to re-tool!^" conclusion, thanks to NSF GS2648, once more. The warning proved somewhat premature. In particular, the semantical theory, though it unlocked many doors, turned out not to be the hoped-for universal master key, and several doors have remained firmly locked, as Anderson and Meyer explain in 'Open problems II' in DRL. Meyer has (accordingly) not re-tooled. 429$

PA. BORING DETAILS

Of DELAYED

PRODUCTION

AND OTHER

PRODUCTIONS

POSTSCRIPT TO THE APPENDICES Largely for historical reasons the two papers making up the Appendices appear in their originally circulated forms (only some references having been updated). As a result, the notation of the papers differs - in easily recognisable ways - from notation that superseded it which is used in the text, e.g. 0 now generally replaces property P, base world T has superseded the former base 0 (which was constantly confused with 0 and also with the value false), and holding values =T and =F have given way to =1 and =0 respectively. As to other notational curiousities, E is intended to represent the boldface 'E' which Anderson and Belnap now use to refer to their system, while what is referred to as ' E' in Appendix 1 is system Et of the main text. Anderson and Belnap's barred symbolism for (relevant) negation is also adopted. Both papers were submitted to the Journal of Symbolic Logic fairly early in 1972. The first paper, by Routley and Meyer, was accepted, subject to revision and shortening. The second paper, by Meyer, was rejected on the (extraordinary) ground that a paper on the topic had already been accepted by the Journal. Meyer did not submit his paper anywhere else. Routley also, but with less excuse, did nothing: he put off revision partly because a key paper upon which proposed revision appeared to depend essentially, Gabbay 73 (then cited as Gabbay 72 1 ), did not become available in legible form in Australia until early 1974, at which stage (he now belatedly confesses 2 ) he did not get around to it, being absorbed by then in other enterprises. But in addition he felt an ideological objection to recasting the paper - in the way the referee proposed within the framework of what has since been called Scott's notion of consequence (e.g. in Gabbay 73, which now accomplishes some of the refitting work proposed), when in fact the research was independent of that framework, and Routley and Meyer had already arrived at a similar notion of consequence3, which, as it happens, has a longer history (for it goes back at least to Smiley 59). In what follows in this Postscript we shall however accede to general requests as regards revision to have SE IV and surrounding works fitted into their historical setting, and do a little more as well. Worlds semantics for system E were found not only by Routley and Meyer, but also, though not independently of the similar earlier semantics for R (of SE I), by Maksimova 73, and subsequently by Fine (see 74) who knew of the first degree theory (of FD) which already included a key idea of the operational semantics, the world-shift in making evaluations.* S o m e of what we were apparently asked to do in revision was subsequently accomplished in this paper, though for different less interesting systems. The material is largely repeated, in similar cryptic form though in a much grander setting, in Gabbay 76, p.203ff. In fact core results are already given by the strong completeness and L-implication theorems (of 4 and 5), which consequence theory re-presents in slightly different garb (see further 12.9). 2

He also takes this opportunity to apologise to Meyer for the 10 year delay in the public appearance, in much less prestigious surroundings, of the paper.

3

See the definition of entailment in SE I, p.207;

cf. also FD.

''As with other intellectual break-throughs, grand or small (e.g. modal logic semantics, the infinitesimal calculus, relativity theory, etc.), so with semantics for relevant logics, the time was ripe, e.g. requisite techniques were sufficiently available, and several people somewhat independently reached similar results at around the same time. For instance, Urquhart also hit upon operational semantics for parts of R and E (see 72) and his results (combined with those of FD) would soon enough have led to full semantics for E.

430$

PA.

THEOREM CONSEQUENCE

VS ENTAILMENTS DIVERGENCE

ANV CONSEQUENCE

THEORY

Rather than revising SE IV to set the 'systems against the background of the consequence relation and semantics of Scott' - a rather unsatisfactory background, as will emerge - we can enlarge our own theory of consequence relations and simply add additional points to the work already accomplished. The main further point (requested) comes to no more than this: whereas systems E and II' have the same theorems, they have different consequence relations. In fact the claim made in SE IV that E and IT are not equivalent under extensions is tantamount to the claim that the systems have different consequence relations. What turns out is that whereas A & ~A jf JJ B, A & ~A /{- ^ B, where |(- is essentially the consequence relation for system L defined in 2.1. (It amounts to entailment as defined in SE I and SE IV, and the properly system-relativised L-implication of 4.5.) System 11, which calls for normal models in order to guarantee rule y, is useless for strongly paraconsistent extensions, trivialising simply inconsistent theories, while E does not. It is then important, in obtaining proper perspective, to contrast the theorem convergence of systems - for wff of E all the full systems, E, E, E + y , IT, IT', IT", have the same class of theorems - with the entailment (or consequence) divergence of the E systems from the E + y and II systems.1 We now begin to elaborate, in a historico-critical way, one setting into which results of this sort fit. But the main elaboration of the theory of consequence largely a Polish-initiated enterprise - will be undertaken in Chapter 12, §9. The Scott-Gabbay account of consequence first appears in Scott 71, Scott's contribution 2 to the 1971 symposium with Meyer and Grice on Entailment. However this contribution bears only obliquely on entailment. Scott's theme (p.788) is that 'some program of formalising a "logical implication" can be presented in a non-confused manner arriving eventually at certain of Lewis's systems. The path is different from Lewis's, but ... the method ... engenders a comfortable illusion' of understanding strict implication: 3 for this is how a fuller title of Scott's 71 would have ended. What are considered are certain Lewis systems (though not the systems S2 and S3 that Lewis emphasized): there is nothing about entailment proper. However, with the semantics and methods of SE I-IV the program can be extended to apply to entailment as well as strict implication, as Gabbay 73 makes a start on showing. In fact however a development of the program in Scott's way as reapplied to entailment leads not to system E but to ES, i.e. E + Strong Distribution (see further chapter 15). The account considers the general principles of deductive inference. In the fashion of Gentzen, a relation (= between sets, or sequences, of statements is introduced: Scott calls it conditional assertion and Gabbay consequence, and Scott reads it in a consequential way: A | = T w h e n 'under the condition of truth for all A e A the truth of at least one B e T follows' (p.796)7^ In conWith respect to wff of E,

and

coincide.

Given the (in retrospect) indir-

ect way in which E was analysed through E, this coincidence is significant. 2

A11 subsequent page references without further indication avert to Scott 71.

3

Scott's setting of his theme suggests that there was something radically wrong with Lewis's original procedure: but what grounds are offered in support of this suggestion amount to no more than the hoary points of Quine concerning use/mention confusions and the like (already met in chapter 1).

''But subsequently (p.894) he warns against calling it 'an entailment relationship'. For, so it is claimed, 'the words 'entailment' and 'consequence' harbour an echo of syntactic shuffling'. 431$

SOME DEFINITIONS

PA.

OF CONSEQUENCE;

SYNTACTICAL

CHARACTERISATION

OF THE

RELATION

trast however to Gentzen's turnstyle, |= is straightaway given a semantical explanation, as follows:A sentential language SL contains connectives (and constants) which can be divided into those that are classical (e.g. in the case of Et, &, v and t) and those that are nonclassical (e.g. ~). A valuation (2-valuation when more specificity is required) is any function a: Wff -»• II, i.e. from Wff to values 1 or 0, which respects the classical assignments (e.g. in the case of Et, a(A & B) = 1 iff a(A) = 1 = a(B), a(A v B) = 1 iff a(A) - 1 or a(B) = 1, and a(t) = 1). With respect to an intended set N of such valuations (N is called a 2-model) a consequence relation f« (more explicitly is defined thus, for finite sequences of wff A and T : ~ A |= T iff, for every a e N , if a(A) - 1 for every A in A then a(B) = 1 for some B in T (2-definition).1 In particular, A B iff ( a e N ) . a ( A ) - 1 > a(B) = 1 . A consequence relation ^ is completely characterised (in the absence of specific connectives) by the general properties of Reflexivity, Augmentation (or Monotonicity) and Cut (or Transitivity). Scott presents this result (p.797) as 'a solution to a big part of Lewis's program of setting up - as a metalinguistic activity - a general theory of deductive inference'! Quite apart from the metalinguistic slanting, neither part of "Lewis's program" nor necessary, this is a substantial overstatement, as the argument of the text from 2.1 on should show: such a characterisation result provides only a beginning - a significant beginning nonetheless - on the task of characterising deductive inference and entailment. The 2-definition is a case of the (potentially) more general world-relativised definition: for non-null sequences (or sets) A and T, A ^ T iff for every v e N and a e K, if v (A) = 1 for every A in A then v (B) = 1 for some B in T (indexed definition). This is essentially the definition used in SE I-SE IV and 2.1 but cut loose (as in 2.1) from the specific connectives of entailment logics and extended to finite sequences (or sets) of wff. For vfl(C) is but v(C, a) in different dress, i.e. indexed valuations are tantamount to world-relativised evaluations. What is important is that |f ties directly to the semantical evaluation of as ^ does not, without shuffling. Yet plainly on their own neither does anything the other cannot. To obtain f» from ||- simply write vfl(C) as a(C), in effect the procedure of 3.2 for obtaining truth-valued semantics for the first degree FD. Conversely, construe a as the vector < v , a > ; then |j- emerges from (= . Not only does |f connect more neatly with intensional connectives such as -+• and ° 3 ; it meshes better with a sensible definition of theoremhood, through 0, l

Such a definition for the sufficiently general case A^, ..., A is given by Smiley 59, p.236. Smiley's definition requires, like Gabbay's, respect of classical connectives, but in fact allows for nonclassical connectives. And Smiley proceeds to set down general properties of the consequence relation which it has 'quite independently of the presence or absence of any particular connective'. The properties are restated by Scott, p.796; and also in 73, p.255; and fuller proofs of adequacy results are given in Scott 74, §1.

2

Cf. 2.1. The adequacy theorem 2.1 (of p.73) is readily extended (in the light of Chapter 4) to embrace multiple consequents. But then Cut is essential in proof of completeness (see lemma 4.3). The results of chapter 2, and for that matter elsewhere in the main text, are entirely independent of the work of Scott and Gabbay.

3

And also many-valued connectives, such as Lukasiewicz's implication: see the definition of the 5-valued implication, where Scott resorts to indexed valuations (73, p.270), and accordingly makes room for the more flexible indexed account of consequence. Similarly in his 74 §2, for 6-valued implication. 432$

PA.

WEAKNESSES

IN SCOTT'S

ALLEGED

RATIONAL RECONSTRUCTION OF LEWIS'S

APPROACH

not, as classically, through a special case of A f= T where A is null, i.e. as A T (whatever that means). So also the nonsense A., ..., A f= (and also f* ) 1 n is avoided, again by simply excluding, what Scott does not, null sequences of wff. Such one-sided occurrences of |{- are properly excluded. For, in particular, from nothing nothing follows. 1 Of course theoremhood can be separately defined, and we can even represent it, if we like, through a one-sided use of the symbol If- . 'Lewis's approach', on Scott's ' "rational" reconstruction of his plan', was 'in effect, to inject a part of the metalanguage into the object language with the aid of his modal operators' (p.800). That certainly was not Lewis's approach, any more than such injection was the intuitionists' approach in working towards intuitionistic logic as an alternative to classical, though it could be (misleadingly) reconstrued in that way (e.g. on the basis of Heyting's proposed interpretation of the calculus by way of proof constructions). Scott (like other modal bashers) finds no analogous confusion in the introduction of implication and happily proceeds to introduce a connective understood 'as some kind of implication' (73, p.252), without any of the de rigeur remarks about use/mention confusion that accompany his introduction of Lewis's implication. 2 If we can 'imagine that [the set SL of statements] is closed under a binary connective -*•', (73, p.251), then we can equally suppose that SL 'is closed under a new binary operation =*' (p.801). But in the latter case (where use of Lewis's hook would have prejudiced matters!) 'the problem ... is to imagine which rules of inference are appropriate for the intended role of =*' (p.801). In the former case (which is without prejudice) the following two orthodox rules are simply assumed A , A f= B (MP) A, A + B |= B (DT) A f5 A + B the first as obvious, the second (which certainly does threaten any thinker of sensitivity with the DTs) as 'almost as obvious', and subsequently as obvious. Neither is obvious. In the context of the other rules or the intended meanings of expressions, (DT) yields B f= A -»• B in one application and the full paradox B A -*• B in two, while (MP) gives Ass, A & (A B) -»• B, a dubious principle eventually rejected (in 3). Contrast the treatment of ("logical implication" 3 ), where 'it seems better not to try to assume ... (MP) A, A B f= B. Stated in this way there is an uncomfortable mixing of levels I prefer to avoid until I have a better grasp of the nature of the statement A B' (p.801). At ^ h e classical idea that logical truths (or theorems) follow from nothing (or the null set) contrasts strangely with the traditional ontological principle that nothing derives from nothing (ex nihilo nihil fit). By way of obtaining some appearance of consistency it can be said that different senses of derivation are involved. 2

The treatment of other nonclassical logics, such as intuitionistic and quantum logics, by modal bashers provides a simple model which demolishes most of their complaints about strict implicational and relevant logics. It also reveals their double standards.

3

And suppose

were just a more rigorous implication.

433$

PA.

WRONG RULES

FOR WRONG REASONS;

RECTIFICATION

BV NOW-FAMILIAR

BACKTRACKING

least however the right qualifications on the implicational rules result, though for the entirely wrong reasons. The basic rules Scott says he guesses, which he might have "guessed" also for implication, are (the inverse pair) |= A =» B

A (= B

(WMP)

(WDT) A f= B

(= A => B

At this stage the treatment given of degenerate cases does become crucial. Scott's notion of consequence, recommended to us, leads directly to suppression in such forms as f= B

B, A |= C (Positive Suppression); A (= C

and so delivers the Lewis paradoxes. For example, whenever ^=A, B ^ A by Augmentation (or by Suppression from A, B ^ A) whence [= B =» A by (WDT) . Thus Scott's notion is worthless, without some modification, for entailmental purposes. With the deducibility relation |j- these suppression steps are correctly ruled out as illegitmate.1 Scott's proposed injection of 'part of the metalanguage into the object language' - one way only of viewing what is proposed - consists in extending each valuation a in N to a valuation a' (in N') which agrees with a where it is defined and which also evaluates statements of the form A B, in fact so as to satisfy an S5 rule: a' (A =*• B) = 1 iff for every 6 e N (i.e. 8' e N') when 8(A) = 1 then 8(B) = 1. What results is but a recasting of the semantics for S5, in the fashion of pure valuational semantics (of 3.2). It is extremely doubtful that such a semantical construction - whatever its appealing features, or scope for variation - is 'pretty much what Lewis wanted' or accordingly 'nearly vindicates him' (pp.806, 807). Moreover the 'more subtle constructions' required for S4 or S3 call for what is tantamount to indexed valuations again. So also do those for entailment proper. But then the details of extensions from the first degree entailment theory concerning ||- to the fuller semantical theory which includes iterated are those already given in the expansion from systems of chapter 2 to chapter 4, and done for systems R and E in SE I and SE IV respectively. We are back where we have already been.

Scott's quite incomplete argument to the metatheoretic result concerning the horizontalisation of deductive inference patterns (p.803) also involves suppression steps in a crucial way. It is largely in virtue of these that he can go on to 'say something good about material implication' (p.805).

434$

REFERENCES

The list of references which follows includes oaly work specifically referred to in the text (some in Part II). For the bast bibliography of work on entailment, conditionality and associated topics the reader should refer to Wolf 76.

ABE

Anderson, A.R,. and Belnap, N.D. Jr., University Press, Princetoa, N.J., 1975.

AflEM

Anderson, A.R., Belnap, N.D. forthcoming.

AL I

Meyer, R.K. and Routley, R., 'Algebraic analysis of entailment et Analyse, 15 (1972) 407-428..

AL II

Meyer, R.K. unpublished.

BP

Routley, R.

BV

Meyer, R.K. 'A Boolean-valued semantics for R', Research Paper No. 4, Logic Group, Research School of Social Sciences, Australian National University, 1976.

CLF

Routley, R., 'The choice of logical foundations: non-classical the ultralogical choice', Studla Logica, 39 (1980) 77-98.

CR I

Meyer, R.K. and Routley, R., 'Classical relevant logics I', 32 (1973) 51-68.

Studia

Logica,

CR II

Meyer, R.K. and Routley, R., 'Classical relevant logics II 1 , Studia 33 (1974) 183-194.

Logica,

DCL

Routley, R. and Meyer, R.K., 'Dialectical logic, classical logic and consistency of the world', Studies in Soviet Thought 16 (1976) 1-25.

DLS

Routley, R., 'Dialectical logic, semantics and metamathematics', 14 (1979) 301-331.

D3

Routley, R., 'Disjunctive Syllogism seen as a fallacy of relevance', Bulletin of the Section of Logic, Polish Academy of Sciences, 1982.

EMJB

Routley, R., Exploring Meinong's Jungle and Beyond, Research School of Social Sciences, Australian National University, 1980.

ER

Routley, R. and Meyer, R.K., (1970) 355-369.

FD

Routley, R. and Routley, V., 'Semantics of first degree entailment', Nous (1972) 335-359. See also Routley [1970], in this list.

FM

Routley, R. 129-153.

FMP

Osnovy marksi3tskoj filosophii (Fundamentals of Marxist Philosophy) Institute of P'hilosophy "of the Academy of Sciences of the Soviet Union, 1
G&

Goddard, L., and R. Routley, The Logic of Significance and Context, vol. Scottish Academic Press, Edinburgh, 1973.

and

Routley,

Jr.

R.,

and

Entailment,

Meyer,

'Algebraic

R.K.,

analysis

vol.

1,

Princeton

Entailment,

of

vol.

I',

2,

Logique

entailment

II*,

and Routley, V., Beyond tha Possible, in preparation.

and Routley,

V.,

'Extensional

'A

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reduction

of

I',

modality',

The

choices

and

the

Erkenntnis,

Monist,

Nous,

3

60

6

(1969)

1,

REFERENCES

IDftL

Routley, R., V. Plumwood, R.K. Meyer, and E.P. Martin, 'The philosophical bases of relevant logic semantics', Research Paper No.11, Logic Group, Research School of Social Sciences, 1983; forthcoming in Topoi.

LC

Routley, R., 'A rival account of logical Mathematical Logic, No. 3 (1974) 41-51.

MD I

Meyer, R.K., and J.M. (1969), 460-474.

NC

Routley, R. and V. Plumwood, 'Negation and contradiction' in Proceedings of the Fifth Latin American Symposium on Mathematical Logic, Marcel Dekker, 1983":

OED

Oxford English Dictionary, Clarendon Press, Oxford, 1971.

PL

Priest, G., and R. Routley, Contributing Editors, and Paraconsistent Logics, Philosophia Verlag, 1983.

PM

Whitehead, A.N., and B. Russell, Principia Cambridge University Press, Cambridge, 1950.

QRI

Routley, R. 'Problems and solutions in the semantics of quantified relevant logics I', in Mathematical Logic in Latin America, A.I. Arruda, R. Chuaqui, and N.C.A. da Costa, "Editors, North-Holland, Amsterdam, 1980, pp.305-40.

SE I

Routley, R., and R.K. Meyer, 'The semantics of entailment I', in Truth and Syntax Modality H. Leblanc, Editor, North-Holland, Amsterdam, 1972, pp.199-243.

3E II

Routley, R., and R.K. Meyer, 'The semantics of entailment Philosophical Logic, 1 (1972), 53-73.

II',

Journal

of

SE III

Routley, R., and R.K. Meyer, 'The semantics of entailment III', Philosophical Logic, 1 (1972), 192-208.

Journal

of

SE IV

Routley, R., and R.K. Meyer, 'The semantics of entailment IV: E, II' and II", formerly forthcoming, Journal of Symbolic Logic; now Appendix 1 above.

SE V

Routley, R. and R.K. Meyer, 'The semantics of entailment V', Australian National University, cyclostyled (1972); now incorporated in this volume.

SE VI

Routley, R., and R.K.

31

Routley, R., 'A semantical analysis of implication system I and of the degree of entailment', Mathematische Anna1en, 84 (1972), 58-84.

SL

St. Louis Collection. K. Collier, A. Gasper, and R.G. Wolf, Editors, The Current Status of Relevant Logics. Proceedings of_ the International Conference on Relevance Logics. In Memoriam: Alan Ross Anderson. Now re-edited by R. Routley and J. Norman under the title Directions in Relevant Logic (DRL), forthcoming (1983).

UL

R. Routley, 'Ultralogic as universal?', Relevance Logic Newsletter, 2 (1977) 50-90 and 138-175. Reprinted as an appendix in EMJB.

Dunn, 'E, R and Y',

consequence',

Journal

of

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58

Ackermann, W. 'Uber die Beziehung zwischen Implikation', Dialectica, 12 (1958) 213-222.

67

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65

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Anderson, A.R., 'Some open problems concerning the system E of entailment', Acta Phllosophica Fennica, Proceedings of a Colloquium on Modal and Many-Valued Logic, Helsinki, 23rd-26"th August, 1962, fasc. 16 (1963) 7-18^

67

Anderson, A.R., 'Some nasty problems in the formal logic of ethics', Nous, (1967), 345-360.

59

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62

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62

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Routley, R., and V. Routley, 'The role of inconsistent and incomplete theories in the logic of belief', Communication and Cognition, 8 (1975), 185-235.

75a

Routley, R., and V. Nous, forthcoming.

74

Routley, R., and R. Wolf, 'No rational sentential logic has characteristic matrix', Logique et Analyse, 17 (1974), 317-321.

67

Routley, V., 'Some false laws of logic', paper read at the Australian Association of Philosophy Conference, and at St. Andrew's University, 1967.

06

Russell, B., Review of H. Mind, 15 (1906) 255-60.

19

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451

IWPEX

Augmentation, 45 Augmentation (principle), 44, 71 Axiomatisability, finite, 401

ABE (Anderson & Belnap, Entailment), 15, 17, 37, 39, 49, 74, 77, 92, 94, 98, 112, 170, 172, 176, 180, 132, 183, 184, 187, 215, 218, 220, 222, 228, 231, 234, 235, 237, 239, 240, 241, 243, 244, 245, 246, 248, 249, 251, 252, 253, 256-57, 258, 259, 263, 265, 267, 268, 271, 272, 274, 275, 290, 340, 343, 349, 350, 357, 353, 391, 394, 395, 396, 405, 407, 408, 425, 428; on Disjunctive Syllogism, 359-60; on Lewis' strict implication, 353-59 Abraham, A., 376, 377, 373 Ackermann's rule of commutation, 263 Ackermann, tf., 3, 4, 37, 60, 103,

Bar-Hillel, Y., 289 Barker, J.A., 228, 232 Basic Contraposition Logic (BCL), 122-24; adequacy theorem, l?2 Bastable, P.K., 128 Belnap, N.D. Jr., 3, 4, 5, 26, 59, 60, 103, 161, 180, 182, 184, 185, 188, 190, 192, 195, 214, 213, 223, 229, 232, 233, 234, 240, 241, 242, 251, 256-57, 258, 260, 261, 271, 289, 290, 307, 340, 349, 358-60, 410, 423 Belnap matrices, 176-79 Bennett, J., 10, 23, 24, 25, 27, 23, 29, 30, 31-32, 34, 44, 45, 46, 74, 77, 148, 156, 216, 360; functor phi, 24 Birkhoff, G., 123, 125 Bivalence, principle of, 153 Boethius, 4, 83, 90, 239 Boethius (principle), 8, 9, 343, 362-63 Boland, J., 250 Boolean Algebra (BA), 92, 117, 118, 119; Stone model for, 119-20 Boolean Logic (BL), 110, 117 Brady, R.T., 156, 252, 254, 290, 336, 341, 346, 381 Briskman, L., 74 Broad, C.D., 74 Brouwer, L.E.J., 64 Burks, A.W., 49

145, 208, 221, 223, 224, 228, 230, 234, 235, 237, 239, 240, 268, 271, 272, 290, 348, 349, 407, 409, 410, 415, 423 Ackermann-Maksimova fallacies, 244 Ackermann fallacies, 37, 39-40, 264 Ackermann property, 37, 243 Adams, E.W., 6, 8, 9, 10, 44, 48 Addition, 96, 97, 101 Adequacy, minimal, 256 Adjunction, 80-82 Affixing, 38, 44 Affixing systems, 39, 225-26 Albert (principle), 237, 246 Alfonso (principle), 239 Algebra, semantic analysis, 117-20 American plan semantics, 319, 319-34; classical-style evaluations, 320; limited success, 334 Anderson, A.R., 3, 4, 5, 26, 60,

CL Systems: completeness, 373-74 CSyll. SEE Conjunctive Syllogism Cantor, G., 55 Carnap, R., 17, 21, 209, 212, 213, 216, 225 C-condition, 373 Charlwood, G., 260 Chiigey, J., 248 Choice, criteria for, 255 Choice of axioms, 234 Choice of system, 230-32, 240; factors, 262 Chrysippus, 83, 159 Church, A., 5, 46, 286, 408, 423, 428 Clark, M., 161, 162 Classical Assumption, the, 295, 295-97, 296

103, 145, 161, 192, 208, 214, 218, 223, 232, 233, 234, 243, 251, 271, 290, 340, 349, 358-60, 363, 410, 423, 429 Anderson-Belnap (principle), 227 Anderson (principle), 247, 346 Angell, R.B., 4, 83, 87, 88, 92, 93 Antilogism, 26 Antinomies, semantical, 368 Apostel, L., 59 &qvist, L., 47, 48 Argument, valid, 141, 213 Aristotle, 4, 83, 90, 141, 160 Aristotle (principle), 90, 159, 343 Arruda, A.I., 39, 59, 153 Asenjo, F.G., 59, 369 Ashworth, E.J., 158, 159 Ass (principle), 278; counterexamples, 280, 281

4 52

INDEX

Classical logic, 10, 295-97, 391; problems of, 50-58; ways of going wrong, 51 Classification of modal logics: non-replacement systems, 39; replacement systems, 39 Closed relevant systems, 367 Closure, 399 Coffa, J.A., 237, 240 Coherence, 258; M-coherence, 259 Cologne commentators, 158 Commitment, 217 Commutation: restricted, 228, 267; restrictions, 265 Commutative logics, 223 Compactness (of SL), 318 Completeness: theorem for B, 323-24 Completeness, world, 332 Completeness of L, 305 Conceptivisa, 4, 96, 97-101 Conceptivist positions, 3 Conditional assertion, 431 Conditionals: character of sought theory, 49; strong inteationality requirement, 49; theory of, 41-49; ultramodal requirement, 49 Conjunction, 73, 194; normal rule,

Contradictoriness, Hegelian, 370 Contraposition, 43, 326 Conventionally normal systems, 388 Cooper, U.S., 8, 9, 14, 42, 47, 83 Counterexample principles, 229 Counterexamples: method of, 8-9 Cresswell, M., 22, 23, 27, 28, 122, 136 Cross-over lemma: completeness, 331; consistency, 332; mated, 333 Crypto-relevant systems, 378-79 Curley, E.M., 358, 359 Curry, H.B., 72, 170 Curry-Moh Shaw Kwei paradox, 367 Curry paradoxes, 281, 289-90, 367 Cut principles, 345

D'Ottaviano, I.M.L., 59 DS. SEE Disjunctive Syllogism Da Costa, H.C.A., 39, 59, 153 Dale, A . F . , 74 De Morgan Lattice Logic (DHL), 110-16; adequacy theorem, 111 De Morgan lattice, 116 Da Soto, 158 Decidability, 402, 404, 405-6 Deducibility, 16, 69, 75, 86, 146, 170-71, 219, 237 Deducibility relation, 69-71; axiomatisation, 72-78; properties, 70, 71 Deduction Equivalence, 258 Deductive argument: "compulsion" feature of, 145 Deductive reasoning, 85 "Definitional retreat", 10 Degree (definition), 170 Derlvability, L-, 307 Derivability, pure, 73 Derlvability relation, 72 Determinability, logical, 130 Dialectical logic, 58-67 Dialectical system M, 121 Disjunction, 95, 195 Disjunction, Intensional, 356 Disjunctive Rules, 336, 338 Disjunctive Suffixing, 335 Disjunctive Syllogism (DS), 11, 25, 26, 28, 33, 53, 152, 153, 155, 165, 166, 279; counterexamples, 156; as a fallacy of relevance, 359-60; intultlveness, 157, 160 Distribution, Strong, 345 Distribution, rule, 248 Distributional principles, 249

80 Conjunctive Syllogism (CSyll), 45-46, 268, 269, 274 Connexive principles, 343 Connexlvlsa, 4, 82-95; ad hoc, 87; classes of, 83; interaction, 92-95; intuitive, 87; naive, 92; separatist, 93 Connexivist positions, 3 Connexivist theories of conditionals, 47 Consequence relation, 431, 432, 434 Consistency, 22, 56; first degree analysis, 209; logic and semantics, 361; relevant, 362-63; strict, 362 Consistency, world, 332 Consistency Hypothesis (CH), 60-62; modal status of, 65 Constant t, 348; modelling conditions, 352; postulates, 352; semantics, 351; t-postulates, 350 Construction of tableaux, 200, 201 Container Thesis, 217, 218 Content: logical, 212; semantical definition, 211 Content inclusion, 24 Content principles, 212 Content semantics, 213 Contraction principles, 229, 278 453

JNVEX

Finite model property, 175, 399, 404 First Degree (FD): American plan semantics, 192; Dunn semantics, 193-98; adequacy of algebraic semantics, 135; adequacy theorem, 173-74; algebraic analysis, 132-90; alternative semantics, 190; applications, 206-19; decilability, 176; deducibility formulation, 171; entailmental formulation, 172; finite model property, 175; matrices for, 176, 177-80; model sturctures, 183; paradox freedom of, 130-32; semantical formulation, 173; suppression freedom of, 180-32; tableau methods, 200-204; topic semantics, 215; weak relevance of, 180 Fission: logic and semantics, 361 Force (of premisses), 269, 273 Formal sentential logic (FL), 286-37 Fra3nkel, A. A., 289 Functors, intentional, 139 Fusion, 365-57, 421; logic and semantics, 366

Distributive Lattice Logic M (DLLM), 104-3, 108; Gentzan formulation, 109-10; adequacy theorem, 109; weak, relevance lemma, 105 Downing, P.3., 47, 83 Duncan-Jones, A.E., 4, 25 Dunn, J.M., 4, 100, 101, 102, 161, 162, 168, 182, 133, 192, 193, 205, 206, 213, 214, 215, 290, 300, 301, 392, 307, 333, 366, 412, 424, 429 Dunn semantics, 193-98 SMingle, 344 ESyll• SEE Exported Syllogism Entailment, 138, 264, 284; Ackermann requirement, 223; adequacy conditions, 221-24; deducibility/necessary implication, 19; intentionality of theory, 15; intuitive concept, 77-73, 151-52; meaning connection, 23; paraconsistency requirement, 67; pure, 220; c-, 317; transitivity of, 74-75 Equivalence, system, 415 Everett-VJheeler theory, 63-64 Expansion principles, 29 Exportation, 11, 33, 270 Exported Syllogism (ESyll), 226, 268, 269, 271, 272, 274, 275, 276, 277; Lewis' criticism, 273

Gabbay, D.M., 47,. 48, 404, 430, 431, 432 Geach, P.T., 4, 48, 74, 75, 156 Gentzen, G., 410, 432 Gentzen formulations, 404 Gentzan methods, 405 Giambrone, 3., 495 Goddard, L., 18, 98, 129, 130, 131, 137, 294, 295 Godel's incompleteness theorem, 402 Godel, K., 56, 64, 65, 279, 296 Goldblatt, R., 123, 124 Goodman, N., 46 Graves, G.A., 45, 47 Griss, G.F.C., 99 Grossman, R., 235

FL (formal sentential logic), 235-87 Factor, 223 FactorLsation principles, 345 Factor principles, 345 Fallacies: Ackermann-Maksimova, 244; modal, 250; modal dependence/necessity suppression, 41; of modality, 15, 37; modal reduction, 41; of relevance, 250 Fallacy, finite valued, 14 Fallacy axclusion, 14 Faris, J.A., 12-13 Feys, R., 35, 38, 39, 40 Filtration: canonical, 403 Filtration methods, 399-405 Filtration of relational model structures, 399 Findlay, J.N., 128 Fine, K., 4, 100, 310, 430 Fine rule, 260

Haack, S., 54, 57, 58 Hallden, S., 4, 212, 392 Hallden completeness, 246 Hallden reasonableness, 41, 284, 343, 392-94, 416-17 Hanson, B., 213, 214 Hare, R.M., 139 Harper, if., 365 Hegel, S.W.F., 60, 51, 183 Hempel, C.G., 350 Heracleitus, 183 fleyting, A., 126, 127, 154

454

INDEX

Higher degree systems, 219-30; selection criteria, 220 Hi'lbert, D., 66: positive logic, 2, 5 Hilbert's epsilon operator, 298 Hintikfca, J., 104, 139 "Holding it", 294 Holism, 81 Homomocphisja, truth-preserving, 187 Hook, 12-13, 52 Hookers, 31, 236 Hughes, G.E., 22, 23, 27, 28, 122, 364 Hunter, G., 6, 8, 13, 45, 47, 95, 407, 424 Huntington, E.V., 118

Jennings, R.E., 241 Johansson, I., 5, 126, 127 Johnson, W.E., 74, 129, 247 Joint force (of premisses), 270 Journal of Symbolic Logic, 296, 430

KL Systems, 378 Kalmar, L., 305 Kadrov, B.M., 65 Kielkopf, C.F., 74, 101, 104, 143, 217, 234, 261, 295, 297 Kleene, S.C., 93, 106, 369 Kline, G.L., 65 Kneale, W. and M., 22, 159 Kolmogorov, 5 Kripke, S., 38, 113, 200, 201, 202, 203, 394, 417, 423

Identity, 71, 275 "If" statements, classification of, 42 Implication, 284, 294; L-, 302; classification, 138; intuitionistic, 126; metalinguistic account, 17-21; positions, 2; strict, 140, 283, 357 Implication, double lemma, 321 Implication-Conditional (IC) transformation, 17-19 Inclusion (principle), 345 Incompleteness, 370 Independence relations, 364-55 Intelligibility, 296 Intensional Lattice Logic (ILL), 165-69 Intensionality requirement, 17 Interchange behaviour of negation, 132 Interpretations, 301 Intuition, 10-11, 164, 165 Intuitionistic System K, 121 Intultionist logic, 154 Inversion, 131 Involution, 121, 131 Iseminger, G., 74

L Systems: compactness, 318; fundamental extension lemma, 307-8; implicational adequacy, 318; model structures, 312; priming lemma, 308-11; semantical postulates, 313-15; semantic completeness, 305-7; soundness, 303- 4; strong completeness theorem, 315-17 Langford, C.H., 13, 14, 15, 17, 28, 31, 34, 35, 36, 37, 106, 272, 349, 357-54 Language, sentential, 285-92 Language SL, 285-86 Lattices: De Morgan, 183, 186; intensional, 182, 183, 184, 187, 189; product, 188 Law of Excluded Middle (LEM), 282 Leblanc, H., 128 Leibnitz, G.W., 268, 298 Lemmon, E.J., 2, 51, 157 Lewis' paradox arguments, 27, 160-65 Lewis' positive argument, 29, 34 Lewis' positive paradox, 151, 152 Lewis, C.I., 2, 4, 5, 11, 13, 14, 15, 17, 21, 22, 23, 25, 27, 28, 31, 32, 34, 35, 36, 37, 38, 41, 44, 49, 58, 83, 87, 96, 106, 117, 209, 228, 235, 271, 272, 273, 285, 349, 356-64; consistency connective, 300; consistency postulate, 210, 364; implication, 433; proof of paradox, 4, 152; strict implication, 431 Lewis, D., 2, 10, 14, 45, 47, 49 Lewis Carroll argument, 30

JL Systems, 336, 337-40; completeness, 337; extension lemma, 338; model structure, 339; priming lemma, 339; soundness, 337; strong completeness, 339-40 JL systems, 336-39; extension lemma, 338; model structure, 339 Jaskowski, S., 59, 60 Javellus, 158 Jeffrey, R.C., 205

455

I MP EX

Metalogic, 294; choice of, 295; consistency, 296 Metavaluations, 259 Meyer, R.K., 12, 13, 49, 54, 76, 166, 170, 192, 195, 200, 208, 228, 232, 248, 249, 253, 256, 258, 276, 284, 289, 290, 295, 338, 343, 346, 348, 376, 377, 396, 397, 399, 405, 406, 412, 423, 429, 430 Minginterp (principle), 344 Mingle (principle), 154, 223, 229, 241, 252 Minglish extensions of R, 340-42 Minglish principles, 341, 344 Minimal logic, 127 Mints, 248, 345, 346 Modalities, 348-49; absolute, 349; characterisation through t, 355 Modal theories of conditionals, 48 Modelling, canonical, 323 Modelling conditions, 313-14 Modellings, unreduced, 336 Model structure: 3-, 319-20; enlarged, 380, 381, 383 Model structure for B, 298 Model structures, 312, 347; normal, 387, 390 Modus Ponens, 166 Monotonicity, 388-89 Montgomery, H., 22, 88, 246, 247, 275 Moody, E.A., 158 Moore, G.E., 149, 272, 360 Morgan, C.G., 343 Mortensen, C.M., 242, 343 Mortensen chain, 243, 245 Myhill, J., 12

Lewis Lewis Lewis Lewy,

and Langford matrices, 106 paradox, 93 paradoxes, 434 C., 11, 23, 25, 27, 30, 74, 75, 77, 78, 140, 149, 150, 151-53, 160-65, 247, 275, 276, 360^ paradoxes, 163, 164 L° and Lt Systems: strong adequacy theorem, 366 Logic: classical relevant, 372 Logic, non-standard, 158 Logical law, 149 Logics: consistent, 387; modal, 227; trivialisable, 229; typology, 224-30 Logics, dialectical relevant, 368 Londey, D.G., 364 Loparic, A., 191, 193 Lt Systems: adequacy theorem, 354; completeness, 353 -Lukasiewicz, J., 54, 59, 83, 152, 153, 194, 344, 369, 432

Hackle, J.L., 63 Makinson, D., 176, 367 Maksimova, L., 237, 240, 248, 430 Maksimova wff, 244, 245 Martin, E.P., 76 Material detachment, 33, 53, 56, 57, 172; objections to, 54 Material implication, 16; counterexamples, 6-8; Inadequacy of, 5 Material-implication, 382 Mateship, 331, 332, 333 Mathematics, 1, 55, 64 Matrices: Belnap, 176-79; for FD, 177; Lewis and Langford, 106; M0, 178, 252; Parry, 102; Smiley: derivation of, 113, 114-16; Wajsberg, 106, 108, 115; derivation of, 194, 248, 250 Matrix: normal characteristic, 394 Matrlx-assessibility fallacy, 14 Maximality, L-, 307 McCall, S., 4, 83, 87, 88, 91, 92, 93, 94, 117, 265, 266 McColl, H., 356 McRobbie, M.A., 122, 206 Meaning, 216; connection principles, 216; inclusion of content, 146 Meinong, A., 100, 270 Mendelson, E., 119, 310, 402 Metacorrectness, 259 Metalinguistic position, 2

Naturalness, 259; simplicity criteria, 260 Neccessitation systems, 227 Necessity: first degree analysis, 208 Negation, 110-16, 127-40; cancellation, 89; central features, 131-32; classical, 127, 127, 133, 133, 135, 135, 136, 136, 381; deletion, 92; determinable theory of, 12930, 137; intractable principles, 346; non-normal rule, 121; star rule, 134; subtraction, 89, 90-92 Negation, classical: added to relevant logics, 371; irrelevance- inducing, 376

456

INDEX

Negation principles, 346 Nelson, E.J., 22, 26, 82, 83, 84, 85, 87, 270, 356 Nelson, J.O., 4, 3 Non-deducibility theses, 219 Non-degeneracy, 317 Nonimplicational principles, 346 Non-normal strict implicational systems, 39-41 Non-replacement systems, 224 Non-transitlvlsm, 4 Normalisation, 337-88, 415 Normalisation requirement, 257 Normalising table, 257 Normality, 256 Normal systems, 388; conventionally normal, 388, 390 Novell Smith, P.H., 161

relative, 209, 211;

210

strict,

Prefixing, 226 Priest, G., 59, 156, 193, 280, 281, 369, 370 Principle of bivalence, 153, 161 Principle of maximum variation, 143, 144 Principles, transmission, 218 Prior, A.N., 4, 36, 63, 99, 343, 344 Properties: content-loading, 218, 219; range-loading, 218 Property P of situations, 407 Propositional Identity systems, 224 Pseudo-Scotus, 280 Putnam, H., 96

Quantifiers, neutral, 294 Quantum logic (QL), 125, 126 Quesada, F.M., 59 Quine, 1, 17, 18, 21, 31, 55, 160, 199, 431

0-C-condition, 374 "Oblgationes" literature, 158 Operation *, 196. SEE ALSO Star connective, Star Rule Orthologic (OL), 123-24

Ramsey, F.P., 2 Range, logical, 213 Rasiowa, H., 1 Reasonableness, Hallden, 41, 284, 392-94, 416-17 Recovery principles, 255; classical theses, 256; enthymematic, 256; general, 255 Reductio, counterarguments, 282 Reductio principles, 229 Reformulability requirements, 261 Regius, 158 Reiteration, 238 Relevance, 232, 242, 244; weak, 222, 232 Relevant logic: choice of metalogic, 295 Relevant logics: characterisation, 153-54; normal, 154 Relevant positions, 3 Relevant strategy, 4 Relevant theories, 33 Rennie, M.K., 46, 344 Repetition, 238 Replacement: AC-, 345; DC-, 345; principles, 225; systems, 225 Replacement principles, strong, 345 Reschar, N., 63, 64, 66 Robinson, A., 348 Rosser, J.B., 279 Routley, R., 14, 18, 19, 20, 21, 22, 23, 52, 53, 54, 59, 63, 66, 80, 88, 98, 121, 129-30, 131, 137, 142, 143, 145, 148, 156,

P-tf problem, 76 Paraconsistency, 224 Paraconsistent logics, 59 Paradox arguments, 152 Paradoxes, 24, 27, 40, 100, 231; Lewy's, 276; avoidance, 144; and consistency, 63; of implication, 25-26; noninnocence of, 28; strategies, 4 Paradoxicalness, 369-70 Parry, W.T., 2, 3, 4, 75, 96, 97, 98, 99, 100, 101, 102, 104, 228, 273, 357; Proscriptive principle, 100, 102 Partial M-0 matrix, diagram, 178 Peirce's "law", 36 Pelrce-type principles, 343-44, 344 Permutation, 71 Philonian position, 2 Piano theorem, 243 Plato, 259 Plausibility, 259; simplicity criteria, 260 Plumwood, V.. SEE Routley, V. Pollock, J.L., 23, 25, 27; defence of strict implication, 26 Popper, K., 27, 212 Portation, 78 Possibility: absolute, 210; first degree analysis, 209;

4 57

INDEX

Routley, R. (Continued),

Set theory: ideal relevant, 290; scepticism about, 55 Sextus Empiricus, 22, 159 Sheffer, H.M., 97 Simons, L., 8, 10, 11, 32, 33 Salification, 82, 84, 87-88 Slanay, J.K., 231, 366 Smiley, T.J., 70, 71, 74, 75, 76, 231, 274, 278, 430, 432 Smiley entailment, 76-78, 162 Smiley matrices, 114, 115, 116 Smirnov, G.A., 102, 106 Soundness of L systems, 302-5 Stalnaker, R., 2, 9, 14, 16, 41, 44, 45, 46, 47, 49, 95 Star: star imitation, 326; star negation rule, 327 Star Rule Lemma, 334 Star connective, 375 Star operation, 111, 378 Star rule lemma, 334 Stevenson, C.L., 6, 7, 8, 10, 42, 44, 47, 48, 83 Stoic logic, 158 Stoics, 4, 83 Stone, M.H., 139 Stone models, 119-20 Strawson, P.F., 74, 83, 89, 90, 91, 95, 163, 356 Strength: criterion of, 240-45; of systems, 256 Strict implication, 22, 31, 35, 38, 357; inadequacy of, 21, 32 Strict position, 2 Substitution principles, 35 Sufficiency, 237, 284; conditions, 222 Suffixing, 226 Summation principles, 345, 345 Superclassical relevant systems, 378 Superstrong relevant logics, 340 Suppression, 71, 94, 139, 140-53, 155, 237, 238, 267, 268, 271, 434; damaging effects of, 144, 145, 148; of necessary truths, 30, 31, 32; negative, 142, 149-50; positive, 142; principles, 344; reasons for excluding, 147, 148 Suppression principles, 344 "Switches" paradox, 6 Syntax: of sentential languages, 285-92 System II', 410; semantics for, 407 System IT, 290, 349, 409-10; first degree, 416; semantics for, 407. SEE ALSO Ackermann, W.

157, 166, 180, 182, 190, 191, 192, 196, 246, 247, 255, 259, 280, 281, 284, 289, 290, 294, 295, 296, 299, 319, 359, 396, 399, 406, 429, 430 Routley, V., 51, 53, 66, 30, 98, 140, 142, 143, 145, 148, 154, 157, 166, 180, 182, 192, 264, 300 Rule "Y", 172, 254, 297, 387; admissibility of, 391, 415 Rule ( R » , 386 Rule antilogism, 84 Rule distribution, 248 Rules and theses: separation, 45 Russell's paradox, 54, 165 Russell, B., 1, 17, 54, 74, 117, 138, 270, 281, 282 SL:

alternative proofs of priming corollaries, 311; compactness, 318; corollaries to priming lemma, 309; extension lemma, 307-8; interpretation, 301-2; non-degeneracy, 317; priming lemma, 308-12; proof of priming lemma using Zorn's lemma, 309-10; soundness, 302-5; strong completeness corollaries, 318; strong completeness theorem, 315-18; t-entaLlment, 317; validity, 302; valuations, 301-2 SL (language), 285-36 Scheffler, I., 356 Schotch, P., 241 Scott, D., 430, 431, 432, 433, 434 Scroggs property, 251 Segerberg, K., 389, 402 Semantics, 319; Americaa plan, 319 (SEE ALSO American plan semantics); content, 213; enlarged, 381; four-valued, 194, 195, 198; informational, 214; pure valuational, 191; range-content, 214, 216; semi-valuational, 190; topic, 215 Semantics for B: metalogical issues, 294-96 Semi-lattice logic (SLL), 79; adequacy theorem, 79 Sentential language, 285; structure, 285. SEE ALSO SL Separation results, 394-95, 397

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IWPEX

strengthenings, 247; first degree, 416; fusion added conservatively, 421, 421; interpretations, 411; model structures, 411; necessity theory, 268; negation, 421; non- normal molal strengthenings, 246-47; normal modal strengthenings, 246; pure Implicational part (EI), 425-29; pure implicational parts, 419-20; pure implicational parts (EI), 419-20; semantics for, 407; soundness, 412-13 strong distributional extensions, 247- 51; validity, 411; valuations, 411; vell-axiomatisation, 421; worlds semantics, 430

System B, 285, 287-89, 347; Double Implication lemma, 321; Hereditariness lemma, 321; Strong completeness theorem, 323; axiom schemes and rules, 287; equivalences, 291- 92; extensions, 288-90, 299, 300-301; formal semantics, 294-96; implication-negation part, 398; modelling conditions for extensions, 300; model structures, 298-99; positive extensions, 325; reduced model structures, 299; removal of 3ffixing rules, 292-93; removal of distribution, 293; soundness theorem, t'i2; theorems and equivalences, 291-92 System BL (Boolean logic), 110 System Bt, 351; extensions, 350 System C, 289, 347; completeness cross-over leoma, 331; consistency cross-over lemma, 332; extensions, 330; mated cross-over lemma, 333; reduced modellings, 329; reduced modellings for subsystems, 340; soundness theorem, 329; strong completeness theorem, 329 System CR, 290 System DA, 289 System DK, 38, 39, 156, 289, 347 System DL, 38, 156, 289, 347 System DLL, 104-9, 153, 154, 170 System DHL, 110-15, 153, 155, 170 System DMLt: adequacy theorem, 166-68 System E, 39, 40, 153, 22>, 230, 233, 234, 240, 264, 290, 334, 347, 349, 350, 394, 397, 405-6, 408, 409, 434; Mingllsh extensions, 252; argument to in ABE (Anderson & Belnap, Entailment), 235-39; axioms of, 234; canonical model structures, 413; canonical valuations, 413; completeness, 412-14 connexive extensions, 242; conservative extension of EI, 428; counterexamples and arguments, 263-77; criticisms of, 233; extensions, 422-23; extramodal

System E&: operational semantics, 418-19 System E+, 418, 425; model structure, 417 System EA, 244 System EI: canonical model structure, 426, 427 System EM, 154 System ES, 250-51; Piano theorem, 251 System Et, 355 System FD, 171-88 System FOB, 112-13 System G, 289, 335, 347, 388; disjunctive suffixing, 335; extensions, 389; normality and modelling conditions, 389 System ILL: adequacy theorem, 168 System KR, 379; crypto-relevance, 379 System L, 302-18, 391, 394; completeness, 302-5 System L-: completeness, 384; soundness, 383 System L>: completeness, 384; rule (R>), 386; soundness, 383 System MCRf, 377 System MCRt, 376-77 System ML, 155 System N, 289, 290 System NR, 248 System OR (orthosystem), 388 System P, 39, 153 System R, 153, 232, 233, 240, 248, 253, 264, 290, 334, 341, 347, 394, 397, 405-6, 428, 434; Mingllsh extensions, 252, 253,

459

INDEX

System R (Continued), 342; Minglish strengthenings, 340-42; adequacy theorem, 334; classically based axiomatisations, 377; counterexamples and arguments, 263-77; criticisms of, 233; downfall of, 266; model structure, 347 System R+, 394-95, 396; decidability, 405 System RK, 254 System RM, 168, 193, 252, 253 System RM+, 154, 254 System S, 290, 391 System SO.5, 39, 208 System SI, 39 System Sl°, 39 System S2, 35, 38, 39, 40 System S2 6 , 35, 38 System 33, 38, 40, 41, 41, 246, 394, 434 System S3°, 38 System S4, 19, 36, 37, 40, 246, 290, 423, 434 System S5, 19, 36, 37, 208, 434 System SL: truth, 302; validity, 302 System T, 38, 153, 232, 290, 347, 397, 405-6, 428; adequacy theorem, 334 System ZV, 102, 103

Trees, coupled, 205 Truth, double, 321 Truth, logical, 70 Truth-copulation fallacy, 14-15 Truth-functional fallacy, 14 Truth-preservation, 382; axiomatisation of its logic, 382; reaxiomatisation of its theory, 385 Typology of logics, 224-30

Urquhart, A., 4, 100, 161, 406, 429, 430 Urquhart (principle), 230 Urquhart systems, 260

Valid argument, 52 Validity, 202 Validity, double, 321; soundness theorem, 322 Valuations, 301 Van Dijk, T.A., 3 Van Fraassen, B., 52, 56, 216 Variable sharing, 24 Von Neumann, J., 123, 125 Von Wright, G., 4, 40, 74, 76, 349

Wajsberg matrices, 106, 108, 115 Wallace, 425 Weak Relevance Criterion (WR), 3 Whitehead, A.N.,17, 117, 281 Wisdom, J., 128, 247 Wittgenstein, L., 128, 212, 213, 247 Wolf, R.G., 14, 258, 259 Woodruff, P., 98 Woods, J., 47 Worlds, 294; empty, 380; incomplete and inconsistent, 151; null, 380; possible, 140; trivial, 380; universal, 380 Worlds, complete, 331 Worlds, possible, 332

Tableau methods: for FD, 2 K)-204; closure, 201-2; construction,

200, 201 Tamburino, J., 369 Tarski, A., 20, 21 Then-transformation, 43 Theorem convergence, 431 Theories: in completeness proof, 306; null, universal, degenerate, 306; prima, regular, consistent, normal, 306 Theory-saving, 64 Thomason, R., 14, 47, 49, 95, 192 Thomson, J.F., 13 Tradition, 159 Transitivity of entailment: requirement of adequacy, 7476

Zeno's paradoxes, 62, 160 Zinov'ev, A., 93, 102, 103, 104, 106 Zorn's lemma, 310

460

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