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The De-Mathematisation of Logic
Polimetrica
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Barry Hartley Slater
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ISBN 978-88-7699-071-7 Printed Edition ISBN 978-88-7699-072-4 Electronic Edition The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
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Preface ....................................................................................................... 7 1. Logic and Grammar............................................................................. 9 2. Completing Russell’s Logic................................................................ 25 3. Logic and Arithmetic.......................................................................... 51 4. Natural Language Sets ....................................................................... 63 5. Namely Riders: an Update................................................................. 85 6. Concept and Object in Frege ............................................................. 99 7. Frege’s Hidden Assumption............................................................. 113 8. A Poor Concept Script ..................................................................... 125 9. Motivation by De Se Beliefs ............................................................. 149 10. Out of the Liar Tangle.................................................................... 159 11. Ramseying Liars ............................................................................. 179 12. Proving that 2+3=5 ......................................................................... 203 13. Harmonising Natural Deduction ................................................... 217 14. Dialetheias are Mental Confusions................................................ 233 Bibliography.......................................................................................... 247
Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Preface The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
This book collects together several articles to which a certain RATIO paper relates, with that paper itself as an introduction to the whole. It consists in 13 further chapters, with 11 of the total being much as they have been, or will be when published in journals, and other collections. Two chapters extend work in another 2 published papers, while the remaining one is quite new. Chapter 1: Logic and Grammar. Published in Ratio [2007], Blackwell Publishing. Chapter 2: Completing Russell’s Logic. Adds material from ‘Epsilon Calculi’ published in The Logic Journal of the IGPL 14.4 [2006], 535-590, to a paper of the same name published in the journal Russell, in an issue celebrating 100 years of Russell’s Theory of Descriptions; Mcmaster University 2007. Chapter 3: Logic and Arithmetic. Published in the LOGICA conference proceedings, LOGICA Yearbook 2004, Academy of Sciences, Czech Republic. Chapter 4: Natural Language Sets. Includes and expands on section II of ‘Grammar and Sets’, in the Australasian Journal of Philosophy 84.1 [2006] 59-73, Australasian Association of Philosophy. Chapter 5: Namely Riders: an Update. Published in the Electronic Journal of Analytic Philosophy [2002] (http://ejap.louisiana.edu/ EJAP/2002/Slater.html). Chapter 6: Concept and Object in Frege. A modified version of the paper published in Minerva [2000] (http://www.ul.ie/~philos/vol4/ index.html). Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Barry Hartley Slater – The De-Mathematisation of Logic
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Chapter 7: Frege’s Hidden Assumption. Published in Critica 38.113 [2006] 27-37 Universidad Nacional Autónoma de México 2006. Chapter 8: A Poor Concept Script. Appends a disproof of The FixedPoint Theorem to the article of the same name published in the Australasian Journal for Logic [2004] (http://www.philosophy. unimelb.edu.au/ajl/2004/2004_4.pdf). Chapter 9: Motivation by De Se Beliefs. Published in Noetica [2002] (http://www2.psy.uq.edu.au/ CogPsych/ Noetica/Articles/mellor.pdf). Chapter 10: Out of the Liar Tangle. Includes and expands on an article with the same title in S. Rahman, T. Tulenheimo, and E. Genot (eds) Unity, Truth and the Liar. The Relevance of Medieval Solutions to Semantic Paradoxes, Springer, Berlin, 2007, Springer Science + Business Media B.V. Chapter 11: Ramseying Liars. Published in Logic and Logical Philosophy 84 [2004] 57-70. Chapter 12: Proving that 2+3=5. Expands on the grammar of ‘that’clauses, and incorporates ‘Hilbert and Gödel versus Turing and Penrose’, from LOGICA Yearbook 2003, Academy of Sciences, Czech Republic. Chapter 13: Harmonising Natural Deduction. This is new, showing how Gentzen-Prawitz’ rules for Classical Logic harmonize, but only through a re-conception of what logical truth is. Chapter 14: Dialetheias are Mental Confusions. To be published by Elsevier in their two-volume work, Handbook on Paraconsistency and Handbook on Negation, edited by W. Carnielli and J-Y. Beziau, [2007].
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I have written a number of articles recently that have a rather remarkable character. They all point out trivial grammatical facts that, at great cost, have not been respected in twentieth century Logic. A major continuous strand in my previous work, with this same character, I will first summarise (for more, see chapter one), to locate the kind of fact that is involved. But then I shall present an overview of the more recent, and more varied points I have made, reprinted in the chapters that follow, which demonstrate the far larger extent of basic grammar that has been overlooked or suppressed. I end this introduction with some remarks about how this phenomenon can have arisen – principally through logicians not being attentive enough to their own language, and occupying themselves, instead, with often quite imaginary languages. The main theme in my research over the past twenty years or so has been the linguistic reading of epsilon terms, and that shows the general line of my interests. Originally, I was very much taken with the work of Geach, on some problems in anaphoric reference. There are extensive structures in natural language – such as linked pairs of sentences like ‘Celia believed there was a woman in the room. But it was a man’ – which clearly cannot be handled using the logic coming down from Frege. Other notable authors working on this problem included Evans, who attempted no formalisation, and, for instance, Saarinen, who tried to work up a quite new one. One hindrance to getting the matter straight is illustrated in the case just given. For the pronoun ‘it’ there involves cross-reference into an intensional construction, which is not possible on Frege’s views about indirect reference in such constructions. A better account, in this respect, was Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Barry Hartley Slater – The De-Mathematisation of Logic
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provided by Russell’s Theory of Descriptions, although the possibility of ‘quantifying in’ with his ‘primary sense’ forms was found difficult to understand by Quine. The Theory of Descriptions, in general, is an integral part of the problem with anaphora, as appeared in the later work of Neale in the area. But it was a variant to Russell’s theory which attracted me, and which eventually led me to the solutions I subsequently proposed to a number of problems in this area. What had caught my eye, even as far back as 1963, was a feature of language that was highlighted more famously by Donnellan in 1966, and which I found in 1982 that Hilbert’s epsilon terms could be used to express. Why does the epsilon account of definite descriptions work? It is largely because of a trivial point about grammar. If we put Russell’s original case in anaphoric form, namely ‘There is a single king of France, and he is bald’, the ‘he’, it is easy to see, is going proxy for ‘The king of France’, and so, firstly, ‘The king of France is bald’ is not the whole remark, but merely its second conjunct. But then, secondly, the referent of its subject term is the object alluded to in the other conjunct, which means, thirdly, that that object exists not just in a world where the antecedent is true, since the cross-reference is secured independently of that. Another way of seeing this last, and major point is to consider the case where the second conjunct is, say, ‘but he might not have been sole king of France’. That shows that the object referred to as ‘The king of France’ does not necessarily have the character attributed to it in that description. By incorporating the antecedent conjunct into his account of ‘The king of France is bald’ Russell came close to confusing reference with description, and objects with facts, and going with this he certainly formed the impression that this proposition could only be about an individual in a world where there was just one king of France. Later ‘pre-suppositionists’, while opposing Russell, held even more firmly to this opinion. But all this was despite it being a known logical truth, in Russell’s logic, that (∃x)(x=a), for any individual a, which entails that such an individual has necessary existence. What was missing in this logic were Russell’s ‘logically proper names’, which he could only describe informally; and it is such names that Hilbert’s epsilon terms formalise. In particular they enable one to replace Russell’s quantificational form Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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(∃x)((y)(Ky ≡ y=x) & Bx) The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
with an equivalent one about an individual, (y)(Ky ≡ y=k) & Bk,
with k=εx(y)(Ky ≡ y=x), and so see the above points formally. For it is ‘Bk’ which then represents the whole of ‘the king of France is bald’, and while it is contingently true that (y)(Ky ≡ y=k), it is necessarily true that (∃x)(x=k). These results enable us to acknowledge that we can talk about fictional objects just as well as factual ones, and indeed that the same logic applies to each of them (Slater [2006c]). Fictional objects simply do not live up to the description of them, in the actual world: the Gold Mountain, for instance, is an individual in all worlds, but only in some other world is it both gold and a mountain. In this world ‘The Gold Mountain’ is a misnomer, like Frege’s example ‘The Morning Star’. It is very ironic that Frege used this ‘Millian’ name so extensively in his illustrations, for its reference is not determined by its sense, as his theory required, because, of course, Venus is a planet. This possibly-non-descriptive feature of referential phrases has many consequences. For instance, it gives an immediate resolution of Berry’s Paradox, and related paradoxes, such as those discovered by Simmons (Slater [2005b]). The detail of the development of the above line of thought, however, is not what I want to concentrate on in this introduction, merely its general character. My concern was with the application of a certain well-established formal logic — Hilbert’s Epsilon Calculus — to making more apparent grammatical structures and mechanisms in ordinary speech that had previously not drawn notice. Not all formal logicians have such a close interest in natural language, and not all students of natural language look at it formally. Hale and Wright, however, are two of the few contemporary philosopher-logicians with comparable interests, and I have written on matters close to their work in a number of recent papers (and see here, chapter two). Thus I have expanded my previous work to discuss the formalisation of mass and count terms, and shown its relevance to Frege’s attempt to provide a logical foundation for Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Arithmetic. In natural language we distinguish ‘it is gold’ from ‘it is a ring’, for instance, but no such discrimination is present in Frege’s concept script. Predicate Logic, therefore, covers both cases, but only in the latter case do numbers arise: there can be an amount of some stuff but not a number of it, and so nothing about numbers follows from Predicate Logic. In particular, ¬(∃x)Px
is not equivalent to Nx:Px = 0,
as Frege thought, since when ‘P’ is not count ‘the number of Ps’ is not defined. Hale and Wright are more aware than most of the count/mass distinction, but have payed no attention to this consequence of it. In Slater [2003], [2005a], [2005c], [2006a], I have expanded on this kind of point to provide a grammatical critique of settheoretic analyses of number, and indeed of mathematical sets in their own right. Thus it is grammatical nonsense to say things like ‘{} = 0’, since what is true instead is that Nx:(x ∈ {}) = 0. And that point also shows that there are more things to be considered in Arithmetic than omega sequences, as Structuralists would conclude, since there are (lo and behold!) numbers, as well. In addition, a pair of apples, for instance, is not a further object besides the apples, as Set Theory has presumed. It is no more a further object than the half is in a half of a loaf of bread. ‘Pairs’ and ‘halves’ denote units of measure, and do not describe objects in their own right. Furthermore, groups of physical things are not ‘sets’ in the sense of Set Theory. For collective nouns like ‘tribe’, ‘shoal’, ‘crowd’, etc. describe mereological sums – scattered objects that move around in space just like their members. Their members are thus physical parts of a further physical whole, and the relation between them is not ‘set-membership’ in the mathematical sense. I develop the consequences of this point in chapter three. The general combination of logical and linguistic interests that I share with Hale and Wright I also shared with Prior, whose work on Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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intensional constructions, for quite some time, was a major inspiration. (I defend his theory of indirect speech in chapter four, showing in particular how it bears against Fodor’s Language of Thought, and is supported by recent work of Asher and Kamp. I there recall Ryle’s analysis of Heterologicality, but broaden the discussion to comparable analyses of other puzzles about self-reference. Such matters have a crucial bearing on the debate between representational and non-representational theories of mind, as will be explained). But my recent work has also departed from Prior’s line of thinking in one grammatical respect. For Prior tried to analyse locutions like ‘a believed that p’ as having the form ‘Bap’. This follows traditional Modal Logic in which ‘It is possible that p’, for instance, is commonly symbolised ‘Mp’. Prior, however, notably resisted the formalisation of such remarks in another way, as involving relations to, or properties of, propositions. Thus he argued against dividing ‘a believes that p’ into ‘a believes’ and ‘that p’, rather than ‘a believes that’ and ‘p’. Davidson also argued for the latter kind of construal. Neither philosopher, therefore, was interested in the non-cleft forms of intensional and modal expressions, i.e. equivalent sentences like ‘that p was believed by a’, ‘that p is possible’. It took Kneale, and Cocchiarella, in more recent times, to provide appropriate symbols for the ‘that’ in ‘that p’, i.e. for the grammatical element that converts a sentence into the kind of noun phrase which is embedded as the subordinate clause in intensional and modal constructions. There is a resemblance between Kneale’s and Cocchiarella’s symbols, namely ‘§’ and ‘λ’, and Montague’s cap symbol ‘^’. But Montague Grammar is too closely wedded to Fregean notions like sentences referring to truth-values, and identicals not being substitutable salva veritate in intensional contexts, to be entirely satisfactory. The possibility of identicals being substitutable salva veritate in intensional constructions becomes evident once the proper theory of descriptions is employed, as I had shown in my previous work. (It is an immediate consequence of the constancy of the domain, because of the necessary existence of its members.) If the ancients believed the Morning Star did not set in the evening, although it does so set, then they believed the Evening Star did not set in the evening. For the ‘it’ in the last sentence refers to the Morning Star, and so also to the Evening Star, since they are the same. Maybe, though, it wasn’t Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Barry Hartley Slater – The De-Mathematisation of Logic
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exactly the Morning Star the ancients believed not to set in the evening, but instead something whose identity is not clear. In that case, of course, there is no possibility of substitution of identicals. And do sentences refer to truth-values? That is a grammatical question that has been the focus of some of my more recent studies. It is taken up in chapter four. Certainly the nominalisation of a sentence is a referring expression – that is just what ‘nominalising a sentence’ is for. But the nominalisation of a sentence is not that sentence itself, and so, in particular, we must be careful to distinguish sentences from ‘that’-clauses. Remarkably, like Kneale, and Cocchiarella, subsequently, Frege did have originally a symbol for something like ‘that’, i.e. his ‘horizontal’, but he still could not see that, while ‘that p’ was referential, ‘p’ alone was not. Maybe that was because he also took predicates to be referential, since he took the reference of the whole sentence to be formed from the references of its parts. But it is the nominalisations of predicates, again, which are referential: it is ‘being a horse’ that refers to a concept, while ‘is a horse’ does not. Thinking otherwise got Frege into his famous problem about the concept of being a horse. For he thought concepts were unsaturated, and were what were referred to by predicates in sentences. Certainly such predicates are unsaturated, needing subject terms to form a complete thought; but they are not referential, and what is referential in the area, namely their nominalisations, straightforwardly refer to certain saturated things, namely those abstract objects which are concepts. What Frege primarily lacked, in this area, therefore, was some symbol for the nominalisation of a predicate. It is the lack of any symbol for sentence nominals, in the mainline tradition that followed him, which, maybe tempts some to still read the ‘⊃’ in ‘p ⊃ q’ as ‘materially implies’, despite Quine’s strictures. For ‘implies’ is a verbal relation requiring noun phrases on either side of it, as in the form ‘that p materially implies that q’, but ‘⊃’ is a propositional connective, requiring, instead, used sentences adjacent to it, as in the case just given. The point demonstrates how elementary, yet how significant the piece of grammar is which is at the heart of this matter, at the same time as illustrating how possible is lack of attention to that piece of grammar. Maybe the distinction between connectives and verbs is often enunciated as a principle, but Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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in practise, in such a common area as the reading of ‘⊃’, this principle sometimes gets forgotten, or not applied. Another, very similar case will show how such inattention to basic grammar was even what led to the most celebrated trouble with Fregean logic. Here, again, many would be likely to acknowledge in principle the relevant bit of grammar, although the full application of that principle has evidently not been pursued. I refer, now, to the difference between identity and equivalence. The expression for an identity, like ‘a=b’, is between two names, whereas the expression for an equivalence, like ‘p ≡ q’, is between two used sentences. Surely everyone knows that! But then why do they not object to Frege’s identification of concepts with functions? Certainly one can say ‘Pa ≡ T’, and ‘Rab ≡ F’, where ‘T’ is some tautology and ‘F’ some contradiction, but these do not have the forms ‘f(a)=T’ and ‘g(a, b)=F’, where ‘T’ and ‘F’ abbreviate the referring phrases ‘the true’ and ‘the false’. For equivalence is not identity (and sentences do not refer). I have dwelt on further matters in this area in several of the chapters that follow. In fact the development of the proper grammar, together with a closer study of reflexive and other pronouns, then leads to a resolution of Russell’s and related paradoxes, as I go on to show, both in chapter six and chapter seven. What has not been noticed with such paradoxes is that there is a pronoun in the predicates ‘is not a member of itself’, ‘does not apply to itself’, ‘yields a falsehood when appended to its own quotation’. Such variable predicates do not have a direct representation in a context independent language. Certainly one can represent in such a language the whole sentence ‘x is not a member of itself’, for instance, since this is just another way of writing ‘x is not a member of x’. But the whole sentence is not itself the predicate, since nominalisation of it, for instance, produces the state of affairs of x’s not being a member of x/itself, rather than the property of not being a member of x/itself. Furthermore, the sentence ‘x is not a member of x’ is a relational one of the form ‘… is R to …’, not a predicative one of the form ‘… is P’ with constant ‘P’. In particular, the predicate in ‘x is not a member of x’ is not ‘is not a member of a’ with ‘a’ an item with a fixed reference, since the replacement for the pronoun, namely ‘x’, still varies with the subject. Nor is the predicate in ‘x is not a member of x’ the form of the whole sentence, namely ‘y is not a Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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member of y’, since predicates are parts of, and not forms of sentences. If one wants to represent the whole ‘x is not a member of itself’ as involving a predicate of the subject x that predicate is not representable in a context insensitive language. Only a relational analysis is possible in such a language. And what has also been little noticed is that there is no paradox if ‘x is not a member of x’ is analysed relationally. One can locate Frege’s crucial false move that led to the generation of Russell’s Paradox on this basis. The false move originates in Frege’s attachment to Mathematics. For it is well known that it was Russell’s glaring contradiction that alerted Frege to the trouble with his system. But it is less well known that there is less trouble when ‘x is not a member of x’ is analysed relationally, i.e. as saying that <x,x> is a member of {
| y is not a member of z}. For substitution of that set abstract for ‘x’ does not produce a similar contradiction. What Russell’s Paradox shows, in fact, is merely that not all relations between a thing and itself can be a matter of that thing falling under a concept, i.e. having a constant property: ¬(R)(∃P)(x)(Rxx ≡ Px).
But clearly, given a two-valued function f(x,y), one can invariably obtain a function of one variable f(x,x) = g(x). So it was Frege’s analogy between functions and predicates, in ‘Function and Concept’, which led him astray. Predicates are not functions in the required way. First, as we have seen, sentences are not referential terms with the same reference as ‘the true’ or ‘the false’. What one has are equivalences like ‘Pa ≡ T’ and ‘Rab ≡ F’ which are themselves equivalent to ‘Pa’ and ‘¬Rab’ respectively, making ‘Pa’ and ‘Rab’ quite unlike mathematical functions, and ‘T’ and ‘F’ nothing like their values. Certainly we say such things as ‘it is true that Pa’ and ‘it is false that Rab’, but these expressions predicate certain properties of thoughts — they might be represented as ‘T§Pa’, and ‘F§Rab’, with ‘T’ now ‘is true’, and ‘F’ now ‘is false’. So here again there is no reference to truth-values, since as we have seen, predicates are not referring expressions. But Frege’s grammatical inaccuracies about concepts and functions, and about predicates and referring phrases, get dramatically Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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enlarged upon the introduction of reflexive expressions. For not all predicates express a constant concept. If each of A, B, and C shaves D they do the same thing — shave D — but if each shaves himself, or say, in a ring, shaves his neighbour on his left, then they only do the same kind of thing, i.e. what they do merely has a common functional expression: shave f(s) where s is the subject. There might be a contingent equivalence between ‘x shaves himself’ and ‘x is either A or B or C’, if all and only A, B, and C shave themselves. But ‘x shaves himself’ is not logically equivalent to any such setmembership expression, because it’s predicate contains a variable, ‘himself’, allowing there to be ‘paradoxical’ cases where there is no set at all. (Reflexive expressions, of course, are just one kind of indexical, and I go on, in chapter eight, to discuss other grammatical misconceptions with other indexicals such as the spatial and temporal locatives ‘here’ and ‘now’. Some recent thinkers have tried to make out that human motivation is related to the possession of a certain category of indexical belief, by David Lewis called ‘de se beliefs’. This chapter looks at how the matter arises in Hugh Mellor’s work on Time. It aims to show how Mellor’s discussion of Time grammatically misconceives the indexicality in McTaggart’s ‘A-series’ of temporal propositions, by mixing up direct and indirect speech, and how that leads to a misrepresentation of the motivation of human action. A better account of indexicality is to be found in the work of John Perry, but recognising that shows that a much larger criticism of the Fregean tradition on intensions, and indirect speech is also immediately available). In other areas the incorporation of symbols like Frege’s horizontal to represent indirect speech also has considerable significance. Indeed, attention to such distinctions enables Tarski’s theory of Truth to be corrected grammatically, and that has the immediate consequence that well-known propositional paradoxes, like the Liar Paradox, and its kin, are then not a problem, as I explain at length in chapters nine and ten. Related to this, similar points can be made with respect to the syntactic theory of provability, where the comparable puzzles lie in Gödel’s Theorems. Here again, in chapter ten, we find that lack of attention to basic grammar is at the heart of the matter. Specifically, there have been difficulties representing Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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propositional referring phrases of the form ‘that p’, and that has meant that what is to be proved in Mathematics has not been clear. Many essays on the Foundations of Mathematics have held that such a formula as ‘2 + 3 = 5’ is true in the standard model of Arithmetic, whereas what is true there is that 2 + 3 = 5; and formal derivations of the formula ‘2 + 3 = 5’ from other formulas do not suffice to prove that 2 + 3 = 5. Using ‘ “p” ’ as an alternative to ‘§p’, confuses syntactic expressions with their standard readings, and removing that confusion reduces the interest in Gödel’s Theorems in several ways, principally by undermining the nominalistic philosophy that selected derivations of quoted formulas as the paradigm of a mathematical proof. Moreover, unlike with Gödel’s derivability predicate, where the formula ‘p’ is not derivable from ‘Bew “p”’, from the fact that it is provable that p it does follow that p, and that shows that the consistency of provability is immediate. These points are intimately related to Wittgenstein’s views on the Foundations of Mathematics, I believe, and not just in connection with his neglect of Gödel’s results. Is it believable, though, when it has such enormous consequences, that inattention to such obvious and trivial pieces of grammar as the difference between ‘ “p” ’ and ‘§p’, is rife, even among learned and careful men? Here are just two recent cases where there has been inattention to this instance of the otherwise well enunciated difference between use and mention. First there is Stephen Read (Read [2007a] sec. 1), who gives Tarksi’s T-scheme as: (T) x is true if and only if p,
where what replaces ‘x’ is a name of a sentence whose translation into the metalanguage replaces ‘p’. But Read then goes on to quote Horwich as affirming (T) as a truism, when saying: for any declarative sentence ‘p’ our language generates an equivalent sentence ‘The proposition that p is true’.
Read adds further Tarski did not propose (T) as a definition of truth, though others, e.g. Horwich, have done so since. They all describe (T) as a truism.
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But Read is not the only learned and careful man who confuses use with mention in this area. There is also John Burgess, for instance, who says (Burgess [2005b], 205): First, truth and falsehood conform to a disquotation scheme, according to which it is true that p if and only if p, and false that p if and only if not p. Second, as an immediate consequence of this first feature plus the classical principle of the excluded middle, according to which we always have either p or not p, truth conforms to the principle of bivalence, according to which it is always either true that p or false that p.
(N.B. ‘it is true that p’ does not involve quotation). How can these kinds of thing have gone on, in such a wide range of places, over such a length of time, and with such a number of learned and careful people? Confusing use with mention, identity with equivalence, predicates with functions, predicates with their nominalisations, sentences with nominals, sentences with propositions, properties with states of affairs, direct speech with indirect speech, reference with description, objects with facts, pronouns with nouns, mereological sums with (mathematical) sets, numbers with sets, count terms with mass terms, amounts with numbers, units of measure with objects — and propositions with functions, propositions with sets, as we shall see — displays a major failure in the disposition of the logicians who have followed on from Frege. But how has this general phenomenon come about? One prime reason surely relates to the mathematisation of Logic in the same period. Not only was Frege primarily a mathematician, but also there has developed since his time a branch of the subject now called ‘Mathematical Logic’, which has generated extensive results in Meta-theory as well as Set Theory. We have seen how it was Frege’s orientation towards Mathematics that led him to confuse predicates with functions, but there is also a well-known general disjunction between mathematical talent and verbal ability, which easily can lead, for instance, to the acceptance of malformed identities like ‘{}=0’ without a moment’s question. Intelligence is clearly not linked to linguistic competence; and the downside of that is that very brainy people can easily have blind spots in areas like grammar. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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The belief that professionalism in Logic requires advanced mathematical capacity is to be found in several ‘theories of propositions’ I have not discussed before in print. An inspection of these ‘theories’ will serve very well to demonstrate why the required capacity is instead the ability just to speak in natural language. One group of these theorists, for instance, says that propositions are functions from possible worlds to truth-values. That might sound impressively professional until one reflects that a possible world cannot be given independently of the propositions that are true in it. Maybe there are other worlds in which the proposition that p is true, but that does not mean one can first give those worlds in some way, and then work out, via some independently presentable function, whether the proposition that p is true in them. Furthermore, if the proposition were the function then that function of worlds would have to be also what was true in the appropriate worlds, when it is ungrammatical to say function is true. For a given proposition one might define a function f such that f(wi)=1 if V(p, wi)=1,
and f(wi)=0 if V(p, wi)=0,
but there is no way, then, that the proposition is f, since ‘V(f, wi)=1’ does not make sense. In another tradition it is said that propositions are sets: a proposition is said to be the set of worlds in which it is true. This kind of definition might be appealing to somebody who wants to believe the Ontology of the world consists solely of sets, a thesis that was widely held, at one time. But the set definition suffers from the same problem as the function one. For the set of worlds in which something is true is not itself something with a truth value which varies from one world to the next, i.e. V({wj | V(p, wj)=1}, wi )=1,
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does not make sense. Indeed there is another major problem in this case, because ‘the proposition that p is the set of worlds in which the proposition that p is true’ is circular, and so defines nothing. A third ‘theory’ of propositions is even well known nonsense. For other set-theorists want to say that an elementary proposition, for a start, is an ordered set with n objects and an n-place relation as members: n
.
This expression evidently has the advantage, for some, that it is in the currently most respected formal language; but it has the overriding disadvantage, should anyone care to notice it, that nothing in that language is in the right category to do the job. Not only, in addition to sets are there numbers; also, in addition to them, there are propositions! Why does one need something in a different category? Because no set can be stated; in particular, one cannot state . One can state, for instance, that Dobbin is a horse, but one cannot state . The problem with the latter entity has even been given a name, it is so celebrated: it is ‘The Problem of the Unity of the Proposition’. But the now century-old question of how, using a list of names, one can state or propose something has not impeded the repeated enunciation of this third ‘theory’. Indeed, from within none of the above mathematical traditions has the reality check of a comparison with expressions of the form ‘that p’ been attempted. Nowhere in them is such an entity as that Dobbin is a horse discussed. Such an entity is therefore not recognised as a possible fact, i.e. as just what the various ‘theories’ are supposed to be about. And so we see why I have put the word ‘theory’ in scare quotes in all of the above. For once what the ‘theories’ are supposed to be about is recognised, it is immediately evident that no ‘theory’ of those things was what was needed, merely a non-mathematical, natural language in which to identify them. That leads us to the most important reason, I believe, why there has been such a massive blindness, in the area. For the main reason, surely, is that the points I have drawn attention to involve certain aspects of natural language, and a division has arisen, over the last 100 years, between a large portion of ‘Logic’ and the study of that. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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The interest in Meta-theory, and particularly Tarskian Semantics, is a major cause of this division, since it has distanced the language studied from the language used to study it, and so has led not only to inattention to that used language, but also encouraged entirely idle attention to quite unusable languages. And that has led to the logicians’ own language, and thereby the most relevant part of their public behaviour, seeming to become immune from inspection, and criticism, even relevance. Isn’t the application of such a form as ‘“p” is true’ required for us to understand ‘the object language’ from our ‘meta-linguistic’ perspective? Not if the ‘object language’ is our Mother Tongue, and the understanding of it is instead provided though use. Davidson used Tarski’s T-scheme as a basis for a theory of interpretation, but when assessing as true something said in one’s own language the meaning is already given. Hence the operative form of the judgement is ‘it is true that p’ or ‘that p is true’. The problem has been formulating reference to the meaning of ‘p’. For it is not anything of the syntactic form ‘ “p” ’ that is being judged, but its immediate semantic reading, ‘that p’. What else, in particular, could be necessarily true? Certainly not some syntactic form. When judging some argument, like Disjunctive Syllogism, to be valid, one is not assessing bare formulas like ‘(¬p & (p v q)) ⊃ q’
for some quality. Instead one is asking about a certain interpretation of them, and it is that interpretation of them which is necessarily true. The use-mention confusion involved in thinking otherwise has been a major feature of recent ‘logic’, as I explain in detail in chapters twelve and thirteen. Many logicians in the twentieth century were very much concerned with what they saw as alternatives to ‘classical logic’, starting with the Intuitionists who said they doubted the Law of the Excluded Middle. Certainly the formula ‘p v ¬p’ is not a provable formula in their system, but that casts no doubt onto the classical law. For ‘The Law of the Excluded Middle’ refers not to the formula, but to its classical interpretation. Likewise with ‘The Law of Non-Contradiction’, which has been extensively misunderstood, even by the bulk of the intellectual world recently in the area (see Priest et al [2004]). One might not know the interpretation of some symbol, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
and for that, or other reasons, have doubts about whether some formula expresses a necessary truth, but that is not a matter of doubting the necessary truth itself. The problem in getting this clear has been, again, the lack, within formal logic, of appropriate denoting phrases for what it is that is necessary, although just such are the truths of Logic. The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
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23 1. Logic and Grammar
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The epsilon calculus improves upon the predicate calculus by systematically providing complete individual terms. Recent research has shown that epsilon terms are therefore the ‘logically proper names’ Russell was not able to formalise, but their use improves upon Russell’s Theory of Descriptions not just in that way. This chapter details relevant formal aspects of the epsilon calculus before tracing its extensive application not just to the theory of descriptions, but also to more general problems with anaphoric reference. It ends by contrasting a Meinongian account of cross-reference in intensional constructions with the epsilon account. In Russell’s theory of definite descriptions there are, it will be remembered, three clauses: with ‘The King of France is bald’ these are ‘there is a king of France’, ‘there is only one king of France’ and ‘he is bald’. Russell used an iota term to symbolise the definite description, but it is not an individual symbol: it is an ‘incomplete’ term, as he explained it, since ‘The King of France is bald’ is taken to have the complex analysis, (∃x)(Kx & (y)(Ky ⊃ y=x) & Bx),
and so it does not have the elementary form ‘Bx’. Russell hypothesised that, in addition to the linguistic expressions gaining formalisations by means of his iota terms, there was another, quite distinct class of expressions, which he called ‘logically proper names’. Logically proper names would, amongst other things, take the place of the variable in such forms as ‘Bx’. Russell suggested that
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demonstratives might be in this class, but he could give no further formal expression to them. Hilbert and Bernays, in their Grundlagen der Mathematik, introduce a kind of complete symbol, by contrast with Russell, defending what would later be called a ‘pre-suppositional theory’ of definite descriptions. The first two clauses of Russell’s definition, (∃x)(Kx & (y)(Ky ⊃ y=x)),
are not taken, by pre-suppositionalists, to be part of what is asserted by ‘The King of France is bald’; they are, instead, the conditions under which one is allowed to introduce into the language an individual term for ‘the King of France’, which then satisfies the matrix of the quantificational expression above, and becomes a proper symbol to replace the variable in such expressions as ‘Bx’. Hilbert and Bernays still used an iota term for this purpose, although it is quite different from Russell’s iota term, since, when it is part of the language, it is equivalent to the related epsilon term. It has been realized, more recently, that epsilon terms, being complete symbols, are the ‘logically proper names’ Russell was looking for, and that their natural reading is indeed as forms of demonstratives. It is at the start of book 2 of the Grundlagen that Hilbert and Bernays introduce epsilon terms. They first go on to produce a theory of non-definite descriptions of the same pre-suppositional sort to their theory of definite descriptions. Thus they permit an eta term to be introduced into the language if the first of Russell’s conditions is met, ‘(∃x)Kx’, this term then satisfies the associated matrix, but it is, in general, an individual, pre-suppositional term of the same kind as their iota one. There is a singular difference in certain cases, however, since the pre-supposition of the eta term can be proved conclusively, for certain matrices. Thus we know, for any predicate ‘F’, that (∃x)((∃y)Fy ⊃ Fx),
since this is a theorem of the predicate calculus. The eta term this theorem permits us to introduce is what Hilbert and Bernays call an epsilon term. Thus we get the epsilon axiom Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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which therefore implies (∃y)Fy ≡ FεxFx.
So an epsilon term is very unlike the generality of eta terms, since it’s introduction is clearly not dependent on any contingent facts about F. It is this that permits completely formal theories using epsilon terms to be developed, because such epsilon terms, unlike Hilbert and Bernays’ iota terms, are always defined, and, as the equivalence indicates, they refer to exemplars of the property in question. The above predicate calculus theorem, in other words, provides the existence condition for certain objects, which the various epsilon calculi then go on to symbolise reference to, using epsilon terms. Copi has explained the theorem’s relation with exemplars very fully (Copi [1973], 110). Kneebone read epsilon terms as formalising indefinite descriptions (Kneebone [1963], 101), and this idea is commonly also found in the work of his pupil, Priest, although strangely Priest himself has pointed out that reading ‘(∃x)(Gx.Fx)’. as ‘GεxFx’ will not do (Priest [1979b], 6), see also (Slater [1989a], 285). Hilbert read the epsilon term in the above case ‘the first F’, which indicates its place in some, otherwise unspecified well-ordering of the F’s — for instance, in connection with arithmetical predicates, that generated by the least number operator. So ‘εxFx’ is not ‘an F’. Moreover, as Copi’s discussion makes very clear, it is possible that an epsilon term refers to something which is in fact not F — it does this, of course, if there are no F’s at all — and that will lead us to theories of reference which materialised only in the 1960s and later, when reference came to be properly distinguished from attribution. If there are F’s then the first F is a chosen one of them; but if there are no F’s then ‘the first F’ must be non-attributive, and so denotes something it cannot connote. It functions like a Millian name, in other words, with no applicable sense. With denotation in this way clearly distinguished from description we can then start to formalise the cross-reference that even Russell needed to link his first two conditions ‘There is one and only one king of France’ with his further condition ‘He is bald’. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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For, by an extension of the epsilon equivalent of the existential condition, the ‘he’ in the latter comes to be a pronoun for the same epsilon term as arises in the former — whether or not the former is true. And such anaphoric cross-reference in fact may stretch into and across intensional contexts of the kind Russell was also concerned with, such as ‘George IV wondered whether the author of Waverley was Scott’. For, of course, he was indeed Scott, and we may all now know very well that he was Scott. So we obtain a formalisation for transparency in such locutions. That puts developed epsilon calculi at variance with Fregean views of intensional contexts — and also the Kripkean semantics that has continued to support Frege in this area. But Fregean intensional logic did not incorporate Millian symbols for individuals, and in particular, as we shall see in detail later, that meant that it could not clearly distinguish individuals from their identifying properties. The addition of epsilon terms provides the facility for separating, for instance, s = εx(y)(Ay ≡ y=x),
and (y)(Ay ≡ y=s),
and so for isolating the proper object of George IV’s thought. It is through missing this distinction, also, that belief in the possible variability of the domain of individuals, between different possible worlds, arises. For the author of Waverley’s existence is conflated with the instantiation of his defining properties, leaving the pronoun ‘his’ just used, and its descriptive replacement, ‘the author of Waverley’, unformalisable as replacements for an individual variable. Indeed, an object gets confused with a fact. But we can suppose that the author of Waverley did not author Waverley, so his possession of that property must be separate from his identity. When one begins to investigate the natural language meaning of epsilon terms, and the way they formalize descriptive replacements for such pronouns as the ‘his’ just used, it is significant that Leisenring, writing in 1969, merely notes the ‘formal superiority’ of Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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the epsilon calculus, comparing some of its pedagogic features with the comparable ones in the predicate calculus (Leisenring [1969], 63). Apparently its main value, in Leisenring’s day, was that it could prove all that was provable in the predicate calculus but in a smarter, and less tedious way. Epsilon terms, for Leisenring, were just clever calculating instruments. Evidently there is much more to the epsilon calculus than this, but until more recent times only the natural language meaning of the above epsilon axiom has been dwelt upon. There are a couple of further theorems within the epsilon calculus, however, which will show its extended range of application: they are about the nature and identity of individuals, as befits a calculus which systematically provides a means of reference to them. The need to provide logically proper names for individuals only became generally evident some while after Russell’s work on the theory of descriptions. The major difficulty with providing properly referential terms for individuals, in classical predicate logic, is what to do with ‘non-denoting’ terms, and Quine, following Frege, simply gave them an arbitrary, though specific referent. The approach was formalised perhaps most fully by Kalish and Montague, who gave the two rules (Kalish and Montague [1964], 242-3): (∃x)(y)(Fy ≡ y=x) / FιxFx, ¬(∃x)(y)(Fy ≡ y=x) / ιxFx = ιx¬(x=x),
where, in explicitly epsilon terms, we would have ιxFx = εx(y)(Fy ≡ y=x).
Kalish and Montague were of the opinion, however, that their second rule ‘has no intuitive counterpart, simply because ordinary language shuns improper definite descriptions’ (Kalish and Montague [1964], 244). And certainly, in that period, the revelations that Donnellan was to publish about non-attributive definite descriptions (Donnellan [1966]), were not well known. But ordinary language does not, we now know, avoid non-attributive definite descriptions, although their referents are not as constant as Kalish and Montague’s second rule requires. In fact, by being improper their referents are not fixed by semantics at all: like demonstratives the referents of logically proper Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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names are found only in their pragmatic use. Stalnaker and Thomason were more appropriately liberal with their complete individual terms. And these referential terms also had to apply, they knew, in every possible world (Thomason and Stalnaker [1968], 363). But a fuller coverage of identity and descriptions, in modal and general intensional contexts, is to be found in (Routley, Meyer and Goddard [1974]), and also Hughes and Cresswell [1968]). With these Australasian thinkers we find the explicit identification of definite descriptions with epsilon terms (e.g. Hughes and Cresswell [1968], 203). Which further theorems in the epsilon calculus are behind these kinds of identification? There is one theorem in particular which demonstrates strikingly the relation between Russell’s attributive, and some of Donnellan’s non-attributive ideas (see Slater [1988]). For (∃x)(Fx & (y)(Fy ⊃ y=x) & Gx)
is logically equivalent to (∃x)(Fx & (y)(Fy ⊃ y=x)) & Ga,
where a = εx(Fx & (y)(Fy ⊃ y=x)). For the latter is equivalent to Fa & (y)(Fy ⊃ y=a) & Ga,
which entails the former. But the former is Fb & (y)(Fy ⊃ y=b) & Gb,
with b = εx(Fx & (y)(Fy ⊃ y=x) & Gx), and so entails (∃x)(Fx & (y)(Fy ⊃ y=x)),
and Fa & (y)(Fy ⊃ y=a).
But then, from the uniqueness clause, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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and so Ga,
making the former entail the latter. The former expression, as we have seen, encapsulates Russell’s Theory of Descriptions, in connection with ‘The F is G’; it involves the explicit assertion of the first two clauses, to do with the existence and uniqueness of an F. A pre-suppositional account like that in Hilbert and Bernays, which was later popularised by Strawson, would not involve the direct assertion of these two clauses: on a presuppositional account they form the precondition without which ‘the F’ cannot be introduced into the language. But both of these accounts forget the use we have for non-attributive definite descriptions. Since Donnellan (and see Slater [1963]), we now know that there are no preconditions on the introduction of ‘the F’; and ‘The F is G’, as a result, may always be given a truth-value. Hence ‘Ga’ properly formalises it. If the description is non-attributive, i.e. if the first two clauses of Russell’s account are not both true, then the referent of ‘The F’ is simply up to the speaker to nominate. But one detail about Donnellan’s actual account must be noted at this point. He was originally concerned with definite descriptions which were improper in the sense that they did not uniquely describe what the speaker took to be their referent. And on that understanding the description might still be ‘proper’ in the above sense — if there still was something to which it uniquely applied. Specifically, Donnellan would originally allow ‘the man with martini in his glass’ to refer to someone without martini in his glass whether or not there was some unique man with martini in his glass. But someone talking about ‘the man with martini in his glass’ can be rightly taken to be talking about who this phrase describes, if it does in fact describe someone — Devitt and Bertolet pointed this out in criticism of Donnellan (Devitt [1974], Bertolet [1980]). It is this latter part of our linguistic behaviour which the epsilon account of definite descriptions respects, for it permits definite descriptions to be referring terms without being attributive, but only so long as nothing Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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has the description in question. Hence it is not the first quantified statement above, but only, so to speak, the third part of it extracted which makes the remark ‘The F is G’. This becomes plain when we translate the two statements using relative and personal pronouns: There is one and only one F, which is G, There is one and only one F; it is G.
For ‘it’ here is an anaphoric pronoun for ‘the (one and only) F’, and it still has this reference even if there is no such thing, because that is just a matter of the grammar of the language. Now the uniqueness clause is required for two such statements to be equivalent — without it there would be no equivalence, as we shall see — and that means that the relative pronoun ‘which’ is not itself equivalent to the personal pronoun ‘it’. So it was because Russell’s logic could not separate the (bound) relative pronoun from the (unbound) personal pronoun that it could not formulate the logically proper name for ‘it’, and instead had to take the whole of the first expression as the meaning of ‘The F is G’. Using just the logic derived from Frege, it could not separate out the cross-referential last clause. But how can something be the one and only F ‘if there is no such thing’? This is where a second theorem in the epsilon calculus is relevant: (Fa & (y)(Fy ⊃ y=a)) ⊃ (a = εx(Fx & (y)(Fy ⊃ y=x))).
For the singular thing is that this entailment cannot be reversed, so there is a difference between the left hand side and the right hand side, i.e. between something being alone F, and that thing being the one and only F. We get from the left hand side to the right hand side once we see the left hand side entails (∃x)(Fx & (y)(Fy ⊃ y=x)),
and so Fεx(Fx & (y)(Fy ⊃ y=x)) & (z)(Fz ⊃ z=εx(Fx & (y)(Fy ⊃ y=x))). Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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By the uniqueness clause we get the right hand side. But if we substitute ‘εx(Fx & (y)(Fy ⊃ y=x))’ for ‘a’ in the initial conditional then the right hand side of it is necessarily true. But the left hand side is then equivalent to (∃x)(Fx & (y)(Fy ⊃ y=x)),
which is, in general, contingent; hence it is different from the right hand side. The difference, however, is not available in Russell’s logic. In fact Russell confused the two forms, since possession of an identifying property he formalised using the identity sign a = ιxFx,
making it appear that some, maybe even all identities are contingent. But all proper identities are necessary, and it is merely associated identifying properties which are contingent. Ironically, Frege used a complete term for definite descriptions in his extensional logic, as was mentioned before. But Russell explicitly argued against the arbitrariness of Frege’s definition, in the case where there isn’t just one F, when setting up his alternative, attributive theory of descriptions, in ‘On Denoting’. Had Frege’s complete term been more widely used, and, for a start, been used in his intensional logic, results like those above might have been better known earlier. Hughes and Cresswell, at least, appreciated that in addition to ‘contingent identities’ there were also necessary identities, and differentiated between them as follows (Hughes and Cresswell [1968], 191): Now it is contingent that the man who is in fact the man who lives next door is the man who lives next door, for he might have lived somewhere else; that is living next door is a property which belongs contingently, not necessarily, to the man to whom it does belong. And similarly, it is contingent that the man who is in fact the mayor is the mayor; for someone else might have been elected instead. But if we understand (‘The man who lives next door is the mayor’) to mean that the object which (as a matter of contingent fact) possesses the property of being the man who lives next door is identical with Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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the object which (as a matter of contingent fact) possesses the property of being the mayor, then we are understanding it to assert that a certain object (variously described) is identical with itself, and this we need have no qualms about regarding as a necessary truth. This would give us a way of construing identity statements which makes ((x=y) ⊃ L(x=y)) perfectly acceptable: for whenever x=y is true we can take it as expressing the necessary truth that a certain object is identical with itself.
There is more hangs on this matter, however, than Hughes and Cresswell appreciated. For now that we have the logically proper names, i.e. complete symbols to take the place of the variables in such expressions as ‘x=y’, not only do we see better where the contingency of the properties of such individuals comes from — just the linguistic possibility of improper definite descriptions — we also see, contrariwise, why constant epsilon terms must be rigid — because identities involving such terms are necessary. Frege, for instance, thought that we could not derive ‘a believes the Morning Star is illuminated by the sun’ from ‘a believes the Evening Star is illuminated by the sun’, even though the Morning Star is the Evening Star. But (see, for instance, (Slater [1992b]), from BaIεxEx
we can derive BaIεxMx,
if εxEx=εxMx;
what we cannot derive is Ba(∃x)((y)(My ≡ y=x) & Ix)
from Ba(∃x)((y)(Ey ≡ y=x) & Ix), Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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even if The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
(∃x)(∃y)(Mx & Ey) & (x)(y)((Mx & Ey) ⊃ x=y).
Russell improved matters somewhat, by distinguishing a primary, transparent sense (∃x)((y)(Ey ≡ y=x) & BaIx),
from the secondary, opaque sense Ba(∃x)((y)(Ey ≡ y=x) & Ix),
since the former, with (∃x)(∃y)(Mx & Ey) & (x)(y)((Mx & Ey) ⊃ x=y),
does entail (∃x)((y)(My ≡ y=x) & BaIx).
But without epsilon terms to provide explicit instantiations of the primary-sense forms, Russell was in no position to detach their second conjuncts. It follows that there is no essential grammatical difference between such an intensional anaphoric remark about someone’s mind, as The ancients believed there was a star in the morning that was illuminated by the sun. But it was a planet.
i.e. Ba(∃x)(Mx & Ix) & PεxBa(Mx & Ix),
and the extensional cross reference, for instance, in There was a man in the room. He was hungry.
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(∃x)Mx & HεxMx.
What has been the problem, fundamentally, has been getting the cross-reference formalised first of all in the purely extensional kind of case. Yet this just requires extending the epsilon replacement for an existential statement, by means of a repetition of the associated epsilon term, as was mentioned with respect to ‘he’ in Russell’s case, before. The only difference in the intensional case is that, to obtain the required cross-referencing one must move from Ba(∃x)(Mx & Ix)
to (∃x)Ba(Mx & Ix)
via Ba(Mb & Ib)
with b=εx(Mx & Ix) to get a public referential phrase for the object. And note that, while the required epsilon term ‘εxBa(Mx & Ix)’ is then defined intensionally, it still refers to a straightforward extensional object — the planet Venus, of course. It is now better understood how the epsilon calculus allows us to do this (see Slater [1986a], Purdy [1994], Egli and von Heusinger [1995], Meyer-Viol [1995], Ch. 6, for instance). The starting point is the possibility illustrated in the theorem about Russellian definite descriptions before, of separating out what otherwise, in the predicate calculus, would be a single sentence into a two-sentence piece of discourse, leaving the existence and uniqueness clauses in one place, and putting the characterising remark in another. The point really starts to matter when there is no way to symbolise in the predicate calculus some anaphorically linked remarks where there is no uniqueness clause, as in the above extensional case. This is what Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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became a problem for the Fregean and Russellian logicians who woke up to the need to formalise anaphoric reference in the 1960s. It can be seen, as before, how it was lack of the epsilon calculus that was the major cause of the difficulty. Thus Geach, in an early discussion of the issue, went to the extremity of insisting that there could be no syllogism of the following form (Geach [1962], 126): A man has just drunk a pint of sulphuric acid. Nobody who drinks a pint of sulphuric acid lives through the day. So, he won’t live through the day.
Instead, Geach said, there was only the existential conclusion: Some man who has just drunk a pint of sulphuric acid won’t live through the day.
Certainly one can only conclude (∃x)(Mx & Dx & ¬Lx)
from (∃x)(Mx & Dx),
and (x)(Dx ⊃ ¬Lx),
within Fregean predicate logic. But one can still conclude ¬Lεx(Mx & Dx),
within its conservative extension: Hilbert’s epsilon calculus. And through inattention to that extension, Geach was entirely stumped later, in (Geach [1967]), when he discussed his famous intensional example (3) Hob thinks a witch has blighted Bob’s mare, and Nob wonders whether she (the same witch) killed Cob’s sow,
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Th(∃x)(Wx & Bxb) & OnKεx[Th(Wx & Bxb)]c.
For he saw this could not be his (4) (∃x)(Wx & ThBxb & OnKxc),
or his (5) (∃x)(Th(Wx & Bxb) & OnKxc).
But a reading of the second clause as Nob wonders whether the witch who blighted Bob’s mare killed Cob’s sow,
(c.f. Geach’s 18) in which ‘the witch who blighted Bob’s mare killed Cob’s sow’ is analysed in the Russellian manner, as Geach’s (20): just one witch blighted Bob’s mare and she killed Cob’s sow,
does not provide the required cross-reference — for one thing because of the uniqueness clause then involved. Of course the descriptive replacement for the personal pronoun ‘she’ in the Hilbertian expression, namely ‘what Hob thinks is a witch that blighted Bob’s mare’, does not have any implication of uniqueness. The inappropriateness of the uniqueness clause in Russellian analyses has been widely discussed. However it did not deter Neale who much later wrote a whole book defending a largely Russellian account of definite descriptions, and cross-sentential anaphora. But he got no further with Geach’s case above than proposing that ‘she’ might be localised to ‘the witch we have been hearing about’ (Neale [1990], 221), thinking in general that definite descriptions merely should be relativised to the context. But a greater change is needed than that. For it is not that, in addition to witches in the actual world, there are also witches we have been hearing about, or in people’s minds, but merely that, in addition to witches in the actual world there are things in the actual world which are said, or believed to be Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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witches. Geach’s ‘the same witch’ is thus also inappropriate, on the same grounds. A large amount of the important, initial work in this area was done by Evans, who was another person very influenced by the Russellian tradition. But Evans also explicitly separated from Russell over the matter of uniqueness, for instance in connection with backreference to a story about a man and a boy walking along a road one day (Evans [1977], 516-7): One does not want to be committed, by this way of telling the story, to the existence of a day on which just one man and boy walked along a road. It was with this possibility in mind that I stated the requirement for the appropriate use of an E-type pronoun in terms of having answered, or being prepared to answer upon demand, the question ‘He? Who?’ or ‘It? Which?’ In order to effect this liberalisation we should allow the reference of the E-type pronoun to be fixed not only by predicative material explicitly in the antecedent clause, but also by material which the speaker supplies upon demand. This ruling has the effect of making the truth conditions of such remarks somewhat indeterminate; a determinate proposition will have been put forward only when the demand has been made and the material supplied.
It was Evans who popularised the name ‘E-type pronoun’ for the pronoun in such cases as A Cambridge philosopher smoked a pipe, and he drank a lot of whisky,
i.e. (∃x)(Cx & Px) & Dεx(Cx & Px).
He also argued at length, in line with the above (Evans [1977], 516), that what was distinctive about E-type pronouns was that such a conjunction of statements as this was not equivalent to A Cambridge philosopher, who smoked a pipe, drank a lot of whisky,
i.e. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Obviously the epsilon account supports this, since the contrast illustrates the point remarked before: only the expression which contains the relative pronoun can be symbolised in the predicate calculus, since to symbolise the personal pronoun its epsilon extension is needed. Some grammarians have tried to handle this sort of issue in intensional contexts by returning to Meinongian ‘intensional objects’, or the ‘counterparts’ of actual individuals in alternative worlds. For example, Saarinen considers the following case (Saarinen [1978], 277): Bill believes that the lady on the stairs (is acquainted with) him, but John knows she is only a wax figure.
About this Saarinen says ‘Both of the attitudes are of the wax lady, and yet all the relevant individuals in the doxastic worlds are not wax ladies but human beings’ (Saarinen [1978], 282). However, as before, in addition to human beings in this world it is not that there are also human beings in people’s minds, merely, also, that there are things in this world which are taken to be human beings, by people. Saarinen supports his judgement with a Russellian, i.e. attributive reading of ‘the lady on the stairs’, but more important is his retention of the Meinongian idea that such an intensional object as the gold mountain has to be made of gold, and even that an impossible intensional object like the round square has to still be both round and square. Thereby, of course, Saarinen misses the possibility that what Bill believes is the one and only lady on the stairs is not really a lady on the stairs. And it is this possibility, exactly, which allows it to be just the plain, and everyday, physical wax figure which is the object of both Bill’s and John’s attitudes. The form of Saarinen’s case, if ‘the lady on the stairs’ is attributive, is Bb(∃x)((y)(Ly ≡ y=x) & Abx) & KjWεxBb((y)(Ly ≡ y=x) & Abx),
but the second conjunct of this entails ¬LεxBb((y)(Ly ≡ y=x) & Abx), Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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since knowledge entails truth, and being a wax figure entails not being a lady. As was mentioned before, it is even contingent that the lady on the stairs is a lady on the stairs, but the source of this non-doxastic, and simply modal contingency, which allows the same object to appear in other possible worlds, cannot be properly seen until we link it with the linguistic possibility of improper, i.e. non-attributive definite descriptions. Seeing the source of this even more radical contingency is thus essentially linked to seeing how there can be de re attitudes. But it is also directly linked with the much more substantial programme of replacing such a metaphysical view as Meinong’s simply with accurate linguistic analysis: ‘philosophical problems arise through misconceptions of grammar’ to paraphrase Wittgenstein. I will now discuss some further work, which is a part of this history, but which in particular, I believe, has helped us to see just why reference in modal and general intensional constructions must be rigid. I will explain how it relates to other formal, epsilon calculus work in this area, by Routley, Meyer and Goddard. It involves a reordering of our views on semantics, as has been evident in a few places already, and it begins outside the area of predicate logic altogether, in a review of the standard semantics of propositional logic. When one thinks of ‘semantics’ one thinks of the valuation of formulas, but it is only since the 1920s that this kind of study has supplemented the traditional kind of proof theory, which before that had dominated Logic. In the truth tables that Post and Wittgenstein introduced, for instance, it may indeed seem we get a quite different thing from any ‘proof theory’. Proof Theory we might, at one time, have associated with axiomatic procedures, for one thing, but certainly, in all its forms, nowadays, with ‘the object language’. The theory of truth which Tarski developed depends centrally upon the separation of object and meta-languages — and in fact we may seem obliged to make such a discrimination because of the threat of contradictions from semantic paradoxes, if there was, by contrast, ‘semantic closure’ (c.f. Priest 1984). Truth may thus appear to be inescapably a meta-concept, and certainly to be a predicate of sentences. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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On the other hand, because of Tarski’s Theorem, we know that what is true, in Tarski’s metalinguistic sense, is not representable. There is a quite different way of looking at these things, however, which is deliberately non-Tarskian, and which others have followed (see Prior [1971], Ch 7, and, for instance, Sayward [1987]). This involves, at the propositional level, taking ‘it is true that’ as the basic truth locution. This is not a meta-predicate of object level sentences, but an object level propositional operator, and its manipulations in truth tables are then just another form of proof procedure - any completeness and soundness results in connection with it merely show that there are two ways of arriving at exactly the same theorems. Use of truth tables proves things, as does use of truth trees; the proofs are not axiomatic ones, but that is another issue. In fact, if there are proofs using the propositional operator notion of truth then this notion is eliminable, producing immediately a non-’semantical’ proof procedure. For operator truth is definable, but eliminable, because of the necessity of p ≡ it is true that p,
since ‘it is true that’ is the null, identity modality. Propositional operators include ‘it is provable that’, which is distinct from Gödel’s provability predicate, as Gödel himself pointed out (Gödel [1969]). They also include ‘it is necessary that’ and ‘it is believed that’, for instance, and the attempt to see these, along with ‘it is true that’, as metalinguistic predicates was a general feature of the philosophy of logic in the early decades of the last century, for instance in Quine’s treatment of modality and intensionality. On that approach, someone would be believing that the Morning Star is in the sky, but the Evening Star is not, if, respectively, they did, and did not assent to the two associated sentences. After Montague’s and Thomason’s work in the 1960’s and 1970’s (Montague [1963], Thomason [1977], [1980]) we are now more convinced that ‘indirect discourse is not quotational’, as Thomason put it, but the earlier generation considered only what words the agent would use. The change requires seeing the subordinate sentence in indirect speech as not in the words of the agent, but instead in the words of the reporter. That is indeed the core of what reported speech is about Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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- putting something into one’s own words rather than quoting them from another source. Thus someone may say Celia believed that the man in the room was a woman,
but clearly that does not mean that Celia believed there was a man in the room who was a woman. And that is because the referential terms in the subordinate sentence are in the mouth of the speaker, and so mean what the speaker means by them. The point shows that Celia believed a woman was in the room, but it was the man there.
contains a transparent intensional locution, with the same object ‘inside’ the belief as ‘outside’ in the room. Here is where the rigid constant epsilon terms come in, in symbolising the cross-reference: Bc(x)(Wx & Rx) & (y)((My & Ry) ≡ y = εxBc(Wx & Rx)).
But to get a full grasp of the matter, we must parallel in the general case the shift from meta-predicate to object language propositional operator that we saw above with truth. Routley, Meyer and Goddard, in 1974, saw that a rigid semantics required treating the ‘x’ in such expressions as ‘BcWx’ as open for quantification. They derived ‘Routley’s Formula’, which in the case of necessity is L(∃x)Fx ⊃ (∃x)LFx.
But now we can see, in a better way than Routley, Meyer and Goddard, how the number of the planets provides a clear illustration of the validity of this formula. The fact that some concepts may not have a number is crucially what must be appreciated, but also it must be remembered how the referential, and attributive senses of individual terms are distinguished. For, with ‘Fx’ as ‘x = εn(ny)Py’ the entailment holds because identities are necessary, making both sides of the formula true. But if ‘Fx’ is ‘(xy)Py’ this is contingent, since it is contingent that there is a number of planets, instead of just planetary material; and that makes both sides of the formula false. Hughes and Cresswell argue against Routley’s Formula, saying (Hughes and Cresswell [1968], 144): Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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... let (Fx) be ‘x is the number of the planets’. Then the antecedent is true, for there must be some number which is the number of the planets (even if there were no planets at all there would still be such a number, viz. 0): but the consequent is false, for since it is a contingent matter how many planets there are, there is no number which must be the number of the planets.
But this overlooks mass terms, which do not involve a unit without some arbitrary nomination, with the consequence, as we have seen, that any number associated with an amount does not number the amount, being related instead to the chosen unit. The attempt to see continua in terms of a fixed number has been central to the development of Analysis, and Set Theory, and it has also influenced advanced logic as well, supporting the set-theoretic semantics of predicate logic. Certainly if the term involved is a count noun then there will be a number (maybe 0) of the F’s then determined. But the number associated with some F does not number any Fs, with the result that the antecedent of Routley’s Formula is not necessarily true. Quine, of course, considered the example of the number of the planets in his famous argument against the possibility of quantification into modal contexts. He said (Quine [1990], 195–197): If for the sake of argument we accept the term ‘analytic’ as predicable of sentences (hence as attachable predicatively to quotations or other singular terms designating sentences), then ‘necessarily’ amounts to ‘is analytic’ plus an antecedent pair of quotation marks. For example, the sentence: (1) Necessarily 9 is greater than 4 is explained thus: (2) ‘9 is greater than 4’ is analytic ... So suppose (1) explained as in (2). Why, one may ask, should we preserve the operatorial form as of (1), and therewith modal logic, instead of just leaving matters as in (2)? An apparent advantage is the possibility of quantifying into modal positions; for we know we cannot quantify into quotation, and (2) uses quotation ... But is it more legitimate to quantify into modal positions than into quotation? For consider (1) even without regard to (2); surely, on any plausible interpretation, (1) is true and this is false: (3) Necessarily the number of major planets is greater than 4. Since 9 = the number of major planets, we can conclude that the position of ‘9’ in (1) is not purely referential and hence that the necessity operator is opaque. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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But Quine fails to distinguish the referential, and necessary ‘the number of the major planets is greater than 4’ from the predicative, and contingent ‘There are more than 4 major planets’. If 9 = εn(ny)Py,
then εn(ny)Py is greater than 4,
but it does not follow that (∃n)((ny)Py & n is greater than 4).
Substitution of identicals in (1), therefore, does yield (3), in its correct interpretation, even though it is not necessary that there are more than 4 major planets. But how does the ‘x’ in such a form as ‘LFx’ come to be open for quantification? In the traditional, meta-linguistic modal semantics (see Hughes and Cresswell [1968], passim) one finds formulas like V(Fx, i) = 1
which are supposed to say that the valuation put on the formula ‘Fx’ is 1, in world i. But there should be quotation marks around the ‘Fx’ in this formula, if that is the intention. By a common convention, they are omitted. To move over to the alternative point of view, one simply reads ‘V(Fx, i) = 1’ as it stands, in which case it becomes the propositional operator form ‘it would be true in world i that Fx’. The ‘Fx’ is then in reported speech rather than quoted speech, and so the ‘x’ is in the words of the speaker, and the meta/object distinction has been discarded. Any variable inside the subordinate sentence can then be quantified over, just like any variable outside it, so there is unproblematic ‘quantifying in’, and all standard predicate calculus, and identity rules apply. The epsilon term ‘εxFx’ is then rigid because, quite generally, p ≡ (V(p, a) = 1), Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Barry Hartley Slater – The De-Mathematisation of Logic
where a is the actual world, and therefore, necessarily The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
(x)(Fx ≡ (V(Fx, a) = 1)),
which gives εxFx = εx(V(Fx, a) = 1),
by a second epsilon axiom. More specifically, suppose that in every world there are F’s, i.e. A: (i)(V((∃x)Fx, i) = 1),
then we know, by the necessary equivalence produced by the epsilon definition of the existential quantifier, that B: (i)(V(FεxFx, i) = 1),
and that means that C: (i)(∃x)(V(Fx, i) = 1),
i.e. that in every world there is an object such that it is F. But that means, using the definition of the quantifier again, that D: (i)(V(Fεx(V(Fx, i) = 1), i) = 1).
Now from D we cannot obtain E: (y)(i)(V(Fy, i) = 1),
since ‘εx(V(Fx, i) = 1)’ is not free for ‘y’ in ‘(i)(V(Fy, i) = 1)’. But we can get E from B, since ‘εxFx’ is a constant, and hence is obviously free for ‘y’ in ‘(i)(V(Fy, i) = 1)’. Nevertheless, if F is individuating, i.e. contains a uniqueness clause, then from B and D we can get (i)(V(εxFx = εx(V(Fx, i) = 1), i) = 1), Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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and so prove that, in all worlds, the F in fact is identical with the F in that world. The respective objects are not thereby identical tout court, but their passing identity can make it seem that what goes for one can hardly be separated from what must go for the other. Yet if we think in terms of the functional epsilon term there will be opacity, while with the now properly identified constant epsilon term there clearly is transparency. And that holds even though, respectively, in each world, they refer to the same thing. A simple example illustrating this involves the winner of a race, and the runners which might have won in other circumstances. If the actual winner is Tom — so he is xFx — but it is supposed the winner is Dick — so he is εx(V(Fx, i) = 1), for the specific ‘i’ — then what would have to be true for those circumstances to obtain would be for Tom to be Dick, i.e. εxFx = εx(V(Fx, i) = 1).
Tom is not in fact Dick, but that does not mean they cannot be identical in other worlds (c.f. Hughes and Cresswell [1968], 190). Certainly if there were several people variously thought to have been the winner, those other winners in the various other supposed circumstances would not provide a constant c such that Fc is true in all worlds. But that is because those other winners are functions of the other possible worlds - they are εx(V(Fx, i) = 1) for other ‘i’s. It still remains that there is a constant, c, such that Fc is true in all worlds — εxFx. Moreover, that c is not an ‘intensional object’: Tom is just a straightforward extensional object, the actual winner. Routley, Meyer and Goddard did not appreciate this last point, wanting, in their rigid semantics, the reference of individual terms to be ‘intensional objects’ (Routley, Meyer and Goddard [1974], 309). And what also might have misled others is that the temporal functionality of ‘Miss America’, even if it is recognised, is of another sort. For if this year’s Miss America is last year’s Miss America, that is only one thing being identical with itself, unlike with Tom and Dick. Also there is nothing which can make this year’s Miss America to be identical with last year’s different Miss America, in the way that the counterfactuality of the situation with the winners forces two actually non-identical things to be the same. Other worlds are Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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categorically different from other times, and other spaces, i.e., in general, other possible worlds must not been seen ‘Realistically’. Hughes and Cresswell describe the model that is misleading all parties (Hughes and Cresswell [1968], 197): Consider, for example, the expression ‘the top card in the pack’ as it occurs in the rules of a card game. The rules may, without ambiguity, specify that at a certain point in the play the top card is to be dealt to a certain player; yet on one occasion the top card may be the Ace of Spades and on another it may be the Queen of Hearts. Thus ‘the top card in the pack’ does not designate any particular card (individual piece of pasteboard), except in the context of a particular state of the pack; yet we can in one sense think of it as standing for a single object, contrasted with the bottom card in the pack, and so forth ... Such ‘objects’ are often called ‘intensional objects’, and ... for a logic in which ... the individual variables range over intensional objects ... L(∃x)Fx ⊃ (∃x)LFx would be valid ...
But it is the top card in the pack at one point in the play which is the Ace of Spades, and the top card in the pack at another point in the play which is the Queen of Hearts. So the logic of such expressions requires a formalisation using another variable: say ‘Fxt’ means ‘x is the top card in the pack at time t’. But ‘at every time there is some top card’ then gets symbolised, (t)(∃x)Fxt,
and while this is (t)F(εxFxt)t,
we cannot use existential generalisation to get (∃x)(t)Fxt.
since εxFxt
is not free for ‘y’ in Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
Hence there are no ‘intensional objects’ as Hughes and Cresswell, and the others, conceived them. The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
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49 2. Completing Russell’s Logic
(t)Fyt.
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We must now go on to see that, since there are non-sortal predicates Frege’s attempt to derive Arithmetic from Logic stumbles at its very first step. For there are properties without a number, and so the contingency of that condition shows Frege’s definition of zero is not obtainable from Logic. But Frege made a crucial mistake about concepts more generally that must be remedied, before we can be entirely clear about those specific concepts that are numbers. Frege thought the predicate ‘is a prime number’ was a denoting phrase, and denoted a concept, i.e. something that is ‘unsaturated’. The concept of being a prime number was then a function that takes the value True when applied to the argument 5, for instance. But while the predicate is incomplete, and so unsaturated — simply because it is just a part of a sentence — it does not denote anything at all. The concept of being a prime number is denoted not by the predicate but by its nominalisation, and so it is ‘saturated’ and cannot be a function — it is an object. Cocchiarella wrote the nominalisation ‘λxFx’, and the associated predicate ‘λxFx( )’ (Cocchiarella [1987], 83); in Kneale’s terminology (Kneale and Kneale [1962], 602), if the predicate is ‘F’ then the referential ‘§xFx’ refers to the concept. Such refined discriminations between referring phrases to concepts and open predicates enable us to oppose Frege’s most basic thought that concepts are categorically distinct from objects. When we are talking about concepts we are nominalising the associated predicates, and it is those open predicates that are distinct from referring phrases to objects. Frege was plainly not too clear about some of these discriminations. For he believed not only that concepts were not objects, but Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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specifically that numbers were objects and so not concepts. Indeed it was Frege’s identification of numbers with objects rather than concepts that supported the specific reasoning that led him to separate objects from concepts, since he thought of numbers as ‘saturated’ while concepts were not. He believed this, i.e. that numbers were not concepts, even though he could formulate no clear way of separating the objects that were numbers from other objects, like Julius Caesar. So we must inspect his reasoning about numbers very closely. Here is what Frege relevantly said (Geach and Black [1952], 24-5): The two parts into which a mathematical expression is thus split up, the sign of the argument, and the expression of the function, are dissimilar; for the argument is a number, a whole complete in itself, as the function is not... For instance, if I say ‘the function 2.x3 + x’, x must not be considered as belonging to the function; this letter only serves to indicate the kind of supplementation that is needed; it enables one to recognise the places where the sign for the argument must go in.
But, by Frege’s own grammatical criterion, the expression ‘the function 2.x3 + x’, being a definite description, ought to denote an object, even though, in his representation of it, it contains some gaps, and so is unsaturated. Moreover, although what might fill the gaps, namely numbers like the number 7, arise in arithmetical statements such as ‘7 + 5 = 12’, this merely uses ‘7’ as a substantive, and Frege elsewhere recognised that the numerals, in their adjectival use, were parts of second-order predicates, which means that these terms also can form parts of unsaturated expressions. The phrase ‘there are (exactly) seven’, for instance, needs a substantive added to it, such as ‘horses’, to make a complete thought. So just what is, and what is not saturated, or unsaturated, needs some careful unravelling. What Frege did not fully appreciate was that corresponding to the referential and descriptive uses of numerals, there are complete and incomplete expressions with all predicates, as above. Thus, following Cocchiarella, there is the functional expression λx(2.x3 + x)( ),
which is not a referential phrase, and so does not denote any object at Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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all, and there is the definite description ‘the operation of doubling the cube of a number and adding it to that number’, i.e. λx(2.x3 + x),
which contains no gaps, and therefore refers to an object – a mathematical operation, which is one kind of abstract object. Being 7 in number (i.e. λQ(7x)Qx) is another abstract object: it is that property of discrete and distinctive things of having a correlation with, for instance, the non-zero numerals up to ‘seven’, while in the predication ‘The Ps are 7 in number’, i.e. ‘(7x)Px’, or its equivalent ‘λQ(7x)Qx(P)’, the same property is not referred to but expressed. There is a related grammatical issue in this area that also must be cleared up. That concerns the putative removal of reference to numbers, through the replacement of expressions involving substantival uses for numerals with certain other expressions, involving adjectival uses. Neil Tennant presented the case very clearly in Tennant [1987], 232: What does the nominalist do if he wishes to eshew reference too particular numbers in statements of number? What, for example, does he do with a claim such as ‘The Number of F’s = n’? The usual answer is that he would rephrase this so as to read ‘There are n F’s’, where now the number-word n is being used adjectivally rather than substantivally. Now for each canonical designation (numeral) n, one can provide a sentence in first-order logical notation expressing precisely the thought that there are n F’s. One uses only quantification, identity, and the logical connectives, along with the predicate F. There is no reference, either implicit or explicit, to numbers as such. The way this goes is as follows: There are 0 F’s = df There are no F’s There is 1 F = df There is an F and every F is identical to it. There are 2 F’s = df There is an F and another F, and every F is one or the other. … and so on in ways left to the reader.
Tennant went on to question whether such a reductive analysis can be provided in all cases that are needed, but wanted to insist that it certainly works in the given set of cases, adding “We must not lose Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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sight of the fact that there being n F’s is precisely the truth condition for the number statement ‘The number of F’s is n’”. We shall see in a moment that the latter assertion is crucially wrong. But what about the absence of numerical reference in the variant re-expressions for ‘There are n F’s’ etc.? The nominalist’s claim is tantamount to denying that one could go on from such a remark, anaphorically, saying things like ‘and there is the same number of G’s’. But clearly one can, and that requires that the former remark allows the numerical place in strict numerical quantifiers, like ‘(nx)Fx’, be open to quantification. Yet it cannot be open in this way, if, for every ‘n’, there is an expression in which the number-word does not occur. So Tennant cannot be right in saying that there is no such reference ‘either implicit or explicit’ to numbers as such, on the right-hand side of his definitional identities. In fact, of course, there is an implicit reference in the cases Tennant gives, since the number of repeats of the term ‘F’, in his right-hand expressions, is always one more than the number on the left-hand side. One can even make the reference explicit, if one subscripts the individual variables with numerals, in the parallel formal re-expressions. Then one would have, for instance, There are 2 F’s = df (∃x1)(∃x2)(Fx1 & Fx2 & (x1 ≠ x2) & (y)(Fy ⊃ y=x1 v y=x2)).
As a result, we can write ‘There is a number of F’s’ as ‘(∃n)(nx)Fx’, a formula of fundamental importance, whose significance will appear shortly. To see its importance we must now proceed to look more closely at one central consequence of the above definition of number – the fact that things with a number must be discrete. The point about discreteness is crucially involved in, amongst other things, the question whether Julius Caesar is or is not a number. According to Wright (Wright [1983], 11), there were three specific considerations that were involved in Frege’s judgement that numbers were objects. One was the use of definite descriptions like ‘the number of the planets’. Another was the currency of numerical identities, like ‘5+7=12’. The remaining consideration Frege appealed to was the contrast between, for instance, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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the number of planets is 9 The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
(Nx:Px = 9) and there are exactly 9 planets
((9x)Px). Only the former represents ‘9’ as a singular referential term, and so Frege took it to be the basis of his formal analysis of Arithmetic. But, as we shall now see, it is the predicative form that has priority, and it is that fact which also shows that numbers, while still objects, are categorically distinct from objects like Julius Caesar, since they then cannot be known independently of predications. By contrast, one does not need to know someone is Julius Caesar before one can be acquainted with him. The priority of the predicative form arises because the foundation for the theory of number is to be found in appropriate definitions of quantificational expressions like ‘(nx)Fx’ (i.e. ‘there are exactly n Fs’), from which expressions like ‘εm(mx)Fx=n’ (i.e. the number of Fs is n) follow quite straightforwardly. For (nx)Fx
entails (∃m)(mx)Fx,
by existential generalisation, and so ((εm(mx)Fx)y)Fy,
by the epsilon definition of the existential quantifier, which equates ‘(∃x)Px’ with ‘PεxPx’. One can then derive εm(mx)Fx=n
because of the uniqueness of the exact numerical quantifier. The reverse entailment crucially does not hold, however, because of the numerical indeterminacy of non-sortals: one can have Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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without the epsilon term numbering the Fs, i.e. without ((εm(mx)Fx)y)Fy,
since there may not be any Fs, but merely some F. The numerical identity then can still arise, but only through the arbitrary specification of a value for the epsilon term, in a case where ¬(∃m)(mx)Fx,
i.e. where the predicate ‘F’ is not count, and so does not discriminate discrete things. Such a case is when ‘F’ is a mass term, and there is only an amount of stuff, in which case ‘the number of Fs’ must be non-attributive. Thus we do not have the epsilon equivalence (nx)Fx ≡ εm(mx)Fx=n,
but merely a one-way implication (nx)Fx ⊃ εm(mx)Fx=n.
Certainly one can have the iota equivalence (nx)Fx ≡ ιm(mx)Fx=n,
but this does not involve an individual term on the right hand side, since it is the same as (nx)Fx ≡ (∃m)((mx)Fx & m=n).
The crucial difference between epsilon terms and iota terms is that epsilon terms are complete terms for individuals, unlike iota terms, which are incomplete terms, as this last point shows. That means epsilon terms may be non-descriptive, and so can formalise Millian names; in fact they are the logically proper names Russell hypothesised, but did not have a symbolism for. The epsilon definition of the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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existential quantifier means that ‘¬(∃x)Px’ equates with ‘¬PεxPx’, so in the present case, even if there is no number of Fs (¬(∃m)(mx)Fx), still ‘εm(mx)Fx’ will refer, although then, like ‘The Morning Star’, for instance, its reference will be given accidentally. Thus, just as Venus is not a star, although ‘The Morning Star’ conventionally refers to it, so ‘the number of F’ when ‘F’ denotes some stuff, does not refer to a number which numbers discrete things. One specific consequence of the possibility of such deceptive, Millian ‘number names’, which dramatises the matter, is that their arbitrary reference might well be, on occasion, a physical object — for instance even Julius Caesar. How can the basis for the theory of number lie in numerical quantification? In fact David Bostock essayed a deduction of Arithmetic from Logic in this quantificational style (see also, more recently, Agustin Rayo [2002]). Bostock defined the numerical quantifiers in the Fregean fashion, and used a generalised theory of quantification, applicable to the numerical place in such expressions as ‘(nx)Fx’, to deduce Peano’s Postulates, with certain further assumptions. Bostock was much more appreciative of the difference between numbers and amounts than other logicians (see Bostock [1974], and Bostock [1979], respectively, as a whole). Nevertheless his analysis is not perfect, since he did not appreciate the above points about the differences between count and mass terms. For, right at the start, he tried to define a weak numerical quantifier with (∃1x)Fx ≡ (∃x)Fx,
and a strong numerical quantifier with (0x)Fx ≡ ¬(∃x)Fx,
(Bostock [1974], pp 9, 10). So the given foundation for Bostock’s deduction was unsafe — as unsafe, as we shall see, as Frege’s. If ‘F’ is a mass term, then ‘(∃x)Fx’
and Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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simply read ‘there is some F’
and ‘there is no F’,
respectively, and even ‘(∃x)(Fx & (∃y)(Fy & y≠x))’
merely reads ‘there is some F, and some more F’.
In none of these cases, therefore, do numbers or pluralisation enter the content. There are guaranteed to be Fs, however, if there are two or more Fs, so there is no difficulty with the strong numerical quantifiers above 1, and a construction of Arithmetic in Bostock’s style remains possible. If ‘n’ ranges from 2 upwards then F is count if and only if (∃n)(nx)Fx
or (∃x)(y)(Fy ≡ y=x) & M(∃n)(nx)Fx
or ¬(∃x)Fx & M(∃n)(nx)Fx,
and abbreviating the latter disjuncts to ‘(1x)Fx’,
and Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
3. Logic and Arithmetic
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respectively gives us the simplified definition that ‘F’ is count ifand only if (∃n)(nx)Fx,
where ‘n’ ranges from 0 upwards. That means that Bostock’s definitions above will hold only on the supposition that (∃n)(nx)Fx. The restriction in the case of the number 1 is required not just because of the possibility of mass terms, but also proper names, since ‘is Peter’, for instance, will hold without ‘is one Peter’ holding. The restriction with the number zero is required because for non-sortals there may be no F without there being zero Fs. Why are there guaranteed to be Fs if not just (∃x)(Fx & (∃y)(Fy & y≠x)),
but (∃x)(Fx & (∃y)(Fy & y≠x & (z)(Fz ⊃ z=x v z=y)))?
Consider two rings of gold, or two atoms of water, say. Since these are both gold, or water, there is clearly a third object that is also gold (water), namely the mereological sum of the previous two objects. So if there are just two ((2x)Fx), then the predicate must be count. The case of 1, as it is standardly formalised (∃x)(y)(Fy ≡ y=x),
still allows the predicate to be a mass term, since if there is just one atom of water, then while portions of that atom are not water themselves, and so only the whole atom is water, that whole atom is still ‘some water’ and not ‘one water’. In a somewhat similar manner, although 2 is the only even prime it is still ‘an’ even prime, not ‘one’ even prime. So always the possibility of there being two items is required before we can start to count with a term. The point even holds when there can be nothing of the kind in question. For if we Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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could say there were no round squares we could rightly say there were zero round squares. But in fact there is merely no round square, from which it does not follow there is zero round square. We can now finally see how the restricted definition of zero, which emerges from such considerations about plurality, undermines entirely all Fregean, and Neo-Fregean attempts to derive Arithmetic from Logic. In the above terms, Frege presumed that ‘(∃n)(nx)Fx’ held for all predicates, and the leading Neo-Fregean Crispin Wright is notable for being amongst the first to publicise the fact that this is not so. But this criticism has more radical consequences than Wright realised for the development of Arithmetic using Hume’s Principle. Boolos and Wright, with others, have demonstrated how most of Frege’s development of Arithmetic can be obtained from Hume’s Principle, starting from Frege’s definition of zero as the number of things which are not self-identical (Nx:x≠x). But in this extensive, and now very elaborate discussion, no question has been raised about whether ¬(∃x)(x≠x),
entails Nx:x≠x = 0.
If the negative existence statement entails the numerical statement, then Nx:x≠x must be determinate, and that is contingent on ‘x≠x’ being a sortal predicate, as Wright has admitted. But what unit is determined by non-self-identity? No argument for there being one has been given, either by Wright, or by any one else within this tradition. Indeed, at one time it was simply presumed that all predicates were sortal. But Wright has recently given a proof that self-identity is not a sortal concept (Hale and Wright [2001], 315). For conjoining a mass term with a count term invariably results in a count term: ‘red socks’ is count because ‘socks’ is count. But conjoining a mass term with ‘x=x’ leaves one with the original mass term, since the addition of ‘x=x’ produces an equivalent. Hence ‘x=x’ cannot function like ‘socks’. As a result (as Wright himself explicitly realised earlier, see Wright [1983], 187), argument is Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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needed to show that non-self-identity is a sortal concept. On the above definition it clearly is not. Much ink has been spilled debating whether Hume’s Principle is analytic, and so whether the Arithmetic taken to be derivable from it can, or cannot, be properly described as a part of Logic. But if Logic does not discriminate between sortal and non-sortal concepts, then there is no way to get from it the other crucial element in Frege’s generation of the number series: its starting point. We say ‘the number of Fs is n’ and can do so whether n is 0, 1, 2, or more; but only count nouns pluralise in the appropriate way. Mass terms sometimes appear pluralised, but not in the same sense: in ‘there are several champagnes’, for instance, we are speaking about glasses of champagne, maybe, or varieties of champagne. In English we can say ‘it is F’ rather than ‘it is an F’, but nothing corresponding to this is to be found in Frege’s language. Maybe there is no martini in a glass. Does that mean the number of martini in the glass is zero? It does not. There is no such thing as a number, i.e. a number of Fs, in this case. There might have been some F, rather than no F, and in both cases as much F as G, but the required plural case, and so the possibility of a number, and the same number, just does not arise. Could we not simply introduce a count noun, and talk about the number of Cs that are F instead? Certainly if there is no gold then the number of ingots which are gold is zero. Wright has discussed this matter more than most, and he has admitted: ‘to number the instances of some non-sortal concept is intelligible only if relativised to a sortal’ (Wright [1983], 3, see also Hale and Wright [2001], 315, 387). But the necessary distinction between substantives and adjectives is just what is lacking. Maybe ¬(∃x)Fx,
is equivalent to (C)(Nx: (Cx & Fx) = 0),
where ‘C’ ranges over count terms, but one cannot say this in Frege’s concept-script.
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Oliver and Smiley have tried to demonstrate how irreplaceable plural predications are, by showing specifically that certain plural predications cannot be handled in terms of sets, or any other kind of singular object. This chapter first points out some grammatical facts bearing on the adequacy of the current mathematical account of sets, and then goes on to inspect the adequacy of Oliver and Smiley’s defense of their account of plurals. It is something wrong with the mathematical theory of sets which leads these writers, first to think of set expressions as referring to abstract objects in some objectionable way, and then to try to replace these ontologically questionable objects with something with a clearly non-abstract reference, like a plural term. Oliver and Smiley have not been thinking in terms of ordinary speech, but in terms of the standard professional theory, and it is something to do with that theory that is making them come to their erroneous judgements. I here describe a conservative extension of standard Set Theory that avoids the abstraction problem with ‘sets’, and end by applying it in the evaluation of Oliver and Smiley’s arguments. As I have pointed out before (Slater [2006a]) the principal philosophical problem with the current mathematical account of sets is the difficulty in identifying singleton sets, as writers like D.K. Lewis, Penelope Maddy, and Michael Potter have seen. For instance, while one can see an apple on a table, where is the singleton set of that apple? Is it in the same place as the apple? But in that case, how is it distinguished from it? The solution to the problem, we shall see, is found by looking first at non-singleton sets, such as pairs. For consider, instead, a pair of apples. The common assumption has been Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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that a pair of apples is a novel kind of individual object distinct from each of the apples. There is certainly a single object in the vicinity, namely the mereological sum of the two apples. But that cannot be the pair of apples, since that whole might be carved up in more than one way, and in some ways with more than two components. The expectation, as a consequence, has been that one must look elsewhere for another object to be the pair of apples. But this supposed other object is a grammatical mirage. The crucial question with respect to a pair of apples is: is that pair of apples a single object? The answer is ‘no’: it is simply a pair of objects, instead. There are, in the area, at least three types of expression, with quite different grammars, and without a close inspection it is far to easy to run them all, or even just some of them together. Thus there are collective nouns like ‘shoal’, ‘herd’, ‘pack’, ‘tribe’, collective nouns like ‘pair’, ‘triplet’, ‘quartet’, ‘dozen’, and plural collective nouns like ‘the fish’, ‘the beasts’, ‘the cards’, ‘the savages’. It is from collective nouns of the first sort that Set Theory drawns the idea of collections of objects, but the distinctive thing about those natural language terms is that they are species specific, to one extent or another, so that they each describe certain mereological sums under a certain aspect. That is to say, the principle of division of the whole mereological sum is provided through the further count noun they are normally associated with — ‘shoal’ with ‘fish’, ‘herd’ with ‘beasts’, ‘pack’ with ‘cards’, ‘tribe’ with ‘savages’, etc. The fact that such collections are mereological sums is also shown by the fact that shoals, herds, packs, tribes, and the like, are located and can move around in physical space, just like their members. There is not the same to be said with regard to collective nouns of the second sort, and not only because a complete description of the intended set must be given – the pair is maybe of fish, the triplet of beasts, etc. For these collective nouns can also be used when no physical objects are involved, and so when there are no mereological sums of the associated objects. They therefore only indicate the number of some things, and we have to remember the general grammar of ‘y is one of a number of Ss’. This is not of the form ‘y ∈ s’, with a singular term in place of ‘s’. A specification of it, for instance, would be ‘y is one of 2 Ss’, which relates ‘y’ to a plural term, and the original is just ‘(∃n)(y is one of n Ss)’. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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John Burgess (Burgess [2004], 197-9, 211) thinks there are two senses of ‘is one of’, so that one might say, for instance, y is a member of the set of apples,
but y is amongst the apples,
even though the expressions are equivalent. That would allow a singular ‘s’ to occupy ‘y ∈ s’, while a plural ‘xx’ occupies ‘y α xx’. Certainly y is a member of a/the/that pair of apples
is the same as y is amongst 2/the 2/those 2 apples,
but ‘is one of’ could be used in both cases, and in the same sense. For the equivalences show that ‘a/the/that pair of apples’ still refer to the same things as ‘2/the 2/those 2 apples’ — the former merely refer to them (sic) in a different manner, namely collectively, i.e. taking them as a unit. One must take care about what is added to the bare y is one of some apples.
Clearly y is one of 2 apples
may speak about the same apples, but is more specific about their number, and y is one of a pair of apples
likewise. But the latter does not invoke a further object, ‘a pair’, in addition to the two apples, it merely introduces a certain numerical measure of the apples, by taking the two as a unit. ‘A pair of apples’, in other words, differs from ‘2 apples’ simply in changing ‘2 times 1 apple’ into ‘1 times 2 apples’. We can count with such units, by counting in pairs, but we are then not counting something other than Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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apples; we are merely not counting the apples singly, i.e. one by one. We say There are 2 pairs of apples,
but this is exactly equivalent to There are 4 (single) apples,
and There is a quartet of apples.
To say there is a set of apples is equivalent to saying (∃n)(there is an n-tuple of apples).
Words like ‘pair’, ‘quartet’, ‘16-tuple’ are thus like ‘ounce’ and pound’ in ‘There are/is 8 ounces/a pound of beef’.
They do not refer to further entities, but are instead the basis for measurements of quantity. Here are two pairs of alphabetic letters: a b c d.
Notice that each of the pairs is indicated without braces, as with ‘{a, b}’, since the latter, i.e. the common ‘set-theoretic’ symbolism, does not represent sets as merely numbers or quantities of objects but as independent objects, distinct from their members. However, the four letters above, which were taken as two pairs, also can be taken as a quartet, while the two ‘sets’ {a, b} and {c, d} cannot be the same as the ‘set’ {a, b, c, d}. In fact no two ‘sets’ can be the same as any one ‘set’, but 2 twos are exactly the same as 1 four. The reification of a unit of measure as a further separate object, maybe arises through forgetting the difference between the two sorts of collective term. For the mereological ‘tribe’ does have an objective reference to an independent object, but ‘pair’ needs supplementing, and then in ‘a pair of apples’, it merely qualifies the following substantive. Maybe focussing on count terms and forgetting mass Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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terms also has something to do with the misconception. For the same matter of change of units even more clearly arises with fractions than with multiples of individuals. In ‘There is a half of a loaf’ there is obviously no reference to something other than bread: there is not, in addition, reference to one of a range of mysterious, further objective entities, ‘halves’, ‘quarters’, ‘parts’, etc. There is merely a specification of how much of a loaf there is, maybe as a prelude to counting half-loaves, or totting up different parts of loaves to find an equivalent sum of complete loaves, etc. We can now have a first try at tackling the question of what a ‘settheoretic’ singleton is, i.e. what ‘{y}’, or ‘{x: x=y}’ might represent. In the natural language locution ‘a singleton S’, of course, ‘singleton’ just describes the S as the only one of its kind, and does not refer to any other object. But another way of representing being the sole S as ‘being one of’ something might well be the prime source of the settheoretic notion of ‘singleton’. For even if the number of things which are S is just 1 then we can still say ‘y is one of those things which are S’, making those things which are S what the sole S is one of. But ‘is one of’ is then again succeeded by a plutral term, not a singular one. If we read set abstraction expressions as such plural terms, therefore, {x: x=y}
would be those things which are y,
allowing y ∈ {x: x=y}
to be y is one of the things which are y.
But then, identification of such ‘singletons’ with their single members, in the manner of Maddy, would clearly be ungrammatical, since ‘y is those things which are y’ does not make sense. Indeed the
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general identification of set abstracts with plural terms could not be thoroughgoing, since not only would ‘y={x: x=y}’
be ungrammatical, so would ‘y={x; Px}’
for any ‘P’, and therefore also ‘{{x: Px}}’,
and ‘{{x: Px}, {x: Qx}}’,
because these latter would have to be ‘{z: z={x: Px}}’,
and ‘{z: z={x: Px} v z={x: Qx}}’
respectively. It might be thought that such points about collective nouns and plural terms presuppose that unless Mathematics can be re-construed back into non-mathematical language then it is unintelligible. But the claim is merely that the ‘sets’ of ‘Set Theory’ cannot be interpreted either in terms of groups of things, or in terms of numbers of things, or by translation into plural expressions. Certainly there is the settheoretic symbolism, and the rules for its manipulation, and maybe it all has some interpretation. But it does not have any of the traditional interpretations, on the basis of which it was developed, and the trouble that Lewis, Maddy, Potter and others still have had with the basic notion of a singleton shows it has yet to be given a clear sense. We can now bring in the specific interests of Oliver and Smiley in this area. Oliver and Smiley have been concerned with, amongst other things, whether collective predications can be construed as predications on sets — with mathematical sets being what they had in Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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mind. They start from the well-known distinction amongst plural predications between those that are collective, and those that are distributive. Thus ‘A and B sum to C’
has a collective predicate, while ‘A and B fight C’,
for instance, has a distributive predicate. As the term ‘collective’ indicates, normally it would be said that the difference was a matter of the first, but not the second being about the collection {A, B}. If so, then both can be mapped onto singular forms, namely ‘{A, B} sums to C’ ‘A fights C, and B fights C’.
Using ‘<’ to mean ‘is a part of’, and ‘∈’ ‘is a member of’, they can be represented {A, B} ∈ {y: y sums to C}, {A, B} < {y: y fights C}.
Of course, the plural terms that can enter into such collective and distributive constructions are not limited to lists; they also include predicative phrases like ‘the men’. So we can distinguish, similarly, singular equivalents of ‘Those men are causing a riot’ ‘The men are bald’,
i.e. {x: Mx} ∈ {y: y is causing a riot}, {x: Mx} < {y: y is bald},
which might read, in the appropriate context ‘That mob of men is causing a riot’, ‘Each man is bald’.
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One further virtue of this mode of analysis is that it allows a representation of multigrade predicates, to which the same writers, Oliver and Smiley, have recently drawn attention. They think, however, that multigrade predicates cannot be handled by standard formal logic (Oliver and Smiley [2004]). For with ‘those men are building a house’, for instance, there would not seem to be a standard representation in logical notation, since all of the obvious candidates, ‘Bxh’, ‘Bxyh’, ‘Bxyzh’, etc. require the number of builders, x, y, z, to be specified. But if we merely say that a group of men is building a house, then the number of men in that group is unspecified, and so is not of any particular size, allowing the plural equivalent to be multigrade in the way Oliver and Smiley want. There are troubles, however, with this singular style of analysis of collective plurals. The main trouble is that it requires groups, and sets of things to take part in the physical world, while the sets of mathematical set theory are generally thought to be abstract objects without any causal effects. The point is made much of in Oliver and Smiley’s original paper on plurals. The largest point Oliver and Smiley repeatedly make concerns the supposed abstraction of sets, and thereby the impossibility of treating plural predication over concrete objects in terms of sets. This would certainly seem to be a strong, and significant point to make. They say, for instance, (Oliver and Smiley [2001], 295): It is easy to ridicule this naïve version of changing the (plural subject into a singular one). Here is Boolos using upper case to great effect: ‘It is haywire to think that when you have some Cheerios, you are eating a set — what you’re doing is: eating THE CHEERIOS’. We can get away with saying that a set of premises implies a conclusion, because we are free to enrich the sense of ‘implies’ to accord with our talk — at least if we are prepared to accept what may be unwelcome consequences. Not so for ‘eat’. Sets are abstract objects, and no abstract object is ever eaten. Collective predicates also cause trouble. Tom and Dick weigh over 200 kg, but the set of the two has no weight.
The ridicule misses its target, however, since anyone who has been to a greengrocer, and on a visit in a hospital, knows one can easily weigh a bunch of grapes, and eat a handful of them. So what is wrong is the idea that bunches and handfuls are abstract objects in Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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some pernicious sense, through being examples of what mathematicians have called ‘sets’. Certainly, as Oliver and Smiley show, plural collective predication cannot be reduced in any systematic way to plural distributive predication. But their favourite example of the former, namely ‘Whitehead and Russell wrote Principia Mathematica’ still means that that book was written by a couple of people. At the time of William and Mary, also, it may be remembered, a married couple ruled England. So that is a couple of examples that immediately come to mind where couples of various descriptions have done things in the world. And that is not to mention the many couples that take to the floor and do the Tango in Buenos Aires, probably every day of the week. As we saw before, ‘the book was written by a couple of people’ says no more than ‘the book was written by two people’, even though the grammar of the former expression is easily misinterpreted to suggest that it makes reference to a further individual object beyond the two people, namely the pair of them. But while it does allude to this pair, that pair is not an individual object: it is simply a pair of individual objects. How could Oliver and Smiley, as well as Boolos, have been so misled with regard to the proper grammar of ordinary speech? What has mislead them, I believe, is that the current mathematical theory of sets is pre-eminently a theory of ‘pure sets’ rather than sets of things, and Oliver and Smiley are thinking of ‘sets’ in terms of this mathematical theory. The obvious thing is that natural language sets, unlike with the common image of ‘mathematical sets’, in many cases involve members that are physical objects. Indeed, we shall see that they are mereological sums of such singletons. This is already plausible with respect to such physical sets as tribes, and shoals, which are located in, and move around in space just like individuals. Here, as we saw, the singletons spanning the mereological sum are determined through a further, associated count noun — ‘savage’ and ‘fish’. Such physical objects are not strictly banned from mathematical set theory, but the nature of them has not been as fully investigated as it should have been. In particular it has not been realised that such individual objects are themselves sets — singleton sets. We shall see not only that, on a revised understanding, sets of such have a place in the world, but also, and perhaps even more surprisingly, that it is principally through its inattention to these matters, and its related Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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concentration on ‘pure sets’, that mathematical set theory got itself into its worst crisis, at the turn of the twentieth century. The theory was founded slightly before that, and much formal work has ensued, of course, but, in the present connection, it was not until the work of Harry Bunt and David Lewis, in the 1980s, that further significant philosophical developments, relevant to a proper understanding of physical sets, were made. By now it is well known that mathematical sets reduce to mereological sums of ‘unicles’, or ‘singletons’, i.e. sets with a single member. That is one important advance these two writers brought about, but the nature of such singletons has remained mysterious, from a philosophical point of view. To make their nature clear, however, it becomes necessary to include a discussion of something not commonly included, namely mass terms, since mereological notions are more commonly brought in in connection with them, and so, for a start, discrimination between the two applications of mereology is required. More to the point, mass terms describe kinds of material, and clarification of natural language singletons only emerges through seeing just how the material in some singleton is distinguished from that singleton itself. For that allows us to see that the singleton is an object with material in it, even while it is not that material itself. Indeed it might be said that ‘abstraction’, if it is present, begins even with singleton individuals, through their separation out from stuff. And it is that, in reverse, which is required to see that sets, more generally, can be said to have a place in the world, since they have as much reality as individuals, which are commonly taken to be part of ‘what is there’. The present analysis has therefore benefited most from Bunt’s work, which, although significantly different in several respects, also gives a theory of mass, or ‘stuff’ terms alongside a theory of count, or ‘thing’ terms — the latter being the only proper basis for a theory of sets. The specific point to start from is that predicates that are not count do not determine discrete things. For it is that which solves the major problem about mereological singletons: how they can have substance and yet still be atomic. The prime grammatical distinction to observe is that mass terms do not pluralise in an appropriate way, like count terms. In addition to a gold ring, for example, there may be other rings, but not other golds in the same sense. Indeed there may only be other kinds of gold. When summing some items describable with the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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same mass term, ‘F’, we therefore cannot grammatically form the term ‘the set of Fs’, since there are no appropriate ‘Fs’. What we can do is introduce an associated count term, and talk about the gold in a ring, for example, or the flesh in some limbs; more generally, the material in some individual, or individuals. Bunt’s unicle-member relation then becomes the relation between an object and the stuff ‘in it’, i.e. the stuff that comprises it. As a result of this, not only can the marble in a statue (m) be distinguished from the statue (s), since s = {m}, but also, even more importantly, the parts of some group can be distinguished from what comprises it. Thus a beast (b) might be a part of a herd (b < h), while the beast consists in some material (n1), which is not a part of the herd (b={n1}, but not n1 < h), even though the herd consists in a set of such bits of stuff, i.e. h = {n1, n2, n3, …}. Again, the flesh of some seed (e) might be part of the flesh of some grape (g), so that e < g. But that does not mean either that the seed is part of the grape ({e} < {g}), or that the flesh of the seed is part of the grape (e < {g}), or that the seed is part of the flesh of the grape ({e} < g). Other discriminations also become available. Thus, if the material in Peter’s arms is d, then d is a mereological sum, b + c, where {b} and {c} are Peter’s arms; and if the material in Peter’s whole body is p, then while Peter = {p}, it is not the case, for instance, that {b} < p, or that {c} < p, or that d < {p}. Certainly the material in Peter’s arms is a part of the material in Peter’s body, i.e. d < p. But neither of Peter’s arms, {b}, or {c}, is a part of the material in Peter’s body, p. Nor is the material in Peter’s arms, b + c, i.e. d, a part of Peter, {p}. Peter’s arms, as a result, constitute a pair of things with only the unicles of each of his arms as proper parts; and Peter is an individual, which is to say he is a singleton, in the sense that he only has himself as a part (an improper part). That shows that Anne’s flesh was human flesh
has the form Anne’s flesh < the totality of human flesh in her time,
but Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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has the form Anne ∈ {y: y ruled Britain in the early 18th century},
where Anne = {Anne’s flesh}. The case brings up the most crucial issue about physical sets, however: their transience. For Anne’s flesh can appear in another form at some other time. That happened, of course, upon Anne’s death, when the material in her body ceased to form a living thing. But it is a dateable matter, also, when the person Anne was formed from the material in her mother’s ovum and her father’s sperm. Hence, the relation between the unicle, x, and its sole member, y, must be recognised to be a temporally qualified relation. The material, y, in an object at a certain time, forms that object, x, but before that time, and afterwards perhaps, it may not have that form, so that then, as one might say, x ≠ {y}. In those terms, whether it is true or not that x = {y} is contingent on the way the world is. But it would be better, in such cases, to suffix the unicle symbol, ‘{y}’, in order to indicate the time at which the material in the object is indeed in it: ‘{y}t’
would then indicate the object formed from the material y at time t. And then quite possibly {y}t ≠ {y}t’.
Of course, in other cases there can be no temporal variation, allowing a return to the usual, unadorned set-theoretic symbolism. But the extended symbolism gives us a way, for instance, of referring timelessly not only to physical sets with material members, but also to physical sets that are no longer, or not yet material ones. For even if {y}t ≠ {y}t’, still {y}t may be referred to at time t’. But note that the exact same object {y}t cannot be formed from different material at a different time, i.e {y}t = {z}t’
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y=z,
so it is possible there may only be, for example, counterpart Ships of Theseus, at times t and t’. That means, for example, that when one looks at old photographs one maybe sees oneself at an earlier age, but never just ‘oneself’. For there are properties of oneself at the earlier age that are different from corresponding properties of oneself at the present time. Formally it must first be remembered that there are collective and distributive predications even with singular expressions. For we must distinguish singular collective predications like That object is a ring,
from singular distributive predications like The material in the ring is gold.
If the object is {r}t then we can symbolise them respectively {r}t ∈ {y: Ry},
and r < g,
where ‘g’ is the totality of all gold. Like all distributive predications the latter admits of variants using quantification, such as All of the material in the ring is gold, Some of the material in the ring is gold,
i.e. (x)(x < r ⊃ x < g), (∃x)(x < r & x < g).
But the main point to note is that, in the collective case we are not considering a different piece of material, merely the same piece of
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material differently, i.e. collectively, and so as a single unit rather than something with parts. The units considered in this case are rings, and one might imagine several such lined up on a display cushion, made of some silky material perhaps. Looking at the display one way it might be seen as a continuous, if lumpy sequence of various sorts of stuff. But by marking divisions between the rings one can be brought to attend to those rings as units, and so, for instance, count them. The function of the lines in the following, however, must be closely noted: |A| |B| |C| |D| |E| |F|
The lines naturally serve to mark off the rings from one another, but also they mark off the ring ‘|A|’ from its material content ‘A’, etc. Likewise when we mark the rings another way: ( |A| |B| ) ( |C| |D| |E| |F| ).
Here we indicate, overall, not individual rings as the unit, but groups, and specifically pairs and quartets of them. But we also distinguish the groups from their contained, individual physical objects, and do so while allowing only the latter to be individual objects. This point about the insubstantiality of units is in line with Frege’s point about number words and their dependence on a count noun. For Frege pointed out that 52 cards are 1 pack, while 4 suits are 1 pack also, and that specific Fregean point was what was extended above to show that such formal set words as ‘pair’, and ‘quartet’ are comparable to units of amount, like ‘ounce’ and ‘pound’. Such units of discrete or continuous measure are not terms with a direct physical reference, although when attached to an appropriate count or mass term a compound is formed with such a reference. Thus an ounce of tobacco might be found in a pouch, and a pair of gloves might be located in a drawer. But, as we have seen, the grammar of ‘a pair of apples’ requires that no further individual object is involved than the apples, even though there is a further singular referential phrase, ‘the pair of apples’, referring, naturally enough, to a pair of individual objects. More significantly still, that pair of individuals can now be seen to be as ‘physical’ as an individual apple, once such individual objects are distinguished from their material content. Certainly a Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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physical group is a mereological sum, as we saw before — thus a shoal is the sum of its fish — but once one realises that sets can be physical, one sees that that does not lead to the abandonment of Set Theory’s applicability to such cases. For one must not miss the possibility that individual fish, themselves, should be sets — singleton sets — allowing Lewis’ mereological analysis in Parts of Classes to apply directly. Specifically, if {f1}, {f2}, {f3}, etc. are the fish, then the shoal, s, is {f1, f2, f3, …},
which means that {f1} < s,
etc., as we saw before with beasts in a herd. One further consequence, however, is that it is now apparent there are no more individuals in a set than in its transitive closure. One might try to distinguish, for instance, the power set of a pair by writing ‘{∅, {A}, {B}, {A, B}}’ in opposition to ‘{A, B}’, thinking of the former as containing four individuals. But the reader is encouraged to try doing the same thing with a pair of apples. The difficulty is that, in the power-set symbol the repeated ‘A’s, etc., are tokens of the same type, but with the actual objects supposedly symbolised there can be no such repetition, since the type-token distinction does not apply. Only a Formalist could miss the difference. With a physical collection of objects all one can do is indicate successively all the single individuals in the collection, and then all the pairs, triplets, quartets, etc. of those individuals. But all this is hand waving in front of the objects, and only the original individuals are involved. Cantor’s Theorem, of course, is not in danger, since one can still distinguish the number of parts of a group from the number of its singleton parts. But its proof becomes an exercise in elementary Permutations and Combinations, rather than the counting of different individuals. The above discriminations may seem to be concerned merely with the application of Set Theory. But several large formal consequences also emerge. Thus, for a start, they show that there is no problem with any analogue of Russell’s Paradox on the present understanding Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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of sets (answering, amongst other things, Oliver and Smiley’s arguments in this connection, see Oliver and Smiley [2001], 301-5). For we now have a principled way of avoiding this paradox, and all related paradoxes, because the fault in connection with the Abstraction Axiom has very evidently come from taking it that (x)(x ∈ {y: Fy} ≡ Fx)
holds for all ‘F’, whereas we now know that it only holds for count predicates. So while substitution of ‘λz(z ∉ z)( )’
for ‘F’, for instance, and then ‘{y: y ∉ y}’
for ‘x’, produces a contradiction, the conclusion merely has to be that the substituted lambda expression is not a count predicate. In fact, as we shall see later, it is not a predicate at all, and refers to a state of affairs rather than a property, but still, if (x)(x ∉ x),
as is standard in Bernays and von Neumann style set theories, then ‘x ∉ x’ is equivalent to ‘x=x’, and it is provable quite independently that this is not a count predicate (Hale and Wright [2001], 315). For conjoining a mass term with a count term invariably results in a count term: ‘red socks’ is count because ‘socks’ is count. But conjoining a mass term with ‘x=x’ leaves one with the original mass term, since the addition of ‘x=x’ produces an equivalent. Hence ‘x=x’ cannot function like ‘socks’. Notice that that does not mean there is ‘Limitation of Size’ in the usual sense, since such an expression as ‘x=x’ now does not define a proper class rather than a set. Rather it relates to some stuff. It means that ‘λx(x=x)(o)’ says not that o is a member of something but simply that it is a part of something. It says that o < u, where ‘u’ denotes the whole of the material universe. The reverse point bears on the possibility of creating a theory of ‘pure sets’, and thereby a theory of what would certainly be entirely abstract objects. For the starting point of this would have to be a Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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definition of the null set, in terms of some paradigm expression that, appropriately enough, cannot have any application. It is common to take ‘x ≠ x’, to do this job, as we have seen, although there are many equivalent others. But the present question, again, is whether any such expression is a count predicate, for only then could it determine a set. No proof of this has been given, to date, or has even been attempted. Indeed, it has simply been presumed that all such expressions are count predicates, and so give rise to sets, or, at least, set-like totalities. But there are also mass terms, and these give rise to urelemente, like the marble in a statue and the gold in a ring, which themselves have no members. That means one cannot define the null set, either, via the Axiom of Extensionality in the form (x)(x ∈ y ≡ x ∈ z) ≡ y=z,
since this would make all urelemente the same, and so must be restricted to the case where y and z have members. Also it must be restricted to sets whose members are not themselves urelemente. For the same material might take different forms at different times, as we have seen, allowing it to be the case that {m}t ≠ {m}t’
even though these two objects have (timelessly) the same single member, i.e. (x)(x ∈ {m}t ≡ x ∈ {m}t’).
That means, in particular, that the sets y and z in the above axiom cannot be physical individuals. But that is quite in tune with standard Set Theory, since the given axiom is commonly applied just to abstract objects, such as numbers. These requirements might still seem strange, but they parallel some previously made points that connect count terms and sets. For, to allow for the pluralisation of count terms, there has to be at least the possibility of two instantiations before a description is count, ruling out standard definitions of the null set straightaway. But also proper names, and definite descriptions like ‘The Queen of England in 1710’, which identify individuals, necessarily are not pluralised, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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and cannot be used for counting, so they can only determine degenerate sets. For while one can say ‘She was Anne’ one cannot say ‘She was one Anne’, even though one can say ‘She was one person called “Anne”’ — but that is different. Thus we see that, while discovery of Russell’s paradox was the worst crisis in Set Theory’s history, it was not until much later that proper progress was made on the question of just what had caused it, and thereby what adjustment was needed to satisfactorily remedy it. Recent writers like Oliver and Smiley, indeed, have gone so far as to abandon this theory in favour of a logic of plurals, on account of the more widespread, but we can now see intimately related fault of its abstraction. But it is, in fact, through seeing how collective plural constructions are exactly equivalent to ones involving set expressions that we start to remove Set Theory’s remoteness from the world. And that removal is completed once we see the significance of the separation between things and stuff, i.e. individuals and material, since we then see that there are collective singular expressions, equivalent to ones that involve set expressions, but also other singular expressions, which are distributive, and have no representation in terms of sets. That clarifies what the material of the world consists in, and leads to a Set Theory that is applicable, where it is applicable, to real life. Was Principia Mathematica written by Russell and Whitehead, or by the pair of them, {Russell, Whitehead}? You can say what you like, as long as you adjust the number of the verb appropriately. If it turns out that Russell and Whitehead have written another book it will be a great surprise, but if it turns out that that pair has written another book it will be exactly the same surprise. More importantly, perhaps, the full analysis of the singular case shows that single individuals are sets already — singleton sets — and so are not the ontological primitives many have imagined them to be. In particular they are not ever some material, even though they might consist in some material, i.e. have some stuff as their only member. With regard to Oliver and Smiley’s justification for a logic of plurals, we have thus seen the invalidity of their main argument for the irreducibility of collective plurals. But there is also a clear formal difficulty with Oliver and Smiley’s plural account, connected with this, as well. For, in line with their idea that plural subjects are not Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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replaceable with singular subjects in collective predications, they would want to say, for instance, that nothing sums (sic) to 10, but instead that what sum to 10 are, for instance, 2 and 8, 3 and 7, 4, and 6, etc., where ‘2 and 8’ and the rest are plural terms. But how can they collect into an appropriate set the relevant values of such terms? Once we remember that some pairs of numbers do sum to 10, we can use standard set theory to collect up {2, 8}, {3, 7}, {4, 6}, etc. into a set of sets each of which sums to 10. One subset of this is, for instance, {{2, 8}, {3, 7}}. But Oliver and Smiley must try to talk about 2 and 8, and 3 and 7 in place of this subset, even though this group of four numbers is the same as 2 and 3 and 8 and 7. And what are they going to say: that 4 and 6 is (sic) a member of the set of things that sum to 10? Surely not, since ‘4 and 6’ is a plural subject. Likewise with The men, the women, and the children are each 10 in number.
In place of the subset {{x: Mx}, {x: Wx}}
of {y: y is 10 in number},
they would have to have the men and the women together, i.e. {the men, the women}
as a subset of {y: y are 10 in number}.
But that subset numbers 20, in this case. And they would have to have the men being a member of something, mixing a plural subject with a singular verb, again. Without the ‘singularisation’ provided by set expressions there is no way that Oliver and Smiley can handle these higher-order idioms, and indeed there is no sign that they have considered them. But clearly, if a set consists in these, and those, and
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those others, then each of these, those, and those others has to be a member of the set, since any member of it has to be singular. That is not the only further problem with Oliver and Smiley’s account. For trying to re-phrase individual cases so that the subjects are irreplaceably plural, to allow for causal effects, also has its problems. Oliver and Smiley, in fact seem to have produced a rather jumbled, and inconclusive collection of arguments against singular, set-theoretic re-phrasings of collective plurals in particular cases. Certainly this is not possible with Tom, Dick and Harry are similar to one another / each other,
but then these are distributive cases, surely: Each of Tom, Dick and Harry is similar to the others
is (x)(x ∈ {T, D, H} ⊃ (y)((y ≠ x & y ∈ {T, D, H}) ⊃ Lxy)).
Likewise with Tom, Dick and Harry are shipmates,
once a preceeding quantified phrase, ‘There is a ship’, i.e. ‘(∃z)(Sz…’, is inserted as an extra conjunct before (x)(x ∈ {T, D, H} ⊃ (y)((y ≠ x & y ∈ {T, D, H}) ⊃ Mxyz))).
And what is the difficulty with Tom, Dick and Harry carry the piano upstairs
being equivalent to something like The gang {Tom, Dick, Harry} carries the piano upstairs?
In another case Oliver and Smiley argue in this fashion: Tom and Dick are two, but the pair of them is one, so Tom and Dick cannot be the pair. But is it even grammatical to say Tom and Dick are two, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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as Oliver and Smiley repeatedly do? As before, Frege pointed out that number words are second-order quantifiers, requiring a firstorder predicate to be supplied before they are determinate: a yard is 3 feet but 36 inches, a year is 52 weeks but 365 days. Certainly, therefore, in Oliver and Smiley’s case, Tom and Dick may be two men, but with the unit of the counting then being specified, we see that those two men can easily be a pair of men, without 2 being 1. None of these cases, therefore, shows that plural collective predications are not equivalent to singular ones about the associated sets. Indeed, at one point, Oliver and Smiley admit that a set analysis is appropriate, but then go on (Oliver and Smiley [2001], 297): Fair enough, but other cases cause insuperable difficulties, such as plural proper names like “the Hebrides” and “Aberedw Rocks”. Please do not say that these names are really descriptions built from the predicates “is a Hebrid” and “is an Aberedw Rock”.
Here they seem to forget that maybe these rather arbitrary groups of things are better listed, than described. So all of this more particular argument by Oliver and Smiley is, at the very least, highly debateable.
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The most remarkable thing about the so-called Heterologicality Paradox is that a straightforward logical analysis of the case shows there is no proper paradox. The appropriate analysis, moreover, has been published more than once, yet even its author did not see that there was no problem. If the reader will take the trouble to consult, for instance, the 4th edition of Irving Copi’s Symbolic Logic (Copi [1973], 301, see also earlier editions of this book), then, in the chapter about the theory of types, he will find the definition of Heterologicality, and then a supposed proof of the contradiction: Het‘het’ ≡ ¬Het‘het’.
But Copi has to explicitly assume ‘‘het’ is univocal’ at one place in this ‘proof’, so his otherwise thorough analysis shows instead that the contradiction is not derivable as a matter of logic, but rests instead on a contingent premise — which therefore Reductio requires we deny. I will have some things to say later about Copi’s blindness to his assumption. Indeed there are comparable blindnesses with many other ‘self-referential paradoxes’ that have impeded understanding of these matters, as we shall see. But it is appropriate first to realise that Ryle (Ryle [1951]), saw the relevant point: ... what we are asked to decide is whether ‘heterological’ and ‘homological’ are themselves heterological or homological...But to ask this is to suppose that ‘heterological’ and ‘homological’ do stand for philological properties...And this supposition is false.
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Ryle then went on to make his well-known point about the unavailability of namely-riders in this case: If unpacked the assertion that ‘heterological’ is heterological would run: ‘Heterological’ lacks the property for which it stands, namely that of lacking the property for which it stands, namely that of lacking the property...” No property is ever mentioned, so the seeming reference to such a property is spurious.
I made a rather similar point myself, in Slater [1973], in an article Ryle kindly accepted while editor of Mind. For although there is trouble with the definition ‘x’ is H ≡ ‘x’ is not x,
there is no problem with ‘x’ is not self-applicable ≡ ‘x’ is not x,
so long as it is recognised there is a variable in the now fully expressed predicate, i.e. that ‘self’ is a pronoun replacing ‘‘x’’. For ‘x’ is ¬x ≡ ‘x’ is not x
is perfectly unobjectionable. It is useful, however, to keep in mind the Copi version of this kind of point, for, from his ‘proof’ we see that what is necessary is just that if ‘het’ just means one thing (Het) then Het‘het’ ≡ ¬Het‘het’,
and that leads us to a clear proof of Ryle’s point in the form: ‘het’ does not just mean one thing.
But the fact that a condition has been suppressed, which must be negated, is a parallel feature in several other cases. Thus, a demonstration that the Liar (and associated) paradoxes do not arise, given a proper logical analysis, was provided in this style by Sayward (Sayward [1987]), following work by Goodstein, and Prior. For the T-schema presumes univocality of the sentence said to be true, so the necessary fact is not that schema itself, but instead Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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and the now explicitly stated assumption shows the paradox does not arise when s = ‘¬Ts’,
without a further premise, that s only means that ¬Ts.
But the contradiction that would otherwise be available means that this further premise must be denied. The key point which maybe obscures this, even for many people today, is that the shift between the two latter indented statements involves a shift from direct to indirect speech. In the historical times, when it was thought there were paradoxes of self-reference, this distinction, significantly, was not formally expressible in the published logics. It only started to become formally available with the development of modal, and general intensional logics in the late 50’s, and early 60’s, since ‘s says that p’ is formally parallel to ‘a believes that p’ etc. Even with the first given formalisation of indirect speech, by Goodstein (Goodstein [1958]), which led to an understanding of grammatical operators, and the development of the propositional attitude logics, blindness to the direct/indirect distinction was present. For in Goodstein’s initial case, if A says that everything A says is false,
then we can prove Something A says is true,
and Something A says is false
and even Prior, in Prior 1958, was puzzled by this kind of result. Thus, if there was a sentence in a book, which said that everything said in the book was false, then Prior originally thought that, by the above kind of proof one could show by logic that there must be a Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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second sentence in the book! Certainly something said in the book must be true, and something said in the book must be false, but that does not require there to be two sentences in the book — since we are dealing with ‘saying that’, i.e. indirect rather than direct speech. I made this point in Slater [1986b], showing that all that is required by logic is that two different things are meant by any sole sentence in the book, i.e. that it is not univocal, as in the previous cases above. Prior himself started to see the distinction himself, in the chapter ‘Tarskian and non-Tarskian Semantics’ in Objects of Thought (Prior [1971], Ch 7). In particular, if the operator form ‘it is true that p’ is written ‘Vp’ then we can say, as he did: Ts ≡ (p)(if s means that p, then Vp).
And from this can be derived Sayward’s conditionalised T-schema — given that Vp is logically equivalent to p, i.e. that operator-truth is the null, identity modality. Now if the non-univocality of the Liar, and Heterologicality were just ordinary ambiguity then one might get a very proper demand for some namely-riders: ‘It is a bank’ has two senses, which can be specified, namely ‘It is a money bank’ and ‘It is a river bank’. But the lack of namely-riders is essential, in some cases. The general demand that we should be able to discriminate meanings, or implicit sayings, in parallel to explicit sayings is, of course, a common feature of compositional theories of meaning, as in, for instance, the Tractarian Picture Theory of Meaning. But meanings must be non-compositional otherwise there would be real paradoxes. This was shown by Thomason, in his and Montague’s arguments for an operator analysis of propositional attitudes, of which ‘it is true that’ above is, of course, just one example. For Montague had shown, quite generally, in Montague [1963], that ‘Indirect Discourse is not Quotational’, as Thomason put it (Thomason [1977]). But Thomason not only supported Montague, he also showed that a comparable argument proved that indirect discourse was not about any structured intensional objects either. Thus it was not, for instance, about some semantic analogues of sentences in a ‘Language of Thought’. This led Thomason (Thomason [1980], see also Slater [1998], Ch 2), to the conclusion that representational theories of mind, such as Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Fodor’s, had to be inconsistent, when fully worked out. As Asher and Kamp said (Asher and Kamp [1989]): To happy-go-lucky representationalists, Thomason [1980] is a stern warning of the obstacles that a precise elaboration of their proposals would encounter...(W)ith enough arithmetic at our disposal, we can associate a Gödel number with each such (representational) object, and we can mimic the relevant structural properties of, and relations between such objects by explicity defined arithmetical predicates of their Gödel numbers. This Gödelisation of representations can then be exploited to derive a contradiction in ways familiar from the work of Gödel, Tarski and Montague.
Asher and Kamp, nevertheless, were at the time trying to re-construct a representational theory of the attitudes despite this, on account of a difficulty they thought they had seen in non-representational theories. Asher and Kamp had produced what they took to be a demonstration that such non-representational accounts must fail (e.g. Asher and Kamp [1989], 94). For if an arithmetisation of the notion of expression is formulated, by means of some relation which says that the formula with Gödel number n means that p, then it seems that fixed points can be generated, and standard difficulties with selfreference lead naturally to contradictions. But Asher and Kamp’s ‘proof’ presumes that the expression relation is 1-1, and there is no guarantee that this is so. Comparable consideration of self-referential forms generated with an expression relation that does not presume univocality lead to no trouble, as above (Slater [1986b], 1991). Indeed there is a simple consistency proof which guarantees such an operator analysis cannot lead to contradiction in Goodstein’s original paper on the operator approach. It follows that different meanings cannot always be discriminated in different sentences as compositional theories of meaning would require. In Goodstein’s original case, for instance, we can go on to talk about that thing which A says which is true,
and that thing which A says which is false,
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but there need not be any way in which we can specify them further. In close connection with this, there is the otherwise largely technical question of just how the quantification involved with operator constructions completely works. What is the quantification over? Propositional quantification is normally taken to be substitutional, but one very forceful argument against such an account is that then there would not seem to be a stateable truth condition in the following kind of case (c.f. Loux [1998], 150): (∃p)(there is no linguistic expression in English for the thought that p).
Indeed, Loux has made this difficulty his entire basis for rejecting Prior’s account. For, of course, in such a case there expressly cannot be any namely-riders, if the statement is true. But it is not the case that the only complements in operator constructions are explicit used sentences, they can also be propositional descriptions. Thus we can say, for example, ‘Tom believed what Peter said’. And such a description can be provided in Loux’ case, even if, necessarily the proposition in question cannot be given directly. Thus we can talk about ‘that proposition there is no expression for in English’, which can be formalised using an epsilon term at the propositional level (see Slater [1989b].) And then the truth condition in the above case is that that proposition there is no expression for in English is indeed inexpressible in English — which follows the standard epsilon definition of the quantifiers. Once the 1-1 nature of Asher and Kamp’s expression relation is removed (c.f. Turner [1991], 11) we must accept that there may be things which can only be gestured towards, and not explicitly said. These points clearly bear, as well, on the possibility of replicating operator accounts with syntactic ones in the manner of des Rivieres and Levesque [1986] — see also Reinhardt [1980], Schweizer [1993]. Isomorphic structures have been constructed which seem to show that non-representational accounts of the attitudes are (materially) equivalent to certain representational ones. But that would, as before, conflate indirect and direct speech. Des Rivieres and Levesque notably do not consider languages which include their own expression relations, and indeed have to explicitly exclude selfreferential cases (des Rivieres and Levesque [1986], 126f), so they Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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have not faced up to all the questions which self-reference and expressibility in a language produce. Neither, in the more discursive philosophical tradition has Stephen Schiffer, for instance. Schiffer, in his book Remnants of Meaning (Schiffer [1987]), argues very tightly, but non-mathematically, against both a ‘sentential’ and a ‘propositional’ construal of attitude reports, which, at a distance, parallels Montague’s and Thomason’s formal work quite well. Moreover, he also defends substitutional quantification for properties and propositions, like Prior (Prior [1971], 35), and a non-compositional theory of meaning, which we have seen is required by the operator account. Schiffer, however, would still prefer a predicative account in terms of some causal ‘Language of Thought’. Thus he wants to assert at one place (Schiffer [1987], 206): (p)(∃s)(Harvey believes that p ≡ R(Harvey, s)).
But this gives, as before, a demonstrably false account of the operator construction. For since s is a syntactic entity the biconditional is falsified directly by Montague’s work showing that operators are not equivalent to meta-linguistic predicates. The central grammatical difference is that operators take sentences to make other sentences, whereas predicates take names to make sentences. And that means that operators do not describe, but merely qualify the sentences they act upon. Furthermore, although Schiffer uneasily settled for the use theory of meaning, he did not see the further central fact about operators in that connection, namely that, even is s means that p the operator form alone incorporates the meaning of the associated syntactic string: it does so simply by using rather than mentioning ‘p’. Fodor crucially forgot this possibility about meaning when arguing for his theory that propositional attitudes are relations to internal representations. For instance, in ‘Propositional Attitudes’ (Fodor [1991]) he argues for relations to internal representations, rather than external speech. Certainly in indirect speech the words in the complementary clause gain their meaning expressly by being internalised. But they are internalised just by not being merely mentioned, and so they are still a form of external speech. Also they are then internalised by the interpretative reporter not the agent, and
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their internalisation is just a matter of adoption and use, not structured representation. Fodor’s type of internalism is explicit in his discussion of ‘Objection 2’ to the above paper, although it is possible he would be happier with propositional attitudes being relations to what his internal representations are supposed to represent, namely propositions and facts. Such items might more easily be thought of as ‘in the world’, and so there is a little remarked quasi-externalism in Fodor’s remark, regarding ‘Objection 1’, that propositions are the mediated objects of propositional attitudes. But given Fodor’s first formulation of his theory above, which is more well-known, Montague’s and Thomason’s work applies, as with Schiffer. Likewise Gödel’s result applies, and also Tarski’s Theorem, forcing a hierarchy of metalanguages, which was shown to be impossible in the late 50’s again (Cohen [1957], c.f. Haack [1978], 144). For a recent reiteration of the impossibility see Turner [1991], 7. None of these issues are addressed by Fodor, yet, as Thomason had previously said, they are final against his language of thought account, on the above common presentation. But there are difficulties also with Fodor’s ‘mediated’ theory, as he presented it at the place above. For he there stressed the need for internal representations on the basis that propositions ‘don’t have forms’, believing that a theory claiming ‘empirical repute’ would have to apply in virtue of the form of the entities in its domain, and so making propositions a problem. He is right that (non-reified) propositions do not have forms — they are structureless, in Stalnaker’s terms (Stalnaker [1976]), for the very reasons Fodor gives — despite his wanting a compositional theory of meaning. But introducing immediate objects with forms does not get over the problem of formlessness (if it is one), since the relation between the immediate object and the proposition still brings in a formless ‘entity’. In short, ‘a believes p’ is maybe a ‘relation’ between a person and something formless, but re-writing it as, say, (∃s)(a Bel s & s means p)
does not eliminate the element which has no form. The point is obscured if the theory is presented as first above, namely as equating Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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‘a believes p’ with ‘a Bel s’. Schiffer argues, as we saw, that the relevant semantics is non-compositional (Schiffer [1987], Ch7), and that, of course gets at this in another way. The extra clause needed, ‘s means p’, like Asher and Kamp’s expression relation, is properly another ‘relation’ with a formless entity. And that also negates Fodor’s ‘decisive’ argument against Schiffer, at the end of Fodor’s review of Schiffer’s book (Fodor [1990]). For Fodor there tries to defend compositional semantics by insisting that there must be distinctive truth conditions for distinct expressions of Mentalese. But given the above points about the sometimes-necessary absence of namely-riders that cannot be guaranteed. Does the above have any further relevance to the other puzzles about self-reference which gripped the minds of philosophers and logicians earlier this century? I mentioned before the blindness that Copi seemed to suffer when thinking he was driven to an inescapable contradiction in connection with Heterologicality. A very similar blindness has also clearly affected many discussions on Truth — those that take Tarski’s T-schema to be unexceptionable. But somewhat similar blinkers would seem to have been worn in other cases of self-reference. This is very clear in the case of Berry’s Paradox, since Graham Priest, for instance, in his derivation of the seeming contradiction requires the denotation relation, as with Asher and Kamp, to be 1-1 (Priest [1983], 162). But if ‘the least ordinal not definable in less than 19 words’ defined, i.e. uniquely referred to some object, then that object both would and would not be definable in less than 19 words. Hence some identifying descriptions cannot have a unique denotation. To think otherwise, indeed, is to forget the function of demonstratives like ‘this’ or purely referential terms like Donnellan’s ‘the man drinking Martini’, whose semantics does not determine, in either case, a unique referent. In such cases there is no ‘namely-rider’ without a further parameter being given. And there is no problem about ‘the least ordinal not definable in less than 19 words’ then not correctly describing what it refers to, since that merely means it is non-attributive, and gains its reference gesturally from the context in which it is used. I have formalised the general point about pure reference in my many discussions of the epsilon calculus, and in Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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particular I have given the specific analysis of Berry’s case in Slater [1989a], 211. Priest, of course, has been so persuaded of the inescapability of contradictions that he has proposed we accept there are true ones. But you might as well say that we should accept there could be composite primes: it is in the definition of contradictories that they cannot be both true, so Priest’s ‘dialetheism’ is easily met with firm resistance (Slater [1995]). And, sensibly enough, more appropriate approaches to the phenomenon of inconsistent information (rather than inconsistent reality) have been presented (Brown, B. [1999]). It was Russell’s Paradox in Set Theory, of course, which magnified interest in the notion of self-reference, and it was that particular paradox which inspired the other main ‘paraconsistent’ logician, Newton da Costa in his development of non-classical logical systems (da Costa, [1986]). Da Costa constructed systems that withdrew from classical logic merely to the extent of not affirming ‘¬(A & ¬A)’. But it is only recently that a start has been made on an approach to sets that makes such an allowance for absurdity itself absurd. Thus Harry Bunt has formulated a general theory of ‘ensembles’ that includes sets as a special case (Bunt [1985], see pages 262-3 in particular). Bunt does not develop the idea quite to the best effect, but given there is a condition on an ensemble which is required to produce a set, then the Abstraction Axiom, like the T-Schema is not unconditional, and traditional problems with self-reference in connection with it then merely return one to the reverse case — namely to where no definite number of discrete elements is determinable, as when (but not exclusively when) the predicate in question is a mass term. That gives a theory of continua which involves indeterminate numbers, not excessively large ones (c.f. Bostock [1974], [1979]), and so one blindness which has held people to many paradoxes in set theory would seem to involve forgetting that with an amount of stuff no number can be specified without an extra parameter — the nomination of a unit. But there may be discrete items that have no total number, as we shall see. As Mary Tiles, amongst the few, has noted, Frege’s logical symbolism does not discriminate between count and mass terms (Tiles [1989], 151). It therefore suggests, even if it does not assert, that the concept of number is applicable to all concepts whatever. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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The point exposes clearly the omitted presupposition in Set Theory, which strictly should allow sets to be formed only when there is a determinate number of discrete elements to be collected together. Cantor himself appreciated that there could not be a number, or a set of all things because of his, Cantor’s Paradox, and for a time the question of the ‘limitation of size’ was a preoccupation of thinkers, wondering which infinities might still be consistently numbered. Cantor could, in the end, only offer as a postulate that Aleph Zero is consistent (Hallett [1984], 175), but a stronger point is made by Tiles (Tiles [1989], 63), who reminds us that there is indeed an inconsistency even in Aleph Zero once we attend not to the ‘cardinality’ or ‘power’ of the natural numbers, but the size or number of them: When two sets (whether finite or infinite) can be put into one-one correspondence with each other they are then said to have the same power or cardinal number...in the infinite case it will mean that an infinite set may have the same cardinality as a proper part of itself. This is, to say the least, paradoxical if one is thinking of number as a measure of plurality, of the size of a discrete collection. The size of a whole should always be greater than that of any of it proper parts. There must surely be more natural numbers than there are even numbers (indeed twice as many)...
In other words, if there were a determinate number of natural numbers it would be the same as the number of the even numbers, because of the 1-1 correspondence, but also it would be bigger, since the even numbers are a proper part of the natural numbers. Hence there is no determinate number of the natural numbers — and therefore no set of them (c.f. Boolos [1998], 306). Here indeed was a paradox about infinity that teased most theorists until the end of the nineteenth century, even if it has been forgotten by succeeding generations of set theorists. But its resolution is in a similar manner to those before: a presupposition of determinacy, now determinacy of the number of some discrete elements, must be negated. What is the number of the continuum? What is the number of natural numbers? There is no namely-rider, in either case. Tiles thinks that Cantor’s proofs of non-denumerability, nevertheless, still require us to discriminate between infinities. She goes on: Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Th(e) consideration (above) by itself would suggest that the fundamental distinction is that between finite and infinite, and that infinite sets are without number not only because they cannot be exhaustively counted but also because even the notion of cardinal number, as a measure of size or numerosity, can get no grip there. And this would have been the correct conclusion to have drawn had it not been for Cantor’s discovery that not all sets have the same cardinality, i.e. there are infinite sets which cannot be put into 1-1 correspondence with each other.
But given there are no completed infinities the facts about nondenumerability have to be looked at rather more carefully. In a constructivist account of real numbers they are represented by functions, and then naturally one cannot have a function such that fm(n) = fn(n)+1.
But this merely shows that we must discriminate amongst the denumerable between what can be enumerated, and what cannot. No difference in size is involved, merely a difference in being able to specify the function in question — give it as a ‘namely-rider’ in an extended sense. Now, if we define ‘real numbers’ not in terms of impossible Platonic limits but merely convergent sequences of rationals, then we are identifying ‘real numbers’ with certain functions, since sequences are functions from the natural numbers. But the name ‘real number’ is then strictly a mis-nomer, since a function is not a number, even if each of its values is one. There is no numerical representation, as a result, of the length of the diagonal in a unit square, for instance. This length is available geometrically, but all arithmetic can do is produce a function which generates a series of approximations. The function is then properly just a representation of the geometric ratio, but naturally is not equal to it. The ‘irrational number’ is not available ‘extensionally’ only ‘intensionally’ it may be said. But the difficulty for arithmetic is strictly more acute, since properly there is no number available at all, in this case. And the functions that should replace them are ‘non-denumerable’ merely because there is no enumerating function of them, as Skolem’s Paradox has otherwise indicated. For while all computable functions of one variable are enumerable, there is no way to Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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specifically enumerate just those that have completely defined values — otherwise the halting problem would be solved. Hence the ordinal numbers of those functions that generate ‘the real numbers’, although denumerable are not enumerable. There is, in other words, a more general kind of expression, which is like that for a ‘real number’ except certain decimal places may be undefined. These are enumerable, but diagonalisation does not produce a further one of them, since neither fm(n), nor fn(n)+1 need equal anything. Amongst the functions that generate these expressions are all the real-number functions, since, to any depth, each string of values, say of 0’s, and 1’s must be taken. But we cannot, in general, determine which these real-number functions are. Even if fn(x) determines a ‘real number’, which function it is is only determinable from its ordinal place amongst all computable functions, not from its ordinal place amongst the real-number functions. And that has the result that, if the latter ordinal place is ‘n’, then fn(x) is not a calculable function of n. Of course if one specifies a sequence just of real-number functions that makes it the case that which function is the nth in that sequence is determinable from n, and fn(n)+1 (modulo the base) will then be a further, distinct real-number function of n. But it is only the specification of such a sequence which makes fn(x) a function of both x and n, and so there is no further diagonal function in the general case, merely a denumerable list which cannot be specified, only gestured towards.
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There has been a lot of work done on Frege’s views on concepts, but there is a good deal of work still remaining to be done, it would appear. My attention was drawn to this fact through reading Antony Kenny’s recent book on Frege. Kenny is a very knowledgeable, highly skilled and widely respected philosopher, yet, remarkably, he finds ample justification for Frege’s contention that the concept horse is not a concept (Kenny [1995], 125). This is in contrast to common opinion, and most previous commentators, who held that Frege’s contention is clearly a paradox. Indeed, that it is considerably more than a paradox was seemingly even proved formally by Milton Fisk. For, with respect to what Frege said in ‘On Concept and Object’, Fisk said ‘I wish to show that these principles, which constrain Frege to deny that the concept horse is a concept, lead to a contradiction’ (Fisk [1968], 382). Clearly something fundamental has not been made clear, if Kenny can return, in the face of this, to a support for Frege. I support previous, learned, as well as everyday opinion in this paper, revealing not only the fallacy in Kenny’s present, and Frege’s original reasoning, but going deeper to unearth an inadequacy in the common logical symbolism for concepts — the lack of second-order nominal expressions, like ‘being a P’. Thus I start by showing that, since the predicate in a sentence is not in such a nominalised form, no reference to, merely expression of the associated concept is made there. The point has been suggested before (c.f. Kneale and Kneale [1962], 585-586, Wright [1983], 21), but re-working and extending it not only improves upon Dummett’s treatment of Frege on concepts, it also shows why some traditional difficulties, like the problem of Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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the unity of the proposition, the paradox of predication, and the Liar Paradox, are not present in natural language. I principally point out that, when we talk about a concept we nominalise a predicate, and so make the concept into a subject, even though the concept is not an object. On the way to correcting Kenny on Frege on concepts, I therefore also show why Frege’s associated belief that numbers are not concepts but objects is mistaken. As Dudman showed (Dudman [1976]), there is considerable difficulty in supposing that predicates have referents. Frege held that the referent of a predicate is a concept, and while, undoubtedly, predicates have a semantic role in determining the truth value of sentences in which they occur - so they do have ‘reference’ in one sense - this is not sufficient to ensure that they have anything which compares with the paradigm referents, i.e. bearers of proper names, and similar first-order individuating expressions. The problem is that a predicate is not a nominal phrase: ‘lived to 100’, for instance, is not ‘living to 100’. The participle refers to living to 100, but ‘John lived to 100’ does not refer to this, even if, in some way, the concept is involved in the proposition. The point, although widely discussed at one time, has not always been taken note of recently, even by those discussing ‘the unity of the proposition’. For the presupposition of one train of thought (see Linsky [1992]) is that while ‘Dobbin is a horse’, for instance, cannot just list the object Dobbin and the concept of being a horse (for how can a list of names say anything?), instead what the sentence says is that the object is related in a certain way to the concept, i.e. is of the form The object is an instantiation of the concept.
But this then itself would have to be analysed in the same way, which would generate Bradley’s infinite regress, and most find that unacceptable, even if one recent thinker (Gaskin [1995]) asserts that such a regress is just what is required to provide for propositional, or sentential, unity. The trouble with this tradition, it ought to be remembered, is its starting point: the idea that the sentence ‘s is a P’ is not only about s, but also, in some way or other, about being a P. What refers to the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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concept of being a P is a nominal expression, but ‘is a P’ is not a name, it is a description. The relation between a predicate and the associated concept, therefore, cannot be reference: it would be better called ‘expression’. But in just that fact lies the everyday power of ‘s is a P’ to say something, in opposition to ‘s, being a P’. For while no collection of names says anything, still a predicate, when appropriately completed, does. In other words, the non-referential, descriptional form is crucial, and is just what is forgotten in the tradition above. ‘Dobbin is a horse’ does not relate Dobbin to some other object, a concept; it simply conceptualises Dobbin in the way described by the given predicate, and so says something about him. Somebody who claims ‘Plato is wise’ has the concept of wisdom, but in that statement he is then applying the concept, not talking about it. This is a distinction that is not taken note of in current logical symbolism. Thus if we move to second-order logic, and start talking about concepts, then from ‘Wp’ one might try to deduce (∃F)Fp
or produce an equivalent (∃F)(F=W & Fp)
But there is something wrong with this way of writing the matter, since the intention is that in ‘(∃F)’ the ‘F’ is a nominal form - ‘there is something/some property/a concept such that’ - whereas in ‘Fp’ the ‘F’ is not nominal, but predicative. If ‘F’ is to be a nominal phrase throughout, then what is wanted is some ‘de-nominaliser’ which will take, for example, ‘living to 100’ into ‘lived to 100’, and more generally participles into appropriate finite forms of the verb. In semi-formal natural language a kind of pluralising is often used: if F is a concept, then maybe x Fs. Hence the sort of thing that is grammatical is s
(∃F)F p
and s
(∃F)(F=W & F p))
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In these terms, if what is expressed when ‘Ws’ is used is the concept of being wise, then there is nothing wrong with deriving these from Wsp, or with it being equivalent to, for instance, There is a characteristic Plato had, namely the characteristic of being wise
i.e. s
(∃C)(H (p, C) & C=W)
But no vicious, hierarchical regress ensues, since an explanation of how ‘Wsp’ says something was provided before, and this then applies to all higher-order predications. Moreover, no paradox then arises akin to the standard paradox of predication, since, if we formulate the predicate ‘x is a property which does not apply to itself’,
i.e. s
‘(∃F)(F=x & ¬F x)’,
then we still need that something is expressed by this predicate, for instance the concept G, in order to produce the propositional equivalence s
s
(∃F)(F=G & ¬F G) ≡ G G,
which is a contradiction. By Reductio, no property corresponds to the linguistic predicate. Again, the temptation might be to try to abbreviate the predicate s
‘(∃F)(F=x & ¬H (x, F))’
as s
‘H (x, G)’,
from which a contradiction could be drawn, by taking ‘x’ to be ‘G’; but that abbreviation equally makes the assumption that there is a property G. The basic difficulty is that, through not making the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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discrimination between expression and reference, the standard logical symbolism, when automatically writing (∃F)(F=x & ¬Fx) ≡ Gx,
provides no way to say that no property is expressed by the predicate. Comparable points hold for propositions, of course (c.f. Kneale [1972], 239f, Slater [2001b]), giving a resolution of other paradoxes. In the equivalence between ‘that p is true’ and ‘p’, we find the connection between reference to and expression of propositions: what is in one place referred to with the nominal phrase ‘that p’ is expressed and asserted in other places, with ‘p’. So, given that it operates on a sentence, ‘that’ is a nominaliser: it operates just like a demonstrative to produce, with the sentence then immediately used, a linguistic phrase which refers to what is then expressed. In connection with the Liar Paradox, for example, one must then consider what proposition is expressed, for instance, by the sentence, ‘what is expressed by this sentence is not true’. But one needs something of the form what is expressed by that sentence is that p
in order to obtain the supposed proposition, before one can derive the contradiction p ≡ that p is not true.
It follows, by Reductio, that the sentence does not express a proposition. Some of the above points about concepts were recognised by Dummett, in his lengthy treatment of such issues (Dummett [1981], Ch 7), but he did not investigate the full consequences for secondorder logic. Thus in place of ... is a concept
which might seem to require ‘...’ to be replaced by a first-order referring phrase, immediately leading to the absurdity that what satisfies it is both a concept and an object, Dummett exploited
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Frege’s own way out of the paradox (at the time unpublished), and constructed what he took to be a second-order predicate ... is something which everything either is or is not,
(Dummett [1981], 216-217) which enabled a replacement of ‘...’ with ‘a horse’. Dummett realised that if we wrote this more formally: (∃F)(...=F & (x)(Fx v ¬Fx)),
then the concatenation ‘Fx’ would have to be shorthand for ‘x is F’, so he was aware of some of the problems with this symbolism. But the major problem with Dummett’s analysis is that it has still totally avoided nominal phrases, like ‘being F’. This is unfortunate, since the formal replacement for specifications like ‘the concept horse’, would be ιF(H=F & (x)(Fx v ¬Fx)),
or its equivalent ‘ιF(F=H)’, with the fact that the concept horse is a concept then following from the evident truth that (∃G)(ιF(F=H)=G).
But Dummett would also have to say, as a result, ιF(F=H)=H,
which, given that he wanted to substitute phrases like ‘a P’ for the capital letters, would imply that the concept horse was a horse! The problem is that ‘a horse’ is not a referring phrase, so Dummett should have started from, for instance, ...is a property which everything either has or has not,
from which he could have derived (∃F)(...=F & (x)(Fsx v ¬Fsx)),
in which the gap is to be filled by participles like ‘being a horse’. Despite this grammatical confusion, however, Dummett’s thoughts here, about second-order logic, clearly show that he is trying Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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to follow Frege in denying there is an overlap between the two categories of concept and object. But many others, perhaps through following Fisk’s second way out of the Fregean Paradox (Fisk [1968], 387), have tried to distinguish between concrete and abstract objects, with concepts being in the latter class, but also, thereby, still clearly taken to be objects of a sort. As should be well known, Frege took numbers to be abstract objects, although he also sometimes took them to be second-order concepts, so, despite his avowed separation between objects and concepts, even he had inclinations the other way. To some extent the issue is a matter of linguistic taste, to be settled by some agreed convention, which, however, should be strictly adhered to. Either one says the only objects are concrete objects (like Julius Caesar), and then ‘abstract objects’ are dropped in favour of concepts of one kind or another; or one allows that some objects may be concepts — the abstract objects. The grammatical facts are that ‘the concept of a P’, ‘the proposition that p’, ‘the number of Ps’ etc. are all referring phrases, like ‘Julius Caesar’, so if referring phrases always denote objects then some objects are concepts. Alternatively one can say that some referring phrases do not denote objects, but simply, instead, concepts. I will (now) adopt the former usage, although several cautions are required. For first one must negate the idea that talk about concepts can be replaced with talk about certain object correlatives associated with them: sets. Fisk, in fact, shows this to be impossible, as Frege understood such things (Fisk [1968], 389, n16). Again, higher-order logicians have commonly tried to distinguish first-order from secondorder logics, by taking the semantics of the latter to be in terms of sets. Quantified property variables are then taken to range over sets, sets of sets, etc. (see, for instance, Gamut [1991], Vol I, 170-171). But any association between concepts and sets obviously cannot hold in general - otherwise the Naive Abstraction Axiom would hold, and we would have Russell’s Paradox. Wright has pointed out (Wright [1983], 2-3) that the association with sets is only plausible for predicates which determine sortal concepts i.e. count nouns, and mass terms, such as ‘yellow things’ ought to be handled differently. As Tiles said (Tiles [1989], 151), Frege’s logical symbolism does not discriminate between those concepts that do, and those concepts that do not, determine a unit, suggesting that the concept of number is Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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applicable to all concepts without restriction. So some symbolisation of the difference between ‘s is a P’ and ‘s is P’ is required before the area where sets arise can be identified. A different discrimination needs to be drawn by those who try to analyse propositions in terms of the sets of possible worlds in which they are true. A proposition isn’t the set of possible worlds in which it is true — because sets aren’t the sort of things that can be true. The general form of a proposition is maybe that the actual world is a member of the set of worlds in which it is true. But propositions can only be given by being expressed — and there is no way to express sets. The proposition is, above all, that so-and-so, so, as before, propositions are displayed using the form ‘that p’. Wright forgot this when he said (Wright [1983], 78): ...what should count, for example, as the ability to identify the referent of an abstract singular term if it is ‘appropriately presented’? Well, evidently, there is no presenting an abstract object save by the use of an expression of which it is the referent. So the ability to identify the referent of an abstract singular term, appropriately presented, cannot but be the ability to recognise the coincidence in reference of that term with another, used to ‘present’ the object to one. Knowledge of the reference of an abstract singular term, if taken to be a recognitional ability at all, thus has to be construed as the ability to recognise the truth or falsity of identity statements involving that term...
But there is one form of referential term that is significantly different from those which Wright seemingly has in mind: that which involves the demonstrative ‘that’ as in ‘that p’. For then the referent is not only referred to, but actually present, by being expressed. The idea that one can get no closer to ‘abstract objects’ than terms which refer to them is perhaps one reason why they are sometimes taken to be ‘abstract’; but there is nothing hidden, distant or non-concrete about an expressed sentence. And by being not just mentioned but used, it is not at all like an external, physical object; it is more like a tool, in both Heidegger’s, and Wittgenstein’s, senses. The point applies to concepts as well as propositions. We can see the planets, but why cannot we see the number of them? Is it that the number of them is still an object, but one that is invisible, except maybe ‘to the mind’s eye’? Well, determining the number of the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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planets, like determining any other property or quality of them is a perfectly straightforward physical process. We can, in particular, count the number of them, and so see that they are nine in number, but that they are nine in number is not a further object which is then directly seen, since ‘seeing that’ is an indirect speech locution, and refers instead to some propositional judgement. If ‘seeing that’ is ‘seeing in the mind’, all right, but there is nothing abstract, or otherworldly, about this activity. Wright, amongst many others, has discussed the epistemological problem with access to abstract objects (Wright [1983], 84f). There are difficulties for Realism on this score; but Empiricism seems no more promising on these matters (Wright [1983], 1f). The fact that Mathematics and Logic, for instance, are skills, and that one’s knowledge of these subjects only comes through practise with symbols and words is the basis for a more satisfying Anti-Realism. But, by the nature of our relation with our skills, one has to be satisfied with a less articulate form of philosophy in support of it. With the above discriminations between concepts and various other objects made, we now start to see why it would, after all, be better to avoid Frege’s main thought, and not distinguish concepts categorically from objects — and clearly, in any case, we could not follow Frege all the way, since when talking about concepts we must then not think we are talking about these concepts’ ‘object correlates’, as Frege tried to have it. Instead, when we are talking about concepts we are merely talking about them as subjects — which simply involves nominalising the associated predicates. But because of that we now have a decisive way of separating out concepts from objects such as Julius Caesar, for instance. For it is not primarily that he is or was a concrete object rather than an abstract one, or that he is the possible subject only of first-order predications, and it is categorically different objects which are the possible subjects of second-order predications. The crucial reason why such higher-order predications are to be distinguished from predications about physical objects is now apparent: Julius Caesar can be presented independently of language, whereas the higher-order subjects cannot. Now Frege was plainly not too clear about some of these discriminations. For Frege had his notorious difficulty with Julius Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Caesar, and he believed that numbers were objects, even though he could formulate no clear way of separating them from other objects (Wright [1983], 111). Indeed, Julius Caesar, the concept horse, the number of the planets, and John’s living to 100 all may be spoken about, i.e. be subjects, but they are subjects all of different orders, and only the first is a physical object. But it was Frege’s identification of numbers with objects rather than concepts which supported the specific reasoning which led him to separate objects from concepts, since he thought of the former as ‘saturated’ while the latter were not. So we must now inspect his reasons for that identification. In ‘Function and Concept’ (Geach and Black [1980], 24-5), Frege says: The two parts into which a mathematical expression is thus split up, the sign of the argument and the expression of the function, are dissimilar; for the argument is a number, a whole complete in itself, as the function is not... For instance, if I say ‘the function 2.x3 + x’, x must not be considered as belonging to the function; this letter only serves to indicate the kind of supplementation that is needed; it enables one to recognise the places where the sign for the argument must go in.
But, by Frege’s own grammatical criterion, the expression ‘the function 2.x3 + x’ must denote an object, even though, of course, such a function is just what would be quantified over in a modern functional calculus, showing that, despite Frege, that function is indeed a function. Moreover, Frege elsewhere, as was mentioned before, took numbers to be second-order concepts, so they, like functions, must be incomplete: ‘there are seven’ for instance, needs a substantive added to make a complete thought, even though we can also say things like ‘7 + 5 = 12’. The arithmetical statement means that seven things plus five things equals twelve things, although the fact that ‘there are seven things’ is only fully completed in sentences like ‘there are seven horses’ is often forgotten in talk about the more abstract statement. In fact, when numbers are thought of as abstract objects, the omission of ‘thing’ is surely crucial. For there is no problem about being acquainted with a number of things, as there is with being Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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acquainted with a number; and one’s acquaintance with the number, also, is no great problem, once it is realised that that is a matter of one’s practised activity with numerals and the like, in such processes as counting, addition, multiplication, etc. According to Wright, however (Wright [1983], 11), there were three more specific considerations that were involved in Frege’s judgement that numbers were abstract objects. One was the use of definite descriptions like ‘the number of the planets’, which we saw above could also be used for functional expressions. Another was the currency of numerical identities, like ‘5+7=12’, but there are identities at all levels, as we have seen; for instance, ‘what John feared was what Mary feared’, and ‘what ‘lived to 100’ expresses is the concept of living to 100’. Neither of these considerations is, therefore, decisive. The remaining consideration Frege appealed to was the contrast between, for instance ‘the number of planets is 9’ and ‘there are 9 planets’. The latter could be given a quantificational analysis, with ‘9’ emerging as a numerical quantifier, thus denoting a second-order concept. The former, by contrast, represents ‘9’ as a singular term. Now certainly if we write Nx:Fx = 9,
then the singular term on the right might induce us to think it is not functional. But it is equivalent to ‘the number of F’s’ on the left, which reveals the general functionality of numbers, and in the exact numerical quantifier (9x)Fx,
the place of the number 9 amongst second-order concepts is clear. The other, equational form was the basis for Frege’s formal analysis of Arithmetic, but, as Wright himself acknowledges, a construction of Arithmetic starting from numerical quantifiers has been developed by Bostock (Wright [1983], 36f). Wherein lies the priority of the seemingly non-adjectival form of expression, then? Frege’s use of this form of expression, of course, is in his generation of the number series, starting with his definition of zero: Nx: x ≠ x = 0 Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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and here he is supplying a specific function ‘Fx’ to fill the gap in ‘Nx: ...’, which means that the latter function is saturated. But that does not mean that ‘0’ is not itself a function, since it still leaves the truth-value of (0x)Fx
a function of ‘F’. Whatever priority Frege’s preferred form of expression therefore has in the development of pure Arithmetic, that does not show that in their application numbers are not concepts. What things are talked about is irrelevant in pure Arithmetic, but that does not mean they are irrelevant tout court. The point, therefore, is that while functions and numbers are unsaturated, they still may be spoken about, i.e. singular subject expressions may be formed which make reference to them, though not as objects. So some grammatical subjects do not denote objects, and Frege’s grammatical criterion for designating objects must be modified: it is first-order definite descriptions, and the like, which alone identify them (c.f. Higginbotham [1998], 6). The concept of seven things is then definitely a concept, simply because ‘thing’ makes it unsaturated. Not only Kenny seems to have been unaware of this, but Kenny’s discussion in particular, we can now see, has evidently not improved much upon Frege’s in this area. Kenny, in fact, does not take any note of the material Dummett cites as Frege’s own way out of his paradox; he treats merely the material published in Frege’s lifetime, notably ‘On Concept and Object’. There, Frege held there was merely an ‘awkwardness of language’ in saying ‘the concept horse is not a concept’. He pointed out (Geach and Black [1980], 46): A similar thing happens when we say as regards the sentence ‘this rose is red’: The grammatical predicate ‘is red’ belongs to the subject ‘this rose’. Here the words ‘The grammatical predicate “is red”’ are not a grammatical predicate but a subject. By the very act of explicitly calling it a predicate, we deprive it of this property.
Taking heart from this, Kenny asks us to consider (Kenny [1995], 125): (18) the verb ‘swims’ is not a verb, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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arguing that the third form is the key to understanding Frege’s contention that the concept horse is not a concept. For certainly, as Kenny says, ‘ “swims” ’ is not a verb.
But Kenny’s explanation of how the third form (20) is to apply to the concept case has no backing in Frege. Kenny says (Kenny [1995], 125): The expression ‘the concept...’ is really meant to serve the same purpose with regard to concepts which quotation marks serve in relation to predicates.
whereas Frege merely relevantly says (Geach and Black [1980], 46): The peculiarity of our case is indicated by Kerry himself, by means of the quotation-marks around ‘horse’; I use italics to the same end.
So Frege took there to be only one pair of quotation marks, in The concept ‘horse’ is not a concept,
even if Kenny’s device would produce the two pairs needed to parallel (20). But, in addition, Frege draws no proper parallel with Kenny’s (19), since, while, undoubtedly ‘The grammatical predicate “is red” ’ is not a grammatical predicate
the controversial claim is not that ‘The concept “horse” ’ is not a concept.
The relevant parallel in Kenny’s collection to remarks not about ‘the concept “horse” ’, or ‘ “horse” ’, but about the concept ‘horse’, is neither (19) nor (20), but (18) — unfortunately for Frege. For then, while certainly
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and so, in Frege’s terms, ‘the concept “horse” ’ denotes an object,
it is still the case that The grammatical predicate ‘is red’ is a grammatical predicate,
and so, likewise, using Frege’s own parallel, it must be that The concept ‘horse’ is a concept.
That means that Frege’s first treatment of the issue is condemned out of his own mouth, and Kenny’s attempt to defend him has lead instead to a clarification of just why he was wrong. The full explication of what concepts are, and how to handle them formally, without contradiction or paradox, must extend Frege’s own amended treatment, as we saw in Dummett. But that extension must include second-order nominal expressions, which have yet to be incorporated into the public symbolism.
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This chapter is concerned with locating the specific assumption that led Frege into Russell’s Paradox. We shall see that his understanding of reflexive pronouns was weak, for one thing, but also, by assimilating concepts to functions he was misled into thinking that one could invariably replace a two-place relation with a one-place property. It is well known that it was Russell’s Paradox that alerted Frege to the trouble with his system, for substitution of the set-abstract for ‘x’ in x ∈ {y | y ∉ y} ≡ x ∉ x,
produces a contradiction. It is less well known that there is not the same trouble with <x,x> ∈ { | y ∉ z} ≡ x ∉ x.
For substitution of the set-abstract for ‘x’ here does not produce a similar contradiction (c.f. Slater 1984). Substituting the set abstract for ‘x’ in the first case yields something of the form ‘a ∈ a ≡ a ∉ a’,
but the comparable substitution in the second case merely produces something of the form ‘ ∈ b ≡ b ∉ b’.
What this suggests is that ‘x ∉ x’ cannot be analysed as involving simply a predicate of x rather than a relation between x and x, the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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larger moral being that not all relations between a thing and itself can be a matter of that thing falling under a concept, i.e. ¬(R)(∃P)(x)(Rxx ≡ Px).
This is defended further in what follows, but it is what might have led Frege to think otherwise which is the main interest in the present chapter. In fact Carnap’s notes on Frege’s lecture course on the Begriffsschrift show precisely where Frege went astray. Other passages in Frege might also be quoted, of course, but these lecture notes, which have recently been published in English, are particularly clear in this regard. For there is an argument in them about concepts and relations, and specifically about the possibility of generating certain (one-place) concepts from certain two-place relations. One can put the problem highlighted by Russell’s Paradox directly in terms of relations and concepts, to bring it closer to these kinds of expression. Remembering the general format of lambda abstraction reduction is λyFy(a) = Fa, the trouble with the Paradox of Predication, (∃P)(x=λtPt & ¬λtPt(x)) ≡ λyQy(x),
is that one seems to be required to equate the ‘λyQy’ on the right with ‘λy(∃P)(y=λtPt & ¬λtPt(y))’,
i.e. with the expression formed by abstracting the ‘x’ from the left hand side of the equivalence, to produce a concept of x. But this does not work. For there is then a contradiction when ‘λyQy’ replaces ‘x’. One gets, because (∃P)(λyQy=λyPy)
is guaranteed, something of the form ¬c(c) ≡ c(c).
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On the other hand, there is no similar paradox if one abstracts from each ‘x’ separately, i.e. with (∃P)(x=λtPy & ¬λtPy(x)) ≡ λyλz(∃P)(z=λtPt & ¬λtPt(y))(x)(x).
In this case, substituting ‘λyλz(∃P)(z=λtPt & ¬λtPt(y))’
for ‘x’ does not produce a contradiction. Hence the left hand side, ‘(∃P)(x=λtPt & ¬λtPt(x))’,
cannot be analysed as involving simply a concept of x rather than a relation between x and x. A paradox only arises when taking the two argument relational expression on the left (with the two arguments identified the same) to be equivalent to a single-subject with constant predicate expression, as when there was just ‘λy(∃P)(y=λtPt & ¬λtPt(y))(x)’
on the right. With respect to the Paradox of Predication, we therefore see that, while ‘x is a property which y does not possess’
expresses a relation between x and y, that does not mean that, if ‘y’ is replaced by ‘x’, the result is a one-place property of x. And it is also very clear why this is so. For the predicate in the diagonal case, ‘x is a property which x does not possess’
is itself something that varies with the subject specified. What is predicated of a would be that it is a property which a does not possess, but what is predicated of b would be that it is a property which b does not possess. In other words, the general predicate can be taken to be ‘is a property which it does not possess’, and this contains a pronoun, which is a contextual item with no direct representation in a context-free language. This pronominal predicate
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is functional, in other words, and the nearest one can get to any concept it expresses, in a language without pronouns, is: λyλz(∃P)(z=λtPt & ¬λtPt(y))(s),
with ‘s’ not entirely free, but limited to repeating the subject of the sentence. Alternatively, by taking the second entry of ‘x’, in the diagonal case, as the subject one could take the general predicate to be ‘does not possess the property it is’, and the nearest one could get to any concept this expresses, in a context independent language would be λzλy(∃P)(z=λtPt & ¬λtPt(y))(s).
The attempt to construe these predicates as expressing a functional concept of their subject becomes needless, however, once the required subject term is actually attached, since the whole sentence is then revealed to be simply, though irreducibly relational, by each time being the analysis that was paradox free, namely: λyλz(∃P)(z=λtPt & ¬λtPt(y))(x)(x).
What Frege first missed with reflexive forms was the functionality of such pronouns. In the Begriffsschrift he says (Frege [1972], 127): The proposition that Cato killed Cato (can be considered in three ways, involving three different functions). Here, if we think of ‘Cato’ as replaceable at the first occurrence, then ‘killing Cato’ is the function. If we think of ‘Cato’ as replaceable at the second occurrence, then ‘being killed by Cato’ is the function. Finally, if we think of ‘Cato’ as replaceable at both occurrences, then ‘killing oneself’ is the function.
But ‘killing s’, with ‘s’ a pronoun, is not expressible in a context-free language, and, in addition, there are, in this case, other functions expressible in such a language which Frege does not mention: the two-place ones where the same name may (but need not) be put in both places: ‘killing’ and ‘being killed by’. So why did Frege think that by abstracting ‘Cato’ from both occurrences in ‘Cato killed Cato’ one obtains a one-place function rather than a two-place one? Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Here is the passage in Carnap’s notes that provides the answer. Frege says (Frege [2004], 155): A function of two arguments, e.g. x-y, can be transformed into a function of one argument in two different ways, either by saturation (x-2), or by identifying the two argument places (x-x). Functions of two arguments that always have a truth-value as value are relations. Therefore we can transform the relation x>y into a concept, e.g. x>0 (the concept of a positive number). Or we can form the concept x>x.’
If Frege could get the reflexive concept at the end, from the relation he started with, then a comparable derivation would produce a concept of x from the relation between x and x, on the left in the Paradox of Predication. So clearly this cannot be done. How could Frege have missed the fact that only a reflexive relation, and not a reflexive concept is derivable? Clearly it was Frege’s background in Mathematics that got him into trouble. Specifically, if there is no derivation of the supposed concepts from the given relations (both in the case of ‘x>x’, and in the case of the Paradox of Predication), then Frege must have been working entirely on the basis of his understanding of mathematical functions — which is also nicely illustrated in the passage above. For there is no doubt that, given any function of two variables, f(x,y), one can invariably obtain a function of one variable, by identifying the two arguments: f(x,x)=g(x). One important case where this undoubtedly happens is in Cantor’s diagonal procedure, for instance. But the seeming parallel case with relations and predicates, which generates Russell’s Paradox, works very differently, as we have seen. So, while it is well known that Frege thought of concepts as functions, the analogy between the mathematical case, and the ‘truth-value’ case must limp just at this point. The point to note is that the ‘truth-value’ case involves an equivalence, not an identity, and we now know, from all logic texts subsequent to Frege, that identity is not equivalence. Frege himself had a curious system, which allowed him to conflate, to some extent, identities and equivalences; but this has not been followed, and for good reason. For the expression for an identity, like ‘a=b’, is between two names, whereas the expression for an equivalence, like ‘p ≡ q’ is between two sentences. Thus ‘if and only if’ has quite a different grammar from ‘is identical to’. Maybe by ‘sentence’ Frege Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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meant ‘nominalised sentence’, since those certainly are referential expressions, and we can, as a result, say ‘John’s being a bachelor is the same as him being an unmarried male’. But since Frege did say ‘sentence’ we have every right to correct him. One cannot say, for instance, ‘John is a bachelor is identical with John is an unmarried male’. Sentences are the sort of expression that enters into equivalences, so they are not referring terms which can enter into identities (c.f. Prior [1971], 35), and specifically, therefore, sentences are not referential terms with the same reference as ‘The true’ or ‘The false’, as Frege thought. If anything at all like ‘Pa=T’, or ‘Rab=F’ holds, it is with ‘=’ as material equivalence, ‘T’ a tautology, and ‘F’ a contradiction. And then one has that ‘Pa ≡ T’ and ‘Rab ≡ F’ are equivalent to ‘Pa’ and ‘¬Rab’ respectively, making ‘Pa’ and ‘Rab’ quite unlike mathematical functions, and ‘T’ and ‘F’ nothing like their values. On the specific question of a reflexive relation being a function of one argument, certainly one might be able to define a function f(x) such that, say, Rxx ≡ (f(x) = 1), ¬Rxx ≡ (f(x) = 0).
But not only does that not make the relation the function, also the right hand sides of these equivalences cannot be figured as involving the same predicate of x. For ‘f(x) = 0’ is not contradictory, but merely contrary to ‘f(x) = 1’. If ‘f(x)=0’ was replaced by ‘f(x) ≠ 1’ there would be the same predicate of x; but no specific function would then be defined. The propositional equivalences above though, namely ‘Pa ≡ T’ and ‘Rab ≡ F’, maybe still suggest that predicative expressions are functions of some sort. So we must delve deeper. The question in Carnap’s case is whether from the relation λyλx(x > y), one can obtain the concept λx(x > x), as well as the concept λx(x > 0). The second reduction is straightforward, since applying the two-term relation to 0 one gets the concept of being greater than 0: λyλx(x > y)(0) = λx(x > 0).
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But the first reduction hits a problem. Proceeding as before one might try λyλx(x > y)(x) = λx(x > x),
but in ‘λyλx(x > y)’ the ‘x’ is a bound variable, and so the whole is equivalent to ‘λyλz(z > y)’, and using that form one merely gets λyλz(z > y)(x) = λz(z > x),
i.e. the concept of being greater than x. Frege talks about getting his second, reflexive, concept by identifying the two variables, but he cannot be thinking that one can produce his second concept from ‘λyλz(z>y)(x)(x)’,
since while that produces the statement ‘x>x’,
it still does not identify the concept he mentions, because that statement is still analysed as a relation between two arguments, not as involving a single subject with a predicate expressing the concept λx(x>x). Certainly if we could form λx(λyλz(z>y)(x)(x))
we could get the desired λx(x>x),
but the abstraction of x in that larger form is just as questionable, given that a relation between a thing and itself is not necessarily replaceable by a concept applicable to that thing. It might be said that it is anachronistic to use closed lambda terms to try explicate what Frege was saying. The Paradox of Predication, for instance, is not obtainable in Frege’s system, since he did not allow the application of a concept to another concept. So he most probably would have resisted the use of closed lambda terms in any explanation of what he meant by transforming the function x>y into the function x>x. But if one cannot talk about Frege in a language he Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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would not use, then one cannot criticise him on a scientific basis. One could not tell him, for instance, that sentences are not referring terms, as was done above, since ‘for Frege’ they are referring terms, and so one’s remarks, it might be said, are not about what Frege was talking about, namely ‘Frege sentences’ which are referring terms, by definition. Popper, however, amongst several others, had a lot to say about this sort of thing in connection with closed societies, and pseudo-science. Frege, in his article ‘Sense and Reference’, wanted the ‘F’ and the ‘a’, in an elementary sentence such as ‘Fa’, to be both referring phrases, taking the reference of the whole — a truth value — to be formed from the references of the parts. But only the singular term is referential: both the predicate, and the sentence as a whole, are merely expressive (c.f. Kneale and Kneale [1962], 585-6, Prior [1971], 35, Wright [1983], 21). They are expressive of a concept and a proposition, respectively. A further argument Frege had for his ‘truth-value’ conclusion rested on what is commonly called his ‘slingshot’ (see Neale, [1995], 765, and 791-5). But the irony with this argument is that it is plainly invalid if complete individual terms are used for referential phrases (Neale [1995], 795f), and Frege’s extensional logic (unlike Russell’s, for instance) did employ such complete referential terms. As it stands that point provides merely an ad hominem argument against Frege, but I made it very plausible, earlier, that a better representation of referential phrases is obtained using certain other complete terms, namely Hilbert’s epsilon terms, and so the inadequacy of Frege’s ‘slingshot’ argument can be argued for much more generally. The most crucial reason why sentences are not referring phrases, however, arises from the more basic fact mentioned above, that predicates are not referring phrases either. ‘For Frege’ they were, but Frege’s thought at that point, of course, was what got him into his paradox about the concept horse. Following Cocchiarella (Cocchiarella [1986]), we can remove that paradox. For we can distinguish, as before, the concept of being a horse (λxHx) from the predicate ‘is a horse’ (‘λxHx( )’), and so see that it is the latter, and not the former, which is unsaturated. That is because ‘being a horse’ is a nominalised predicate, which hence is a referring phrase, while ‘is a horse’ is not nominalised, and contains a gap that needs to be Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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filled before a complete thought can be expressed. If the predicate was a referring phrase, and referred to the concept, then that concept would certainly be unsaturated, but also the phrase ‘the concept horse’ would not refer to it, since that is saturated. Hence there would be Frege’s paradox. But the predicate ‘is a horse’, while unsaturated, is not a referring phrase, and what does refer in the area is the nominalisation of a predicate, such as ‘being a horse’. Indeed Frege himself, in his informal language, used nominalised predicates to refer to concepts, as when talking about ‘being killed by Cato’ and the like, in the first quotation above. But his ‘official’ position was that non-nominalised predicates had this purpose, so his theory was not in tune with his practise. Frege lacked a symbol for nominalised predicates in his formal language, which is what fundamentally led him to the conclusion that there is an inadequacy in natural language at this point, when it comes to expressing the semantical facts. For there is no inadequacy in natural language when it comes to expressing the associated natural language semantic facts, and Cocchiarella has provided a symbolisation separating out predicates from their nominalisations, so natural language in this area can now be represented in a properly formal manner. Adopting Cocchiarella’s symbolism we thereby move over to a clearer formal language without Frege’s paradox, and with a clear distinction between predicates and their nominalisations, for a start. But Cocchiarella’s language naturally contains nominalisations of zero-place predicates, i.e. closed sentences, and misconceptions about such nominalisations also got Frege mixed up about truth. For it is not a sentence ‘p’, but its associated ‘that’-clause (‘λp’, see Cocchiarella [1986], 217) which is referential, and the subject of judgements of truth and falsity. Thus it might be judged true or false that 5 is a prime number, for instance. But while ‘that 5 is a prime number’ therefore refers, it refers to a proposition, not a truth value, and so judgements of truth and falsity do not equate something with a truth-value, but instead predicate a truth-value of a proposition. That is to say, such judgements are not in the form of referential identities like ‘λP5=T’ and ‘λP5=F’, with ‘T’ and ‘F’ ‘The true’ and ‘The false’. They are instead predicative remarks like ‘TλP5’ and ‘FλP5’, with ‘T’ and ‘F’ ‘is true’, and ‘is false’, where the lambda Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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expressions obey the propositional schemas: Tλp ≡ p, Fλp ≡ ¬p. Certainly ‘is a prime number’ is then a function taking as values propositions which are true or false, but that means it is a propositional function, not a truth-value function like those in Frege’s ‘Function and Concept’ (see, e.g. Frege [1952], 28). As a result, the focus has to be on what sentences express. So look at a reflexive case again. If A, B, and C each shave D then they do the same constant thing — shave D — but if they each shave themselves, or, in a ring, shave their neighbour on their left, say, then they do not do the same constant thing, since what they do merely has a common functional expression: shave f(s), where the variable s is the subject. That means that a reflexive predicate is never, in itself, equivalent to a constant one-place predicate — although, contingently, of course, such a pair may be equivalent. They will be, for instance, if the number of objects involved is finite, since they then can be listed, and do not need to be described. Thus if all and only A, B, and C are self-shavers, then ‘x is a self-shaver’ is materially equivalent to ‘x is one of A, B, and C’. But it is not logically equivalent to this disjunction, i.e. it does not say the same thing. For the variable in the predicate ‘is a self-shaver’, namely the pronoun ‘self’, prevents the whole expressing a fixed property of its subject. Of course one can put ‘A shaved D’ differently. In line with the point made right at the start, one can say ‘A and D are a shaving (i.e. shaver-shaved) pair’ in place of ‘A shaved D’, and the predicate in that formulation is not functional even in the reflexive case: ‘A and A are a shaving pair’. But the latter does not predicate a fixed property of just A, since instead it predicates a fixed property of the ordered pair consisting of A and himself. Focussing on what they express, therefore, we see that a relational expression like ‘Rxy’ invariably generates thoughts about the two objects x and y, and the reflexive, or diagonal expression ‘Rxx’, as a result, generates thoughts about x and itself. Certainly the latter thoughts can be taken to be thoughts about the single subject x, as Frege saw with respect to ‘Cato killed Cato’. But what Frege missed there was that the two ways in which this can be done each can be expressed with a pronoun, since each of those two thoughts about the subject involve a further function of it. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Thus what is thought about Cato could be that he killed Cato, i.e. that he killed himself. More generally Rxx ≡ λyRyx(x) (≡ λyRys(x)),
and it could be that he was killed by Cato, i.e. that he was killed by himself. More generally Rxx ≡ λyRxy(x) (≡ λyRsy(x))).
Frege had no way to differentiate ‘killing himself’ from ‘being killed by himself’, but in both cases the pronoun is simply a context dependent replacement for the immediate subject, allowing the two expressions to be differentiated in a partly context-sensitive language as ‘λyRys’ and ‘λyRsy’. The tradition in modern logic has followed Frege in this respect, making a difference from the case with identities and equivalences. But the objective in all cases has been to give a representation of natural language structures and arguments, so these points about pronouns are in the same category as those about predicates and their nominalisations, and sentences and their nominalisations. It needs more than such a device as a lambda term, however, to adequately formalise reflexive pronouns. For pronouns are context-sensitive elements, and so a whole context-sensitive formal language is required to symbolise them. As before, there is no way to represent reflexive pronouns as such in a completely context-free language. Certainly a pronoun with its antecedent can be represented in a context free way — thus ‘a is not a member of itself’ is the same as ‘a is not a member of a’. But without that antecedent, the relevant context is unspecified, and the pronoun in the predicate ‘is not a member of itself’ is revealed to be a limited variable, i.e. the referent of the pronoun is seen to be functional upon the subject supplied. Frege’s handling of pronouns was therefore at fault, but also, when assimilating concepts to functions he was misled by a presumed affinity between identities and equivalences, and this had much larger consequences. Thus it was his move from ‘x-y’ to ‘x>y’, in Carnap’s passage above — and, of course, similar moves in other passages — that led him astray. Certainly ‘x-y’ is a mathematical function of x and y, but ‘x>y’ is not. The former has two arguments, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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the latter two subjects, i.e. things the whole is saying something about. As a result the latter is a propositional, or logical function, of the form λzλt(z>t)(y)(x),
and identifying the appropriate variables in it merely turns this into another propositional function with two subjects, λzλt(z>t)(x)(x),
not a function of one variable λz(z>z)(x),
as with ‘x-x’. Returning to the original case of set-abstraction, we see that the predicate ‘is not a member of itself’ does not collect its subjects into a set, because there is a variable, ‘itself’, in this predicate, and so those things that are not members of themselves do not thereby have a common property — not even the common property of being members of the same set. Nevertheless, each one, paired with itself, is a member of a set of pairs. The Set Abstraction Axiom, in the case of elementary sentences, viz (Rn)(∃S)(x1)(x2) … (xn)( Rnx1x2 … xn ≡ <x1, x2, … xn> ∈ S),
thus requires that none of the variables are repeated in the relation. Of course that still allows other equivalences to hold contingently in some cases, and even the given one when some variables are repeated — in the latter case they simply must be repeated, as well, in the ordered set on the right. Clearly similar revisions are necessary not only with n-ary relations, but also with second and higher-order ones.
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The formal structure of Frege’s ‘concept script’ has been widely adopted in logic text-books since his time, even though its rather elaborate symbols have been abandoned for more convenient ones. But there are major difficulties with its formalisation of pronouns, predicates, and propositions, which infect the whole of the tradition that has followed Frege. It is shown first in this chapter that these difficulties are what has led to many of the most notable paradoxes associated with this tradition; the chapter then goes on to indicate the lines on which formal logic — and also the lambda calculus and set theory — needs to be restructured, to remove the difficulties. Throughout the study of what have come to be known as first, second, and higher-order languages, what has been primarily overlooked is that these languages are abstractions. Many well-known paradoxes, we shall see, arose because of the elementary level of simplification that has been involved in the abstract languages studied. Straightforward resolutions of the paradoxes immediately appear merely through attention to languages of greater sophistication, notably natural language, of course. The basic problem has been exclusive attention to a theory in place of what it is a theory of, leading to a focus on mathematical manipulation, which ‘brackets off’ any natural language reading. I first remind readers that reflexive and personal pronouns only arise in context sensitive languages, and so do not have a direct representation in the context insensitive language of recent formal logic. That leads to resolutions of Russell’s Paradox, the Paradox of Heterologicality, and related antinomies.
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Fregean predicate logic has no representation of reflexive pronouns, as such, nor therefore of any predicate in which they occur. Instead it has tried to make do by relying on such equivalences as that between ‘x is not a member of itself’ and ‘x is not a member of x’, taking the whole of the latter as a ‘predicate’, whereas it is merely the form of a sentence. Like ‘is not a member of itself’ in the former, ‘is not a member of x’ is the predicate in the latter, even though it contains a variable. The confusion between forms of sentences and predicates goes along with a confusion between facts and properties, which are commonly distinguished in natural languages, but which have come to be conflated in lambda abstraction and set abstraction languages. Isn’t the property in question λx(x is not a member of x)
which is a constant? No: the predicate is ‘is not a member of itself’,
so it is the functional ‘λx(x is not a member of s)’,
where ‘s’ is a pronoun for the immediate subject, and not the constant term ‘λx(x is not a member of x)’
that expresses the property. It is not a property but a fact if x is a not a member of x. Certainly those things with the property in question are those things for whom the same form of fact holds, i.e. (x)(x is not a member of itself ≡ x is not a member of x).
But that does not mean that each of those things has the same constant property, i.e. that the property in question is not functional. Indeed, the equivalence itself is what gives the form of the function in this case. So one merely gets ‘x is not a member of itself’ Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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‘x is a member of its complement’
i.e. ‘x ∈ {y: y ∉ x}’,
or ‘x is one of the sets it is not a member of’
i.e. ‘x ∈ {y: x ∉ y}’,
not ‘x is a member of R’
for some fixed set R. There is a set spoken of, but it is a variable one dependent on something else, i.e. (x)(∃y)(x ∈ y ≡ x ∉ x),
not (∃y)(x)(x ∈ y ≡ x ∉ x),
and there is no contradiction obtainable from (x)(x ∈ {y: y ∉ x} ≡ x ∉ x),
or (x)(x ∈ {y: x ∉ y} ≡ x ∉ x).
A predicate is a part of a sentence, not the form of one; and it is then complemented by the subject. So what was missing in Fregean abstracts was the identification of a subject place where the variable is bound. Without a subject place being nominated in a sentence there is no way to isolate the remaining part which is the predicate; and in sentences with multiple possible subjects, therefore, there will be Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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several possible predicates associated with them: they are commonly distinguished by their voices, i.e. by whether they are active or passive. ‘John shaves John’ involves not the single ‘property’ λx(x shaves x),
but the property of shaving John, i.e. λx(x shaves John),
if the first ‘John’ is taken as subject, and the property of being shaved by John, i.e. λx(John shaves x),
if the second ‘John’ is taken as subject. In ‘John shaves himself’ the predicate is ‘shaves himself’, but in this case the men who shave themselves do not thereby share a constant property, since this predicate is context sensitive, and so it does not collect up a set of things with a constant property. If each of A, B and C shave C, they do the same thing — shave C — but if each of A, B, and C shaves himself, or say, in a ring, shaves his neighbour on his left, then they only do the same kind of thing, i.e. what they do merely has a common functional expression — shave f(s), where s is the subject. Of course, those men who shave themselves might still be listed, but being a member of the set {A, B, C} clearly does not identify the men in question either as self-shavers, or as left-neighbour-shavers. We have seen, in the last chapter, how to formulate an embracing set in connection with all the sets which are not members of themselves, but that again, of course, cannot be a set of those non-self-membered sets, on pain of paradox. The need to consider functional predicates re-appears with respect to the paradox of Heterologicality. For well-known contradictions are produced when the reflexive pronoun in ‘is not applicable to itself’ is taken not to have a variable reference, so that the whole is read as having a fixed sense: then ‘x is not applicable to itself’ seemingly comes out equivalent to ‘x is h’. But the resultant confusion is eliminated when one allows the pronoun to have its referent fixed through its association with the local subject term. In the sense, if not Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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the syntax of ‘x is not self-applicable if and only if x is not applicable to x’ there are four occurrences of ‘x’ and not just three, and so the predicate on the left is not replaceable by any constant one. And if one places ‘is not self-applicable’ at each of these four places there is no paradox. Replacing it at just the three places where there is an ‘x’ produces the notoriously puzzling contradiction: ‘is not self-applicable’ is not self-applicable if and only if ‘is not self-applicable’ is not applicable to ‘is not self-applicable’.
But replacing it in all four places produces: ‘is not self-applicable’ is not ‘is not self-applicable’-applicable if and only if ‘is not self-applicable’ is not applicable to ‘is not selfapplicable’
which is paradox free. Likewise in Quine’s famous reformulation ‘yields a falsehood when appended to its own quotation’: here ‘its own’ has a reference dependent on the subject to which the predicate is attached, and so the predicate is only properly expressible in a context sensitive language. In “‘yields a falsehood when appended to its own quotation’ yields a falsehood when appended to its own quotation” the second ‘its own’ has a reference determinable from the context, but the first ‘its own’ has no determinate reference at all, since it has been abstracted from any context by merely being part of a mentioned expression. As a result, in the required full sense, that mentioned expression, without a determinate reference for the pronoun, can never get appended to its own quotation, since when appended it is not mentioned but used, and thereby a portion of it necessarily has a different sense from when it is not appended, through the context sensitivity, which gives a determinate reference to the pronoun in it. In addition to this we must now see how Frege’s formalisation of elementary predicates has led to problems with ‘The Unity of the Proposition’, and ‘The concept horse’, and contributed to the Paradox of Predication and problems with an objectival reading of second-order quantifiers. For the difficulty with seeing all of the above compounds if one is attached to Frege’s ‘Px’ symbolisation of Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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elementary sentences. This form of symbolisation arose because of Frege’s assimilation of concepts to functions. Concepts were not objects, he believed, making ‘the concept horse’, which Frege agreed was a referring phrase to an object, paradoxically not a referring phrase to a concept. Instead, concepts, like P, were functions, and with the subject x taken as the argument of the function, the value of the whole subject-predicate expression was supposedly a truth-value, in parallel with ‘f(x) = y’. But this construal of truth as the value of a function is erroneous, as we have seen: it was just that which led to reflexive relations being reduced to unary properties, engendering Russell’s Paradox. More to the present point, if we wrote for the form of an elementary predication, not ‘Px’, but ‘x has P’, where ‘P’ is a referring phrase to the property or concept, it would be more clear that abstracting the subject leaves not a referring phrase to that property or concept, but merely the predicate in question, ‘has P’. In fact, in the symbolism ‘Px’ the ‘P’ is doing double duty for a predicative expression involving a finite form of the verb, like ‘is a horse’, along with a referring phrase to the property alone, like ‘being a horse’, which would allow objectival second-order quantification. But, of course, on the latter reading of ‘Px’, ‘x, the property P’ is not a coherent sentence, giving rise to the problem of how a proposition is unified, and is not just a list of names. It characterises the ‘bracketed’ approach to Fregean Logic which is more prevalent, that writers on the unity of the proposition, and Frege’s problem with the concept horse, for instance, do not commonly remember in their own practise the difference between ‘is a horse’, and its nominalisation ‘being a horse’, so that the absence of predicate nominalisations in Frege’s symbolism is not brought up as an evident difficulty with it. There is no lack of unity in ‘Dobbin is a horse’ like there is in ‘Dobbin, being a horse’, and the predicate in the former clearly does not refer to any concept, since it is not a referring expression at all, not being nominalised, as in the latter. As we have seen, it is a predicate like ‘is a horse’ which is functional, needing a subject term to be inserted to form a complete thought, and that leaves the associated concept phrase ‘being a horse’ to be unproblematically the designator of an (abstract) object. That, of course, relieves us of one difficulty with the Paradox of Predication: Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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is ungrammatical. The objectival quantification ‘(∃P)(x=P & ¬(x has P))’,
in which ‘P’ is a referring phrase to a property, is not ungrammatical, by contrast. Alternatively (c.f. Kneale and Kneale 1962, 602), if one introduces the nominaliser ‘§’, so that ‘§yPy’ refers to the property while ‘P’ remains descriptive (and so equivalent to ‘has §yPy’), that would allow a further expression, ‘(∃P)(x=§yPy & ¬Px)’,
to be well formed, in which the quantification is substitutional, with ‘P’ then an open predicate. In either case, however, we would have the question, as with Heterologicality, and Russell’s Paradox, whether the resulting predicate expressed a single, unambiguous property. Indeed, if (∃P)(x=§yPy & ¬Px) ≡ Qx,
then there would be a contradiction when x=§yQy. That contradiction proves indirectly that there is no single property, but what is the direct proof of this? How is it that ‘(∃P)(a=P & ¬(a has P))’,
and ‘(∃P)(b=P & ¬(b has P))’,
do not say a and b have the same property? The matter is now plain. Certainly these may seem to say the same thing about a as about b, but the sameness is only in the linguistic expression, which is systematically ambiguous. The common predicate ‘is a property which it does not possess’
contains a pronoun which refers back to the given subject term, so while the same syntactic item is involved, it is context sensitive with Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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respect to its referent, which means that the whole is replaceable by The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
‘is a property which a does not possess’
and ‘is a property which b does not possess’,
respectively, in the two cases. Geach once considered this kind of possibility, in connection with Heterologicality, but he thought one could remove the variability in other re-formulations (Geach [1972], 90). Certainly one can move the context sensitive element in ‘is a property which it does not possess’
to another place, as in ‘is a property, but does not possess that property’,
or ‘is, but does not possess the same property’,
but the referent of ‘that property’, and ‘the same property’ is variable as before. Some of the above confusions in Frege’s ‘concept script’ are replicated at the level of sentences. Frege said (Geach and Black [1952], 64): One might be tempted to regard the relation of the thought to the True not as that of sense to reference, but rather as that of subject to predicate. One can indeed say: ‘The thought, that 5 is a prime number, is true.’ But closer examination shows that nothing more has been said than in the simple sentence ‘5 is a prime number.’ The truth claim arises in each case from the form of the declarative sentence, and when the latter lacks its usual force, e.g. in the mouth of an actor upon the stage, even the sentence ‘The thought that 5 is a prime number is true’ contains only a thought, and indeed the same thought as the simple ‘5 is a prime number.’ It follows that the relation of the thought to the True may not be compared with that of subject and predicate. Subject and predicate (understood in the logical sense) are indeed elements of thought; they stand on the same level for knowledge. By combining subject and predicate, one Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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reaches only a thought, never passes from sense to reference, never from a thought to its truth-value. One moves at the same level but never advances from one level to the next. A truth-value cannot be part of a thought, any more than, say, the Sun can, for it is not a sense but an object. If our supposition that the reference of a sentence is a truth-value is correct….
But it is painfully evident, to anyone who does not bracket off his or her own practise, that Frege’s supposition is not correct, since sentences, unlike their nominalisations, simply are not referring expressions. Moreover, as Frege himself admits in this passage, the referent of ‘that 5 is a prime number’ is a thought, and it is that sort of thing which can be said to have a truth-value. Frege did have an expression for such thoughts — with a horizontal line replacing ‘that’ — but regrettably they were not taken as subject terms, allowing him no natural expression not only for modal remarks like ‘that p is necessary’, but also more general comments of the same kind, like ‘that p is obvious’, ‘that p is surprising’ etc. With thoughts allowed a subject place in this way it becomes clear that truth is a property of thoughts, and it is the sentence nominalisation ‘five’s being a prime number’ which refers to its being true that 5 is a prime number. Frege’s judgment stroke, which was put before selected thoughts, is possibly best seen as performing this sentence nominalising function, with the consequent marking of something’s being true, since certainly, as the passage above shows, Frege did not take it to mean the predicative ‘it is true’, or even ‘it is asserted’. Is the concept of being a prime number, then, a function that takes the value True when applied to the argument 5? No: the concept of being a prime number, like the concept of being a horse, is an object (an abstract object), since predicate nominalisations are straightforwardly referential. A concept is thus ‘saturated’ and so cannot be a function; it is the associated predicate, ‘is a prime’, or ‘is a horse’, which is incomplete, and incomplete expressly because it is merely a part of a sentence. Thus we may complete such a sentence by inserting ‘5’ or ‘Dobbin’ in the remaining gap, but when asserting the whole we then do not transform 5, or Dobbin into the True, we merely express a thought, which has truth as one of its properties. In the mouth of an actor on a stage certainly the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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same thought might be entertained rather than asserted, but that is just a matter of the associated sentence being quoted, and so being merely mentioned, not used. Use requires no assertion sign, merely the dropping of quotes, and the use of a sentence rather than the quoting of it is crucially what is involved in truth assessments like ‘that p is true’. The nominaliser ‘that’ has been symbolised ‘§’ as we have seen. By prefacing a used sentence ‘p’, which expresses a thought, it makes ‘§p’ into a denoting phrase that refers to the thought. The referential combination ‘§p’ is closely related to, although distinct from a plain sentence nominaliser: Tarski’s being wrong, for instance, is the same as its being true that Tarski is wrong, because of the propositional truth scheme ‘p ≡ it is true that p’. There are still many, of course, who accept Tarski’s sentential Tscheme, thinking truth is primarily expressed by a predicate of sentences, instead of a predicate of ‘that’-clauses (c.f. Prior [1971], Ch. 7). The tradition at this point has departed from Frege, in accepting that ‘is true’ is a straight predicate, but has still held onto the Fregean dismissal of ‘that p’ as a possible subject term. But in selecting, instead, mentioned sentences as the subject of judgments of truth it has run into a host of troubles. For, notoriously, a paradoxical form of self-reference is obtained in connection with identities like “t = ‘t is not true’”, if truth is a disquotational property of sentences. Then ‘t is not true’ is true if and only if t is true (on account of the identity), but also true if and only if t is not true (by disquotation). Revisiting the points made above, about Quine’s reformulation of the heterologicality predicate, helps one to see where the fallacy lies in this disquotational argument, however. For in such an identity as “t = ‘t is not true’”, while the referent of the first ‘t’ is thereby given, there is no referent provided for the second ‘t’, since it is merely mentioned. So there is crucially no implication that (∃x)(x = ‘x is not true’), which means there is no real self-reference — the relation is merely a nominal one between t and something which includes the letter ‘t’. Hence when disquoting the ‘t is not true’ above to obtain the implication that t (sic) is not true, one is illicitly assuming a reference for an item lacking a reference, and thereby producing a transposition, from mention to use, as invalid as that which would be involved in quantifying over a quoted place. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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By contrast, not even the appearance of a contradiction is obtainable if truth is primarily expressed by an operator on sentences, i.e. a predicate of ‘that’-clauses, making truth a property of thoughts. For there is then no opportunity for paradoxical substitution of the above identity: one says that t is not true, since sentences are no longer the bearers of truth, and that simply means that ‘t is not true’ is not true. In saying that t is not true, isn’t one asserting that the sentence one then uses (‘t is not true’) is true? No: one is not saying ‘p’ is true when one says it is true that p, because use is not mention, and operators are not predicates, and so, a fortiori, not meta-linguistic predicates. The sentence ‘t is not true’ is not said to be true when one says that t is not true, since one is then saying, instead, that the referent of the ‘that’-clause is true. There is no self-reference available with operator constructions, of course: mereology prevents ‘it is not true that p’ from itself being ‘p’, since a whole cannot contain itself as a proper part. So there is no way that appropriate self-reference can arise, if the subject of the judgment of truth is of the form ‘§p’. Can’t we get self-reference, though, using other subjects in such judgments? What if Tarski utters at time t just ‘What Tarski states at time t is not true’? Then, it may seem, what Tarski states at time t is that what Tarski states at time t is not true, and we have a characteristic impasse. But can we be so positive about what Tarski states at the given time? It is here we must remember the possibility that the language is context sensitive, with the consequent distinction between sentences, statements and propositions being required to identify the bearers of truth (c.f. Haack [1978], Ch. 6). Reading the statement made by the use of a sentence off the face of that sentence is not guaranteed if the language is context sensitive — only the proposition expressed is obtainable that way. And what makes the given sentence context sensitive (and similarly for other like sentences) is that what its subject term refers to depends on what Tarski states at time t in the world spoken about. In particular, therefore, it means that the statement made by the sentence varies with who utters it (or when, or where it is uttered). If someone else utters it, for instance, and Tarski says something else at time t, then a different statement is made from any statement made when Tarski utters it at time t. And that latter case is most like the case where Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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someone else utters it, and Tarski says nothing at all at time t, since if he states no single thing with his words, then there is no way to identify what he states, and so there is no paradox. Writing ‘what Tarski states’ as ‘εrStr’, certainly then Tarski states that what he states is not true, i.e. St§¬TεrStr,
but only supposing further that only one statement is involved can one obtain a contradiction. For St§¬TεrStr
entails (∃r)Str,
i.e. that there is something Tarski states, so we have StεrStr,
i.e. that Tarski states what he states, by the epsilon definition of the quantifier. But one needs the further assumption that (∃!r)Str
to get (s)(Sts ⊃ s=§¬TεrStr),
and only on this basis does it follow that εrStr=§¬TεrStr,
i.e. that what Tarski states is that what Tarski states is not true. This identity then gives TεrStr ≡ T§¬TεrStr,
and so TεrStr ≡ ¬TεrStr, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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which is the contradiction. But all that follows is that the further assumption is false, i.e. that not just one statement is involved. As in the previous paradoxes with predicates, the paradox with Tarski’s sentence (and the like) is resolved through recognising it is ambiguous: we have StεrStr
and St§¬TεrStr,
where εrStr≠§¬TεrStr.
But if the above demonstrates that many classic paradoxes arose because of the level of abstraction in the languages studied, the question subsequently arises of what form a more sophisticated formal logic might take. Maybe it is clear that attention to the context sensitivity of pronouns clears up many classic paradoxes, attention to predicate nominalizations clears up other notorious difficulties, and attention to sentence nominalizations yet others, even so, it may be asked, what is the general form, and, in particular, the syntax and semantics of any reformed logical symbolism to be? The main problem here is accepting that it must be radically different in certain respects, even if in others the significant differences are fairly minor. But some of the major changes merely take the form of surgery on what is old, and do not add much that is new, so the final result is not too strange. Thus, for a start, there is no pressing requirement to add pronouns to the symbolism, since pronouns are an accidental element in language, and in any context of use are replaceable with the nouns they abbreviate. One minor, but still significant change occurs in the case of propositions and operators on them, like ‘it is necessary that p’, ‘it is believed that p’, and so on. The study of such propositional operators has developed since the 1960s without access to a generally accepted expression for the equivalent, non-cleft forms ‘that p is necessary’, ‘that p is believed’ etc., and clearly the inability to formalise such Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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equivalences has arisen because of the deliberate absence in the heritage from Frege of subject terms of the kind ‘that p’. Thus what has been written, in modal logic, ‘Lp’, with ‘L’ a propositional operator, could be written ‘N§p’, with ‘N’ a predicate, making ‘N§’ merely a more structured form of ‘L’. But without ‘§’, or the like, in the symbolism no such equivalence can be expressed. Prior, who promoted at great length the study of propositional operators, in his posthumous book Objects of Thought, did so without the advantage of Kneale’s formalisation of ‘that’, and this led directly to the wellknown difficulties with Prior’s treatment of propositional quantification, notably the difficulty of giving an objectival reading of his propositional quantifiers. Moreover, Prior also missed the fact that, with the more articulated grammar for modal and general intensional constructions, their logic becomes directly a form of firstorder predicate logic, with the appropriate subject and predicate terms — although this is best extended by adding propositional epsilon terms, as in the example above, in parallel with the common conservative extension of the predicate calculus to the epsilon calculus. Epsilon terms are particularly valuable in formalising the descriptive replacements for certain anaphoric pronouns, as we have seen. Thus the ‘it’ in ‘Tarski stated something. It was that what he stated was false’
has the replacement ‘what Tarski stated’, so the whole becomes ‘(∃r)Str & εrStr=§¬TεrStr’.
Evidently there are more basic mental pre-occupations which might need to be overcome, before allowing into logic referential terms to thoughts, notably Quinean fears of ‘intensional entities’ like propositions, and Quinean quotational theories of belief, for instance, which involve the same fear, and try to start from properties of sentences, instead. But the latter pre-occupation, as with Tarskian quotational theories of truth, ignores the difference between two senses of ‘say’, namely ‘utter’, and ‘state’ (c.f. Kneale [1972], 230); and the former ignores related aspects of ‘that’-clauses. We say, for instance, ‘That John was already there was surprising’, and so predicate something of a subject of the form ‘that p’, but the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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nominaliser ‘that’ is then clearly not a quotation maker, since what was surprising was not the sentence ‘John was already there’, but instead the fact that John was already there. This leads us to the most radical change, which involves a fundamental re-assessment of what semantics is about; indeed, in one sense, it becomes quite inappropriate to ask for a semantics of the more sophisticated language. For meta-linguistic forms like “‘p’ is true”, in terms of which semantics traditionally has been expressed, entirely lose the significance they had. Are not such forms needed to give us an understanding of the object language? Not if that language is our Mother Tongue, and understanding is obtained just through use. Prior saw that, because of this move to use, the move to the operator approach spelt the end of Tarskian semantics (c.f.Prior [1971], Ch. 7), but that does not mean it is the end of formal semantics entirely. For when using the propositional operator form ‘it is true that p’, instead of the meta-linguistic form “‘p’ is true”, a truth locution is still involved, only now in the ‘object’ language itself. But that means it has to be fully appreciated there is no problem with that, i.e. that a language can be semantically closed without contradiction, since the tradition from Tarski has come firmly to believe that a language cannot be semantically closed without being defeated by The Liar Paradox. But, in addition to the above resolution of this paradox, the consistency of the epsilon calculus is readily shown (Leisenring 1969, 48), and Goodstein proved the consistency of his operator theory at the end of the first formal paper on the subject (Goodstein [1958]). In addition to this consistency, the now propositional T-scheme ‘T§p ≡ p’ means, for instance, that the Principle of Bivalence (it is true that p or it is true that ¬p) is logically equivalent to the Law of the Excluded Middle (p v ¬p), and lines in a truth table such as ‘If it is true that p and it is not true that q, then it is not true that p & q’ are equivalent to propositional theses like ‘(p & ¬q) ⊃ ¬ (p & q)’. So the move to a Priorian semantics not only produces a semantically closed language but also collapses, amongst other things, the usual soundness and completeness proofs in propositional logic. The grammar of second order logic needs to be revised, of course, once nominalised predicates are incorporated. But, in fact, this has Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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already has been done, to the required extent, by Cocchiarella (Cocchiarella [1986], Chs IV, V, VI). Thus Cocchiarella, as we have seen, writes ‘λxφx( )’ as a complex predicate, when ‘φx’ is a wff, with ‘λxφx’ as the corresponding singular term (Cocchiarella [1987], 161). These lambda expressions are in place of the everyday ‘has P’ and ‘P’, with ‘P’ a referring phrase, and Kneale’s ‘P’ and ‘§yPy’, with ‘P’ an open predicate, mentioned before. But Cocchiarella did not point out the consequent relief provided for the problem of the unity of the proposition, and the problem about the concept horse, or the problem about concepts being functions, and the resultant problem about truth-values being the referents of sentences. Cocchiarella was merely concerned with Russell’s Paradox and the Paradox of Predication, in the application of his enriched language. As we have seen, however, the basic problem with the latter two paradoxes lies elsewhere, in problems about the formalisation of pronouns. The resolution of these problems about pronouns principally impacts on the allowable abstracts in the Lambda Calculus, and Set Theory, showing there are wider repercussions of this matter than those just in Formal Logic. Take the case of ‘John shaves John’ which we looked at before. Since abstracting just the first ‘John’, or the second ‘John’ from this sentence leaves a predicate still involving ‘John’, it certainly follows that we must abstract both ‘John’s to obtain a predicate which lacks ‘John’ entirely. But, as we have seen, the crucial question is whether that predicate is still just a predicate of ‘John’. It is with respect to the subject of this predicate that the traditional Lambda Calculus, and Set Theory misconstrued the grammar of the matter, and thereby the possible subject-predicate divisions of ‘John shaves John’. For if one wants a predicate that lacks ‘John’ entirely, one must talk not about some property of John, but a property of the subject pair <John, John>: the property of the subject pair <John, John> being a shaving (i.e. shaver-shaved) pair. More generally, as we have seen, repeated variables have to be handled by means of ordered sets: Rxx ≡ (<x,x> ∈ {: Ryz}).
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Forgetting this, the received view of reflexives, within the Fregean tradition, has been that they turn binary relations into unary properties by identifying their two argument positions. But that there is something amiss with this line of analysis is very evident, since, notoriously, there is a severe problem with anything of the form ‘x ∈ {y: Py} ≡ x ∉ x’
in which there is taken to be a unary property, while there is no corresponding problem with ‘(<x,x> ∈ {: y ∉ z}) ≡ x ∉ x’
in which a binary relation is involved. So this relation between x and itself provably cannot be configured as a unary property of x, and it cannot be presumed, as a result, that abstracting ‘John’ entirely from ‘John shaves John’ leaves a predicate just of ‘John’. Abstraction separately, i.e. with distinct variables, from any or all of the individual subject places in an elementary sentence is entirely possible, but repetition of the same individual subject must respect the undoubted fact that ¬(R)(∃P)(Rxx ≡ Px). That is why ‘λx(x is not a member of x)’ does not denote a property: because, in the first place, ‘x is not a member of x’ is not a predicate with a single subject ‘x’, but instead a relation — the relation ‘is not a member of’ — with subjects ‘x’ and ‘x’. So the relation must be identified independently of the repetition of ‘x’, and have a pair of argument places: Rxx ≡λyλzRyz(x)(x).
So we see the full dimension of the major tangle Fregean Logic got itself into, through failing to understand the workings of such reflexive pronouns. In its attempt to construct a formal language without any contextual elements, it lost track even of what a property is, and specifically confused properties with states of affairs. Being unable to express, as a separate entity, the context-dependent pronoun in such a diagonal predicate as ‘is an instance of itself’, it has taken the subject added to this to be part of the property expressed. It has tried to reproduce the effect of the contextdependent predicate by introducing and repeating the subject term, in Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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abstracting from the sentence form ‘x is an instance of x’. The ‘property’ that being F has if it is an instance of itself, is then not being F, but an abstraction from being F’s being F. You might as well say that the property John has if he is happy is not the property of being happy but of his being happy, which is a state of affairs, or, more specifically, of something’s being happy, which is a generalised state of affairs. In Grelling’s case the ‘property’ the predicate ‘Fv’ has is then taken to be not the property of not applying to itself, i.e. the property of not applying to ‘Fv’, but the generalised state of affairs of something’s not applying to itself — so that is what Heterologicality is, a kind of state of affairs. In Russell’s case, the ‘property’ the set x has if it is not a member of itself is taken to be not the property of being a non-member of x, i.e. λy(y ∉ x), but the generalised state of affairs of something’s being a non-member of itself, i.e. λy(y ∉ y). These results echo William Kneale’s earlier conclusion ‘there is no predicate … “x is heterological”’ (Kneale [1971], 228), drawn not from a study of pronouns, but from a formal analysis of what a predicate is. For the parallel point in connection with Russell’s Paradox see Kneale [1971], 335-7. But there are even deeper consequences than that. For we shall now see that these results fundamentally undermine the fixed-point theorem having any analogue in natural language. Indeed, the fixedpoint theorem has figured centrally in professional studies of selfreference. It has the form: q ≡ A(gn‘q’),
and equates certain propositions with propositions about the Gödel numbers of sentences that express them. But while the theorem is certainly necessary, there is no necessary connection between the number and the sentence in question, since that depends on the system of Gödel numbering that is chosen. So that is where pragmatics still comes in: the right hand side of the equivalence is about a certain number, but does not itself make reference to the sentence in question, since the association with the sentence is made only through the interpretation of that number in terms of an external system of Gödel numbering. More important, however, is whether Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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there is any analogue of this fixed-point theorem when natural language is in use. The theorem clearly arises in connection with the meta-linguistic attachment of Gödel numbers to formulas in formal systems, and it is that which is principally lacking in connection with natural language use. We find the following very full definitions, for instance, in Goble 2001, 77: In particular, for any predicate H and any expression X, the expression H(X) is a sentence that informally asserts that the set named by H contains the Gödel number of X. To each pair of numbers x and y is associated a number denoted by x*y such that if x is the Gödel number of an expression X and y is the Gödel number of an expression Y, then x*y is the Gödel number of the expression X(Y). Now, to the important notion of diagonalisers. For each predicate H is associated a predicate H# called the diagonalizer of H. Informally, for any number n, the sentence H#[n] expresses the proposition that n*n lies in the set named by H. Thus H#[n] expresses the same proposition as the sentence H[n*n]. Also, therefore, for any expression X, the sentence H#(X) expresses the same proposition as H(X(X)).
Thus there is a series of equivalent expressions, H#[n], H#(X), H(X(X)), H(X[n]), and H[n*n]. But consider only the sentence The formula with Gödel number n is not satisfied by n,
i.e. ¬X(X), or ¬X[n]. If this has the form ‘M[n]’, with a predicate ‘M’ independent of ‘n’, then it should be equivalent to The formula with Gödel number m is satisfied by n.
That leads to a contradiction, of course, upon substituting ‘m’ for ‘n’. Does that mean there is no predicate ‘M’ of the required kind? The traditional lesson drawn from this kind of case is instead that the Gödel number of the predicate ‘M’ cannot be defined, but, as we have seen, the deeper point is that there is no constant, one-place property involved in such a case, anymore than there was with ‘is heterological’ before. To see that deeper point again, consider, for instance, Haim Gaifman’s recent short proof of the fixed-point theorem (Gaifman Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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2006, 710, see also Boolos and Jeffrey 1985, 173). Gaifman’s proof can be condensed, even, into two lines. Given a predicate ‘Av’, he proposes we let A*(gn‘Fv’) ≡ A(gn‘F(gn‘Fv’)’).
It then follows that A*(gn’A*v’) ≡ A(gn‘A*(gn‘A*v’)’),
i.e. that p ≡ A(gn‘p’),
for a certain ‘p’. In particular, if there was a ‘truth predicate’ ‘T’ of Gödel numbers, such that for all ‘q’, q ≡ T(gn‘q’),
then there would also be the negative of that predicate, ‘not-T’, and therefore a specific ‘p’ for which p ≡ not-T(gn‘p’).
From this contradiction Tarski concluded that there could be no truth predicate of the required kind, in formal systems that were consistent, so any truth predicate had to be in a ‘meta-language’ rather than the object language studied, and ‘semantic closure’ within the same language was impossible. The formal tradition has therefore not taken such consequences to suggest that some premise on which the fixed-point theorem is based is questionable. But given the above proof’s brevity the only possible question there can be is revealed: can one just stipulate ‘A*’ to have the sense Gaifman gives it? If predicates and sentences are being talked about in a meta-language, using Gödel numbers, then one can certainly make that stipulation, since from any function of two variables, like ‘x*y’, a function of one variable can be obtained, when the two variables are identified. But if it is not predicates and sentences being spoken about in a metalanguage, but the properties and propositions they express being talked about in a used object language, the situation is different. In Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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particular there can then easily be semantic closure, using a truth predicate of propositions in that used object language. For if one talks about properties then one cannot just stipulate some ‘A*’ to have a sense comparable to that it is proposed to have above, anymore than one can stipulate, in Grelling’s case, that ‘is heterological’ is to have the sense of ‘is not applicable to itself’. Certainly there is a predicate ‘is not applicable to itself’ which describes all cases, but that predicate contains the pronoun ‘itself’, and so the property it attributes, when used, varies with the subject it is attached to, which means that no constant predicate can express it. In fact the Grelling case is directly derivable from the fixed-point theorem, taking ‘A’ to be ‘is not true’, using quotation rather than Gödel numbering, and applying Tarski’s disquotational T-scheme. For then we get that not-true‘F“Fv”’
if and only if not-F‘Fv’,
making the stipulation amount to saying that not-true*‘Fv’
if and only if not-F‘Fv’.
The option of taking the predicate ‘not-true*’ to express the variable property of being not-F, however, is not commonly considered, and the resolution of this and related puzzles is more commonly thought, as above, to lie in some modification of Tarski’s T-scheme. A deeper meaning appears, however, when transferring from the Nominalist framework in which Tarski worked to a Realist one. For one is then not talking meta-liguistically about a given language, but using a language to talk about what it expresses. So no Gödel numbers, or quotation is involved, and the previous formal results are therefore not available as stated. The nominal for a predicate is then something of the form ‘being F’, while the nominal for a sentence is Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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a ‘that’-clause — the former expression referring to a property, the latter a proposition. In these Realist terms an attempted proof of the fixed-point theorem would go: let us say being F is A*
if and only if that being F is F is A.
Then being A* is A*
if and only if that being A* is A* is A,
and so p if and only if that p is A,
for a certain ‘p’. One could then derive that q if and only if that q is not true,
for a certain ‘q’, since there is no doubt that a truth predicate (of propositions) is available in natural language. But there is no way to escape the propositional truth scheme. For, as before, ‘that q is not true’ is simply the non-cleft form of the propositional attitude expression ‘it is not true that q’, and ‘it is true that’ is the null, or vacuous modality in the modal system T, i.e. the modality for which is it necessary that Lp if and only if p. What the putative, Realist fixed-point theorem shows is therefore that one cannot have a property of not being true* such that a given property is not true* if and only if that the property is an instance of itself is not true. Reductio demonstrates this indirectly, of course, but we also know from the previous discussion the sort of observation that demonstrates it directly: to say that a property is or is not an instance of itself is not to say it has a fixed property, but a variable one, since the pronoun ‘itself’ is involved in the diagonal predicate ‘is/is not an Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
instance of itself’. It follows that there is no analogue of the fixedpoint theorem in natural language use. The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
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147 8. A Poor Concept Script
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Some recent thinkers have tried to make out that human motivation is related to the possession of a certain category of indexical belief, by David Lewis called ‘de se beliefs’. Rather, it relates to the use of certain indexical expressions for certain beliefs. The misconception of grammar involved characterises a very popular approach to indexicality that has been current since the 1970s, stemming from the work of Casteñeda, and Kaplan. Gareth Evans was inclined to allow, for instance, that one could say ‘“To the left (I am hot)” is true, as uttered by x at t if and only if there is someone moderately near to the left of x such that, if he were to utter the sentence “I am hot” at t, what he would thereby say is true’ (Evans [1985], 358). But not only does this disturb the proper relation between direct and indirect speech, it continues a Fregean tradition that these very cases show to be quite mistaken about the logic of intensions. In this chapter, however, I want primarily to point out how this misconception of grammar has distorted our view of people. For some of the above thinkers have tried to make out that human motivation is related to the possession of a certain category of indexical belief, by Lewis called ‘de se beliefs’. I shall look here at how the matter arises in Hugh Mellor’s work on Time. In connection with Time, indexicality arises in McTaggart’s ‘A-series’, and Mellor treats this indexicality in parallel with Evans’ language. First, therefore, I aim to show how Mellor’s discussion of Time grammatically misconceives the situation, and leads to a misrepresentation of the motivation of human action. But a larger conclusion about Fregean intensions is also then immediately available.
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Mellor recently said (Mellor, [1998], 50, see also Mellor [1981], 62): Thus although here in Cambridge London is sixty miles south of here, that is not London’s A-place everywhere. Forty miles north of here, London is a hundred miles away to the south; and eighty miles west of here, it is a hundred miles away to the south-east. While in London itself, sixty miles south of here, London is here, and Cambridge is sixty miles away to the north.
The difficulty with this kind of language has been discussed before (Lowe [1987]), but it clearly led to no appreciable change between 1981, and 1998. The only point needed to correct Mellor, however, is one more widely found in philosophical logic. What one needs to remember are the differences between sentences, statements and propositions (Lemmon [1966], Haack [1978], Ch 6), and the related difference between direct and indirect speech. For while, said in London, the sentence ‘London is here’ makes a true statement, that does not mean that what is true is that London is here. In the indirect locution, giving the content of the sentence, the ‘here’ is in the mouth of the reporter, and so relates to his or her location — which may not be London. You might as well say that, for Mellor, I am Mellor, when what is correct is that, for Mellor, ‘I am Mellor’ expresses a truth — the truth that he is Mellor. Likewise in Evans’ case: if ‘I am hot’, uttered by y to the left at some time, expresses a truth, it could be said that, at that time, to the left, y is hot, but not that I am hot (unless, of course, I am y). Where the tradition followed by Evans has gone wrong is thus in taking it that the sentence ‘I am hot’, said by y at time t is what might be true. What might be true is that y is hot at time t, but ‘that y is hot at time t’ is not a sentence, it is a referring phrase to a statement (form). The point, of course, was ignored first by Tarski, in connection with nonindexical expressions, for ‘it is true that snow is white’ likewise predicates nothing of a sentence. Sentences need an interpretation, so it is not a sentence alone which is true; interpretations are given in indirect speech, so it is what is expressed in that which is true; and while the interpretation is commonly homophonic when no indexicals are involved, this is rarely so when they are, forcing attention to the difference. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Mellor’s intended point, in connection with the above passage, in some ways reflects the proper grammar of the situation. For he wants to say, in the end, that his ‘A-places’ are not real, that real space is described in B-sentences only, just like real time. Thus he wants to say that there is no property just of being here, and so no fact just of London being here. But while this is true, his not seeing that these items are ruled out straightaway, if one speaks normal English, also blinds him to a more fundamental possibility: that (real) properties and facts can be described both token-reflexively (i.e. ‘A-ly’) and non-token-reflexively (i.e. ‘B-ly’), and so that distinction merely reflects on the mode of expression and not on what is expressed. As we shall now see, there are no distinctive A-facts, or, alternatively, A-facts are just the same as appropriate B-facts. Mellor debates whether A-statements are translatable into Bstatements (Mellor [1998], 3): Besides pressing objections to A-theorists, we B-theorists must dispose of their objections to us. One, dealt with in Real Time, is that A-statements cannot be translated by B-statements. Thus ‘e is past’ said at t, cannot mean that e is earlier than t...But this does not stop the fact that e is earlier than t making ‘e is past’ true at t. Compare, for example ‘I live in Cambridge’ said by any person x (where ‘x’ is a name like ‘Hugh Mellor’, not like ‘me’). This cannot mean that x lives in Cambridge...Yet no one can deny that the truthmaker for ‘I live in Cambridge’ said by any x is that x lives in Cambridge.
But Mellor does not clearly distinguish the sentence ‘e is past’ from the statement it might make when uttered at a certain time — and both from the proposition involved, i.e. the general meaning of the sentence. Certainly ‘e is past’, said at t, does not mean that e is earlier than t, but it does state that e is earlier than t, when said at t. There is no translation between A-statements and the relevant B-statements, as a result, since they are even identical. It is sentences which might be subject to translation, and then into other sentences with the same meaning, i.e. expressing the same proposition. The statement made by the A-sentence ‘e is past’, when said at time t, is the same statement as that made by the B-sentence ‘e is earlier than t’, when said anytime. Certainly there are no A-facts if such relate to what ‘I live in Cambridge’ expresses independently of who says it, but that Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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still leaves what statement it makes in the mouth of Mellor to be what ‘Hugh Mellor lives in Cambridge’ says more generally. The tokenreflexivity, in this way, merely relates to the mode of expression and not to what is expressed. Mellor needs the untranslatability of his A-statements, however, for the further purpose that we are mainly interested in (Mellor [1998], 4): My B-theory...not only...does it show why A-statements are untranslatable, it also shows why we need the A-beliefs these statements express. The reason is that we need such beliefs to make us act successfully when our success depends, as it nearly always does, on our acting only at certain times. I know for example that it is no use shopping when the shops are shut. So I shop only when, and because, I believe they are open, i.e. open now. If this A-belief of mine is true, I will succeed in shopping; and if it is false I will fail.
But if Mellor believes the shops are open at a certain time, he can have that belief at any time. He needs to believe they are open at the time he is expressing the belief for the appropriate action to ensue, in the manner he is suggesting. For instance, if he said to himself the sentence ‘The shops are open now’ that might initiate the envisaged action — and, if so, it would do so in the quite unproblematic, physicalistic sense in which one material event can causally lead to another. Mellor overlooks this possibility, that the expression of appropriate words (given normal training in the use of them) will do all the work. The belief that the shops are open at a certain time is in quite a different causal category: it is not even an event, for one thing, and Mellor could well have that belief before, or after the time in question. The belief expressed at 2pm that the shops will be open in one hour is the same as the belief expressed at 3pm the same day that the shops are open then (sic), since, as Mellor would put it, the truthmaker for both is just the shops being open at 3pm. Of course, only at 3pm can the belief be expressed ‘The shops are open now’, but its distinctness does not make it express an ‘untranslatable’ belief: this form of expression merely has a distinctive causal place because of the psycho-dynamics of words such as ‘now’, ‘here’, ‘I’ etc.
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The point was better appreciated by Perry when defining the notion of a ‘belief state’, which Mellor’s view misinterprets as a kind of belief. He said (Perry [1988], 99): Anyone can believe of John Perry that he is making a mess. And anyone can be in a belief state classified by the sentence ‘I am making a mess’. But only I can have that belief by being in that state.
Here Perry realised that what Lewis called the de se was merely a matter of an attitude expressed a distinctive way, not a distinctive attitude. Lewis, also, recognised that the de se involved a property of the subject not expressible as a relation to a proposition, though he still talked about ‘de se beliefs’ (c.f. Lewis [1983], 138). Mellor, by contrast, took ‘de se beliefs’ to be distinctive beliefs, despite all Perry’s, and Lewis’ lengthy arguments against that way of portraying the matter. Mellor’s original confusion lies in thinking that one can have a belief that p when ‘p’ is, in effect, not a used but a mentioned sentence. So did Perry make a mess? There certainly is a mess, but the person ultimately responsible is not Perry. For we now start to see better the source of the mix-up with Mellor’s original ‘here’. He said ‘While in London, sixty miles south of here, London is here’ trying with the second ‘here’, unlike with the first, to refer to where some further judgement is made. And likewise, he is trying to make the ‘now,’ within the above belief report, to refer not to the time of the report (its natural referent), but instead the time of expression of the belief. Even with that intention it should have been ‘then’ (as above), but Mellor is trying to bring in the especial force of the word ‘now’, as well. Perry, of course, was concerned with a certain ‘messy shopper’, who was leaving a trail of sugar on the supermarket floor. After some while he realised the trail was coming from his own trolley (Perry [1988], 83): My beliefs changed, did they not, in that I came to have a new one, namely that I am making a mess? But things are not that simple. The reason they are not is the importance of the word ‘I’ in my expression of what I came to believe. When we replace it with other designations of me, we no longer have an explanation of my behaviour, and so, it Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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seems, no longer an attribution of the same belief. It seems an essential indexical. The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
At this early point in his paper, Perry acknowledged the distinction lay in the indexical expression of a certain belief while still wanting to say that the belief he expressed by saying ‘I am making a mess’ was different from the belief he would express by saying, for instance, ‘John Perry is making a mess’. Later he came to discriminate beliefs from belief states, as we have seen. But his resolution of the problem of the essential indexical has a wider application: it applies not only to expressions of beliefs involving ‘I’, but also to those involving any other referring phrase. An immediate conclusion therefore follows about Frege’s idea that, within a belief context, referring phrases may shift from their customary referents. The Fregean tradition has been principally impressed by the fact that I might sincerely assert some sentence using one designator and sincerely deny the result of substituting some other co-referential designator: ‘The Evening Star is a planet’, ‘The Morning Star is not a planet’, etc. Equally I might self-reflectively report this by saying, e.g. ‘I am entirely convinced that the Evening Star is a planet; I am not entirely convinced that the Morning Star is a planet’. The fundamental question, however, is whether I would (necessarily) be correct in so moving from direct to indirect speech. If one follows the Fregean tradition one is thinking that direct expressions give access to the subject’s mind, although in that there is a lingering Cartesianism one might elsewhere want to deny. Moreover, there is then a difficulty with saying what objective facts are, since they would then be related to intensional objects, and not extensional ones. Frege’s hyperintensional approach leads to problems with verifiability. For if, as faithful Fregeans must believe, beliefs are not about states of affairs, containing objects of direct reference, but simply propositions containing Fregean senses (c.f. e.g. Dodd [2000], Chs 3, 4), then there cannot be any objective grounds for criticising any properly comprehended belief. Unless the belief that The Morning Star is not a planet is about The Morning Star, there is no factual basis on which to assess it. Without that basis it can only be, at best, an object of aesthetic delight, as Frege himself appreciated. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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The anti-Fregean view therefore starts from the simple insistence that words in reported speech have their direct reference. Indeed, they are the reporter’s. But that means that the subject does not necessarily have immediate authority to say what is on his or her mind, since this is only available through a reporter’s interpretation. Davidson, of course, has set out this point of view, although within a much broader anti-Cartesian tradition, including Freud, Wittgenstein, and Dennett, for instance. There are even logical theorems that defend this point of view; thus Thomason showed ‘indirect discourse is not quotational’ in 1977, and that there is no ‘Language of Thought’ in 1980. Because of this, like the circle-squarers before them, the recent generation of ‘indirect quoters’ have continued to batter their heads against what are, in fact, logical impossibilities. Even though Perry might not say ‘I believe John Perry is making a mess’ still he does believe John Perry is making a mess, when he says ‘I am making a mess’. Certainly if Perry were to say ‘I am making a mess’ appropriate action might be taken, but the special force of indexicals like ‘I’ does not bring in any special belief, merely a special expression for one. Such token-reflexive expressions directly motivate appropriate actions, therefore, not the beliefs they then express. But that means we must revise the view of intensional contexts coming down from Frege. Referential terms in belief contexts are, against Frege, transparent, since co-designative ones can be substituted salva veritate. What are non-transparent are, in Perry’s terms, ‘belief states’, i.e. states in which beliefs are expressed in distinctive ways. The link to direct speech is integral to such states, and so there is no ‘translatability’. But cannot we continue to talk in Mellor’s ‘use=mention’ way? Mellor certainly did, in the face of Lowe’s criticisms. Evans said of his proposed operator (Evans [1985], 359) ‘though I am doubtful whether English incorporates any examples of the new form of embedding, it seems perfectly clear to me that a language could coherently do so’. He even went on to sketch how ‘To the left (I am silent)’ might be correctly uttered, using a notion of ‘potential utterance’ found in Davidson. Speaking of ‘propositions of limited accessibility’ Perry considered the possibility that there might be a class of propositions that can only be expressed in special Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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circumstances, with what he expressed by ‘I am making a mess’ then being one such. He concluded (Perry [1988], 96): Such a theory of propositions of limited accessibility seems acceptable, even attractive, to some philosophers. Its acceptability or attractiveness will depend on other parts of one’s metaphysics; if one finds plausible reasons elsewhere for believing in a universe that has, in addition to our common world, myriads of private perspectives, the idea of propositions of limited accessibility will fit right in. I have no knockdown argument against such propositions, or the metaphysical schemes that find room for them. But I believe only in a common actual world. And I do not think the phenomenon of essential indexicality forces me to abandon the view.
So the ‘indirect quoters’ are not in the same class as the circle squarers? What Perry, and Evans, and by implication Mellor overlook is that the impossibility of squaring the circle took a lot of time, and highbrow mathematics, to finally prove. The issue reduces to the possibility of replicating operator accounts of the attitudes with syntactic ones, in the manner of, for instance, des Rivieres and Levesque [1986]. Isomorphic structures have been constructed which seem to show that non-representational accounts of the attitudes are materially equivalent to certain representational ones. But not only does that, as before, conflate indirect and direct speech, the above writers notably do not consider languages which include their own expression relations, and indeed des Rivieres and Levesque have to explicitly exclude self-referential cases (des Rivieres and Levesque [1986], 126f). So they have not faced up to all the questions which selfreference and expressibility in a language produce. It has been realised for some time, by contrast, that operator treatments of truth, knowledge, provability, and the like, are immune to the standard paradoxes of self-reference. Thus it is provable that, within any extension of KT, the following is a simple theorem: ¬L(p ≡ ¬Lp).
Hence there cannot be negative self-referential propositions in connection with any of the associated operators. More generally, operator treatments of semantic notions do not allow self-reference at all, simply because operators are completed with a used sentence, not Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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a referential phrase. Thus ‘it is unprovable that p’ cannot include itself in place of ‘p’, since that would require a whole to include itself (or, at least, a copy of itself) as a proper part. Asher and Kamp explained the issue, and its broader context, in the following way (Asher and Kamp [1989], 94): …the familiar systems of epistemic and doxastic logic, in which knowledge and belief are treated as sentential operators, and which do not treat propositions as objects of reference and quantification, seem solidly protected from this difficulty.
The difficulty was (Asher and Kamp [1989], 87): Suppose that a certain attitude, say belief, is treated as a property of ‘proposition-like’ objects — let us call them ‘representations’ — which are built up from atomic constituents in much the way that sentences are. Then, with enough arithmetic at our disposal, we can associate a Gödel number with each such object and we can mimic the relevant structural properties of and relations between such objects by explicitly defined arithmetical predicates of their Gödel numbers. This Gödelisation of representations can then be exploited to derive a contradiction in ways familiar from the work of Gödel, Tarski and Montague.
The singularity of operator logics thus stands out against at least two other major traditions. There is, first, the extensionalist tradition associated with the original formal work of Tarski and Gödel, which took semantic notions like truth and provability to be about sentences. But there is also, secondly, a certain intensionalist tradition that would variously take the attitudes to be about other, related objects. The move away from the first tradition was forced by some of Montague’s work, amongst other things, but Thomason notably showed that arguments like Montague’s could also be successfully aimed at intensionalist theories of a certain sort: he had in mind, as before, Fodor’s Language of Thought (Thomason [1980]). Asher and Kamp, however, used a modified Thomasonian argument to show that even Montague’s own Intensional Logic suffers the same drawbacks. Talking about Thomason’s general approach Asher and Kamp said (Asher and Kamp [1989], 87):
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Thomason argues that the results of Montague (1963) apply not only to theories in which attitudinal concepts, such as knowledge and belief, are treated as predicates of sentences, but also to ‘representational’ theories of the attitudes, which analyse these concepts as relations to, or operations on (mental) representations. Such representational treatments of the attitudes have found many advocates; and it is probably true that some of their proponents have not been sufficiently alert to the pitfalls of self-reference even after those had been so clearly exposed in Montague (1963)...To such happy-go-lucky representationalists, Thomason (1980) is a stern warning of the obstacles that a precise elaboration of their proposals would encounter.
But intensional objects do not need a structure, Asher and Kamp realised, for the same sort of result to hold (Asher and Kamp [1989], 93-4): in order to obtain a contradiction ‘It sufficies to have a mechanism for correlating the attitudinal objects with the sentences by which they are expressed...when the correspondence is representable within the theory itself’. By avoiding attitudinal objects entirely traditional operator theories immediately escaped contradiction; more sophisticated operator theories allow there to be attitudinal objects, while denying there are always sentences by means of which they can be expressed (Slater [2001b]).
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We can now go into more detail about propositional paradoxes like the Liar and Strenthened Liar. There are some seemingly small points to be made, first of all, about use-mention confusions, as I will illustrate primarily with regard to a paper by Stephen Read (Read [2007a], see also Read [2007b]). But underlying them is a grammatical point that has much wider repercussions. For it generates, on its own, a more straightforward way of understanding what gets people into a tangle with Liar and Strenthened Liar sentences, and that leads to a much fuller, critical assessment of the line of approach to these matters that Read derives from Bradwardine. Difficulties with representing propositional referring phrases of the form ‘that p’ are shown to be what have made Liar and Strengthened Liar sentences seem paradoxical. Using “ ‘p’ ” as an alternative, confuses syntactic expressions with their immediate readings, and leads to misunderstandings about the necessity of the T-schema. Frege’s content stroke, i.e. the horizontal line that he sometimes used to indicate the thought expressed by a sentence, has not been incorporated into the generality of logic texts that have followed his formal work. I set out here how provision of an explicit content marker gets one out of Liar and Strenthened Liar ‘paradoxes’, and what similarities and differences there are with Read’s account of Truth, as a result. When discussing Tarski’s T-schema, at the beginning of his paper, Read also mentions a similar schema in Horwich’s work. He even thinks of them as the same, saying at one point ‘Tarski did not propose (T) as a definition of truth, though others, e.g. Horwich, have Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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done so since’. But there is a crucial difference between Tarski’s and Horwich’s schemas. Horwich’s is a propositional truth schema, viz the proposition that p is true if and only if p,
whereas Tarski’s is a sentential schema: x is true if and only if p,
when what replaces ‘x’ is a name of a sentence whose translation into the metalanguage replaces ‘p’. The difference is most pointed in the homophonic sentential case, which parallels very closely the propositional one. For what replaces ‘x’ then is a quotation-name of the sentence that replaces ‘p’, not that sentence itself. As we shall see, there is a formal difficulty with distinguishing the two schemas clearly, since there is no agreed, distinct symbolisation for predications of the form ‘that p is true’, and something more like the sentential predication “ ‘p’ is true” is often used in place of it. Certainly there would be less need to make the distinction if all sentences were unambiguous and non-indexical, i.e. had just one interpretation, since then facts about propositions could be mapped 1-1 onto facts about sentences. But, as we shall see later, the central question is whether Liar sentences are indeed univocal in the required way. It has to be said that the difference between ‘that p is true’, and “‘p’ is true” may not be completely appreciated even in Horwich’s informal work, since he thinks there are still paradoxical cases of his propositional schema. But ‘it is true that’ is the null, or vacuous modality in the modal system T, i.e. an ‘L’ for which it is necessary that Lp ≡ p, and so one cannot have p ≡ ¬Lp, since T is consistent. Going with that, there is no way to construct self-referential propositions parallel to sentential constructions that, for instance, produce identities like t = ‘t is not true’.
For ‘t’ is there the name of the sentence, and not the sentence itself, and one cannot have anything of the form ‘p’ = ‘that p is not true’, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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since nothing can be a proper part of itself. This shift in grammar illustrated here already is revealing about Read’s core concerns. So we must first look directly at problems with certain Liars in the light of it, before turning to a wider assessment of Read’s paper. The main question, of course, centres on what is wrong with the following kind of argument. From t = ‘t is not true’,
there follows t is true ≡ ‘t is not true’ is true.
But from the T-schema there follows ‘t is not true’ is true ≡ t is not true,
so it seems we can derive t is true ≡ t is not true.
There is no difficulty finding true identities like that expressed on the first line, and the second line follows necessarily from the first, the last from the preceding two, hence the problem must be with the third line itself. Read agrees, but what is causing the problem? As we shall see, it is the lack of a clear expression for the thought that t is not true which is keeping people gripped with the third line. One can see there is a problem with getting a clear expression for such a thought not just by attending to the sort of grammatical difficulties in Read’s paper discussed above. The difficulty is much more widespread, as appears if one attends even to the common description of sentences like t as ‘self-referential’. One must remember, first of all, that the original identity does not itself show that some sentence is about itself, since it does not entail that (∃x)(x = ‘x is not true’),
by existential generalisation, and neither can ‘t’ be replaced throughout by a quotation name for the supposedly self-referential sentence, since nothing of the form Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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is possible: again, nothing can be a proper part of itself. By contrast, if one has a statement about the content of t, in a sentence where there is not direct quotation, but indirect speech, such as t says that t is not true,
there follows unproblematically that (∃x)(x says that x is not true),
and so that something is self-referential. And there is an equal possibility of providing a quotation name for a sentence of the required kind, since there is nothing against cases of the following form being true: ‘p’ says that ‘p’ is not true.
So properly there is no syntactic self-reference: no sentence in itself refers to itself. That, indeed, may be granted quickly by many, saying that, of course, a sentence needs an interpretation before it can be said to be about anything, since it itself is just a syntactic string. But the rapidity of such an admission hides the crucial difficulty. The difficulty comes in various forms, and is at least threefold. First, in the nonprofessional area, people commonly do not attend closely to usemention issues, so the syntactic string ‘t is not true’
gets conflated with its prima facie interpretation, that t is not true,
with the consequence that the expression “‘t is not true’” gets used both for the uninterpreted string, and for the interpreted expression. Second, in the professional area of semantics, this conflation gets institutionalised when there is talk about sentences being ‘true in models’. Is ‘Pa’ true if ‘a’ refers to Socrates and ‘P’ expresses the property of being wise? Why not say instead that what is true in the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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specified model is simply that Socrates is wise? The answer to that leads to the third, and major point enunciated before: in the professional area of formal logic there is considerable difficulty with saying the latter, since there is no standard expression, in the symbolism coming down from Frege, for the word ‘that’ in such expressions as ‘that Socrates is wise’. It is all this that combines to be the cause of the problem people have had with those Liar problems which involve direct quotation. There is another class of Liar problems involving indirect speech, which we shall look at later, but first we can clear up the problem with ‘t is not true’ is true ≡ t is not true.
For, because of the above, a question now arises about the referent of the “‘t is not true’” on the left hand side. Either the quoted expression is a syntactic object, or it is an interpretation of that sentence which is meant, in which case the quoted expression is not a Tarskian syntactic object, but a Horwichian semantic one. In the Horwichian case, there is then an ambiguity, since in t is true ≡ ‘t is not true’ is true,
the right hand side definitely involves a syntactic object, which means one cannot go on to get the traditional contradiction t is true ≡ t is not true,
by combining the two equivalences. The matter is even clearer if one removes the use-mention conflation in this case, and writes the first equivalence as the indubitable that t is not true is true ≡ t is not true.
Then it is very evident that there is no way to get the contradiction. But in the Tarskian case, i.e. if it is a syntactic object in the T-schema, that again shows that we do not get a contradiction. For we would need, in addition, as was pointed out before, a proof of the univocality of the sentence ‘t is not true’, showing there is no other interpretation of it possible beyond the superficial one. Homophonic Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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examples of the sentential T-schema are in the required way disquotational, but they cannot arise with semantically ambiguous sentences like ‘There is a bank’, or indexical ones like ‘She is pretty’. So the question first has to be answered whether or not the same holds with ‘t is not true’ in the special case where t=‘t is not true’. Once they are separated from their prima facie interpretations, however, it becomes clear that such sentences can have more than one interpretation. In the case of any sentence like ‘t is not true’ what I have called above its prima facie interpretation, drawn from reading what is in its face, is that t is not true. But that depends on taking the ‘t’ in it to refer to t, which, while certainly possible, is not necessary – as we saw before, there is no such thing as syntactic self-reference. Maybe at the top of some page there is the sentence ‘the sentence at the top of the page is not true’, but if a preceding sentence was, say, ‘Once upon a time a wise old man could be found reading a certain page in a book’, then the reference of the following subject term ‘the sentence at the top of the page’ is not the sentence at the top of the page that sentence is on, but a sentence in another possible world. And it is no good replacing that sentence with anything like ‘the sentence at the top of the actual page this sentence is on is not true’ since that involves an indexical, which could be given a variety of references, in different possible worlds. In these, and similar cases there is, or can be, therefore, a second pragmatic reading of the subject phrase, beyond the superficial one, showing no semblance of a paradox need be around. Of course, there will be if the quoted ‘t’ is given the same reference as the unquoted ‘t’, but in that case, we must primarily remember that giving it that reference is a matter of choice. There is no syntactic self-reference, but also any pragmatic self-reference is not obligatory. That means there is no way to get the T-schema absolutely, when the quoted expression on the left is a bare syntactic object, with the result that a contradiction does not necessarily follow this way, either. Isn’t, at least, the sentence t, on the self-referential interpretation, paradoxically both true and false? No, for what one is involved with, if one chooses the self-referential interpretation is not directly the syntactic identity, but the kind of statement about content which was introduced before, namely Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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t says that t is not true. The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
This is clearly true only on a certain pragmatic interpretation, and so it is not just an extensional remark about the syntactic object t, i.e. ‘t is not true’. The distinction makes it plain that what is true is not the sentence ‘t is not true’, but the proposition that t is not true. As was pointed out before, the inability of the logical tradition to represent such a propositional referring phrase as ‘that t is not true’ has made it seem that what is true on the given interpretation is still the (mentioned) sentence, but only the sentence in use, preceded by ‘that’, refers to the item that has the truth value. In fact, as a result, sentences, in themselves, are neither true nor false, and that t is not true is definitely true, in this case. It follows that there is no longer any problem with Strengthened Liars. Certainly the question naturally arises about what to say in ‘strengthened’ cases, where, for instance, s = ‘s is neither true nor false’.
But first of all there is no need to take the sentence thus defined to speak about it itself being neither true nor false. For sentences, by themselves, have no voice. If anyone chooses to interpret this sentence self-referentially that is therefore an additional (intensional) matter beyond the direct speech identity. And what is true in that selfreferential case is simply that s is neither true nor false, which is not paradoxical in any way, since to say that is not to say that what is true is s, i.e. ‘s is neither true nor false’. The above supports very forcibly Read’s arguments against the Tarskian T-schema, and his move towards a calculus of indirect speech, in Read [2007a] and Read [2007b]. But, as the above points stand, they lead to a positive account of truth that, while very similar to Read’s, is crucially different from it. Read ‘s primary case, for instance, is one of those covered, in a very similar way, in Goodstein’s original paper on the formalisation of indirect speech (Goodstein [1958]), which led directly to Prior’s theory (Prior [1971]), and then eventually to its subsequent modification using Kneale’s ‘that’ (Slater [2001b]). But there are some grammatical problems with Read’s symbolisation of this same kind of material, and even more importantly, he also introduces, or at least relies on, a Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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further axiom that rules out the kind of ambiguity we have seen to be necessary to allow an escape from the above syntactic paradoxes. What are the grammatical difficulties when we move over to Liar problems arising with the indirect speech form ‘x says that p’, if this is symbolised as Read symbolises it: ‘x:p’? For a start, the previous difficulties with formalising ‘that p is true’ affect this style of symbolisation, in formulas such as Read’s (A), i.e. Tx ⇔ (∀p)(x:p → p).
This expression employs quantification over propositions, which Read glosses with a reference to Church’s type theory. This kind of symbolism was used, at length, by Prior, and indeed this thesis (A) is to be found in Prior’s book ‘Objects of Thought’, on page 104. But there are major problems with reading Prior’s propositional quantifiers, some of which he himself struggled with in chapter 3 of his book (see also Haack [1978], 130), and others of which have been presented more recently by Loux (Loux [1998], see also Slater [2001b]). These difficulties are also present in Read’s account. Centrally, there is no provision, within such a system of propositional quantification, for truth predications on propositions. In ‘x:p ∧ p’, for example, the ‘p’ is not a referential phrase, and so any ‘p’ in a quantifier before it equally would have to be not a referential phrase; also when reading ‘x:p ∧ p’ as ‘x says that p and it is true that p’, there is no representation of the ‘it is true that’ in the formalism. What is wanted, first of all, as we have now repeatedly seen, is a nominaliser, which will produce, in a formalisation of ‘that p’, a referring phrase to a proposition expressed by ‘p’. In addition one then needs a truth predicate of such nominalised phrases, and, when generalising expressions like ‘x says that p, and that p is true’, one must employ variables which range not only over ‘that’-clauses, but also other propositional referring phrases, like ‘what x says’, for instance. The form of the given case is then ‘x says r, and r is true’ over which one can quantify objectivally, with quantifiers which read quite straightforwardly ‘for some/for all propositions r’, etc. The analysis of Read’s (C), i.e. (C) (∀p)(C:p → ¬p), Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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then proceeds as follows. First C says that everything that C says is false, i.e. Sc§(∀r)(Scr ⊃ ¬Tr),
so suppose that everything C says was indeed false, i.e. (∀r)(Scr ⊃ ¬Tr),
then, contradictorily, something C says would be true, i.e. (∃r)(Scr & Tr),
since from the propositional T-schema there follows in particular that (∀r)(Scr ⊃ ¬Tr) ≡ T§(∀r)(Scr ⊃ ¬Tr).
Hence not everything C says can be false, i.e. ¬(∀r)(Scr ⊃ ¬Tr),
and so, equivalently, (∃r)(Scr & Tr),
but then something C says is false, namely §(∀r)(Scr ⊃ ¬Tr),
and so, also, (∃r)(Scr & ¬Tr).
It follows that C must say (indirectly) at least two distinct things, one true and the other false. Read says similarly ‘However, although C:∀p(C: p → ¬p) this may not be all that C says’ obtaining, as well, (∃p)(C:p ∧ p). Indeed, there is a close similarity with another matter raised before, since from his (A), i.e. Tx ⇔ ∀p(x:p → p),
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there follows a conditionalised T-schema, requiring ‘singleness of saying’, i.e. something like univocality, or uniqueness of interpretation, before there can be truth assessments of sentences in the traditional Tschema form, viz: (x:p & (q)(x:q → q=p)) ⇒ (Tx ≡ p).
So both the necessity of this conditionalised T-schema, and its consequences, are endorsed by Read, and they are to be accepted, once grammatical corrections have been made. What is there to be said, by contrast, in substantive criticism of Read’s treatment of Liar paradoxes arising in indirect speech? The main point which needs to be made relates to this issue of univocality, since the contrary line of analysis we have been following allows for ambiguous sentences, and even requires them in certain places, while Read explicitly dismisses ambiguity, and holds that if x both says that p and says that q, then x says that p and q. Certainly this latter rule, namely Adjunction, is not made explicit in Read’s axioms, but he still employs it, making, for instance, his transposition, in section 4, from L:(TL ∨ ¬q),
to L:((TL ∨ ¬q) ∧ q),
given L:q, not something actually enforced by his stated rules. Read uses Adjunction, implicitly, in this derivation in section 4, and in his discussion of (C) in section 5. Thus, after the remark about C:∀p(C:p → ¬p) possibly not being all that C says, quoted above, he goes on: ‘Suppose C also says that q, that is C:(∀p(C:p → ¬p) ∧ q)’, adding the extra ‘q’ as a conjunct to the other thing C says. But, for a start, given that so much can be obtained without the use of this rule, Ockham’s Razor, if nothing else, suggests strongly that it should be dispensed with. One is reminded, on that score, of the discussion of Moore’s Paradox in the ‘Budget of Paradoxes’ chapter of Prior [1971], 81-84. This is first analysed using Hintikka’s axioms for
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belief, but then an entirely adequate explanation of its ‘logical oddity’ is produced using nothing more than quantification theory. There is a stronger argument against Adjunction, however, than the need for axiomatic economy. For we have seen, when dealing with Liar paradoxes arising in direct speech, that it is necessary to grant ambiguity a place, in connection with identities like t = ‘t is not true’.
So why is Read against ambiguity? Read does debate the matter, in Read 2007a, but only briefly, and his conclusions are not well supported by his arguments. Thus he is concerned that the point of formalisation is to remove ambiguity, but seems to forget that that concern still allows entirely precise statements to be made about the ambiguity of certain sentences. If x is ambiguous, it can still be completely unambiguous that x has two meanings, i.e. means that p and means that q without meaning that p and q. More specifically, Read says, about ambiguous expressions such as ‘Visiting relatives can be boring’, that ‘we do not require that both senses be the case for x to be true — either will suffice’. But such a sentence is then true in one sense and false in another, not true simpliciter either time, making Read’s ‘Tx’ better read ‘x is true in all senses’, not ‘true simpliciter’. Only starting from the conditionalised truth schema identified above is the truth of sentences limited to truth simpliciter, and so to truth as more generally understood within the Tarskian tradition. And there is another notion that is more generally part of that tradition. For the discussion before, for instance, involved the notion of meaning, which is a notion not found in Read’s paper, but crucial to the Davidsonian tradition following on from Tarski. Read glosses his ‘x: p’ in various ways, such as ‘x says that p’, ‘x implies that p’, but there is this other expression in the area, ‘x means that p’, and that behaves rather differently. Thus ‘John is a bachelor’ means that John is an unmarried male of marriageable age, and the latter implies that John is unmarried. So the meaning of something is its entire content, or, at least, something from which its entire content can be deduced. Writing ‘x means that p’ as ‘Mx§p’, and using ‘I’ for the
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relation of implication, we can in fact then define ‘x says that p’ as ‘something x means implies that p’, i.e. (∃r)(Mxr & Ir§p).
Given this, the other main axiom Read employs, (K), i.e. (∀p,q)((p ⇒ q) → (x:p ⇒ x:q)),
becomes, with appropriat`e grammatical corrections, just a matter of definition also. Thus we can get ‘x says whatever is implied by what it says’, as the schema: (p ⇒ q) → ((∃r)(Mxr & Ir§p) ⇒ (∃r)(Mxr & Ir§q)).
This follows from the transitivity of ‘if’, using some quantification theory, and attending to the appropriate grammar. It must be remembered, in this connection, that ‘implies’ is a verb and relates two propositional subjects given by referring phrases, as in ‘that John is a bachelor implies that he is unmarried’ and ‘what Peter stated implies that John is unmarried’. The symbol which Read uses, ‘⇒‘, is, by contrast, a propositional connective, which can be defined via the equivalence: I§p§q if and only if p ⇒ q.
Lack of a propositional nominaliser, of course, is what inclines many (indeed most) to read the connective as ‘implies’. That confusion only deepens if one says, like Read, that sentences imply things; indeed one is then involved in the full grammatical confusion with which we started, in this chapter. In conclusion against Read, I have supported Read’s arguments against Tarski, pinpointing more accurately the fallacy that holds people to his schema — a certain conflation of syntax, semantics, and pragmatics. However, the core grammatical insight needed to be clear of Tarski is still widely ignored in Read’s system, and there are superfluous and restrictive axioms there that either can be reduced to definitions, or do not allow for the full extent of the cases that can be encountered. Logic is an exact science, but there are many problems with being as exact as it requires, and so Logic must allow for inexact Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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and ambiguous expressions. Only quantification theory, at the propositional, i.e. indirect speech level, is required to work through the ‘paradoxical’ cases that result. Read’s rejection of ambiguity and his consequent endorsement of Adjunction, are the central features of his account which prevent him from seeing that only quantification theory is needed. But his lack of appreciation of the place of ambiguity is also directly linked to his confusion over the sentential versus propositional nature of the T-schema. And we have seen, at length, that a clarification of that distinction is what gets one out of standard Liar paradoxes in direct speech. The crucial point to realize is just why the interpretation of the relevant sentences is a matter of choice. We have previously seen how ambiguity arises in Quine-type cases like: “‘appended to its own quotation expresses no true proposition’ appended to its own quotation expresses no true proposition”. The ambiguous term in it is the pronoun ‘its’, which gains its reference from some preceeding expression, but which is left dangling without such in the doublyquoted place. Quite generally, one can give a number to, or individuating description for, some sentence or phrase ‘....its....’, but if the pronoun’s antecedent is not within that sentence or phrase then the sentence or phrase will not have a fixed meaning, nor therefore, if it is a sentence, a truth-value. The case is like the case of heterologicality, where the predicate ‘is not self-applicable’ contains another pronoun, ‘self’, which is without its antecedent in the predicate. Hence that predicate does not express a fixed property. Certainly ‘appended to its quote is grammatical’ can be used to describe a property, but only a variable, or context dependent property. In Grelling’s case the error arises, as we have seen, when people replace ‘is not self applicable’ with ‘is heterological’ since the latter masks the variable ‘self’ in the former. We have: a is not self applicable if and only if a is not a applicable, b is not self applicable if and only if b is not b applicable,
so, given the subject is a, there is one property — being not a applicable — and given the subject is b, there is another property (if a is not b) — being not b applicable. There is no paradox substituting anything for ‘a’, or ‘b’, so long as it is also susbstituted for ‘self’. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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These two cases, Quine’s and Grelling’s, along with the elementary ‘Liar’ case involving a demonstrative, ‘this sentence expresses a proposition that is false’, where ambiguity is showable by direct inspection of the sentences, clearly give good inductive grounds for the belief that there is ambiguity in all cases. But the most valuable, additional benefit of looking at them is that they show what kind of thing has been missed by people who have been blind to the ambiguity in them: features of language, like demonstratives and pronouns, which were not studied within logic until recently. Indeed, at one time, Quine’s own ‘eternal sentences’ were supposed to be able to eliminate all such contextual items from sentences that could enter into logical forms. Sentences that do not have just one meaning, as we have seen, include pragmatically indexical ones, like ‘that flower is pink’, and semantically ambiguous ones, like ‘There are flowers in the bank’. Common examples of self-referential sentences, such as ‘this very sentence is false’ fall directly into the first category, since the referent of the ‘this very sentence’ has to be determined with a gesture, and might be to some other sentence. If one gives names to sentences there is the same problem, since it is not in ‘sentence 1001 is false’ itself that it is sentence 1001, if it is, and that very same sentence might make a different statement using a different numbering system. Here are the sentence-statement facts regarding ‘open pairs’, sometimes written 1: 2 is not true, 2: 1 is not true.
First, if ‘1’ and ‘2’ are names of sentences then the proper expression is: Sentence 1=‘sentence 2 is not true’, Sentence 2=‘sentence 1 is not true’,
but since sentences on their own are neither true nor false (they need to be interpreted), neither sentence 1 nor sentence 2 have truth values, indeed they are not the sort of thing that can have truth values. Certainly something sentence 1 says (namely that sentence 2 is not true) is true, but one must distinguish sentence 1 from what it says, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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i.e. what statement it makes. What is true is not sentence 1 but the statement made by it, namely that sentence 2 is not true. As we saw in the strengthened case above, what is true is not s, i.e. ‘s is neither true nor false’; what is true is what s says on a self-referential interpretation, i.e. that s is neither true nor false. On the other hand, if what is being talked about are statements or propositions, then one must make it explicit that ‘that’s are involved instead of quotation, so the proper, and full expression for what is intended is: Statement 1 is that statement 2 is false, Statement 2 is that statement 1 is false.
But this case leads to an infinite regress, since one can make substitutions from one expression into the other, unlike in the sentential case where quotation is involved. Notice also that one cannot just nominate a self-referential propositional identity, in the same way as a ‘self’-referential sentential one. There is no barrier to naming the sentence ‘sentence 1001 is false’ as ‘sentence 1001’, since the identity of the sentence is independent of what the phrase ‘sentence 1001’, in it, refers to. Not so with any attempt to name the statement that statement 1001 is false, as ‘statement 1001’. For to know the identity of the statement on the right hand side of Statement 1001 is that statement 1001 is false,
one needs to already know the referent of the ‘statement 1001’ used in making that statement. So its referent is settled, and one is not free to nominate the referent of the expression subsequently. Of course, no referent of the expression is given by this identity as stated, since it is circular. So one must be careful to notice indexicality in these cases, but where does it reside in other cases? The major problem with seeing the general indexicality is that the contextuality of things like ‘the sentence at the top of page n of book B does not express a true proposition’ has not been apparent. For in this case the variability of truth-value of the associated statements is with respect to different possible worlds. It is not an accident, therefore, that paradoxes like those above arose in logicians minds before possible worlds, and Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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context dependence, were studied fully, and indeed if one keeps to that early twentieth century mind-set one will still find the paradoxes puzzling, particularly followers of Quine on Modality, and eternal sentences, of course. Contrariwise, moving out of that mind-set, and in particular moving away from the influence of Tarski, the answers become extraordinarily simple — contrast the above with, for instance, the strenuous efforts of a whole range of people recently, like McGee, Soames, Simmons, and Priest. The crucial point is that it is not possible for a sentence s to specify which world it is used to speak about, since that is a matter of pragmatics. A code might be provided to give the various referents, in different possible worlds, of things like ‘the sentence at the top of page n in book B’, etc (supposing there were just a denumerable number of worlds). But still what world that sentence is used to talk about is a matter of pragmatics. So the code itself shows that such sentences are transworld, if not ‘contextually’ indexical, and so ‘ambiguous’, along with standard referential indexicals like demonstratives, and pronouns. In the case of the original type of syntactic identity t = ‘t is not true’,
the main point is that just ‘t’ is involved on the right hand side, without reference to a world, and so no specific referent can be involved. But the referent of ‘t’ in this world is involved on the left hand side, because we are speaking in this world. But, again, if one tries to improve on this by constructing something like t = ‘t when uttered in in this world/ in the actual world is not true’
then one either has brought in an explicit indexical, or an equivalent to one (remembering David Lewis on ‘the actual world’ in Lewis [1986]). One has to also remember that other possible worlds are not entirely abstract objects, since we can imagine entering them, which is a process that takes place in this world. In the case of linguistic fictions, as was pointed out before, this commonly involves certain context markers, like ‘Once upon a time’. Maybe once upon a time an old man was reading page n of a book, B, the first sentence on that page (i.e. page n of book B) being ‘The first sentence on page n of Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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book B is not true’. In this case, at its unquoted place the subsequently quoted referential phrase refers to the page the old man was reading in the fiction, i.e. the possible world. Without the ‘Once upon a time’, the story starting ‘a man was reading…’ might be fiction or non-fiction, although the same kind of linguistic cross-reference would still occur, from the referential ‘the first sentence...’ back to the previous introductory description of the old man. In non-anaphoric uses of referential phrases, i.e. when they are ‘deictic’, there is no context marker like ‘Once upon a time’, or even explicit introductory description. But, now, the absence of such a context marker is not part of the sentence(s) that follow, so, even when there is direct reference to the actual world, that is a matter of the pragmatics, not the semantics of the utterance, and so not something in the sentence alone, in itself. One is nowhere near appreciating any of this if one is at all closely attached to Russell’s Theory of Descriptions. For what otherwise would be a ‘referential phrase’ is taken there to include some introductory descriptive material, and so to have a quantificational, and thereby nondemonstrative, analysis. Finding a better representation of properly referential phrases is first required, therefore, such as that given by epsilon terms rather than iota ones, as I have shown before. One other misleading aspect of the Formal Logic scene before WWII, besides the sidelining of context, and possible worlds, and the engrossment in Russell’s theory, was the close connection, then, between Logic and Mathematics, especially the Foundations of Mathematics. It is quite plausible to believe that one can have a truth predicate of elementary mathematical sentences such that, e.g. T’2 + 3 = 5’ if and only if 2 + 3 = 5,
and so that sort of context-free case came to be taken as the paradigm. The realisation that the cases where this kind of equivalence has a chance of holding are very special cases has taken a long while to dawn — even given Tarski’s own Indefinability Theorem showing there cannot be such a ‘T’ in general, even in Arithmetic. But outside the Formal Logic tradition there are plenty who have insisted, on other grounds, that it is not sentences but
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statements and propositions that are truth bearers. Thus there is no objection to a truth scheme in either of the forms: p if and only if it is true that p, p if and only if that p is true.
Here, any referential term in ‘p’ is used (in this world), on both sides of the equivalence, unlike when it is just mentioned on the right hand side in the corresponding sentential form: p if and only if ‘p’ is true,
and that leads to a lack of paradox in the former case. The trouble with developing such a propositional theory of truth formally, however, has been the fact that standard twentieth century logic has had no way to write a sentence when interpreted the standard way as a referential phrase referring to an object which might be the subject of a truth predication. Quite a lot hangs on this. What is true, in Gödel’s First Incompleteness Theorem, for instance, as we shall see in more detail later, is not ‘(x)Fx’, for a certain predicate ‘F’, but that all natural numbers are F. Humans certainly can, like machines, utter sentences like ‘(x)Fx’ — sentences which are then put in quotes — but they also do something a machine cannot, namely use sentences like ‘(x)Fx’ to state things about models (in this case the standard model) — the sentences are then not in quotes. In terms of the operation of a machine it would have to not only utter a sentence, but mean by it one thing rather than another. But a machine lacks any capacity to mean anything, and thereby any capacity to prove that Fn for any number n, even if it can generate the sentence ‘Fn’ for every numeral ‘n’. The inability of the logical tradition to represent such a propositional referring phrase as ‘that (x)Fx’ has made it seem that what is true or false on the standard interpretation is still the (mentioned) sentence ‘(x)Fx’, but only that sentence in use, preceded by ‘that’, refers to the item that has the truth-value. Another, related case will illustrate how further confusions between indirect and direct speech can come in, if one is not careful. It might be said, for instance:
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Suppose “this very sentence cannot be proven true” or “the current sentence cannot be proven true” succeeds in achieving selfreference. Let us call it “K”. Under that condition K surely expresses no proposition, for if it expressed a proposition then it could be provable only if it could not be proved, making it incapable of proof, but thereby proving it.
But if we are talking about an interpretation of the sentence, then we are involved not in syntactic self-reference, but the only possible form of self-reference, namely pragmatic self-reference. That is why a calculus of indirect speech is required, to formalise remarks like ‘s says that p’. It is very uncommon to find people considering indirect speech in connection with Liar, and Liar-like Paradoxes, and yet it must be brought in because of the impossibility of syntactic selfreference. In the given case what is wanted is to say about the sentence, ‘k’, that it says that it cannot be proven that it is true, i.e. that it says that that k is true (§Tk) cannot be proven, i.e. Sk§¬P§Tk.
But the sentential T-scheme is conditional on the univocality of the sentence, as we have seen, so all we can say is: (r)(Skr ≡ r=§¬P§Tk) ⊃ (Tk ≡ ¬P§Tk),
and any contradiction derivable on the right therefore merely leads to a denial of the left, which means the sentence does not express a single proposition. The argument might go on: K cannot be proven true, for if K were proven true it would be simultaneously true and false. But if K expresses a proposition, then what K says has just been proven true, so that K must be both true and false. Hence it does not express a proposition.
Here the thought is that, since we know that P§Tk ⊃ Tk, and also that Tk ≡ ¬P§Tk, we know that ¬P§Tk. But that again depends on the sentential T-scheme holding, which is not itself proven. Certainly something comparable to the second claim would hold if the story started ‘if K expresses just the proposition that K cannot be proven true …’. For if Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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(r)(Skr ≡ r=§¬P§Tk)
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then one would definitely be able to get that ¬P§Tk, and so Tk, because the sentential T-scheme would hold, but also thereby P§Tk, and so ¬Tk, again from the T-scheme. But the second claim misses out the uniqueness clause needed to get all this. It is to a fuller discussion of cases like these, in indirect speech, to which we now turn.
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Despite the volume of discussion on the Liar Paradox recently, there is one stream of largely British thought on the matter that is hardly represented in the wider literature. If one formulates the Liar, and associated paradoxes, in terms of propositions it is easy to see why Kneale thought there was a trivial resolution of paradoxes in the Liar family. Kneale put the matter like this (Kneale [1972], 242): ...anyone who pronounces the sentence ‘What I am now saying is false’ appears to use the opening phrase to designate what is expressed by his whole utterance. If, however, he succeeds in designating any proposition by use of that phrase it should be possible in principle for him to designate the same proposition by means of a ‘that’-clause. ... But obviously this is impossible; for an attempt to carry out the prescription merely results in ‘It is false that it is false that it is false ...,’ and so on ad infinitum.
Kneale adds a footnote to a paper by Gilbert Ryle: in Ryle’s terms there is no ‘namely rider’ to the phrase ‘what I am saying now’, in the above context. This chapter points out salient aspects of the history of this tradition, from its origin in forms of propositional quantification found in Ramsey, through to more precise symbolisations that have emerged more recently. But its purpose is to exposit, with respect to a number of contested cases, the ensuing results. Thus it goes on to apply the analysis to several other well known paradoxes, including one rarely discussed, which reveals more fully the consequent consistency and completeness of natural language.
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There is another way of putting Kneale’s point, which illustrates this consistency, for a start. For it might be false of a certain subject s that it is P, and so, in particular, that it is false might be false of the proposition that p. But for that it is false to be false of itself would be for it to be false of the proposition that it is false, i.e. something whose expression still contains a waiting pronoun. So there is no way to specify a self-referential proposition by just using a ‘that’-clause. But why even talk in this way of operator constructions when there can quite clearly be self-referential sentences? That is because talk about such constructions is required to avoid the well-known paradoxes associated with self-referential sentences. Can it be said of the sentence ‘The sentence at the top of the page is false’, that it is false of anything — in particular that it is false of itself, if it is the sentence at the top of the page? No: for there is no place waiting in that sentence for the name of any thing it might be false of to be inserted. That it is false is the sort of thing which might be false of something — the sentence at the top of the page, and even that sentence when it is the sentence previously given — but that is a possibility only because ‘that it is false’ has a space waiting to be filled in the appropriate way. In fact, that it is false is false of the said sentence (no matter where it might be), because sentences are neither true nor false. What might be true or false of s is that it is P, but ‘that s is P’ is not a sentence; it is a noun phrase — a demonstrative referring phrase to a proposition. The use of such phrases most forcefully appears if the sentence at the top of the page is a sentence like ‘The sentence at the top of the page is false, or neither true nor false’. Then — since no sentence at the top of the page is either true or false, and so any there is either false, or neither true nor false — wouldn’t we, contradictorily, have to say that the sentence actually there was true, since it states the facts? No: because what is true is simply that the sentence at the top of the page is false or neither true nor false, not the sentence ‘the sentence at the top of the page is false or neither true nor false’. The identification of the referents of the ‘t’s in such identities as ‘t=“t is not true”’ produces confusion, if truth is a taken to be a disquotational property of sentences, because then ‘t is not true’ is true if and only if t is true (on account of the identity), but also true if and only if t is not true (by disquotation). The same confusion is not Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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obtainable if truth is taken to be a property of propositions, as above. For there is no opportunity then for paradoxical substitution of the sentential identity: one says that t is not true, since sentences are no longer the bearers of truth, and that simply means that ‘t is not true’ is not true. In saying that t is not true, isn’t one asserting that the sentence one then uses (‘t is not true’) is true? No: one is not saying ‘p’ is true when one says it is true that p, because use is not mention, and operators are not predicates. The sentence ‘t is not true’ is not said to be true when one says that t is not true, since one is then saying, instead, that the referent of the ‘that’-clause is true. As a result, there is a very plausible case to be made for accepting such banalities as the above as a resolution of the problems in this area. But there are some very significant consequences. Thus Prior realised that the operator approach spelt the end of Tarskian semantics, if not semantics entirely (c.f. Prior [1971], Ch 7). With propositional operators, a truth locution is involved, but it is in the object language, which is semantically closed without contradiction. That trivialises the usual soundness and completeness proofs in propositional logic, but it is not only that which becomes elementary. For we can proceed to show quite generally that there cannot be a proposition saying that it itself is not true, i.e. an r such that r=§¬Tr.
All that is needed are propositional epsilon terms. If r=§¬Tr then Tr would be equivalent to T§¬Tr, and so to ¬Tr, through application of the propositional T-scheme, which equates T§p with p. But the resulting fact that ¬(∃r)(r=§¬Tr)
still leaves the associated epsilon term ‘εr(r=§¬Tr)’ with a referent, and so we can still speak about ‘the proposition which says about itself that it is not true’. The referent of this phrase, however, is semantically arbitrary, since it is a misnomer. We know that (∃x)Fx ≡ FεxFx,
making Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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so although we can refer to the lying proposition, it does not satisfy its description, and that means we cannot identify it. The proposition which says of itself that it is not true thus may be true or false depending on choice. As we shall see, however, further use must be made of propositional epsilon terms before all doubts are expelled in this area. For Kneale’s own example provides us with a case where reference is made to a proposition, but not in the form of a ‘that’-clause. Is it so certain then that no form of self-referential proposition can be constructed to parallel the case with self-referential sentences? If ‘r’ is a referring phrase to a proposition, we must allow it to be a noun phrase like ‘what I am saying’, as well as a ‘that’-clause like ‘that the sentence at the top of the page is false or neither true nor false’. And isn’t it evident that what Kneale was saying was that what Kneale was saying was false? A little history will clear up some preliminary matters. The study of the Liar and related paradoxes, in something like the above manner, started in the late 1950s, with a series of papers in the Journal of Symbolic Logic by Cohen, Goodstein and Prior. Goodstein originally showed, amongst other things, that if A says that everything A says is false, i.e. Sa(p)(Sap ⊃ ¬p),
then something A says is true, i.e. (∃p)(Sap & p),
and something A says is false, i.e. (∃p)(Sap & ¬p).
Prior developed this form of ‘protothetic’, in which there can be quantification over propositions, extensively, in his posthumously published book Objects of Thought of 1971. In particular he proved four similar theorems on page 105. But a major problem with this specific kind of symbolization is whether the quantification is objectual or substitutional. Prior and Goodstein’s symbolisation Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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followed the quantified propositional approach deriving from Ramsey, but Ramsey’s formulation was problematic, as Haack pointed out (Haack [1978], 130). For, as we saw in connection with Read, there is a bare ‘p’ in the matrix ‘Sap & p’, and so the ‘p’ in the quantifier ‘∃p’ would have to be equally not a referential phrase, but a full sentence, leading to difficulties in reading the quantifier (Prior [971], Ch 3). These difficulties enlarge when reading ‘Sap & p’ as ‘A says that p, and it is true that p’ since there is no explicit representation of truth, in the symbolism. Kneale introduced the above symbol ‘§’ for the nominaliser ‘that’ which turns an expressed proposition ‘p’ into a referring phrase ‘that p’ which designates the proposition, and Haack, following him, used it for ‘the statement that...’ (Haack [1978], 150). But in Haack there remained not only the problem with reading propositional quantifications, and the associated problem with an expression for truth. She also, for instance, purported to derive a paradox with her notion by first producing a sentence: 1. (p)(c =§p ⊃ ¬p),
and then letting ‘c’ abbreviate ‘the statement made by sentence numbered 1’. She went on to say, as a result, that ‘it can be established empirically’ that c = §(p)(c = §p ⊃ ¬p),
deriving a contradiction analogous to one Tarski produced. But we can escape this paradox in Goodstein’s way, by admitting that while any statement made by the sentence numbered 1 is certainly any statement made by ‘(p)(c=§p ⊃ ¬p)’,
that does not ensure that any single, identifiable statement is made by the latter in this context. A sentence may be ambiguous, after all, although the form of semantic indeterminacy in paradoxical cases is not simply a double meaning, as we have seen. Goodstein’s original proof of non-univocality lacked Kneale’s statement forming operator, and the related propositional truth Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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predicate, so to produce a proof which is fully satisfactory we must introduce ‘§’, where appropriate, and include the above predicate of ‘that’-clauses, such that T§p is equivalent to p. Then we get: Given Say, where y=§(x)(Sax ⊃ ¬Tx), assume, first (x)(Sax ⊃ Tx),
then, since Say only if Ty, and Say, we must have Ty, and hence (x)(Sax ⊃ ¬Tx),
but since also (x)(Sax ⊃ Tx)
and (∃x)Sax,
that is contradictory; hence ¬(x)(Sax ⊃Tx),
i.e. (∃x)(Sax & ¬Tx).
But assume, second, (x)(Sax ⊃ ¬Tx),
then Ty, but also Say, hence (∃x)(Sax & Tx),
and a contradiction; so ¬(x)(Sax ⊃ ¬Tx),
i.e. (∃x)(Sax & Tx).
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In the modified version of Goodstein’s conclusions, therefore, The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
(∃x)(Sax & ¬Tx), (∃x)(Sax & Tx),
one must first read the variables as nominal variables (unlike the ‘p’ above), but, second, one must be careful not to take the ‘T’ to be an operator; it is a predicate. And that means that there is no certainty that it can be eliminated. The replacement of ‘Tx’ with ‘x’ was always criticised on this basis, in discussions of the RamseyGoodstein-Prior calculus, but attending to that criticism is, remarkably, just what gets us out of the semantical liar paradoxes even more clearly. Correcting the calculus, however, also means accepting that not every proposition referred to can be independently expressed. From Goodstein’s conclusions it follows that, in one sense in which what A says is false, it is false, i.e. ¬Tεx(Sax & ¬Tx),
and that in one sense in which what A says is true, it is true, i.e. Tεx(Sax & Tx).
So εx(Sax & ¬Tx)≠εx(Sax & Tx),
and there is not just one proposition involved. But these propositions are only referred to, and there need be no way to express the alternative senses just linguistically. What A says that is false can be taken to be that everything A says is false, but no linguistic candidate is determinable for what A says that is true. More significantly, even though an identity like ‘εx(Sax & Tx)=§p’ is available, using further referential terms, it is still indeterminate what a says, i.e. there still is a choice about what ‘εxSax’ refers to. It is here that we get the needed amplification of the point that led Kneale to his resolution of the Liar Paradox. For in the central indirect speech case where one takes it that a certain sentence says that what it says is not true i.e. that Sa§¬TεxSax, where ‘a’ refers to
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the sentence, then we might suppose it was automatic that what it says was that what it says is not true, i.e. εxSax=§¬TεxSax,
from which it would follow that TεxSax if and only if T§¬TεxSax, where the latter is equivalent to ¬TεxSax, producing a contradiction. But the Reductio then available would show that while a says something ((∃x)Sax), and so says what it says (SaεxSax), there is a choice between two distinct things for what a says, each of which contradicts the other: εxSax, and §¬TεxSax. In Goodstein’s ground-breaking analysis above, we can now look at just why the alternative senses required by his proofs need not be expressible linguistically. Specifically, in the case there analysed, a candidate for what A says which is true can only be given in epsilon calculus terms, namely §(Sau ⊃ ¬Tu),
where u=εrSa§(Sar ⊃ ¬Tr), and this has no equivalent in the predicate calculus. A more elementary case will illustrate more clearly the kind of thing that is going on, say where A says that something he says is not true, i.e. Sa§(∃r)(Sar & ¬Tr).
What A says here is true, but what it says which is not true is not so readily locatable. But a speaker of A has an instantiation of the existential remark in mind, and it is that which is a candidate for what is not true. One needs the epsilon calculus to realise fully, for a start, that making an existential remark does involve referring, or at least alluding to an instance, since the predicate calculus hasn’t got an automatic placeholder for what individual is being spoken about when someone makes such a remark. But the problem is that the existential quantifier in this case is within the scope of the intensional operator, so any instantiation which a speaker of A has in mind is up to him to say, and remains a feature of the pragmatic context. Nevertheless there is a clear choice a speaker of A might make that
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would identify what he says that is not true. Here is the proof. First, given Sa§(∃r)(Sar & ¬Tr),
then, if (r)(Sar ⊃ Tr)
then T§(∃r)(Sar & ¬Tr),
and so (∃r)(Sar & ¬Tr)
absolutely, and so (∃r)(Sar & Tr),
since we have the instantiation Sa§(∃r)(Sar & ¬Tr) & (∃r)(Sar & ¬Tr).
To find a possible instantiation of ‘(∃r)(Sar & ¬Tr)’,
by contrast, we must proceed as follows. We know that Sa§(∃r)(Sar & ¬Tr),
so Sa§(Saz & ¬Tz)
for a certain epsilon term ‘z’=‘that thing which a says which is not true’, and so (∃r)Sa§(Sar & ¬Tr),
by existential generalisation, and Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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for another epsilon term ‘y’=‘that thing about which a says that he says it and it is not true’. But ¬(Say & ¬Ty),
i.e. ¬Say v Ty,
if the only two options are the ones above, namely y=§(Say & ¬Ty),
and y=§(Saz & ¬Tz).
In the first case we get Ty ≡ (Say & ¬Ty),
and that yields ¬Say. In the second case we get Ty ≡ (Saz & ¬Tz),
i.e. Ty ≡ (∃r)(Sar & ¬Tr),
and so Ty, since (∃r)(Sar & ¬Tr).
Hence if Say then Ty, providing us with the fact that Sa§(Say & ¬Ty) & ¬(Say & ¬Ty),
which is therefore one possible instantiation for ‘(∃r)(Sar & ¬Tr)’. In Goodstein’s original case above a similar conclusion is available starting from
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Sa§x)(Sar ⊃ ¬Tr). The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
For from this we get that (∃r)(Sar & ¬Tr),
and (∃r)(Sar & Tr),
since if (x)(Sar ⊃ ¬Tr)
then, by substitution of what was first given, ¬T§(x)(Sar ⊃ ¬Tr),
giving (∃r)(Sar & Tr),
and so (∃r)(Sar & ¬Tr),
with the instantiation of the latter being provided by Sa§(x)(Sar ⊃ ¬Tr) & (∃r)(Sar & Tr).
To get a possible instantiation of the former, i.e. what specific thing A says which is true, we first obtain from what was first given, Sa§(Sav ⊃ ¬Tv),
for v=εr¬(x)(Sar ⊃ ¬Tr), by the epsilon definition of the universal quantifier, and then from this (∃r)Sa§(Sar ⊃ ¬Tr),
by existential generalisation, and so Sa§(Sau ⊃ ¬Tu), Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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for u=εrSa§(Sar ⊃ ¬Tr), again by the epsilon definition of the existential quantifier. But T§(Sau ⊃ ¬Tu),
i.e. ¬Sau v ¬Tu,
if the options are either u=§(Sau ⊃ ¬Tu)
or u=§(Sav ⊃ ¬Tv).
In the first case Tu ≡ (Sau ⊃ ¬Tu),
which gives ¬Sau. In the second case Tu ≡ (Sav ⊃ ¬Tv),
i.e. Tu ≡ (r)(Sar ⊃ ¬Tr),
and so ¬Tu, since (∃r)(Sar & Tr). It follows that Sa§(Sau ⊃ ¬Tu) & T§(Sau ⊃ ¬Tu),
providing a possible instantiation of (∃r)(Sar & Tr). Having thus defended Kneale’s specific conclusion more fully it becomes evident how the process can be generalised quite readily. Turning to where there are two speakers, Jones and Nixon, say, and the first says ‘What Nixon states is not true’, consider first the telling case where the second speaker in fact utters nothing. We saw in the simple case where Jones states that what he is stating is false, that the crucial difference is that between
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‘Sj§p’ The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
and ‘εxSjx=§p’.
Here it is even more evident why we can have the former without the latter. For the phrase ‘what Nixon states’ now refers to a fiction, and so the epsilon analysis simply leaves it with an indeterminate referent. We know that Sj§¬TεrSnr,
so we have a representation for what many have called the ‘proposition’ Jones expresses §¬TεrSnr,
and that necessarily contains a term for ‘what Nixon states’ i.e. ‘εrSnr’. But there is no way to specify the statements that either Jones or Nixon makes, because there is nothing to determine what ‘εrSnr’ refers to. Notably, the distinction between propositions and statements was made by Strawson and Lemmon (Haack [1978], Ch 6). The case of fictions was originally understood to provide a case where no statement is made, as with Russell’s ‘The King of France is bald’ said at the present time. But an epsilon representation of such definite descriptions as ‘The King of France’, as we saw in chapter one, allows them to be complete individual terms, with merely an indeterminate referent in the fictional case. Even if Nixon says ‘What Jones states is true’, there are fictions around. We have Sj§¬TεrSnr,
and now also Sn§TεrSjr,
so if (∃!r)Sjr Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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(∃!r)Snr,
we get that εrSjr=§¬TεrSnr,
and that εrSnr=§TεrSjr,
which means, using the truth scheme, that TεrSjr ≡ ¬TεrSnr,
and that TεrSnr ≡ TεrSjr;
and those are together contradictory. It follows that at least one of the speakers does not make an identifiable statement, in which case the other is speaking about a fiction. It is this indeterminacy of meaning that distinguishes the present approach from, for instance, the logic of ambiguity that Bryson Brown described as an equivalent alternative to standard systems of paraconsistency for handling the paradoxes (Brown, B. [1999]). One of the features of operator theories is that they do not incorporate 1-1 expression relations between sentences and propositions — so they allow for ambiguity. But if only plain ambiguity was involved then in a disambiguated language, which restored the 1-1 relationship, paradoxes could still be generated. The further, crucial feature of operator theories is that they allow for ineffability, i.e. reference to meanings that cannot be independently identified. Once the 1-1 nature of the expression relation is rejected we must accept that there may be things, as Kneale said, which can only be gestured towards, but not otherwise expressed. This is illustrated again in other well-known paradoxes, which fall into the same category as those discussed above: The Knower Paradox, The Strengthened Liar, Curry’s Paradox, and the Paradox of Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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the Preface. For the record, solutions to The Strengthened Liar, Curry’s Paradox, and the Paradox of the Knower will be given now. The Paradox of the Preface was dealt with, using Goodstein’s system, in (Slater [1986b]), and it is easily adjusted, as above. The Strengthened Liar is sometimes said to be a difficulty for propositional solutions to the Liar: if sentences can lack meaning, then what about ‘this sentence is false, or has no truth value’? Seemingly if it is false it is true and has a truth-value; if it is true, it is false (since it then must have a truth value): and if it lacks a truthvalue then it is true (since it says so). The formulation presumes that it is sentences that are true or false, so consider instead: ‘this sentence either expresses a false proposition, or does not express any proposition’. Using the modified Goodstein calculus, one starts from Sa§((∃x)(Sax & ¬Tx) v ¬(∃x)Sax).
One can then obtain (∃x)Sax,
and so, if we had (x)(Sax ⊃ Tx),
then we would have (∃x)(Sax & ¬Tx),
because then T§((∃x)(Sax & ¬Tx) v ¬(∃x)Sax).
Hence, absolutely, ¬(x)(Sax ⊃ Tx),
i.e. (∃x)(Sax & ¬Tx).
But that means, also, that Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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and so, by existential generalisation, (∃y)(Say & Ty),
where y = §((∃x)(Sax & ¬Tx) v ¬(∃x)Sax). Hence A does not express a single proposition. Curry’s Paradox is significant, since Priest, for instance, admits that his dialetheism is not sufficient to resolve it — he also wants to abandon ‘Absorption’. But there is no trouble with ‘if what this sentence states is true then q’ (for arbitrary ‘q’). Certainly if we had anything like p ≡ (p ⊃ q),
we could deduce the arbitrary q, by reasoning that given ¬p
we would have (p ⊃ q),
by elementary logic, and hence p, so we must have p, and so (p ⊃ q), and so q. But if instead we start from Sa§(∃x)(Sax & (Tx ⊃ q)),
and suppose, for instance, (x)(Sax ⊃ ¬Tx),
that means ¬T§(∃x)(Sax & (Tx ⊃ q)),
and so ¬(∃x)(Sax & (Tx ⊃ q)).
From this it follows, amongst other things, that Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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(x)(Sax ⊃ Tx), The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
and since we also have (∃x)Sax,
we must say ¬(x)(Sax ⊃ ¬Tx),
and so (∃x)(Sax & Tx).
No further conclusion can be drawn, however, about whether T§(∃x)(Sax & (Tx ⊃ q)),
since there is no guarantee that A is non-ambiguous. Comparable results follow if we start from Sa§((∃x)(Sax & Tx) ⊃ q),
Or Sa§(TεxSax ⊃ q).
The Paradox of the Knower was influential historically, since it motivated Montague towards his support for an operator theory of the attitudes. If the objects of attitudes like knowledge are syntactic then, by considering ‘this sentence is not known to be true’, for instance, we can derive a paradox, since if it is known to be true then it is true, and so not known to be true; but if it is thereby shown to be not known to be true, not only is it shown to be true (because that is what it says), also it has come to be known to be true. Hence, by Reductio, there cannot be a syntactic account of the objects of knowledge. Consider, however, a version that does not presume it is sentences that are known to be true, but instead the facts which they may state, viz: Sa§(∃x)(Sax & ¬K§Tx).
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(x)(Sax ⊃ K§Tx),
then K§T§(∃x)(Sax & ¬K§Tx),
and so (∃x)(Sax & ¬K§Tx).
Hence (∃x)(Sax & ¬K§Tx),
absolutely, and so, from the latter, we can say ¬K§Tεx(Sax & ¬K§Tx),
and also K§T§(∃x)(Sax & ¬K§Tx).
But there is no way to get from any of this that εx(Sax & ¬K§Tx) = §(∃x)(Sax & ¬K§Tx),
i.e. that that thing which A says which is not known to be true is that there is something A says which is not known to be true. So when we move away from the syntactic account of the attitudes there is no trouble. There is another paradox that fits in here, although it is not usually considered in the same class: if the sentence ϕ is defined to be ‘ϕ is (informally) unprovable’, then this sentence has been said to generate ‘Gödel’s Paradox’. The problem it posed was made prominent in Priest [1979a], although it had occupied the same writer since at least 1971. Priest said (Priest [1973], 130): We are assuming that English can be turned into a formal axiomatic system and that the truth of its Gödel sentence ϕ can be proved in English. Hence ϕ is assertible (i.e. provable) in English. Gödel’s Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Theorem states that any such system can prove its own Gödel sentence if and only if it is inconsistent. It follows that English is inconsistent.
But that conclusion only follows if English can be turned into a formal axiomatic system, and if it is not formalistic but contentful, because it employs a provability operator, then Priest’s conclusion does not follow. The expectation that any proper reasoning process should be formalistic is part of the practical grip that the preoccupation with computers has had in recent decades. But it has a further consequence that must be remarked: that seeing English is not inconsistent is a very difficult mental challenge for people with that general formalistic expectation, despite the proof’s immediacy. Even if there were no difficulty in finding such a sentence as Priest supposes, still, on an operator analysis, such sentences have no relevance to anything to do with truth — or therefore provability — in natural language. Used English is interpreted, and there is no comparable item leading to paradox in that area, simply because of that matter of interpretation. For what is (informally) provable is some fact, not some sentence, or formula, as with knowledge before. Forgetting the difference between facts and formulas is presumably the reason why, in common representations of the objects of formal proofs, a proposition is indicated, rather than a mentioned sentence: ‘|— p’. For only if the turnstyle symbol is used to represent an operator notion of provability, is it certainly ‘p’ that is involved, since “‘p’” would be the proper way of symbolising what is derived in formal proofs. By a common convention the quotation marks are omitted, but that only obscures the categorical difference between the two cases. The proved fact in question is then certainly what some sentence is standardly taken to express, i.e. what is true in its standard model. But even then, remembering the arguments against Tarski before, it is not the sentence which is what is true in that standard model: what is true is that 1+1=2, not ‘1+1=2’, and ‘that 1+1=2’ is not a sentence, it is a noun phrase referring to a proposition. The relevant locutions are not predicative locutions like ‘“p” is derivable’ but operator locutions like ‘it is provable that p’, and we have seen that it is trivial that there can be no self-reference with such locutions. For there is no ‘q’ such that ‘q’ = ‘it is unprovable that q’ Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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— that is ruled out by mereology, since nothing can be a proper part of itself. In addition there can be no ‘q’ such that it is necessary that q ≡ it is unprovable that q,
since ‘it is provable that’ at least obeys the rules for the modal operator in the system T. That means that, with ‘it is provable that’ as ‘L’, Lq would entail q, and so ¬Lq, giving ¬Lq absolutely, and so q likewise, and hence, by the Rule of Necessitation, contradictorily Lq. Certainly we can say such things as ‘that proposition where the proposition is that it itself is not provable, is provable’, but there is no independent namely-rider expressible with a ‘that’-clause, as we saw when developing Kneale’s point at the start of this chapter. In addition, ‘that proposition where the proposition is that it itself is not provable’ cannot refer to anything properly so described, as an epsilon analysis will demonstrate with complete formal accuracy. The proposition this phrase refers to is indeterminate, and so also must be its truth, and its provability. Thus while we can construct the referring phrase ‘that proposition where the proposition is that it itself is not provable’, once we remember the possibility that propositional referential terms are misnomers, the conclusion in connection with this phrase is merely that no proposition with this character can exist. That is to say, although (∃s)(s=εr(r=§¬Pr)),
we can show that ¬(∃r)(r=§¬Pr).
For if the reverse of the latter were the case, then, for some ‘s’, we could say that s=§¬Ps,
and so, if Ps, then, because then Ts, since provability in natural language entails truth, we would also have T§¬Ps, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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giving, by the T-scheme, ¬Ps, a contradiction. Hence ¬Ps, absolutely, and that means it is provable that ¬Ps, i.e. P§¬Ps,
which gives Ps, and so another contradiction. But the conclusion to be drawn is just that (∃r)(r=§¬Pr)
is not the case. The crucial difference between self-reference with sentences, and this attempted self-reference with propositions concerns the possibility of displaying the items they are respectively about. In connection with the former, the normal quotation names for sentences such as ‘this sentence is not provable’
immediately identify what objects they are names of. But there is no comparable, grammatical process for automatically identifying what is referred to by ‘that proposition where the proposition is that it itself is not provable’. Propositions are displayed using the demonstrative form ‘that p’, and not only does mereology prevent ‘it is unprovable that p’ from being itself the same ‘p’, other referring phrases besides ‘that p’ do not display the proposition, and so may lie i.e. be misnomers. In short, one can have that q ≡ ¬|— ‘q’ (where ‘|—’ is a predicate of mentioned sentences) but not that q ≡ ¬|— q (where ‘|—’ is an operator on used sentences), and so there is no proof of incompleteness, following Gödel’s proof, in the second case. Moreover, no induction to the first epsilon number, or anything remotely like it, is needed to previously show that the natural notion of proof is consistent. Because of the T-axiom again, there is a simple one line proof of this fact: one can no more have that Lp & L¬p than one can have that p & ¬p. Priest saw things differently (Priest [1979a], 223):
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Thus we see that our naive notion of proof appears to outstrip the axiomatic notion of proof precisely because it can deal with semantic notions. Of course, we can formalize the semantics axiomatically but then naively we can reason about the semantics of that system. As long as a theory cannot formulate its own semantics it will be Gödel incomplete, i.e. there will be sentences independent of the theory which we can establish to be true by naive semantic reasoning.
The principal corrections needed are, first, the removal of the idea that it is sentences which one can establish to be true, but then that idea’s main consequences, namely that the semantics of a language cannot be expressed in the same language without contradiction, and that a consistent language therefore must be incomplete. Of course a consistent language must be incomplete in another sense, for as we saw it must contain referential terms to propositions that cannot be expressed. If, in connection with the sentence ‘this sentence does not express a true proposition’,
for instance, we could formulate a true identity of the kind what that sentence expresses is that p,
then we could derive a contradiction of the form: p if and only if it is not true that p.
It follows, by Reductio, that we cannot formulate a true identity of the given kind, which means the sentence either has no sense, or more than one sense, as in the cases described in detail above. It seems clear, therefore, that there is no intractable difficulty with Liars. It is Tarski’s original T-scheme that is the trouble. For T‘p’ if and only if p,
clearly does not hold in general. The hope was, of course, that in a perfect language such a scheme would hold, and natural language, under that expectation, was inevitably thought not to allow consistent semantic closure. But clearly it does not hold for ambiguous sentences, and many classic paradoxes, it turns out, contain a similar semantic indeterminacy, as in fact Prior already saw (Prior [1971], Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
106). Human language is not so honest, it might be said: what it states cannot always be read off its face. That point, also, provides a further explanation for why The Liar has remained unsolved for so long. For plenty of highly intelligent people have put all their minds onto the task, but maybe they were too trusting. The problem with solving the Liar, in a large part, has been realising how thoroughgoing the lying really is. The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
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This chapter concerns more basic inadequacies in Hilbert’s Formalist Philosophy of Mathematics than those revealed by Gödel’s Theorems, but exposed in some of Wittgenstein’s remarks about mathematical proofs. It is not widely realised just how singular Wittgenstein’s contribution was to the Foundations of Mathematics. In a very clear sense, the mathematical studies called ‘foundational’ prior to Wittgenstein’s contributions were not at all concerned with providing a basis, or support, for existing mathematics. In fact they merely produced further, quite separate mathematics. This is especially, and perhaps most importantly true with regard to central aspects of Hilbert’s programme of Meta-Mathematics. Wittgenstein’s remarks, by contrast, are undoubtedly foundational in the above sense, and in particular, when contrasted with previous approaches, lead to a distinct re-assessment of the place of Gödel in the history of the subject. There are some general errors that crept into studies in the Foundations of Mathematics even before the specific developments that are the main concern of this chapter. It is plausible to believe that the inadequacies in Hilbert’s Formalism to be exposed later were not pressed before (though they were evident to Frege) because those earlier errors had not been corrected, and yet were very prevalent. A mathematical research programme in the late nineteenth century sought to provide a ‘foundation’ for Arithmetic, and Analysis, yet just produced more, different mathematics. Specifically it sought to define, for instance, real numbers in terms of rational numbers, rational numbers in terms of integers, integers in terms of natural numbers, and then, in some cases, natural numbers in terms Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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of purely logical notions. But none of these definitions work. The generality of working mathematicians, then and now, have taken little notice of this research, for very good reason, it appears, but it has occupied most philosophers dealing with the Philosophy of Mathematics, ever since Russell’s Paradox brought to light some completely inescapable problems with the last above step. It is the earlier steps in this ‘foundation’, however, which ought to be questioned more thoroughly. This is easy to see with regard to definitions of the reals in terms of equivalence classes of Cauchy sequences of rationals, or Dedekind cuts in the rationals (see Slater [2006a]). For, while there is no doubt that examples of a complete ordered field can be provided using such equivalence classes, or Dedekind cuts, it is quite erroneous to identify any such class, or cut, with a number. For the equivalence class of the Cauchy sequence consisting in the number 2 iterated, for instance, cannot be the number 2 itself, since the definition would be circular, in that case; and likewise with the set of rationals less than 2. In addition, the set of rationals less than 2 is neither less than nor greater than 2, so it is not even the kind of thing which is comparable with that number. What was going on, more generally, may be illustrated in the case of the ‘definition’ of the integers in terms of the natural numbers. How was this done? Well, we would ordinarily say, for instance, that 4-7 = -3. But also, of course, 11-14 = -3. So the negative integer -3 is associated with certain ordered pairs of natural numbers, <4, 7>, <11, 14>, etc. On that basis one can define equivalence classes of ordered pairs of natural numbers, so that is a member of the same equivalence class as so long as a-b=c-d. But it turns out that one can define operations on these equivalence classes comparable to the ordinary arithmetical operations of addition and multiplication, etc. Thus if A, B, and C are appropriate equivalence classes one can say that A+B=C if and only if, so long as belongs to A and belongs to B, then belongs to C. The final step is to identify, for instance, the equivalence class with <4, 7> as a member, with the integer -3, and we supposedly have provided a ‘foundation’ for the integers in the natural numbers – using some set theory. Alas, what -3 is identical to is 4-7, i.e. it is the difference between two natural numbers, not a class of ordered pairs of them. Certainly Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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there is a 1-1 correlation between the integers and such equivalence classes, a correlation that mirrors, in the equivalence classes, the arithmetical structure of the integers in many ways. But 1-1 correlation is not identity, so it is not the integers that are defined in the process. Likewise it is not addition of integers which is defined in the process, since ‘+’ is different from ‘+’; indeed ‘+’ is used in the definition of ‘+’, so again the definition would be circular if they were identical. These errors, note, were not confined to the nineteenth century writers, since this tradition in the ‘foundations’ of mathematics continued completely unquestioned up to at least Suppes’ classic text (Suppes [1960]), for instance. A more recent writer prefers to call the various ‘definitions’ of real numbers, integers, and the like, merely ‘surrogates’ or ‘proxies’ (Potter [2004]). But that brings out only too clearly their irrelevance in any study properly called ‘foundational’. If it is not the integers that are defined in the above way, but merely some other things with some similarities to them, then no foundational basis for those integers has been provided. A very similar point can be made with respect to Dedekind’s omega sequences, which many would now take to provide the nearest there could be to a ‘foundation’ for the natural numbers. Dedekind gave some axioms that defined simply infinite systems, and proved these axioms were categorical, i.e. that any objects satisfying them had to have exactly the same structure. Does that mean that we therefore have a definition of the natural numbers? Not at all, because the natural numbers are just one series of objects that form an omega sequence, and so they still have to be discriminated from others. Frege, at least, realized that more was needed, but the failure of his attempt led, first, to the identification of selected omega sequences of pure sets with the natural numbers (different sequences of sets were selected by Zermelo, and von Neumann, for instance), and then, after Benacerraf’s criticism of such identifications, to the idea that it is purely the structure of simply infinite systems which matters in Arithmetic, allowing any such system to do. As a result the fact that the natural numbers are just one sequence of objects with the required structure was lost sight of, and no definition specifically distinguishing them is now attempted. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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The period when selected omega sequences of pure sets were identified with the natural numbers was dominated with the idea that, as a result, the whole of mathematics could be represented in Set Theory. But not only is no set a natural number (for more on this see Slater [2006a]), also the further definitions leading supposedly to a set-theoretic basis for the integers, and the rest, are not to the point, as we have seen. So the idea that the Foundations of Mathematics lie in Set Theory was a considerable misconception. The resultant question ‘How can these things have gone on, in such a wide range of cases, over such a length of time, and with such a number of learned and careful people?’ I have addressed in the introduction to this book with respect to other, related matters. At this point we now can turn to that other train of thought over much of the same period that is our main interest. That is Hilbert’s meta-mathematical programme, which held first that it was all and only axiomatic structures that were the proper subject of mathematical study. This programme had something like the same failing as above; the error in it being particularly akin to the error in taking the natural numbers to be given (at least to the extent that they can be given at all) by means of axiomatically defined omega sequences. But this further train of thought has had an even wider ‘foundational’ impact, not only in Mathematics but also in Logic. And it had a singular difference in character from earlier views of axiomatic structures, since Hilbert’s approach was explicitly meta-linguistic, i.e. concerned just with the language, and formulae which appeared in the axioms. Objects satisfying the axioms have come, as a result, to be said to form a model of them, and reference to any such objects is in what was later called the ‘object language’, which must be contrasted with the ‘meta-language’ in which the meta-mathematical analysis is expressed. Hilbert established the plausibility of his line of research with his axiomatisation of Geometry, which dispensed with Euclidean figures, and proceeded entirely by means of logic from completely explicit geometrical postulates. The removal of diagrams took foundational studies away from ‘intuition’ in the philosophical sense. More plainly, it takes one away from what the language in the axioms is about. As a result, despite wanting to say he had provided a foundation for ‘Geometry’ Hilbert had nothing to say about the lines Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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and points in Euclid. Certainly the words ‘line’ and ‘point’ appear in Hilbert’s axioms, but they were taken to apply simply to anything that satisfied the axioms. So the fact that those axioms did apply to Euclid’s elements was quite incidental to Hilbert’s interests, and remained something Hilbert did not attempt to provide any foundation for. The basic error in Hilbert’s programme was therefore that it gave no account at all of what is true in a model of some formulae, being deliberately concerned entirely with the formulae themselves. Hilbert considered the consistency of his formal systems to be very important, but it would require more than consistency to establish results about numbers from proofs in the kind of arithmetical formal system he was concerned with. There would need to be some proof of the soundness of that formal system, on the standard interpretation, before one could show even that 2+3=5, for instance. This follows from the character of Hilbert’s meta-mathematics just in itself, it is important to note: there was no need to wait on Gödel, for instance, to point it out. Gödel’s Theorems do not show, that is to say, that while some standard arithmetical truths are provable metamathematically, others are not. In fact none are, since any derivation within a formal arithmetical system must be supplemented with a demonstration of its soundness, on the standard interpretation, before any of the standard arithmetical facts can be proved on its basis. It is important to underline this, since current received wisdom seems to be that the faults in Hilbert’s Formalism were only revealed once Gödel’s Theorems were proved, whereas the present kind of error was there right at the start. The dominance of Gödel’s Theorems in current thought about the Foundations of Mathematics presumably arose as a result of believing that, before Gödel proved his theorems, Hilbert’s Formalism had no serious arguments against it. But that, we can now see, was most likely because the earlier errors, in settheoretic ‘foundations’, had not been corrected, and so the similar correction that needed to be made to Hilbert’s ‘foundations’ was not attempted. One of the grammatical points I have made repeatedly before, now stated about meta-languages and object languages, is crucial to seeing the detail of the needed correction to Hilbert. That point concerns expressions of the form ‘that p’, and their distinctness from Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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expressions of the form ‘ “p” ’, i.e. sentences. ‘That p’ is not a sentence, like ‘ “p” ’, but instead a noun phrase — it is the sort of noun phrase that is the propositional complement in ‘propositional attitude’ expressions like ‘That John had a BMW was surprising’, ‘That Potter is fair to Wittgenstein in the 1940s was denied by Rodych’
etc. Traditionally, such expressions have been symbolised using sentential operators: ‘It was surprising that...’, ‘Rodych denied that...’, etc.
but the other, ‘non-cleft’ form turns these operator sentences into subject-predicate sentences with propositions as subjects. Kneale in (Kneale [1972]) made moves in this area, but the present writer only awoke to their significance in (Slater 2001). The broader point that emerged then was that reference to propositions in this way rules out self-reference, and so not only is the Paradox of the Liar removed, also analogues of Gödel’s theorems are unavailable in the realm of propositions. More immediately relevant to the present question, once one has propositions with other propositions as subjects it becomes much easier to distinguish them from propositions with sentences as subjects. A proposition about a sentence is about a purely syntactic form, i.e. it is about some symbols independently of their meaning. That kind of remark has dominated the Philosophy of Mathematics since Hilbert’s meta-mathematics got its grip — illicitly because of the above points. But it has quite properly entered into the theory of computing, since a computer, of course, cannot take account of the meaning of any of the symbols fed into it. The bulk of mathematics, however, is not a meta-linguistic study of symbols, and is instead concerned with propositions about other things: proving the proposition that 2+3=5, for example, rather than deriving the sentence ‘2+3=5’. The point shows that it is probably not an accident that most working mathematicians, to this day (like Wittgenstein), give so little time to Gödel’s Theorems. For, specifically, they are not Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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relevant to the Foundations of Mathematics, if that is concerned, amongst other things, with the basis for what is true in the standard model of axiomatic arithmetics. So they are not relevant to Arithmetic, as it was conceived before axiomatic studies came into vogue, and, with them, non-standard models of such structures. The Wittgensteinian point we shall come to is that a full proof of the fact that 2+3=5, for instance, is not available axiomatically. Instead it can be drawn from such illustrations as the paradigmatic stick figure with five lines that Wittgenstein considers (Wittgenstein [1978], 58f). The fact that the required foundations are in praxis rather than theory goes against many very strong expectations in the subject but only in a practical case like that, where the numerical concepts are applied (and so the symbols are used and not just mentioned), does one get beyond numerals, and other symbols, and begin to work with their meanings. No string of sentences in a formal proof can get to anything in the right category, even; no computer can do so either, therefore, since no computer can give an interpretation to the symbols it processes. In part that is because, as we shall see in more detail later, an interpretation is a matter of the Will — showing Lucas on Gödel was right, although not in the way Penrose has taken him. The common convention of not showing quotation marks around formulae does not help people remember what it is that is ‘provable’ — one should really say ‘derivable’, as above — in a formal system. Within a formal system like Principia Mathematica, for instance, one might be able to derive some formula like ‘Con(PM) ⊃ G’ (notice the quotes), where ‘G’ was Gödel’s undecidable sentence. But then, if we knew that the system was sound, on the standard interpretation, we would be able to know, on that interpretation, that if Con(PM) then G (notice the lack of quotes, and the use of ‘that’ just then). If we know that the system is sound, however, then we know that it is consistent, so we can further deduce that G, and so obtain a result PM cannot achieve. But not everyone compares our capacity to know that G with the system’s incapacity to derive ‘G’, and in the general confusion the system’s fulsome capacity to derive formulae easily leads to its total incapacity to prove facts being overlooked. Putnam suggested that we were in no better position than PM in being able to demonstrate its consistency (and hence did not really know that G). But if we were in no better position, then we would not Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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even know that 2+3=5, either. At this point, anyone with any common sense should raise their hands, just like Moore did against the disbelievers in the External World. There is certainly no ‘proof’, involving just a series of formulae, that Peano’s postulates are true on the standard interpretation. For ‘that Peano’s Postulates are true on the standard interpretation’ is not a sentence, and so, a fortiori, it is not the last sentence of any rule-governed series of sentences. Neither, of course, can any arithmetical fact be in this position, since ‘that 2+3=5’, for instance, is equally not a sentence. The proof of this arithmetical fact, therefore, has to be non-formal, at least at some stage, and can even proceed entirely in this way, as Wittgenstein has illustrated in several cases such as that above. The further work that has to be done in this area, therefore, is to make more apparent how Wittgenstein’s remarks relate to what is true in the standard model of axiomatic arithmetics. Clearly similar points can be made, also, about the standard (i.e. Euclidean) model of, for instance, Hilbert’s axiomatic Geometry, so examples in each area can be presented. Books on Wittgenstein’s Philosophy of Mathematics have not dwelt on these matters overmuch; indeed in some such books proofs involving the required physical exemplars are not even mentioned. That is despite it being known that Wittgenstein said the application of mathematics was crucial, as a way out of the standing difficulty with Platonic views of abstract numbers, and ideal geometrical figures, which leads to pressing questions about how knowledge can be gained of objects which are, by definition, non-causal. Frascolla is one of the few commentators who adresses the required issues at all closely. Even then, while he has over 14 pages on ‘Mathematical Proofs as Paradigms’, only in one half-page does he discuss an appropriate example: Wittgenstein’s proof that a the fingers of a hand and the vertices of a pentacle are the same in number (Diamond [1976], 71). Frascolla realises the place of ostensive definitions in this, for he says (Frascolla [1994], 133): The first step along the path that leads from experiment to proof, is the ascription of a paradigmatic function to the sequence of strokes and the star-shaped figure that have been manipulated in the course of the experiment. This is realised by means of a sort of initial baptism, i.e. through the adoption of the two following grammar Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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rules: call every figure that is the same shape as this (namely the sequence of strokes occurring in the experiment construction) “hand”, and every figure that is the same shape as this (namely, the star employed in the experiment construction) “pentacle”. Thus the sequence of strokes in the construction becomes a hand by definition, or a hand-paradigm (and the star in the construction a pentacleparadigm).
The addition to this of a further grammar rule, defining ‘correct comparison of the numbers’ then makes the accompanying picture ‘a distinguishing mark of the concept of correct comparison of the number of elements in a hand with the number of vertices in a pentacle’. The incorporation of such physical paradigms into the language, in other cases, such as the Standard Metre, and colour samples, is a well-known part of Wittgenstein’s later analysis of ‘simples’ (Fogelin [1976], 108f). What Frascolla does not do, however, is apply the same insights to point out the failings in the non-Wittgensteinian tradition on ‘foundations’. To do that, however, all that is needed is to analyse in the same way the case which shows directly what numbers of things are: the simpler case of the five sticks formed from two sticks and three sticks mentioned before (c.f., for example, Resnik [2000], 220, and the papers by Parsons there referenced). For the five-stick picture gives definitions of certain numbers of things, and so provides just what is wanting in axiomatic arithmetics, which by their nature cannot get hold of any truths in the standard models of their axioms. Thus, with respect to the five sticks, one can say that things like this (indicating the two bracketed sticks) are two in number, that things like this (indicating the three bracketed sticks) are three in number, and that if the two things and three things are separated like this (indicating that the brackets are distinct, so no stick is counted twice), then things like those sticks in all are five in number. Ostensive definitions not of abstract numbers, but of numbers of things are thereby obtained, which make it necessary that two things with a further three things add up to five things, that being the natural content of ‘2+3=5’. Of course, one cannot equally survey 1629 things, or a 429-acle, but one crucial arrangement we have with larger numbers, for example, which enables us to be as certain with operations on them as with operations on small numbers, is the use of Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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a convenient radix in the representation of them. Thus, in the decimal notation, addition, multiplication, division, and the like only ever involves operations on the first 10 digits. It is ironic that Gödel believed in ‘intuition’, even though he was so much a Platonist that he believed there was another world of abstract objects accessible to a specifically mathematical intuition. But the traditional philosophical description for the particular use of ostension above, was that it was a matter of applying one’s ‘intuition’ — although that far more in the Kantian sense, which involved intuitions just of the spatio-temporal world, leading to synthetic a priori truths rather than trivially verbal, analytically a priori ones. So much is quite well known, maybe. What is not so well known is that it is also highly ironic that the philosophical problems Hilbert overlooked in his meta-mathematical programme have a formal resolution in the improved predicate logic he himself introduced — the Epsilon Calculus. In that calculus Hilbert provided a technology that most clearly shows, in connection with Gödel ‘s Theorems, what the difference is between the mathematics that humans do and the ‘meta-mathematics’, or ‘symbol manipulation’, which is the job of computers. The main point is that proving Gödel’s Theorems using the Epsilon Calculus as the logic shows more exactly where a Kantian ‘intuition’ is applied even in Gödel’s own case. It comes in in our knowing the fact, above, that G, which PM cannot even derive the expression for, i.e. ‘G’. What the spatio-temporal objects are in this case are the symbols in the expression ‘G’, and what our ‘intuition’ supplies, amongst other things, is their meaning, on the standard interpretation, i.e. we read ‘G’ as saying that G. The case also illustrates some specific ways in which Wittgensteinian notions like paradigms, decisions, surveyability and finitude are essential to the case. First we must have clear some preliminaries about the Omega Rule, and the Epsilon Calculus. The Omega Rule would allow us to infer a universal conclusion about all natural numbers — (x)Fx — from the infinity of instances F0, F1, F2, etc. This infinity of premises, of course, is unsurveyable in the extreme, so there is no way we can apply the rule, as it stands. The situation changes, however, if we employ epsilon terms in our logic, since then, as we Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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have seen, individual terms of the form ‘εxFx’ are defined for all predicates in the language, satisfying (∃x)Fx ≡ FεxFx.
Consequently we have: (x)Fx (≡ ¬(∃x)¬Fx) (≡ ¬¬Fεx¬Fx) ≡ Fεx¬Fx.
So if it were the case that y = εx¬Fx, for some natural number y, then if it were also the case that Fy, that would, alone, imply that Fεx¬Fx, and so that (x)Fx. But the (meta-)mathematical analysis of derivations in systems like PM, which Gödel set out, showed that there was no way to apply the Omega Rule within such systems, in any form at all, not even if the Epsilon Calculus is used as the logic. For in the case he constructed, what were derivable were sentences of the form ‘Fy’ for a certain ‘F’, and every numeral ‘y’, while what was not derivable (if PM was consistent) was the sentence ‘(x)Fx’. But that means that, even if it is true that y = εx¬Fx, for some number y, substitution of the identity into ‘Fy’, for an appropriate numeral ‘y’, is not available, since the quotation makes the context referentially opaque. Also, even if it is true that y = εx¬Fx, the sentence expressing this, namely ‘y = εx¬Fx’ cannot be derivable, for any numeral ‘y’, since the referent of the epsilon term, within any model, is a matter of choice — an alternative name for the Epsilon Calculus is ‘The Choice Calculus’, since the semantics of epsilon terms is given by means of choice functions. PM is therefore incomplete, and specifically ‘omega incomplete’. As Wittgenstein is reported to have said, upon hearing of Gödel’s results, they mean that there are non-standard models of formal arithmetics, i.e. the formal number variables in them can take values other than the natural numbers. It also follows that what standard, or non-standard model we go on to provide for the axiomatic arithmetic is a matter for our decision and choice: PM cannot do this, since PM is a purely syntactic system, and so is incapable of providing itself with a semantic interpretation. Other semantic matters that are in our hands, of course, are ensuring the axioms of PM are true in any Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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chosen model, and determining that its deduction rules preserve truth. These matters, too, PM is disallowed from having any opinions about, although, armed with our normal reading of PM’s symbols, we can (against Putnam again) easily verify that PM is sound on that reading. But that means that not only can one derive all sentences ‘Fy’ with ‘y’ a numeral, by using PM, it is also true, on the standard reading, that Fy, for any natural number y (notice not only the ‘that’, but also the reference to numbers rather than numerals just then). In addition, since we have selected the standard model, the universe of discourse of the quantifiers is thereby taken to be restricted to the natural numbers, and so it is true that y = εx¬Fx, for some number y. It follows that Fεx¬Fx, and so that (x)Fx (on the standard interpretation of the range of the variables). If we chose some other, non-standard model we would not be able to say, for sure, that the epsilon term referred to a natural number, so it is entirely through our selecting the standard model that we make the epsilon term refer in this way, and through that obtain a result which PM not only cannot obtain as a fact (since it cannot prove any fact at all) but also which PM cannot even obtain the symbolic expression for: ‘(x)Fx’. In all this we see how choice, surveyability, finitude and ‘intuition’ in the sense previously indicated, come into this specific case. Our proof that (x)Fx is another piece of mathematics, in other words, with many of the characteristics of Wittgenstein’s proof that 2+3=5. Its distinction is that it is a piece of mathematics about symbols rather than numbers of things, or pentacles, but there is no way that any such result, in any of these areas, can be proved simply axiomatically, as the ‘foundational’ tradition thought before Wittgenstein. They all involve, in other words, synthetic a priori rather than analytic a priori truths. But what of paradigms in the Gödel case? How did they come into the above analysis? They came in directly as the referents of the epsilon terms. For even Hilbert gave as an illustration of the prime epsilon axiom (∃x)Fx ⊃ FεxFx
the case Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
Aristedes is thereby an exemplar of justice, he is ‘the one who is just if anyone is’. So likewise εx¬Fx is ‘the one that is not F if anything is’, and that explains why, if εx¬Fx is not not F (i.e. is F) then nothing is not F (i.e. everything is F). The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
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215 12. Proving that 2+3=5
If anyone is just then Aristedes is just.
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Prawitz proved a theorem, formalising ‘harmony’ in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, then we can extend the theorem to the classical case. Properly arranged there is a thoroughgoing ‘harmony’, in the classical rules. Indeed, as we shall see, they are, all together, far more ‘harmonious’ in the general sense than has been commonly observed. As this chapter will show, the appearance of disharmony has only arisen because of the illogical way in which natural deduction rules for Classical Logic have been presented. Prawitz’ treatment of Natural Deduction presents Intuitionistic Logic, and then Classical Logic as being developed from Minimal Logic, with the addition of rules for Absurdity (⊥). Thus in (Prawitz [1965], 20-21), he gives the ‘and’ rules as: &I: A, B / A&B &E: A&B / A A&B / B,
the ‘or’ rules as vI: A / (A v B) B / (A v B) vE: A v B, ((A) / C), ((B) / C) / C,
and the ‘if’ rules as Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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⊃I: ((A) / B) / (A ⊃ B) ⊃E: A, (A ⊃ B) / B. The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
From the latter, Gentzen’s rules for Negation may be derived, upon defining ‘¬A’ as ‘A ⊃ ⊥’, viz ¬I: ((A) / ⊥) / ¬A ¬E: A, ¬A / ⊥.
To get Intuitionistic Logic one adds the Intuitionistic Absurdity rule, ⊥ / A,
and to get Classical Logic one adds the Classical Absurdity rule, ((¬A) / ⊥) / A.
Adding the Classical Absurdity rule to the other rules is not conservative, since Peirce’s Law, for instance, which is a purely implicational thesis, is not derivable without it, even though it does not itself contain ‘⊃’. But it is another exceptional feature of the Classical Absurdity rule that is of concern here. Prawitz, in formalising a notion of ‘harmony’, proves his ‘Inversion Theorem’ (Prawtiz [1965], 34): If Γ / A then there is a deduction of A from Γ in which no formula occurrence is both the consequence of an application of an I-rule, and major premiss of an application of an E-rule.
This formalises ‘the inversion principle’ which says that nothing is ‘gained’ by inferring a formula through introduction for use as a major premises in an elimination. Prawitz then goes on to demonstrate that this principle fails in Classical Logic, showing specifically that, even if the Classical Absurdity Rule is replaced with a further pair of introduction and elimination rules for negation, then a proof of the Law of the Excluded Middle has to use a formula occurrence both as a consequence of an application of an I-rule and as a major premises of an application of an E-rule. But with the Classical Absurdity Rule unamended, look again at the rules above involving ‘¬’, namely the last one, and the ones labelled the introduction and elimination rules for ‘¬’, according to Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Gentzen. There are three of these in all, not just two, and they are symmetric with respect to ‘¬A’ and ‘A’, so they as much involve the introduction and elimination of ‘¬’ as they involve the introduction and elimination of a bare space before ‘A’. That means nothing distinctive about the presence rather than the absence of ‘¬’ is displayed in these rules, mirroring the fact that not only is ‘¬A’ the contradictory of ‘A’, but also ‘A’ is the contradictory of ‘¬A’. As a result, they are merely about the contradictory of a formula: they are introduction and elimination rules for Contradiction, and they show how forming the contradictory of a formula can be done in two ways: by adding, or, when appropriate, deleting the negation sign. On that understanding, the contradictory of a formula is introduced in both of ((A) / ⊥) / ¬A
((¬A) / ⊥) / A,
while the contradictory of a formula is eliminated in A, ¬A / ⊥.
The consequence is that Prawitz’s Inversion Theorem becomes exceptionless, and his various theorems about the normal forms of classical deductions become much more understandable, and symmetric. The latter theorems are formulated about ‘maximum formulas’, which, in Prawitz’s terminology, concern formula occurrences that are the consequence of an application of an introduction rule, or the Classical Absurdity rule, as well as being the major premise of an application of an elimination rule (Prawitz [1965], 34). But the Classical Absurdity rule, on the revised understanding, is not a deviant rule difficult to insert into the regular system of rules, but simply another introduction rule, which means, not only that the statement of the theorems is neater, but also that the reason why that rule is included in its given place is more evident. The Inversion Theorem likewise involved an anomaly which now vanishes. As before, that theorem shows that corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. So how do the rules for Contradiction now fit in? Taking either premise in the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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elimination rule for Contradiction to be a major premise, here are the two cases in which a formula occurrence is both the consequence of an introduction rule, and a major premise of the elimination rule: ((A) / ⊥) / ¬A A/ ⊥ , ((¬A) / ⊥) / A ¬A/ ⊥.
Clearly to the same effect are simply A/ ⊥, ¬A/ ⊥,
so the detours through the respective formula occurrences of ‘¬A’ and ‘A’ can be removed, establishing the associated induction step in the proof of the Inversion Theorem (c.f. Prawitz [1965], 35-38). As a whole, therefore, the theory of classical logic becomes much more regular. What Gentzen seems not to have noticed is that ‘¬’, while commonly called a ‘propositional connective’, is very different from the other propositional connectives, especially in the classical case. Partly this is because it is a unary connective, but mainly it is because, on account of that, in Classical Logic it symbolises, through its presence and absence, a certain relation between propositions: the relation of Contradiction. In Intuitionistic Logic there is not that relation between ‘¬A’ and ‘A’, since formulas such as these, there, all fall merely into the weaker relation of contrariety. But ‘¬’ there does not symbolise this weaker relation, since not all contraries of the same thing are equivalent. As a result, when developing logic Gentzen’s way one may even lose sight of the fact that relations between propositions are involved at all, other than those given by the deduction sign. There is an even more fundamental reason, also, why this is so. Gentzen divided his inference rules according to whether they introduced, or eliminated propositional connectives. This corresponds, in most cases, with them introducing, or eliminating the associated formulas, namely formulas in which there are ‘and’s, ‘or’s, or ‘if’s between others. But it does not correspond to all cases where what is introduced, or eliminated, are formulas, since it does not cover all cases where the formulas are contradictories of others. The conjunction of two propositions can be expressed by using a symbol between them, and likewise the disjunction of them, and the implication of Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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one by the other. But the same is not the case with respect to the contradiction between two propositions. For that is shown by the one possessing a sign that the other lacks the negation sign. Gentzen’s emphasis on the connectives, as a consequence, easily gives the impression that different sets of rules merely allow one to deduce less or more formulas involving the connectives, with, amongst them, the negation sign. But what is true instead, in the latter respect, is that the restricted rules in Minimal Logic, and Intuitionistic Logic do not allow one to deduce all formulas about contradictories. Certainly one cannot have, in Intuitionistic Logic, that A & ¬A, but that theorem applies to all contraries, and so does not single out just contradictories. For a contrary to be a contradictory it must also be the case that A v ¬A, i.e. that the two formulas are subcontraries. We shall see later that forgetting this kind of matter has had further consequences in connection with misunderstandings about classical logical truths like the Law of the Excluded Middle, and the Law of NonContradiction. It is also involved in understanding why the subformula principle does not hold in the classical case. As Prawitz explains this (Prawitz [1965], 42), it is a corollary of his major theorem on the form of normal deductions: Every formula occurrence in a normal deduction … of A from Γ has the shape of a subformula of A or of some formula of Γ, except for assumptions discharged by applications of the (Classical Absurdity Rule), and for occurrences of ⊥ that stand immediately below such assumptions.
The problem which gives rise to the exception is that, in ((¬A) / ⊥) / A,
the ‘¬A’ is not a subformula of ‘A’. Of course, it is a subformula of ‘¬¬A’, which is equivalent to ‘A’ classically. So the subformula principle obviously only arises in forms of ‘logic’ that concern themselves with signs and not with what those signs signify. But what are we to make of the fact that there are three Contradiction rules? Does it not make Contradiction some kind of special case? Here we come to notice the always-evident fact that Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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there are three natural deduction rules for every basic logical symbol in classical propositional logic. There is not just one elimination rule for ‘and’, but two, and not just one introduction rule for ‘or’ but two. The two elimination rules with ‘and’ are needed to harmonise with the fact that two premises are involved in its introduction rule, and so, given that a conjunction can only be arrived at from these two premises together, it must entail all they separately entail. The elimination rule for ‘or’ harmonises with its two introduction rules in a similar manner, since a disjunction can only be arrived at from one of its disjuncts, and so it must entail only what they both entail. Hence, in the elimination rule a common implication of the two further premises is required besides the ‘or’. It is not the case, of course, that there are three rules in the case of ‘if … and only if …’, for instance, or ‘if … then … else …’. But these can be defined in terms of the elementary ones. The same holds in Predicate Logic, with complex quantifiers like ‘for all … there exists…’. But for the basic quantifiers we can find three rules, so long as we rise above standard Predicate Logic and consider instead its conservative extension, the Epsilon Calculus. So there is nothing at all anomalous about three rules being involved for Contradiction. It is already well known that the epsilon treatment of standard Predicate Logic enables considerable simplifications to be introduced. For while the rules of Universal Quantifier Elimination (UE), and Existential Quantifier Introduction (EI) are straightforward, the rules of Universal Quantifier Introduction (UI), and Existential Quantifier Elimination (EE), in Gentzen’s and comparable formulations, are subject to several qualifications regarding the arbitrary names in them (c.f. Prawitz [1965], 20). But these qualifications can be completely removed if the appropriate epsilon terms are used in place of the arbitrary names (see, e.g. Hazen [1987]). This arises because, in the Epsilon Calculus individual terms of the form ‘εxFx’ are defined for all the predicates in the language, and ‘εxFx’ then refers to the F alluded to in the existential generalisation ‘(∃x)Fx’, while ‘εx¬Fx’ refers to the strongest putative counterexample to the universal generalisation ‘(x)Fx’. To present the whole of quantification theory available when using epsilon terms, however, one must have enough natural deduction rules to match a standard axiomatic base for the Epsilon Calculus, such as the axiom Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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and the quantifier definition (∃x)Ax ≡ AεxAx.
That means we can take as the rules for the existential quantifier: EI: Ay / AεxAx AεxAx / (∃x)Ax EE: (∃x)Ax / AεxAx,
the first two being introduction rules — using either the second alone, or the pair together — and the third being the elimination rule for it. The required, extra induction clauses in the proof of the Inversion Theorem for this case are again trivial (c.f. Prawitz [1965], 38), since a chain of inferences in which an existential generalisation is the conclusion of an introduction rule, and the premise of an elimination rule, viz Ay / AεxAx / (∃x)Ax / AεxAx,
or simply AεxAx / (∃x)Ax / AεxAx,
clearly allows the existential quantifier step to be removed. A full discussion of the theory of Epsilon calculi in Natural Deduction settings may be found in several other places (see Leisenring [1969] Ch V, Meyer-Viol [1995] Ch 2, Wessels [1977], [1978], and Yasuhara [1982]). But all that it is relevant to note here is that the elimination rule given above harmonises with the two introduction rules because, if the latter are together the only means by which ‘(∃x)Ax’ may be introduced, then this must be via ‘AεxAx’ or something which implies it — and so neither by something which implies ‘(∃x)Ax’ without implying ‘AεxAx’, or through some other, quite independent, route. Note also that what might seem sufficient natural deduction rules for the existential quantifier, namely EI: Ay / (∃x)Ax, EE: (∃x)Ax / AεxAx,
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do not similarly harmonise, and in fact do not suffice as an explicit basis for the Epsilon Calculus, since there might be a subsidiary rule forbidding the substitution of epsilon terms for ‘y’, as, for instance, in Mints’ Intuitionistic Epsilon Calculus (Mints [1982]). The situation is somewhat similar to omitting the second ‘if’ introduction rule to be discussed shortly. For the rules for ‘if’, within Classical Logic, also display this same threefold nature — although that fact is hidden in the GentzenPrawitz approach based on Minimal Logic. Indeed Stephen Read has proposed a harmonization of the classical rules using a multiple conclusion logic with this feature (Read [2000], 145), which, in place of the ⊃I rule before has ⊃I’, viz ((A) B, X) / (A ⊃ B), X.
In single conclusion form, this adds to ⊃I the further introduction rule ((A) B v X) / (A ⊃ B) v X.
Multiple conclusions, of course, are not single propositions, and so the entailment of a proposition, in the first case, is not symbolised by ‘/’. The transcription into a single-conclusion form regularises this, but also demonstrates that Read’s second ‘if’ rule is an impure introduction rule, since it essentially involves both of the connectives ‘⊃’ and ‘v’. There is a pure second ‘if’ introduction rule, however, besides the first one given before. That is, of course, B / A ⊃ B.
This is a derived rule in Minimal Logic as presented before, because of additional, subsidiary rules about premise discharges, which allow chains of inferences like this: A B / A&B, A&B / B,
which give a deduction of ‘B’ from ‘B’ together with a redundant ‘A’, at the same time as allowing that redundant ‘A’ to be discharged Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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in a further application of the ‘if’ introduction rule. So, in effect, one gets simply: B / (A ⊃ B).
But there is no formal difficulty in adding this derived rule as a primitive rule — it introduces the ‘if’ connective, after all — or adding the appropriate further clause in the proof of the Inversion Theorem (c.f. Prawitz [1965], 37). More to the point, unless it is taken as a separate, explicit introduction rule, the full harmony of the introduction and elimination rules for ‘if’, in the classical case, is not fully apparent. For the single introduction rule given at the start makes ‘A ⊃ B’ invariably reflect a deduction of ‘B’ from ‘A’. But, because of the additional rules about premises discharges, deductions of ‘B’ from ‘A’ include ones, like that above, where the truth of ‘A’ is irrelevant, hence ‘B’ might follow independently of ‘A’, since it was from ‘B’, essentially, that the conditional was derived. By contrast, once one embraces the second introduction rule, it is quite explicit that the truth of the conclusion of the elimination rule could arise independently of the truth of the first premise, ‘A’, and some derivation of ‘B’ from ‘A’. In fact ‘A’ never combines with ‘A ⊃ B’ to generate ‘B’ (like ‘A’ and ‘B’ combine to generate ‘A & B’), for if ‘A ⊃ B’ is present because there was a derivation of ‘B’ from ‘A’, in which ‘A’ was not redundant, then ‘B’ already follows from ‘A’ alone. On the other hand, if ‘A ⊃ B’ is present because there was a derivation of ‘B’ from ‘A’ in which ‘A’ is redundant, then ‘B’ follows entirely because it was on this, essentially, that ‘A ⊃ B’ was based. Incorporating the second introduction rule shows explicitly that ‘A ⊃ B’ could have come about either directly from ‘B’ itself, or, alternatively from ‘A’ because there was a deduction of ‘B’ from ‘A’ in which ‘A’ was not redundant. Either way, of course, because ‘A’ is a further premise, ‘B’ follows as in the elimination rule. There is more to be said about the significance of this second ‘if’ introduction rule, however, since, as we shall see, Classical Logic, by incorporating this rule, recognises part of the natural language behaviour of ‘if’, while certain other logics, by not incorporating it, are the logics of imaginary languages. The ‘harmony’ of the above rules, though, after that emendation, we can now see, extends well Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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beyond the ‘harmony’ defined in historical discussions of Natural Deduction. For there is, evidently, a remarkable threefold ‘harmony’, in a much more general sense, all the way through the whole structure of first-order logic, once we move over to the Epsilon Calculus for a formalisation of Predicate Logic. In this respect the above rules far surpass those that Milne obtained, for instance, which while ‘harmonious’ were, by his own admission, ‘frankly unnatural’ (Milne [2002], 499). Indeed, what is also most striking, surely, is how banal and obvious the full harmony is, once it is revealed. That raises the further question of why the extensive, yet simple, threefold structure revealed above has not been noticed before. This is a very important question, I think, and it is not just a historical or sociological one, but also a formal one. For countering the lines of thought that have led people to doubt the harmony of Classical Logic is itself a logical matter. There are a number of lines of thought that need to be countered, ranging from ignorance of the Epsilon Calculus through to misconceptions about the relation between Contradiction and ‘not’, and Implication and ‘if’. In connection with ignorance of the Epsilon Calculus, for instance, there is obviously the weight of the entrenched text-book tradition to contend with, setting out, again and again, just Fregean Predicate Logic. That is a hindrance to perceiving the common, threefold, and harmonic form of the above rules, since, for one thing, it encourages tolerance of the messy restrictions there have to be on Fregean Predicate Logic’s Universal Introduction and Existential Elimination rules. In addition, the further applications of epsilon terms are in areas of language not dealt with in this text-book tradition, namely, the formulation of certain anaphors, and their related, purely referential definite descriptions. But even amongst scholarly experts on the epsilon calculus, its larger illumination of logical theory is not commonly recognised, so those people who might have been expected to see the harmonisation this calculus provides, in the areas highlighted above, have not woken up to its central virtues. This point may be made with respect to advanced classical logicians themselves, but there is also the extent of the pre-occupation, within learned circles, of Intuitionistic Logic, which does not sit easily with the Epsilon Calculus. There are major problems with adding epsilon terms, and the epsilon axioms, to Intuitionistic Propositional Logic Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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(Meyer-Viol [1995], Ch3). The natural consequence of such incompatibilities, given the tradition of academic interest there has been in Intuitionistic Logic, is no doubt one reason why the purity and simplicity of the Epsilon Calculus, within its original Classical Logic setting, has tended to get overlooked, and its general harmony with other parts of Logic has been left uninvestigated. Likewise also, of course, with the internal harmony of the classical Negation rules, once these are all set out, and seen to be presented better as rules about Contradiction. Intuitionistic Logic lacks the Classical Absurdity rule, and so, directly against Dummett (Dummett [1991], 291f) there is a clear sense in which both the absurdity and the negation in that logic remain ‘disharmonic’. Specifically, as we saw in part before, Intuitionistic Logic perpetuates the Conventionalist idea that there have to be rules about the negation sign rather than the relation of Contradiction. This is even present in its claim to have denied the Law of the Excluded Middle, by not having a formula of the form ‘A v ¬A’ as a thesis, when that law (which is therefore necessarily true) says instead that either a proposition or its contradictory is true. There is no way to symbolise this law simply as a formula involving an undifferentiated negation sign, since it is not that sign, but the relation it classically symbolises, that is involved. So denial of that law cannot consist in finding something merely of the symbolic form ‘A v ¬A’ which is not a thesis. As a result, many proponents of Intuitionistic Logic even need to be questioned about whether they know what it is that is classically true. It is common in describing different logical systems to indicate by means of a suffix on the inference sign which system of inferences is involved. The same kind of relativisation should properly be employed with the propositional connectives if the conditions on their use are significantly different. Then it would be clear that what is necessarily true is that A vc ¬cA, while what is not necessarily true is something different, namely that A vi ¬iA. Alternatively, using the Gödelian interpretation of the Intuitionistic connectives (c.f. for instance, Beziau [2005]), what is necessary is simply that A or ¬A, while what is not necessary is that LA or L¬LA, for a certain modal operator sometimes read ‘it is assertable that’ or ‘it is provable that’. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Either way that shows that the Law of the Excluded Middle is not denied in Intuitionistic Logic. It also shows that the Classical Absurdity Rule, with ‘¬’ as ‘¬c’ does not fail in Intuitionistic Logic. Indeed it cannot fail at all. What fails is merely of the same form, but with ‘¬’ as ‘¬i’. We therefore see that the root cause of the seeming lack of harmony in the classical natural deduction rules was a misunderstanding about logical truth itself. Conventionalism does not give a correct account of what is logically necessary, i.e. what cannot be otherwise, and progressive series of rules, such as Gentzen’s, should therefore not be seen as simply accumulating rules about the same signs, but as developing different concepts, which need different expressions to avoid ambiguity. This does not only happen with the rules for negation, since, without the second ‘if’ introduction rule, the one given by Gentzen is ambiguous. For that single rule on its own does not explicitly define a concept of ‘if’, and needs subsidiary rules about premise discharges to resolve its intended sense, and the major problem has been understanding the classical sense. In that regard, calling the associated rules ‘implication’ rules is what is primarily misleading, since what is involved in the classical case are merely rules for ‘if’, and ‘if’ does not invariably indicate an implication, as is well known. Thus we can say ‘There is jam in the cupboard, if you want some’, and also ‘If there is jam in the cupboard, then I am a Dutchman’, and the like, and in neither case is there any claim that the consequent is implicated in the antecdent. In fact, by far the biggest area of thought which needs correction, in coming to see the fully harmonious formal structure there is in Classical Logic must be the set of traditions that found difficulty with the second ‘if’ introduction rule unearthed before. That applies both with regard to the narrow notion of ‘harmony’ connected with the Inversion Theorem, and also the broader notion that we have seen also arises. Without the second ‘if’ introduction rule in its natural place — and in place not just as a consequence of a definition of ‘if’ in terms of ‘or’, but directly because of a feature of ‘if’ itself — the broader notion of harmony displayed above can hardly be suspected, and the Classical Absurdity Rule is more likely to be seen as anomalous. For the idea that there is a threefold structure within the rules of Natural Deduction is then not very apparent even at the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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propositional level, since the fact that it is present in the ‘and’ rules and ‘or’ rules easily gets dismissed, given their repetitive form, and also the two-fold structure of the Fregean quantification rules. The traditions that have had this more basic difficulty have all taken off from what have been called the Paradoxes of Material and Strict Implication, and perhaps the most substantial is that formalised within Relevance Logic, where the rules about premise discharges are tightened up in various ways. In particular, there may no longer be any derivations of ‘B’ from ‘A’ in which ‘A’ is redundant. The idea is that an ‘if’ or ‘only if’ statement, such as ‘A ⊃ B’, should give conditions under which its consequent is true. Asserting this merely on the basis of its consequent, ‘B’, makes no attempt at presenting such conditions, since then, with an arbitrary antecedent, should one remember it, the ‘if’ or ‘only if’ statement is part of an affirmation ‘(A ⊃ B) & (¬A ⊃ B)’ saying that its consequent is true come what may, i.e. true whether or not its antecedent is true. Likewise in the reverse case, when such an expression is uttered on the basis that its antecedent is false. To say ‘A ⊃ B’ then, with a quite arbitrary consequent, is to view it as irrelevant to the current point what might follow from the antecedent, since all that matters, in that case, is that that antecedent is false, i.e. emphatically and unconditionally false, and so false whatever else might be the case. An ‘if’ statement, in other words, can be used to express certainty, and assert that there are no conditions. In the discussion of C. I. Lewis’ proof that a contradiction entails everything, therefore, relevance logicians (e.g. Read [1988], 31f) see an ambiguity that is not there in natural language. From A & ¬A
one can derive A,
and ¬A;
while from Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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one can derive A v B;
but can one then use Disjunctive Syllogism to derive B
from ¬A
together with A v B?
The question raised is whether ‘A v B’ involves the right sense of ‘or’. Supposedly, given the way it was introduced, it’s ‘v’ is just an ‘extensional’ connective, and not the ‘intensional’ one, ‘fission’, which would enable one to derive, from the ‘or’, the conditional ‘A → B’ with relevant ‘→’ (note the distinctive sign). Only ‘fission’, we are told, can figure in valid Disjunctive Syllogisms, on account of this. Where ‘→’, ‘fission’, and particularly its conjunctive mate ‘fusion’ are to be found in natural language is the main problem. But that is doubly a problem, since it is not first recognised as relevant, within this tradition, that these connectives should even be looked for there. Natural language, it seems, is only there as the meta-language in which to talk about these connectives, which are in some other, distant, object language. Inspection of the banal piece of natural language that halts this tradition in its tracks is thus inhibited, and by far the larger percentage of time and attention is given to the admitted-to-be dreamt-up semantics, and the consequent need to present the associated soundness and completeness proofs. Is it all a fiction, or does it deal with facts? That is not a worry, so long as one fills one’s mind with the mathematics of telling the fine details of the story. Contrariwise, semantics, completeness, and soundness are not a worry if one attends to natural language, since then ‘object’ and ‘meta-’ language are exactly the same. We say, in natural language, Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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to focus on a fact, things like ‘B is true, whether or not A is’, and ironically the ‘whether or not’ is there to remove irrelevancies to the focus, which is not on why, or how, or because of what B is true, but simply on the certainty that B is true. Hence, if the logic is not to be the logic of some imaginary ‘object language’, but instead of the ‘meta-language’ in which it is talked about, namely the logician’s Mother Tongue, then the second ‘if’ introduction rule must be incorporated. Without a recognition that ‘if’ and ‘only if’ statements can express certainty, and do not always present necessary or sufficient conditions, there are, in addition, ambiguities about whether Logic might be an empirical science, where such conditions have a central, causal place. One telling sign of this is that the previously discussed derived rule, where an arbitrary formula ‘B’ is deduced from a contradiction ‘A & ¬A’, is commonly given the name ‘Explosion’. This rule is abandoned not just within the Relevance Logic tradition, but also in the general study of Paraconsistent Logics. As the name of the rule indicates, supposedly large-scale devastation would follow, if a contradiction were true. But that description embraces totally the wrong image for the logical sense of ‘following’, since contradictions are not elements in this world, and so they can have no causal or temporal consequences. Given ‘A & ¬A’, in its classical sense, one cannot obtain ‘B’ in the manner of some chemical experiment, or the like. Rather, ‘B’, for arbitrary ‘B’ would have to be already true, for ‘A & ¬A’ to be true. As before, it is not a matter of why, or how, or because of what ‘B’ is true, but simply, now, the certainty that ‘A & ¬A’ is not true. What the derived rule implies, and what is made explicit in the elimination rule for Contradiction, is that Absurdity can only arise per impossibile, so Absurdity is something that, while it may be envisioned, cannot in fact arise. The two Contradiction introduction rules fully harmonise with this elimination rule in the same sense: they each prevent a contradiction from arising in fact. For each introduction rule also shows that if Absurdity would follow, given a formula, then one can deduce the contradictory of that formula. So, by denying the respective premises from which Absurdity would follow, the two introduction rules ensure that Absurdity does not arise amongst the possibilities. The elimination Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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rule for Contradiction, therefore, does not state some condition under which something will in fact occur, but rather states that under no actual condition can its absurd consequence occur. So there is no bomb set to go off, in the related rule ‘Explosion’, and hence no need to remove its fuse. In the formal terms described before, ‘⊥’ cannot occur at the end of a deduction without there being undischarged assumptions; indeed, it is on this basis that the consistency of Classical Logic can be established. In Paraconsistent Logic these facts about Contradiction are not expressed, but that does not show they are not facts. For the crucial point made before about the Law of the Excluded Middle has to be made about the Law of Non-Contradiction, ‘Explosion’, and the like: they only fail in a nominal sense in Paraconsistent Logic, through the illicit use of the same sign as in the case of Classical Logic. Logic is not about signs, but about the concepts expressed by those signs. The negation sign in Paraconsistent Logic is used so that ‘A & ¬A’ is not a logical falsehood, and the move from ‘A & ¬A’ to ‘B’ is not a valid inference. Hence it does not represent the relation of Contradiction, and should be symbolised differently. In fact the ‘¬A’ in Paraconsistent Logic is merely subcontrary to ‘A’, rather than contradictory to it, and again its ‘¬’ does not symbolise the weaker relation then involved, since not all subcontraries of the same thing are equivalent. For total clarity (see Beziau [2005] again) the negation in Paraconsistent Logic should have a modal expression dual to that appropriate for Intuitionistic Logic. So the failure of neither of the above verbal forms goes against it being the case that a contradiction cannot be true, which also means, because of the previous points about conditionals, that a contradiction entails anything. One cannot escape the force of logical necessity.
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I show in this chapter that the ‘true contradictions’ favoured by certain paraconsistent logicians are better seen as contradictory thoughts. I start by reminding readers that a similar transformation of an extensional logic into an intensional logic is equally appropriate in the case of Intuitionism. The matter involves a close examination of the nature of negation. For, finding themselves short of a word to translate their various ‘negations’ into, many Intuitionist, and Paraconsistent mathematicians have perforce used ‘not’. Lax attention to natural language has encouraged this. But strict attention shows it is not possible. Intuitionistic ‘ ¬α’ merely entails ‘not α’ since that ‘¬’ is a contrary-forming operator, as is now, quite widely, well known. But not even that can be said for paraconsistent ‘¬‘, as we shall see, since paraconsistent ‘¬α’ does not oppose ‘α’; that ‘¬’ is merely a subcontrary-forming operator. In both cases the ‘negation’ involves a modal notion, and so should be symbolised using an intensional operator. Discussions of the philosophical import of Intuitionistic Logic have been mostly concerned with Brouwer’s school’s historic reasoning, and what can be said either for or against it. But historical studies are not confined to discussing authors just in terms of the concepts they were conscious of, since further researches may well point to external facts that are still highly relevant. This is illustrated in the many epistemic notions Brouwer used, unsymbolised, in his work, about undecided properties of decimal expansions. If the postulated difference between ‘not-not-α’ and ‘α’ is properly one between a modal expression such as ‘not demonstrated that not-α’ and ‘α’ then it should be symbolised differently. Gödel’s well known Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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translation of Intuitionistic Logic into S4, McKinsey and Tarski’s different one, and the semantics Kripke provided for Intuitionistic Logic in terms of stages of knowledge, are even more decisive external facts of the same kind, since they show explicitly how most, or all of Intuitionistic Logic can be seen as not an alternative propositional logic, but instead a modal, or intensional logic. Certainly no-one wants to assert ‘Lα v L¬Lα’, but that does not show in any way that one should not assert ‘α v ¬α’. Moreover, if the former is represented as ‘α v ¬α’ then, amongst other things, the ‘¬’ should not be read simply as ‘not’. We now turn to the detail of how a similar situation arises in the dual, paraconsistent case. I shall talk mainly about Graham Priest’s work: it is mainly Priest who has insisted that there are not only contradictory thoughts but also contradictory truths, i.e. dialetheias. It is the latter extension of other, more valid ideas within the paraconsistent tradition that is argued against here. Graham Priest agrees that Intuitionistic Logic is not a rival to classical logic, but still wants to maintain that Paraconsistent Logic is a rival. He says (Priest [1999], 110): In the case of Intuitionism, where truth and falsity are not exhaustive, I argued that intuitionist negation is not a contradictoryforming operator, and that we can define a genuine such operator by the condition: ¬α is true iff α fails to be true. It is natural to suppose that a similar objection can be made here. Dialetheic negation is merely a subcontrary-forming operator. The same clause still defines the genuine contrary-forming operator.
The objection that dialetheic negation is merely a subcontraryforming operator was considered by Priest before, and rejected on other grounds (see Slater 1995). But the elementary point is that if valuations are functions from formulas to the values 1, 0, -1, then some value’s being greater or equal to 0 is subcontrary, and not contradictory to its being less than or equal to 0. What is contradictory to the value being greater or equal to 0 is its being -1. Some subsequent debate has taken off from this defense of the subcontariety point, but here Priest tries to make out that dialetheic negation is still a contradictory-forming operator, first because it captures so much of the classical account, and second because any Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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argument against it ‘can rest solely on the fact that the truth of ¬α does not rule out that of α’. Indeed it does rest solely on this, so how does Priest try to show that, nevertheless, dialetheic negation is a contradictory-forming operator? He does not consider the proof of the subcontrary point, just given, that his α is true
and α is false
are equivalent to V(α) ≥ 0
and V(α) ≤ 0,
respectively, whereas they should be equivalent to, say V(α) ≥ 0
and V(α) < 0
(or their equivalents), if they are to be contradictories. Instead he tries to show that Boolean negation likewise involves an operator for which the truth of ¬α does not rule out that of α. But, even if this was true, it would merely show that Boolean negation was not a contradiction-forming operator, it would not show that, somehow, dialetheic negation was one. So the argument is a red herring. How is it that Priest does not show that the truth of a Boolean negative does not rule out the truth of the correlative Boolean positive? His argument runs like this (Priest [1999], 110): Suppose we define an operator, ¬, such that ¬α is true iff α is not true, and let us say, false otherwise...The behaviour of ¬ requires...careful Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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examination... In particular, why should one suppose that we can never have both α and ¬α? The natural argument is simply that, if α and ¬α are true, then α is both true and not true, and this cannot arise. But we cannot argue this way without seriously begging the question. If, as the dialetheist claims, some statements and their negations are both true, maybe α can be both true and not true...The Boolean properties of ‘Boolean negation’ may therefore be an illusion.
Boolean negation, i.e. contradiction, however, was not traditionally characterised as Priest supposes, but instead: contradictories cannot be true together, or false together. On that basis Priest’s argument for the possibility that contradictories might be both true collapses immediately: he has himself begged the question, by presuming that contradictories are defined otherwise. But even if they were defined otherwise, what sort of a case would Priest have got? He says at the same place: Indeed, if a dialetheic solution to the semantic paradoxes is correct, and α is ‘α is not true’ then α is both true and not true.
But he does not inspect his supposed identity at all closely. For, first, the fact is that one never can get the supposed identity α = ‘α is not true’,
if it is meant to be of the form (with ‘p’ a sentence) ‘p’ = ‘“p” is not true’,
since that would require a whole to be a proper part of itself. And if what is involved instead is a referring phrase ‘s’ to a sentence such that, for instance, s = ‘s is not true’,
then the T-scheme is required to generate any paradox, which straightforwardly leads to the denial of that scheme, by Reductio, as we have seen. Priest has considered several ways in which a ‘classical logician’ might define a contradiction forming operator, and is quite right that we could prove everything, given only that we have the T-schema Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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and some way of forming self-referential truth-bearers. But the latter is not possible in the propositional case, i.e. there are no selfreferential propositions comparable to self-referential sentences, since ‘‘p’=‘that p is false’’ is ruled out by mereology — something like ‘5=5+0’ is possible, but not anything like ‘‘5’=‘5+0’’, since the right hand expression is longer than the left hand one. And notice that this argument against self-referential propositions is not that they would have to be part of themselves. Propositions are abstract objects, and it is not at all clear that what it would be for them to contain themselves as parts. Instead the argument is about the impossibility, for any ‘p’, of ‘it is not true that p’ being itself ‘p’, i.e. it is about the mereology of certain concrete objects, sentences. Priest has pursued his contrary view of the matter, however, trying to show, on the same basis as above, that the traditional law ex contradictione quodlibet, (ECQ), does not hold for Boolean negation. But his argument is incorrect just because of this issue of what is true: he thinks a formula, i.e. a mentioned sentence, might be what is true, but instead it is the referent of a ‘that’-clause which includes that formula as a used sentence. He goes on (Priest [1999], 110): We can now deal with another law of negation: ex contradictione quodlibet (ECQ): α.¬α |— β… Given the present discussion, it can clearly be seen to fail. For we can take an α which is both true and false, and a β that is not true. This instance of the inference is not truth preserving, and hence the inference is not valid (truth preservation being at least a necessary condition for validity). For good measure, the equally contentious inference of Antecedent Falsity (AF) ¬α |— α → β, must also be invalid, for exactly the same reason (modus ponens holding); as again, and more contentiously, must be the disjunctive syllogism (DS): α, ¬α v β |— β.
But clearly, by using ‘α’, and ‘¬α’, so that such may be true together, none of this goes against the classical rules ECQ, AF and DS, since they deal with contradictories, i.e. things which cannot be true together. Disjunctive Syllogism, for instance, isn’t just the formula ‘α, ¬ α v β |— β’,
otherwise there would be no chance that it was necessary. Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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Disjunctive Syllogism is this formula interpreted a certain way, in particular so that the truth of ¬α does rule out that of α. As a result, while, for instance, the formula ‘α & ¬α |— β’,
fails on a dialetheic interpretation, the explanation is that the ‘¬’ then is just a subcontrary-forming operator. Priest wants to say it fails on a Boolean interpretation also, but that would again only show that his definition of Boolean ‘¬’ did not make it a full contradiction-forming operator. If one says that the conjunction of a formula and its negation entails everything, then what one then says, i.e. the proposition one then expresses, is necessarily true, if one is using ‘negation’ to mean Boolean negation. It is only one aspect of the matter, however, that Priest’s logic (LP) concerns itself with subcontraries. For in (Priest [1999], 102) he said: How could a contradiction be true? After all, orthodox logic assures us that for every statement, α, only one of of α and ¬α is true. The simple answer is that orthodox logic, however well entrenched, is just a theory of how the logical particles, like negation work; and there is no a priori guarantee that it is correct.
Orthodox logic, however, is inevitably correct about what the traditional logical connectives, such as Boolean negation, consists in, since that is just a matter of definition. And what kind of ‘theory’ is Boolean negation? Boolean negation is merely one relation amongst others that propositions may fall into. As a result, if one chooses propositions which fall into this relation, and use the standard ‘and’ and ‘or’, all the above classical laws will hold. Moreover the relation of Boolean negation was what was traditionally called ‘contradiction’, so if ‘¬’ symbolises that relation there is no way that a contradiction can be true. Priest is more likely using the term ‘contradiction’ not for this relation, however, but just for the sign ‘¬’, since certainly ‘α & ¬α’ might be given a non-traditional interpretation on which what it expresses was not true. But that, of course, would in no way conflict with the necessary fact expressed before.
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If Boolean negation comes into some theory about ‘the logical particles’, then that theory must be about how one does or should use words and symbols like ‘not’, and ‘¬’: for instance, that we do or should use ‘not’ or ‘¬’ to symbolise what was classically called ‘contradiction’ rather than some other relation. But that is quite a different matter; it is a conventional, or legislative matter about how we do or should use words, not directly a logical issue. Some people cannot say ‘no’, it is well known, or if they do say this they do not mean it, as their children might say. Such people are not really denying anything when they say ‘no’, and might be said to have a theory about how to use the word — so that it does not oppose, or contradict. But it is just a matter of choice to use the negative particle in this, or some other way; for instance, as normally, in the traditional sense involving denial. So this is quite a separate question from the question whether contradictories may be both true: that question is about a certain propositional relation, and its connection with truth. And that question, as was pre-figured at the start, is also to be crucially distinguished from the question of whether contradictories may be thought to be both true. Priest says (Priest [1999], 114): In explaining their views, people often assert contradictions, unwittingly. In this way, they discover — or someone else points out to them — that their views are inconsistent. In virtue of this, they may wish to revise their views. But in asserting ¬α in this context, they are not expressing a refusal to accept α, i.e. denying it. It is precisely the fact that they accept both α and ¬α that tends to promote belief revision. It may even be rational sometimes — as a number of classical logicians have suggested — to hang on to both beliefs, and continue to assert them: consider, for example, the Paradox of the Preface. If to assert ¬α is to deny α mooting this possibility would not even make sense.
One point arising from this is that Priest has had difficulty saying what denial is, if it is not asserting the reverse — he is like the above weak father with ‘no’. A further point which needs to be emphasised is that the modalities in the above are not the usual ones, and quite deliberately so. Normally, we expect people to be consistent, and the criticism that someone is being illogical by contradicting themselves involves a considerable censure. Priest is trying to make us more Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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tolerant, it might be said: it’s not that people ought to revise their views, if they are inconsistent, it’s that people ‘may wish’ to, he says. And Priest’s view is that such inconsistencies ‘tend to’ promote belief revision, whereas a traditional logician does not see the matter in these terms, as if it was some natural, maybe causal process. The traditional logician would insist that the muddled get out of their muddle and become consistent, taking that action to be a choice the muddled simply have not made. Certainly it is quite natural for people to be muddled, and to stay muddled — but only until they meet a traditional logician. For, unlike Priest, who does not want to be intolerant, the traditional logician will not be so easy going. He will impose upon his pupils the requirement of determinateness of sense, since only in that way can anyone have clear thoughts. But such facts about the social value of being consistent, and the forces which might insist on this, or contrariwise might be more slack, like the question over the use of ‘not’, are not themselves matters of a priori logic. They are matters to do with getting the rigour of a priori logic at least to some extent into people’s lives. How is it then that they enter the discussion of paraconsistent logic? They enter it, surely, because, as with the give-away, but unacknowledged mention of what is demonstrated with Brouwer, they show what Priest has in mind. He has in mind the undoubted possibility of mental confusions. In place of ‘α & ¬α’,
in his Paraconsistent Logic, what is really meant is something like ‘Tα & T¬α’,
where ‘T’ is not ‘it is true that’ but ‘it is thought that’. Thus formulated Paraconsistent Logic is about the possible contents of people’s minds, and in particular the above passage is reminding us of the possible contents of people’s minds before they meet a strict teacher. Priest also mentions ‘belief revision’ in the above passage, but only one way of looking at this is ‘classical’, in the sense that the belief set has to be consistent, and deductively closed. This is the line taken in the standard AGM account (Gärdenfors [1988]). Priest has Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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essayed, instead, a non-classical, paraconsistent theory of belief (Priest [2001]); but it stays within the ‘belief revision’ tradition, from the present point of view, because it focuses on the propositions in a belief set, without symbolising them as beliefs. It thus represents the matter in terms of an alternative extensional logic: the logic that supposedly operates on the propositions in the believer’s mind. What is wanted, by contrast, is an intensional logic, which makes explicit the fact that the propositions concerned are in someone’s mind through the introduction of an intensional operator that expresses this. If we produce the logic of that operator, we produce a logic which is different from classical propositional logic, certainly, but merely because it supplements it, instead of being a rival to it on its own ground (c.f. Haack [1996], Ch. 1). One such intensional logic, in fact, is quite well studied already, in the standard, Bayesian semantics for belief. It was Ramsey, of course, who linked beliefs to subjective probabilities; later writers have constructed more complete Bayesian theories, using conditionalisation as the guide to belief revision. The formal consequences of treating belief in terms of probability without that dynamic aspect are detailed in (Hawthorne and Bovens [1999]). It is worth reminding ourselves what laws hold, in that case. Within this kind of logic we do not get ex contradictione quodlibet in the form ‘Bα & B¬α |— Bβ’,
or ‘Disjunctive Syllogism’ in the form ‘Bα & B(¬α v β) |— Bβ’,
when ‘Bα’ means, at least, that pr(α) ≥ 1/2. And while we do still get ‘Antecedent Falsity’, in the form ‘B¬α |— B(α ⊃ β)’,
we do not in the form ‘B¬α |— (Bα ⊃ Bβ)’,
and ‘Modus Ponens’ fails in this way also, i.e. we do not have Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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which means we do not necessarily have deductive closure of the belief set. The Paradox of the Preface, which Priest mentioned above, is then resolved immediately, since ‘Bα & Bβ & Bγ...B(¬α v ¬β v ¬γ...)’,
is quite consistent. And going with that, ‘Adjunction’ does not hold in the relevant sense, i.e. we do not have ‘Bα & Bβ & Bγ... |— B(α & β & γ...)’,
which shows dramatically how it can be that people ‘do not put 2 and 2 together’. More fundamentally, since it is evidently quite possible that Bα & B¬α,
it is this which most closely represents a ‘true contradiction’, although we now see it is just a matter of mental confusion — whether allowable or not. The Bayesian treatment, however, does not parallel Priest’s LP very closely. Certainly if ‘Bα’ is true if and only if pr(α) ≥ n, for some threshold n, that allows both Bα and B¬α to be true for certain values of ‘n’. But LP only had three values, whereas the probabilistic one is continuously valued. This asymmetry, however, can be removed if we treat Belief as also three valued, say, Belief (V(α) = 1), Wonder (V(α) = 0), Disbelief (V(α) = -1).
We can construct the matter simply using the operator ‘it is thought that’ as before, with the three categories then being ‘Tα & ¬T¬α’, ‘Tα & T¬α’, ‘T¬α & ¬Tα’,
where Tα if and only if V(α) ≥ 0. In Priest’s logic the same semantic Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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relation holds with ‘T’ read in terms of truth, and he accepts ‘A is true if and only if A’ (Priest [1979a], 238). It follows that there is an interpretation of LP in terms of mental confusions. Q.E.D. But while this reformulation of LP may be acceptable to many paraconsistent logicians, and even provable, Priest is definitely committed to denying it. Priest has expressly developed his Dialetheism as a doctrine about contradictions being in things rather than in thoughts, i.e. in the extensional world, and not an intensional one. This is not just because of a resistance to moving away from extensional logic, with the consequent resistance to symbolising human confusion not extentsionally, but with an intensional operator. The logical paradoxes centrally have troubled Priest, as we saw, in part, with the previous discussion of propositional self-reference. The fact that a paraconsistent view of the Liar involves adherence to the disquotation-scheme, however, shows that the contradictions Priest thinks inevitable are in fact avoidable. And the first thing needed to show that any contradictions are in a person’s mind and not reality is just that: that there is an alternative reading of the facts which does not find them paradoxical. We can see this crucial matter again in Priest’s treatment of Heterologicality. On this matter, Priest significantly does not mention an assumption that Copi, for instance, makes explicit. In place of Copi’s formalisation of ‘x is not self-applicable’, which, in Priest’s terms, is equivalent to ∃Y(xRY & (Z)(xRZ ⊃ Z=Y) & ¬Yx),
and contains a uniqueness clause (Copi [1973], 301), Priest merely writes (Priest [1995], 163-40): ∃Y(xRY & ¬Yx),
for the relevant predicate. Abbreviating this to ‘Fx’, he goes on to argue that, given ‘F’RF,
from ¬F‘F’ Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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∃Y(‘F’RY & ¬Y‘F’),
i.e. F‘F’. He also argues that from F‘F’
one can get ¬F‘F’,
just because F‘F’
entails ∃Y(‘F’RY & ¬Y‘F’).
But even given ‘F’RF, ¬F‘F’
does not follow from ∃Y(‘F’RY & ¬Y‘F’),
i.e. the fact that some Y which ‘F’Rs does not apply to ‘F’,
since one also needs a proof of the uniqueness clause before one can say that the only Y which ‘F’Rs is F,
and therefore ¬Y‘F’
for that Y. No contradiction is obtainable without a uniqueness proof, which would exclude non-univocality in ‘F’. Priest is therefore Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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confused if he thinks that the contradiction follows as a matter of demonstrated fact. The necessary truth of a contradiction is not established by Priest’s argument, simply because there is a hidden premise, which it is open to anyone to deny, to free their minds of the contradiction. Priest chooses not to deny this premise, putting the contradiction in his mind simply by an act of his own will. It might be said, against the possibility of ambiguity in such cases that, of course, merely to moot the possibility that a sentence expresses more than one proposition is not to solve the paradox: one needs to give reasons to suppose the claim to be true. Indeed this might be thought to illustrate a quite general strategy for attempting to solve the self-referential paradoxes: take some premise, A, in the argument involved, claim that this is true only on condition B, and then deny B. And certainly, if that was the strategy then it would be worth very little unless there was independent reason for supposing that B fails. The boot, however, is on the other foot, if anyone wants to maintain there is provably a paradox in such cases. If A leads to paradox, and A is true only if B, then to prove there is actually a paradox one must show that B definitely is true. The onus of proof is on the opposition, therefore, to demonstrate conclusively that there is no ambiguity in such a case as that above. In addition, it is an undoubted fact that the relevant predicate ‘is not self-applicable’ is ambiguous. Certainly the predicate ‘is not selfapplicable’ is not like ‘is a bank’ in allowing semantic disambiguation into ‘is a money bank’ and ‘is a river bank’, but it still is dependent on the context for its sense, to pick up the appropriate referent for ‘self’, so its meaning is not constant. Here is an independent reason for the conclusion that there is ambiguity in connection with Heterologicality, and it shows that a defence of the uniqueness assumption explicit in Copi, and hidden or ignored in Priest, just is not obtainable. Notice that, above, I gave two formulations of ‘x is not self-applicable’, one by Copi, involving an explicit uniqueness clause, and one by Priest, lacking such a clause. I pointed out that, as a consequence, there was an undischarged assumption, and specifically an unproved supposition of the above kind, in Priest’s (and also Copi’s) derivation of a contradiction. My claim was not that Priest’s formal expression might itself be ambiguous; the claim was merely that Priest’s formal expression lacked an explicit uniqueness clause, and so proof of the Barry Hartley Slater – The De-Mathematisation of Logic ©2007 Polimetrica International Scientific Publisher Monza/Italy
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uniqueness expressed by that clause was needed before a contradiction could be conclusively obtained. It is common to exclude indexicals such as ‘I’ and ‘now’, from formal work, but here the same caution has not been used with ‘self’. In ‘x is self-applicable’ the ‘self’ is just an anaphoric back reference to x. No wider aspect of the context is needed to determine the value of ‘self’, but still, the value of ‘x’ is needed to make determinate the property expressed by ‘is xapplicable’. As it stands this predicate expresses a functional property, i.e. a property that varies with the value of ‘x’. There is also a further thought that hinders seeing this, as we have seen, amongst the learned: the common reading of lambda abstraction expressions as invariably referring to properties. Isn’t y is not self-applicable,
equivalent to λx(x is not applicable to x)y,
and so doesn’t the lambda term, in which the ‘x’ is bound, denote the required constant property of Heterologicality? The given lambda abstraction expression merely isolates the syntactic form of a sentence, however, and clearly whether a single concept is associated with the predicate in that sentence is not deducible from that — as ‘is not applicable to x’, again illustrates. As we have seen, the given lambda term does not denote a property, but a state of affairs.
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in
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The electronic edition of this book is not sold and is made available in free access. Every contribution is published according to the terms of “Polimetrica License B”. “Polimetrica License B” gives anyone the possibility to distribute the contents of the work, provided that the authors of the work and the publisher are always recognised and mentioned. It does not allow use of the contents of the work for commercial purposes or for profit. Polimetrica Publisher has the exclusive right to publish and sell the contents of the work in paper and electronic format and by any other means of publication. Additional rights on the contents of the work are the author’s property.
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