Absolute Necessities

Absolute Necessities Bob Hale Noûs, Vol. 30, Supplement: Philosophical Perspectives, 10, Metaphysics, 1996. (1996), pp. 93-117. Stable URL: http://links.jstor.org/sici?sici=0029-4624%281996%2930%3C93%3AAN%3E2.0.CO%3B2-3 Noûs is currently published by Blackwell Publishing.

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Philosophical Perspectives, 10, Metaphysics, 1996

ABSOLUTE NECESSITIES

Bob Hale

University of Glasgow

1 Relative and epistemic modality

'

Often, when we assert it to be necessary that p, or possible that q, we incorporate an adjective qualifying the kind of necessity or possibility we mean. Sometimes our qualification signals that we are asserting what it seems reasonable to regard as a species of relative necessity. By saying that a notion of necessity-+-necessity-is relative I mean that there is some body of statements @ such that to claim that it is +-necessary that p is to claim no more and no less than that it is a logical consequence of @ that p. This leaves it open whether the statements in @areto be (all) true. We may accordingly distinguish stronger and weaker notions of relative necessity, depending upon whether or not such an additional condition is imposed. If so, +-necessity will conform to a version of the usual Law of Necessity, i.e. 'it is +-necessary that p' will always entail 'p'; otherwise not. Evidently for both weak and strong relative notions, the members of @ themselves will automatically qualify as +-necessary. The relativity of +-necessity consists in the fact that, whilst no +-necessary statement can be false, provided that all the members of are true, it is not excluded that there are other senses of 'possible' in which the members of @ may be false; if so, +-necessity is a merely relative notion. Thus in one quite common usage, to claim that it is physically necessary that p is to claim that it follows from the laws of physics that p, while to assert that it is physically possible that p is to assert that it is logically consistent with the laws of physics that p. If we mean to exclude the possibility that it is physically necessary but false that p, then this notion is, though relative, a strong one; by the laws of physics we mean the laws as they actually are, rather than the laws as we currently take them to be. Since we shall almost certainly allow that there is a sense in which those laws could have been otherwise, the necessity we claim is merely relativein particular, what is physically impossible may nevertheless be logically possible. Fairly clearly, these remarks apply, mutatis mutandis, to most (not necessarily all) other notions of necessity the characterisation of which

+

94 / Bob Hale

involves reference to some more or less well-defined discipline or body of theory. In other cases, as when we speak of its being logically necessary, or again, of its being conceptually necessary that p, it seems equally clear that we do not think of the kind of necessity asserted as merely relative. We may distinguish between narrowly or strictly logical necessity and broadly logical necessity; I take the former to be a special case of the latter, and make no distinction between that and conceptual necessity. Hereafter, when I speak of logical necessity without further qualification, it is broadly logical necessity that I intend.' There may, of course, be other kinds of necessity which are not happily conceived as relative. In particular, whilst some philosophers still regard the notion with a degree of suspicion or puzzlement, many appear confident that it is sometimes correct to hold that it is metaphysically necessary that p, thinking of this as a further kind of nonrelative necessity. One task in the philosophy of modality is to provide an account of the relations, including relations of relative strength, among the various kinds or notions of necessity and possibility. One kind of necessity, Ell, may be said to'be stronger than another, 0 2 , if 'Olp' always entails '02p9but not conversely. Assuming the usual relations between necessity and possibility, this relationship will obtain if and only if 0, is weaker than O,, i.e. ' 02p' always entails ' 0 ,p' but not conversely. I shall also say that 0, is at least as strong as 0, if the first half of this condition is met, i.e. ' 0 , ~ always ' entails 'U,p'. Where +-necessity is any relative notion-strong or weak-it can easily be seen that logical necessity is at least as strong as +-necessity. If it is logically necessary that p, it is logically necessary that q -;,p, for any q, and so logically necessary that Q, -+ p. Hence it is a logical consequence of @ that p, and so +-necessary that p. It is both natural and very plausible to think that where +-necessity is any merely relative kind of necessity, logical necessity is (strictly) stronger than +-necessity, and logical possibility (strictly) weaker than +-possibility. As noted, it seems clear, for example, that what is physically impossible may nevertheless be logically possible, so that logical necessity is stronger than this merely relative notion. Plausible as it may be, however, it is by no means straightforward to see, much less to prove, that it is so. The hypothesis that +-necessity is merely relative ensures only that there is some sense of 'possible' -0 *-in which the members of @ may be false, i.e. that for some A in @, 0 * 1A,but not 0 "IA. It is not, however, given by the assumption that +-necessity is merely relative that the falsehood of members of Q, is logically possible. We We could close the gap, if we have that ' 0 *p' does not always entail ' 0 9'. could show that logical possibility is at least as weak as any other kind of possibility (so that in particular, ' 0 *p' entails 'it is logically possible that p'). We shall return soon enough to the question whether such a result-or equivalently, the result that logical necessity is at least as strong as any

Absolute Necessities / 95

other (not necessarily relative) kind of necessity- can be secured. First, I need to make some brief observations about epistemically modal notions. Epistemic necessity and possibility could be understood as relative notions, i.e. letting K denote the sum total of what we know, we could define 'it is epistemically necessary that p', or 'Oep', by 'O(K + p)' and ' 0ep' by ' 0(K & p)'. Clearly logical necessity entails epistemic necessity so defined, and epistemic possibility entails logical. Some care is needed here, however, since-as has often been observed-we may, using 'possible' in an epistemic sense, express a truth by saying that it is both possible that the sequence of twin primes is infinite and possible that it terminates, even though-on some views at least-whichever of the twin prime conjecture and its negation is false is necessarily so. This shews, I think, that there is an epistemic notion of possibility which is not relative in my sense: when we say "For all we know, there is a last twin prime pair", we are not making the rather exposed claim that it does not follow from anything we know that there is no last pair, but at most that this is not known to follow from anything we know, and perhaps only that we simply do not know that there is no last pair. Epistemic possibility in this sense plainly does not entail logical possibility. I propose, as I believe I may do without foreclosing on any important issues, to set aside such non-relative2 epistemic notions for the purposes of this discussion. 2 Absolute necessity

The considerations rehearsed thus far prompt some questions about logical necessity. Is it the strongest kind of necessity? (i.e. stronger than every other kind of necessity?) Is logical necessity at least as strong as every other kind of necessity? (or, as I shall say, is it a kind of absolute necessity?) If all other notions of necessity besides logical necessity were relative notions, then , as we have seen, an affirmative answer to the second question would be hard to resist. And, provided epistemic notions are set aside, an affirmative answer to the first would seem scarcely less compelling. But, as we have remarked, there is-or at least, it is widely supposed that there is-another notion of necessity-metaphysical necessity- which is nonepistemic but which certainly does not appear to be a relative notion. Friends of this notion are committed, at the very least, to returning a negative answer to the first of my questions. For they would want to hold that when it is metaphysically necessary that p, there is no good sense of 'possible' (except, perhaps, an epistemic one) in which it is possible that not-p. Metaphysical necessities hold true at all possible worlds without qualification or exception (in contrast, perhaps, with e.g. physical necessities, which are true of all physically possible worlds, while there are allowed to be at least logically possible worlds at which they fail, through suspension of some of the actual laws of physics).

96 / Bob Hale

The notion that there are such metaphysical necessities is what gives my questions their philosophical interest and urgency. An affirmative answer to the first would seem, as noted, to be squarely at odds with widely received views about metaphysical necessity. I want to approach the issue, however, by considering an argument which purports to establish an affirmative answer to the second question. It may seem that an argument for the weaker thesis-that logical necessity is absolute-must be of far less interest than one for the stronger thesis that logical necessity is the strongest kind of necessity, and could carry no very significant implications for claims about metaphysical necessity. I shall try to show, on the contrary, that a successful argument for the absoluteness of logical necessity would leave the friends of metaphysical necessity facing a serious, if less immediately obvious, problem. But first, let me set forth the argument to which I have alluded. The argument is closely related to one put forward by Ian McFetridge3, but I wish to avoid involvement in complications arising from his presentation of it, so I shall give my own formulation.4 The conception of logical necessity which informs the argument is one which sees the concept as receiving its fundamental exemplification in the connection between the premiss and conclusion of a deductively valid inference. If anything deserves to be regarded as logically necessary, according to this view, it is the conditional corresponding to such an inference. Indeed, on this view, the validity of an inference 'p so q'-equivalently, the truth of the claim that p entails q-simply consists in its being logically necessary that if p then q. Accordingly, what the argument seeks to show, first, is that if such conditional is logically necessary, its necessity is absolute, i.e. there is no sense of 'possible' in which it is possible that its antecedent should be true but its consequent false. Here, and subsequently, various different modal notions will be in play, which it is obviously crucial to distinguish. I shall use the dotted box 'El' exclusively to represent the logical necessity operator; ' 0 ' is an arbitrary possibility operator, and ' -+ ' represents the material conditional. We make five assumptions about q and 0 : Al. A2. A3. A4. A5.

If O(A -;,B) then O(A & C -+ B)

O(A -+ A)

If O(A -+ B) and D(A -;, C) then O(A -+ B & C)

If 0 A and D(A -+ B) then OB

iO(A &iA)

1 1) O(A-+ B) 2 2) 0 (A & i B ) 1 3) D(A & i B -+ B) 4) O ( i B -+ i B ) 5 ) O(A & i B -+ 1 B )

assumption

assumption

1, by A1

A2

4, by A1

Absolute Necessities / 97

1 6)m(A&1B+B&iB) 1,2 7) O(B & 1 B ) 8) 1 0(B & TB) 1 9)lO(A&lB)

3,5,byA3

2,6 by A4

A5

2,7,8 reductio

By McFetridge's Thesis I shall mean the thesis which this argument, if sound, establishes-that if the conditional corresponding to a valid inference is logically necessary, then there is no sense in which it is possible that its antecedent be true but its consequent false. Since the argument is valid, its conclusion can be resisted only by rejecting at least one of the five assumptions deployed. A1-3 correspond to familiar principles of entailment. I propose immediately to discount the options of rejecting A2 or A3, since it seems to me that no reasonable notion of entailment will be nonreflexive, or fail to sanction conjoining conclusions which are both separately entailed by the same premiss. A1 corresponds to the principle that an entailment cannot be disrupted by strengthening its premiss, and to the , equivalent principle that a valid argument cannot be rendered invalid by adding an extra premiss. These principles, though both classically and intuitionistically unimpeachable, are, of course, rejected by advocates of relevant logic(s). To that extent, at least, A1 is more controversial. But I would regard it as a result of some significance, if it should prove that the difficulties shortly to be discussed are avoidable only by going relevant, and anything but clear that that would be the right reaction to the situation.5 Since, as I have already indicated, I believe there to be no reasonable sense of 'possible' in which it is possible for a contradiction to be true, I shall likewise discount the option of rejecting A5. There remains the final option of rejecting A4, which amounts to the assumption that any reasonable notion of possibility will sustain transmission across entailment. This holds for any relative notion of possibility, and for logical possibility itself. We shall need to consider in due course whether there may, even so, be good grounds to reject it. Although McFetridge's Thesis is strictly concerned only with putatively logically necessary conditionals, it is very hard to see what good grounds there could be for refusing to accept its obvious generalisation to cover statements of any form whatever (i.e. without resisting the thesis itself). In any case, a quite straightforward argument for the generalised McFetridge Thesis can be given, on the assumption that the thesis as stated is proved. We first prove two lemmas. Lemma 1Eip iff m((p -+ p) -+ p) Proof: Left-right: by the principle that any conditional with a logically necessary consequent is itself logically necessary. Right-left: by A2 together with the principle that 13 distributes over the conditional.

98 / Bob Hale

Lemma 2 0 7 iff O((p

-+

p) & 7 )

Proof: Left-right: by A4 together with the principle that addition of a logically necessary conjunct to the consequent of a logically necessary conditional preserves logical necessity, which gives E i ( 7 -;, ((p -+ p) & 7)). Right-left: by the principle that 0 distributes over &. Now if Elp , then by Lemma 1,Ei((p -+ p) -+ p), whence by McFetridge's thesis, 1 0((p 4 p) & 7 ) for any sense of 0 , so that 1 O l p , by Lemma 2. Thus we have: (Generalised McF): If Elp then there is no sense in which 0 7 3 A dilemma for metaphysicians, and a possible response

The arguments just presented establish (at best) that logical necessity is absolute-that is, that there is no stronger notion. They do not, and could not, establish that logical necessity is stronger than every other notion. They thus leave open the possibility that there are other absolute notions of necessity, one such being metaphysical necessity. For this reason it might be supposed that McFetridge's Thesis presents no especial threat to the friends of metaphysical necessity, such as clearly would be presented by a demonstration of the stronger thesis that logical necessity is the strongest kind of necessity. On reflection, however, this assessment appears unduly sanguine. For the argument for McFetridge's Thesis, if sound, does establish that if it is logically necessary that p, then it is in no sense possible that not-p, and so, in particular, that it is not metaphysically possible that not-p, i.e. that it is metaphysically necessary that p. But then either the converse entailment holds quite generally, or it does not. If not, then it can be metaphysically necessary that p but logically possible that not-p, so that metaphysically necessity is not, after all, absolute. If, on the other hand, whatever is metaphysically necessary is also logically necessary, then even if we have here two notionally distinct kinds of necessity, both of them absolute, they coincide in extension. Neither alternative is-or so it seems-congenial to the friends of metaphysical necessity. To accept that metaphysical necessity is not absolute is to acknowledge that while it is, say, metaphysically necessary that heat is mean kinetic energy of molecules, there are possible worlds-logically possible worlds-in which this is not so. But what the metaphysicians wanted to maintain is that, given the heat is mean kinetic energy of molecules, there are no possible worlds in which heat is not so constituted.6 But the alternative, of accepting that metaphysical and logical necessity are extensionally coincident, is scarcely less unattractive. How could it be held that e.g. the identity of heat with mean kinetic energy of molecules is logically necessary? Furthermore, such supposedly metaphysi-

Absolute Necessities / 99

cally necessary truths are typically held to be knowable only a posteriori, while logical necessities are (or are anyway standardly taken to be) knowable a priori -so there cannot, on pain of contradiction, be extensional coincidence. The argument thus appears to confront the metaphysician with a potentially lethal dilemma. How should she respond? Before we try to answer that question, we should get clear about the role played by McFetridge's Thesis in generating the dilemma. It may seem that the dilemma is not conditional upon acceptance of that thesis at all, so that the metaphysician has no interest in rejecting it, or finding fault with the argument I gave for it. Surely, it may be said, the metaphysician faces the question on which the dilemma turns-whether metaphysical necessity entails logical-irrespective of the answer given to the converse question. The first horn option-accepting that metaphysical necessity is not absolute-is no less unpalatable if the absoluteness of logical necessity is denied than if it is accepted. And while the second horn as formulated assumes the absoluteness of logical necessity, it is clear that the objections to taking this option lose none of their force if it is supposed, instead, that metaphysical necessities are properly included within logical necessities. But this overlooks an important point, and with it, the role McFetridge's Thesis really plays in engineering the dilemma. In the absence of an argument for the Thesis, one might hold that there are different kinds of necessity-metaphysical necessity among them-which are incommensurable in strength with logical necessity: that, for example, some propositions are logically necessary but not metaphysically so, while others are metaphysically but not logically necessary-so that there is no saying that one kind is stronger than the other, or that they are equal in strength. If this were the case, the notion of absoluteness, as I have defined it, would have no application-non-absoluteness would be no more a shortcoming in metaphysical necessity than it would be in logical necessity. It is, therefore, pertinent to consider whether the metaphysician can refuse the dilemma altogether, by way of rejecting the argument for McFetridge's Thesis, or perhaps the supplementary argument for its generalisation. If we continue to discount-as I propose to do-the extreme course of rejecting A l , then it seems to me that, although several further assumptions are in play (including the additional principles involved in the proofs of Lemmas 1 and 2), there is really very little room for manoeuvre. The only assumption we might seriously think of rejecting is the closure principle A4. Not that this is, prima facie, an especially promising candidate for rejection-on the contrary, it clearly holds for many familiar kinds of possibility: logical possibility itself is easily seen to be closed under logical consequence, as is any relative notion of possibility. Furthermore, whilst it would be an exaggeration to say that, were we denied the use of the relevant closure principle, reasoning about a given kind of possibility

100 / Bob Hale

would come to a standstill, it is certainly very natural and plausible to suppose that we may employ the principle in exploring the consequences of the supposition that such-and-such is, say, a metaphysical possibility-in working out, in particular, what other things should then be metaphysically possible. It can hardly, then, be simply rejected out of hand. On reflection, however, it appears that counter-examples to closure are very easily located. Let us say, for example, that it is austerely logically possible that p if it is not a first-order logical truth that l p . Let p be the proposition that there are male vixens, so that it is austerely logically possible that p. It is, however, a broadly logical consequence of p that there are vixens which both are and are not female. But this last is plainly not an austerely logical possibility. So closure fails. Evidently counter-examples of this genre are legion: the recipe for generating them is simply to select some proper subclass Qj of logical necessities, define the +-possibilities as the propositions whose negations are not in Qj,and pick a p such that p is possible but entails some q whose negation is in Q.7 The very natural-and I believe correct-reaction to such counterexamples is that they are, if technically correct, nonetheless quite spurious. Broadly logical necessities are propositions whose truth derives entirely from the concepts involved in them (together, of course, with relevant structure). By taking a restricted range of concepts (say those of negation, conjunction and first-order existential quantification), we can circumscribe a proper subclass of logical necessities, each of which owes its truth entirely to these particular concepts. It is then hardly surprising that if we take the correlated species of logical possibility (i.e. the species whose nondegenerate instances lie in the complement of the restricted subclass of necessities), we are able to find possibilities of that kind, some of whose logical consequences are not merely logically impossible, but impossible in this restricted sense. But those 'possibilities'-such as the austerely logical possibility that there are male vixens-are possibilitiesin name only, not real or genuine possibilities at all. There is, to be sure, no overt-or first-order extractable-contradiction involved in the supposition (so expressed) that there are male vixens. But while absence of first-order-or more generally, purely proof-theoretic-inconsistency is certainly a necessary condition for real possibility, it is clearly insufficient. It affords no guarantee that there could be a situation in which the supposition would be realised. We should accordingly refuse to countenance any attempt to discredit the closure principle (and therewith the argument for McFetridge's Thesis) by appeal to such 'possibilities'. The dilemma remains, so far at least, in force. But now it seems that there is opened up a new way to counter it, by blunting its first horn. If it is legitimate to brush aside apparent counterexamples to closure based on austere logical possibility and its kin on the ground that the possibilities in question are merely formal, not real, why should the metaphysician not make a parallel response to the charge that if

+-

Absolute Necessities / 101

she accepts that whatever is logically necessary is metaphysically so, but further agrees that the converse does not hold, she is constrained to accept that metaphysical necessity is after all not absolute? The moral of our discussion of austere logical possibility and its kin is, in effect, that we should recognise a kind of necessity as absolute iff there is no real possibility that a necessity of that kind should be false. But then why should it not be maintained that, just as certain austerely logical possibilities are not genuine possibilities, and so should not be taken as showing that the associated broadly logical necessities are not absolute, so not every broadly logical possibility is a real possibility, fully apt to destroy the claim to absoluteness of a corresponding metaphysical necessity? The absence of conceptual grounds (conceptual inconsistency) for denying that p can be true does not automatically ensure that p expresses a real possibility. There is no overt or implicit conceptual incoherence in the supposition that heat is other that mean kinetic energy of molecules, say, or that John Major is a fairly cleverly disguised automaton. But given that heat is motion of molecules, or that Major is a member of the human species, it is-or so it may be claimed-in no real sense possible that matters should have been otherwise in these respects. 4 The need for a unifying account There is, thus, a quite simple way in which peaceful co-existence between logical and metaphysical necessity may be secured, without denying the absoluteness of either. It enjoins replacement of the overall bipolar picture of modality which goes with the conception of logical necessity as the strongest kind (i.e. with logical necessity occupying one pole and logical possibility, as the weakest kind of possibility, occupying the other, and all other kinds of necessity and possibility lying in between). In the rival replacement picture that emerges-supposing there to be no other significantly different kinds of absolute necessity-absolute necessity is the union of broadly logical and metaphysical necessity, while real-or absolutepossibility is the intersection of broadly logical and metaphysical possibility. It is absolutely necessary that p iff it is either logically or metaphysically necessary that p, and really possible that p iff it is both logically and metaphysically possible that p. Metaphysical necessity can rank alongside logical necessity as absolute, because while what is metaphysically necessary can be denied without conceptual inconsistency, there is-on this picture-no sense in which the negation of a metaphysical necessity is really possible. This may well be a consummation devoutly to be wish'd- but if it is to be enjoyed, it must be earned. Just because, on the new picture, a truth may qualify as absolutely necessary in either of two ostensibly quite different ways, we have further work to do, if we are to attain to a

102 / Bob Hale

satisfactory overall account of modality. For even if we should be content to embrace the new picture, it does not seem right to suppose that the concept of absolute necessity is a merely disjunctive one-the adventitious union of two fundamentally different kinds of necessity. It seems rather that it should be possible to provide some kind of unifying account of modality, in the light of which it is intelligible how a truth may qualify as absolutely necessary in either of these different ways. Such and account would, it seems, have to take seriously the idea that necessities may derive from different sources. What, in general terms, is needed is an overall conception which allows us to see some common pattern or structure-characteristic of absolute necessity in general-exemplified in different specific ways by necessities deriving from diverse sources. Can such an account be provided? 5 The Essentialist Theory of Modality

An account which certainly does take seriously the idea that there are different sources of necessity has been developed in recent work by Kit Fine.8 In the absence of any well-developed alternative, I want to explore the prospects for a unifying treatment of modality along the lines Fine suggests. Fine's primary concern is to achieve a better understanding and regimentation of the notions of essence and essential property than can be accomplished-in his view-by adhering to the more usual approach which tries to account for them in terms of de re modality. According to the modal account, an object x has a property P essentially if it is necessary that x has P (i.e. OPx), or-in a conditional version-if it is necessary that, provided x exists, x has P.9The inadequacy of this approach shews up, Fine argues, in its inability to do justice to certain intuitively compelling asymmetries arising in connection with de re necessities of a relational sort.10 Thus it is, he claims, an essential property of the singleton set {Socrates) that it has Socrates as a member, but not an essential property of Socrates that he belongs to that, or any other, set. But the modal account fails to reflect such asymmetries: it is necessary that if Socrates exists, he belongs to {Socrates) (since it is necessary that his singleton exists, if Socrates does, and necessary that he belongs to it, if they both exist). Or again, it is (or at least is often taken to be) an essential property of Elizabeth I1 that she was the daughter of George VI, but it is not an essential property of George VI that he begat Elizabeth 11. The problem, in general terms, is that where a statement of de re necessity mentions two or more individuals (i.e. has the shape 'O+xy'), it may be that for one of them, x, standing in the relation to the other(s), y, is an essential property of it (i.e. of x), while y's standing in the converse of that relation to x is not an essential property of y. There is, however, nothing in logical form of the de re modal statement to capture

+

Absolute Necessities / 103

this, since it is, as it were, symmetric in its terms-no distinction is marked between x's necessarily bearing 4 to y, and y's necessarily having x bear 4 to it, so that the distinction is lost between what properties essentially belong to x and what to y. Fine does not dispute that it is a necessary condition of P's being an essential property of x that UPx. What he denies is that this is sufficient. The inadequacy of the modal account derives, he plausibly suggests, from its neglecting the fact that when P is an essential property of x, the necessity of x's being P has its source in what he calls the 'nature' or 'identity' of x: it is, as he puts it, 'true in virtue of the identity of x that Px'.ll That Socrates is man has its source in the identity of Socrates, whereas that {Socrates}has Socrates as (its sole) member has its source in the identity of the singleton, not the man. It would take me too far from present concerns to assess the cogency of Fine's case against attempts to explicate the notion of essence in terms of de re modality. I need not do so, since even if-as I suspect-that case is not irresistable'z, we have-if the argument of the preceding section is sound-independent reason to be interested in whether the positive alternative he proposes is a viable one. What especially merits attention here is the suggestion he goes on to make: that if we do take the relation expressed by 'p is true in virtue of the identitylnature of x' as basic, then we can use it to explain many of the more familiar concepts of necessity:

. . . far from viewing essence as a special case of metaphysical necessity, we should view metaphysical necessity as a special case of essence. For each class of objects, be they concepts or individuals or entities of some other kind, will give rise to its own domain of necessary truths, the truths which flow from the nature of the objects in question. The metaphysically necessary truths can then be identified with the propositions which are true in virtue of the nature of all objects whatever. Other familiar concepts of necessity can be understood in a similar manner. The conceptual necessities can be taken to be the propositions which are true in virtue of the nature of all concepts; the logical necessities . . . the propositions . . . true in virtue of the nature of all logical concepts; and, more generally, the necessities of a given discipline, such as mathematics or physics, can be taken to be those propositions which are true in virtue of the characteristic concepts and objects of the discipline.13 Let us set aside for the present questions of detail-such as how we are to understand the result of filling the subscripted argument place in Fine's operator 'El,', not with an expression designating a particular object or concept, but with a reference to all objects, or all concepts, or all concepts falling within a specified range-in order first to get the broad picture, and its potential bearing upon our earlier question, into clear focus. That picture generalises the leading idea-that necessities which reflect individual essences have their origin in the identities or nature of the 'objects' they

104 / Bob Hale

concern-by introducing the further thought that whole ranges of necessary truths may be grouped together in accordance with the kind of entities to whose natures their truth is owed. At the most general level, we have those necessities whose truth derives from the nature of all entities whateverFine's metaphysical necessities.14 Conceptual necessities then constitute a (presumably proper) subclass of metaphysical necessities, comprising all and only those truths deriving from the nature of all concepts (as opposed to nonconceptual entities of whatever kinds there may be). These clearly coincide with broadly logical necessities in the sense used throughout the present discussion. Logical necessities in the narrower sense employed by Fine then form in turn a proper subclass of conceptual necessities. Thus far, then, we would seem to have a (partial) taxonomy of necessities like this: metaphysical necessities [true in virtue of the nature of all entities whatever] I

I non-conceptual necessities [true i n virtue of the nature of all non-conceptual entities ]

I

conceptual necessities [true in virtue of the nature of all concepts ] I

I

logical necessities [true in virtue of the nature of logical concepts]

?

I

I

?

There is, it seems to me, some uncertainly about how the classification is intended to continue beyond this point. Indeed, Fine does not explicitly provide for a category of non-conceptual necessities co-ordinate with conceptual necessities, as I have done-though its existence does seem to be implied. I am supposing that this would include such putative necessities as heat's being molecular motion and the identity of Hesperus with Phosphorus. His characterisation of mathematical and physical necessities as 'those propositions which are true in virtue of the characteristic concepts and objects of the discipline' [my emphasis] is at first sight somewhat puzzling, and may at first appear to square ill with my broad division of necessities into the conceptual and the non-conceptual, since necessities of these kinds are billed as having their source in both concepts and objects. There is, however, an obvious and quite natural way to bring this into line-though I am not sure that it accords with Fine's intentions. Any truth whatever will be owed at least in part to the concepts involved in its articulation. What will be distinctive of conceptual necessities is that their truth derives entirely from concepts. Fine's characterisation of mathematical and physical necessities may reflect a conviction that conceptual considerations alone do not suffice for

Absolute Necessities / 105

their truth, which must be put down in part to the nature of the (nonconceptual) entities with which these disciplines deal (numbers, sets, etc., in the former case, and particles, fields, forces, etc., in the latter). As far as mathematics is concerned, this is somewhat controversial, since it involves taking a stand against a kind of generalised version of logicism which sees mathematical truths as deriving entirely from logical and mathematical concepts, even if not from the former alone. But the corresponding claim seems indisputable, as far as physics and other empirical scientific disciplines go. This suggests that the necessities belonging to the latter disciplines should appear for the most part in subdivisions of my broad category of nonconceptual necessity, which is to be seen as comprising all propositions whose truth derives entirely from the nature(s) of entities of some kind or other, but never exclusively from the nature of concepts. There may, of course, be truths which hold in virtue solely of the nature of, say, physical concepts (as distinct from the nature of the entities to which they apply)-if so, these will fall within a subdivision of (purely) conceptual necessities. For present purposes, however, it is unnecessary to defend this particular way of filling out the detail of Fine's picture, or to speculate about its further elaboration. Provided that my gloss on its broad shape is right, we have-at least in outline-an account of modality which allows us to see the different kinds of absolute necessity as species of a single genus. The class of absolute necessities may be identified with Fine's metaphysical necessities. Falling within this, we have the subclass of conceptual or broadly logical necessities. Both it and the co-ordinate class of nonconceptual necessities (metaphysical necessities in the narrower sense of our earlier discussion) constitute disjoint species of absolute necessity. In particular, McFetridge's Thesis is accomodated without prejudice to the absolute character of metaphysical necessity in the narrower sense. The unity of the superclass resides in a common broadly structural feature: while each particular absolute necessity has its own distinctive source, each and every one conforms to this general pattern-it is a truth which holds in virtue of the identity or nature of some entity or range of entities. We have, then, the makings of a competitive answer to the question I earlier claimed we must tackle, if we are to see how absolute notions of metaphysical and broadly logical necessity can sit comfortably alongside one another. Or at least, we do, if each absolutely necessary truth can be regarded as true in virtue of the nature of x, for some suitable choice of x. But is that condition met? I want now to rehearse a line of thought which has it that the account runs into trouble at this very point. 6 Another fine regress we've gotten ourselves into?

We can approach the difficulty by returning to a question shelved a few paragraphs back: how should we understand the result of filling the sub-

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scripted argument place in Fine's operator '!I,not ', with an expression designating a particular object or concept, but with a reference to all objects, or concepts, or concepts falling within a specified range? Consider, for definiteness, the class of conceptual necessities, i.e. the class of propositions true in virtue of the nature of all concepts. How, precisely, is this to be understood? There will, presumably, be very many true propositions, each of which is true in virtue of the nature of some single concept-such truths as that vixens are female, that dogs are male, and the like. That is, we will have instances of Fine's basic schema such as:

Eliiren Vixens are female !Idog Dogs are male.15 And we may take it that what holds in virtue of the nature of some single concept thereby ranks as true in virtue of the nature of all concepts (on the principle that the nature of any single concept is part of the nature of all concepts). This might tempt one to suppose that the class of propositions true in virtue of the nature of all concepts can be taken as the union:

where K is the class of all concepts, and {A:O,A) is the class of propositions true in virtue of the nature of the concept C. On reflection, however, this is clearly inadequate. A minor irritant is that it takes no account of conceptual necessities which depend upon not one but two or more concepts. But there is a more serious weakness. Consider the conceptual necessities: Vixens are female Ewes are female which qualify as such, on the present account, because:

!Iviren Vixens are female

Oe, Ewes are female.

How about the proposition that both vixens and ewes are female? Clearly this should rank as a conceptually necessary truth; but why, exactly? Obviously the proposition is not true in virtue of vixen (on its own), nor is it true in virtue of ewe (on its own). The overwhelmingly natural answer is, of course, that our simple conjunctive conceptual necessity is so in virtue of its being a logical consequence of other, more basic conceptual necessities. To accept this answer is to recognise that the class of concep-

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tual necessities is not just a flat, unstructured collection, each one qualifying as such independently of all the rest. When we recognise that there are no end of conceptual necessities, the view may well seem to be imposed that, besides a base class the members of which qualify directly, as true in virtue of the natures of particular concepts, we have an infinite remainder of derivative conceptual necessities, each of which counts as such in virtue of its being a logical consequence of others already in the bag. The point obviously generalises to other types of necessity. Thus it is, we may stlppose, metaphysically necessary that gold has atomic number 79, and also that copper has atomic number 29. Presumably, then, it is metaphysically necessary that gold has a larger atomic number than copper. And again, it seems that this last necessity should be derivative: it is metaphysically necessary that gold has a larger atomic number than copper because (i) it is metaphysically necessary that gold has atomic number 79, (ii) it is metaphysically necessary that copper has atomic number 29, (iii) it is mathematically necessary that 79>29 and-crucially, for present purposes-(iv) it is a logical consequence of (i)-(iii) that atomic number(go1d) > atomic number(copper). If this is right, the class of absolute necessities exhibits a structure of dependency, reflected in the logical consequence relations among its members. But why, if it all, should that be thought to give rise to any sort of problem? Well, that there may be a problem here can be seen by noting that, corresponding to each instance of the consequence relation, there will be a matching necessarily true conditional. The necessity of each such conditional will, it seems, have to be absolute-since anything less than an absolutely necessary conditional cannot be relied upon to sustain transmission of absolute necessity from its antecedent to its consequent-and, in all (or anyway nearly all) cases, it will be derivative.16 But these conditions bear an uncomfortable structural resemblance to those in virtue of which a wellknown version of conventionalism about necessity generates an equally wellknown, and seemingly vicious, infinite regress.17 The objection to conventionalism may be stated as follows. The conventionalist holds that all necessary truths are so by convention, and that each necessary truth is so because true by convention, i.e. that its necessity consists in its being secured as true by convention. The radical conventionalist holds that each individual necessary truth is secured by its own special conventional stipulation, and so must hold that there are only finitely many necessary truths (since we cannot have performed an infinity of such stipulations). To avoid this implausibility, the modijied conventionalist holds there to be a base class of necessities, U, each of which is secured by direct conventional stipulation. All the infinite remainder are to qualify as true by convention indirectly, in virtue of being more or less remote consequences of U. Let q be some necessary truth not in U, which the modified conventionalist therefore claims to be necessary in virtue of being a conse-

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quence of U. The content of this claim is that if all the statements in U are true, q must likewise be true, briefly:

We may assume that (a) @ U, so that q is not directly true by convention, since if the conventionalist says that it is in U, he can be asked whether he will make the same claim about all consequence conditionals-if so, then his position collapses into that of the radical conventionalist; if not, then we may just consider some q of which he will not claim that its consequence conditional is in U, and run the argument in terms of that. But if the conventionalist is to stay in play, he must hold that (a) is true by convention, and so must claim that it is indirectly so, i.e. that (a) holds because the embedded conditional is a consequence of U. But this last claim is again a claim of necessity, namely that:

Thus, to justify taking q to be true by convention (and so necessary), the conventionalist has to establish (a). But to establish (a), given that (a) $Z U, he must establish (b). But (b) $Z U, so to establish it, the conventionalist must show that it is a consequence of U, i.e. must establish

Clearly we are in a regress. The regress seems to show that the conventionalist cannot recognise as necessary any necessary truth lying outside U, because in order to do so, he would need to perform an infinity of acts of recognition-before he can recognise q as necessary, he must recognise (a) as so, but before he can do that, he must recognise (b) as so, and before he can do that, he must recognise (c) . . .-and this he cannot do. Observing that the embedded conditionals U + q, U + (U + q), U + (U + (U + q)), . . . are all logically equivalent (since the conditional is material), the conventionalist may retort that we do not, contrary to appearances, have an infinite sequence of distinct acts of recognition, but one single act: in recognising that Nec(U + q), we eo ipso recognise that Nec(U + (U + q)), etc., since the contents of each act, and so the acts themselves, are identical. It is not uncontestable that acts of recognition with necessarily equivalent contents are identical. But even if that is granted, it does not appear, on reflection, that this reply is open to the conventionalist. For to make it, he must be justified in taking the biconditionals linking the conditionals in the sequence to be necessary. But that necessity must again be derivative, and thus, in his view, a consequence of U. The reply therefore merely re-instates the regress at one remove.ls

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Let us now consider Fine's essentialist account. We may, for simplicity, confine attention to conceptual necessities. These comprise, we are told, "the propositions which are true in virtue of the nature of all concepts." But how precisely is this to be understood-what does the claim involve? Some among these true propositions will be propositions each of which owes its truth to the nature of some one particular concept directly-thus we presumably have things like these (1) Oconjunction if a conjunction is true then so are its conjuncts ( 2 ) O,,junc,ion a disjunction is true if at least one disjunct is. But there will be other truths-such as the truth that if a conjunction is true, then so is any disjunction which shares a component with it-which ought to rank as logical necessities, and so as conceptual necessities, but which are not true in virtue of the nature of any one concept. This particular truth, one supposes, is true in virtue of both the concept of conjunction 'and that of disjunction; I suppose this situation might be represented: if a conjunction is true, so is any disjunction (3) Oconjunction **junction which shares a component with it. But how are we to construe this, except as saying something like: it is true in virtue of the nature of conjunction that a conjunction cannot be true without both its conjuncts being true, and it is true in virtue of the nature of disjunction that a disjunction is true if either disjunct is, so, if a disjunction has as one disjunct one of the conjuncts of a true conjunction, it must be true. That is, it seems that we ought to construe (3) as saying that the cited proposition is true in virtue of its being a logical consequence of other necessities ((1) and (2)) which are themselves true in virtue of the natures of certain concepts. Thus the essentialist, like the modified conventionalist, appears committed to holding there to be a base class, B, of propositions immediately true in virtue of the nature of concepts, and an infinite residue of propositions mediately true in virtue of the nature of concepts, through being consequences of B . But now, it seems, we can repeat the argument deployed against the conventionalist. We consider some conceptual necessity p @ B, which the essentialist holds to be conceptually necessary in virtue of being a consequence of B. The content of this claim is that (a') Nec(B + p) (a') is, presumably, itself conceptually necessary. We may assume that (a') @ B. So the essentialist must hold that (a') is indirectly conceptually neces-

110 / Bob Hale

sary because its embedded conditional is a consequence of B. But the content of this last claim is that: (b') Nec(B + (B + p)) Thus, to justify taking p to be conceptually necessary, the essentialist has to establish (a'). But to establish (a'), given that (a') $L B, he must establish (b'). But (b') $2 B, so to establish it, the essentialist must show that it is a consequence of B, i.e. must establish (c') Nec(B + (B + (B + p))) We are again in a regress. Much as before, the regress seems to show that the essentialist cannot recognise as necessary any necessary truth lying outside B, because in order to do so, he would need to perform an infinity of acts of recognition-before he can recognise p as necessary, he must recognise (a') as so, but before he can do that, he must recognise (b') as so, and before he can do that, he must recognise (c'). . . . Although the argument just given is framed in terms of the essentialist account of necessity, it appears to be quite independent of the detail of that account-that is, it appears that the argument may be generalised so as to apply to any position which offers a certain constitutive account of certain basic necessities and then seeks to view all other necessities as enjoying that status in virtue of their being consequences of those in the base class. Thus suppose it is held that what makes for necessity is a certain property F. Some but not all necessary truths-those in the base class-have F directly; the remainder inherit the property in virtue of being consequences of F-truths. The account then immediately confronts a question about the involved necessities of consequence: how is their possession of F to be explained? They will, in general, not be primitively F, but will again inherit the property in virtue of being consequences of other F-truths . . . and the regress is under way, precluding any finitary model of our recognition of non-basic necessities. If the generalised argument is sound, it appears to dash all hope of precisely the kind of unifying account of modality which I claimed is needed, if we are to make satisfactory sense of there being distinct kinds of absolute necessity and thereby to accomodate McFetridge's Thesis without prejudice to the claims of the friends of metaphysical necessity. Unless it can somehow be argued that the demand for such a unifying account is not after all imposed by the absolutist integration of metaphysical and broadly logical necessity suggested as means of accomodating McFetridge's Thesis, there would seem to be no option but to accept that metaphysical necessity is not absolute. It is therefore of the first importance to determine whether the argument really is sound.

Absolute Necessities / 11 1

7 Possible responses

There are two kinds of response possible: those which grant the parallel between the conventionalist and essentialist accounts, but deny that the regress-based objection is good in either case, and those which seek to disrupt the parallel. I shall conclude by saying something-not enough, but a start-about each. Someone may object: why can't the conventionalist establish q (and the essentialist p) quite simply, by just giving a deduction of it from statements in the base class? The same-or a closely linked-objection may be put: "There has to be something wrong with your arguments, since an exactly parallel argument will 'prove' that we cannot characterise the theorems of a given formal system as the deductive closure of the axioms!" To see why these objections are ineffective, take the second first. In regard to any given formal system S, what is meant by deductive closure of the axioms can be articulated quite precisely by a definition of formal proof, along familiar lines-.rr is a proof in S of A iff .rr is a finite sequence of wffs of S, the last of which is A and every member of which is either an axiom, or is an immediate consequence of earlier members by one of these inference rules . . . To establish, for example, that p + p is a theorem of the system of propositional logic with the axioms

and the inference rules of uniform substitution and detachment, it suffices to construct the following sequence of wffs:

That this sequence conforms to the definition of proof for the system is verifiable in a small number of steps, by observing that 1 is A l , 2 comes from 1 by uniform substitution of + p for q and p for r, 3 is A2,4 comes from 3 by uniform substitution of p for q, 5 comes from 2 and 4 by detachment, 6 is A3 and 7 comes from 5 and 6 by detachment again. Note that while we do in fact believe that all instances of A1-3 are necessary truths, when + and 1are given their intended meanings, and

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that the inference rules preserve truth (and so preserve necessary truth, in particular), these facts-together tantamount to the claim that the system is sound-play no part in establishing that 7 is a theorem. And that is, of course, just as well, since to establish soundness we have to go outside the system into its metatheory. Herein lies a crucial difference between the situation of the conventionalist and the essentialist on the one hand, and more modest theorem provers on the other. When the conventionalist, for instance, characterises the class of necessary truths as U together with all its consequences, he must be understood as meaning by consequences all those propositions which must be true, given that those in U are true. But the propositions which must be true, if those in U are, are not just those propositions which may be derived from U by means of some rules of inference or other, but those which (if derivable from U at all) are derivable by sound rules of inference. And this is why it is not enough, for the conventionalist to establish q, simply to produce a deduction of q from U. He has, in effect, to establish in addition that he has done so by means of sound rules (necessarily truth-preserving ones). And precisely because his characterisation of the class of necessary truths was intended to capture all of them, there can be no question of stepping up into a metatheory to provide such a proof-as we can do, if called upon, in case of deductive systems such as that of Lukasiewicz used as example above. The fact that there will be necessary truths provable in the metatheory of, say, propositional logic which cannot even be formulated, let alone proved, in the object-system is no problem for the ordinary theorist-but the conventionalist cannot allow that there are any such without abandoning his claim to have provided what he set out to give us, i.e. a comprehensive conventionalist account of necessary truth. And the same goes, mutatis mutandis, for the essentialist. The argument depends upon the assumption that the essentialist, like the modified conventionalist, must regard the class of necessary truths as exhibiting a structure of dependency akin to that of a deductive system, in which some truths qualify immediately as necessary in virtue of their possession of a certain characteristic, while the rest qualify only mediately, in virtue of being logical consequences of others possessing it. As far as the modified conventionalist is concerned, the assumption is indisputable, since it effectively defines her position. Conventional truth can be secured by direct stipulation for at most finitely many propositionsany advance beyond radical conventionalism therefore requires that it is transmitted ,to other truths across the consequence relation. But the essentialist may try to repudiate the assumption. Consider, for example, the class of all propositional tautologies, say in the formulation taken as example above. The essentialist will hold that those dignified as axioms are true in virtue of the natures of the concepts of negation and the conditional. But she will resist the idea, enjoined by the assumption, that

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each of the infinite remainder of theorems is true in virtue of being a logical consequence of the axioms, and so depends upon them for its necessity. Axioms and theorems alike are all on a level, each one true just in virtue of the natures of the primitive truth-functional concepts.19 When q is a logical consequence of propositions true in virtue of the nature of such and such concepts, it is not that being a consequence of those propositions is one of the determinants of q's truth-so that the necessity of this consequence has to be (shown to be) due to conceptual natures if q's is to be so-rather, the obtaining of the consequence relation ensures that q is true in virtue of the nature of just those concepts in virtue of which the premisses are true. q's being a consequence of those propositions may be what mediates our recognition of this fact; but that, the essentialist will hold, is a quite separate matter. It is, in this view, a mistake to think of necessary truths as exhibiting dependencies reflecting the relations of deducibility among them. It allows recognition of structure of a different and more partial kind. There will be (conceptual) necessities which de' pend for their truth on the nature of single concepts, others which depend upon groups of concepts, etc. So the truths of first-order logic will all be true in virtue of the nature of the concepts of, say, negation, the conditional and existential quantification, while those of elementary arithmetic will be true in virtue of the nature of these concepts together with those of identity, zero, immediate succession and natural number, and similarly in other cases. But within such blocks of necessities, there is no futher structure, mirroring deductive liaisons. While it is tempting to think that a reply along these lines can be made to work, it raises difficulties which I cannot adequately address here. It relies, clearly, upon repudiating the idea that there is any constitutive structure among necessary truths corresponding to the deductive interconnections through which we acquire knowledge of them, and, more generally, upon a sharp separation of constitutive/explanatory questions from epistemological ones. Neither component is unproblematic. The first draws some plausibility from the difficulty, which I have thus far passed over, of making good sense of the idea that some necessities depend upon others for their truth (as distinct from their recognition as true): we cannot very well explain this in counterfactual terms, since-on standard semantical accounts at least-the relevant counterfactuals, having impossible antecedents, will be vacuously true; and it is so far unclear what alternative explanation should be given. But this difficulty is probably a distraction from the central issue, which concerns how to understand the essentialist's basic notion of a proposition's being true in virtue of the nature or identity of some entity. Fine says of this that "we should understand the identity or being of the object in terms of the propositions rendered true by its identity rather than the other way round."20This is somewhat opaque, but it at least quite strongly suggests that we should think of the identity or nature of an

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entity as given or determined by a privileged collection of true propositions involving it. Being true in virtue of something's nature will then, it seems, consist in being a member of its privileged collection of propositions. But if this is so, it is difficult to see how the class of conceptual necessities, for example, can be anything other than the closure under consequence of the union of the sets of propositions true in virtue of the identity of particular concepts-which would seem to reinstate precisely the structure of dependence on which the regress-based objection feeds. There is some awkwardness, too, in the sharp segregation of constitutive from epistemological matters. There would, perhaps, be no cause for concern, if it could be maintained that every necessity is, at least in principle, knowable noninferentially-we might then view our seemingly unavoidable reliance upon proof as a means of recognising necessity as merely reflecting contingent limitations on our epistemic capacities. But if that very substantial assumption is not made, the essentialist surely owes an account of why the uniformity of constitution she claims to discern is belied by a radical discontinuity in epistemology. To conclude: I have defended a form of McFetridge's Thesis, according to which logical necessity, if not the strongest kind, is absolute. I have argued that this thesis is in serious tension with claims about metaphysical necessity, as usually understood. Resolution of this tension appears to demand an overall unifying account of modality, in terms of which different kinds of absolute necessity can be viewed as exhibiting structural uniformity whilst differing in their specific sources. Whilst the essentialist approach, as developed by Kit Fine in recent work, promises to answer this need, there remain substantial obstacles to be circumnavigated, before we can reckon ourselves in possession of an adequate perspective on the issue.Z1 Notes

1. This is of course controversial; I am assuming familiar sceptical attacks on analyticity, etc can be rebuffed. 2. 'non-relative' is, of course, somewhat inaccurate-such claims are relative to speaker and time; but this is distinct from relativity in my sense. 3. McFetridge 1990, Essay VIII: "Logical necessity: some issues." 4. Interestingly, McFetridge actually presented his argument as establishing that logical necessity is the strongest kind of necessity. Here is his argument:

,The argument rests on two assumptions: first, that is a distinctive and important feature of deductive validity (one in which it contrasts with inductive strength) that adding extra premisses to a valid argument cannot destroy its validity. In particular, then, if the argument 'p so q' is valid then so is the argument 'p,r so q' for any r. The second assumption is that there is this connection between deducing q from p and asserting a conditional:

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that on the basis of a deduction of q from p one is entitled to assert the conditional, indicative or subjunctive, if p then q. Suppose then that 'p so q' is valid, i.e., by the hypothesis of the above claim, it is logically necessary that if p then q. But suppose that the consequent of that claim is false. Suppose that it is possible that p and not-q in another sense of 'possible'. But if that is a possibility, we ought to be able to describe the circumstances in which it would be realised: let them be described by r. Consider now the argument 'p and r so q'. By the first assumption if 'p so q' is valid, so is 'p i n d r so But then, by the second assumption, we should be entitled to assert: if p and r were the case then q would be the case. But how can this be assertible? For r was chosen to describe possible circumstances in which p but not-q. I think we should conclude that we cannot allow, where there is such an r, that an argument is valid. Logical necessity, if there is such a thing, is the highest grade of necessity. (1990, p.138)

q'.

Earlier, however, McFetridge glosses the claim that logical necessity is the strongest kind as follows: That is, if it is logically necessary that p, then it is necessary that p in any other use of the notion of necessity there may be (physically, practically, etc.). But that the converse need not be the case. Something could be eg physically necessary without being logically necessary. Equivalently, then, logical possibility is the weakest kind of possibility. If something is practically, physically, (etc.) possible then it is logically possible. But the converse is not the case. [op cit, pp.136-71 This seems clearly to acknowledge that to establish that logical necessity is the strongest kind, two things must be shown: that where possible* is any notion of possibility other than logical (i) it cannot be logically necessary that p but possible* that not-p, but (ii) it islmay be necessary* that p but logically possible that not-p. Since McFetridge offers no argument, here or elswhere in his paper, for (ii), he is not entitled to his conclusion as stated. 5. It is, in fact, not entirely clear that considerations of relevance do require rejection of A l . Relevant logicians deny that it suffices for the validity of an inference that it be impossible for the premiss to be true but the conclusion false, but they do not-so far as I am aware-deny that it is necessary. It is true that in the text, I identify validity with logical necessity of the corresponding conditional. But that identification is not actually required for the argument-so long as it is allowed that the latter is at least a necessary condition for validity, A1 should be acceptable, and if the argument is not otherwise objectionable, it will show that there is no sense of 'possible' in which a valid argument can have a true premiss but a false conclusion. 6 . cf Kripke 1980, p.99: ". . . characteristic theoretical identifications like 'Heat is the motion of molecules', are not contingent truths but necessary truths, and here of course I don't mean just physically necessary, but necessary in the highest degree-whatever that means."

116 1 Bob Hale To each such counter-example to closure there will, of course, be a corresponding direct counter-example to McFetridge's Thesis. Thus it is logically necessary that no vixens are male, but there is, after all, a sense in which it is possible that some vixens are male, since it is austerely possible. Kit Fine 1994 and 1995. Fine actually considers another conditional variant on the basic account, according to which necessary possession of P is conditional upon identity with x. But he argues, convincingly in my view, that this collapses either into the basic categorical account, or the existential conditional account. Fine produces several other objections to modal accounts. For example, given any necessary truth, p, it will be necessary that if Socrates exists, then p. So on the modal account, it will be part of Socrates essence that there are infinitely many primes. Further, since all statements of essence will be among the necessary truths, it will be of the essence of any one object that every other object has the essential properties it has-I cannot resist quoting Fine's delightful comment: 'Oh happy metaphysician! For in discovering the nature of one thing, he thereby discovers the nature of all things'.(1994, p.6) However, the objection mentioned in the text is the most interesting from the present perspective, as being most directly suggestive of Fine's alternative approach. I shall follow Fine in writing this kind of claim with a subscripted necessity operator 0 , . To amplify a little, the de re modalist may insist that his account does register a difference between 'a essentially 4s b' and 'b is essentially 4ed by a', provided the former is construed as 'O(if a exists, it 4s b)'-i.e. in all worlds containing a, 4ab-and the latter as 'O(if b exists, it is 4ed by a)'-i.e, in all worlds containing b, 4ab. It is true that if a and b are necessary existents, these conditions are equivalent-but in that case, the intuition is much less secure that it is an essential property of a that it bears 4 to b, but not an essential property of b that a bears 4 to it. If, alternatively, 'a essentially 4s b' is construed as above, but 'b is essentially @d by a' as 'b is @d by a in every world containing a', then there is no asymmetry to explain. 1994, pp.9-10 Note that Fine clearly means 'object' to be taken in a very comprehensive way-not in contradistinction to 'property' or 'concept' (in the Fregean sense)-so as to cover 'concepts or individuals or entities of some other kind'. Further, 'concept' in his usage clearly means something like (Fregean) sense, as opposed to concept = reference of a predicative expression. Italicized subscripts denote concepts There being, we may take it, no more than finitely many basic absolute necessities, so that even if some of the relevant conditionals lie in the base class, an infinite remainder will not. The best known version of the regress-based objection is that first developed in Quine(1936). A closely related objection is presented in Dummett (1959). Wright 1980, p.351 makes what I take to be essentially this point against a similar attempt to make out that the regress in which the conventionalist is ensnared is harmless. cf Wittgenstein 1921,6.1262: "Proof in logic is merely a mechanical expedient to

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facilitate the recognition of tautologies in complicated cases." 6.127: "All the propositions of logic are of equal status: it is not the case that some of them are essentially primitive propositions and others essentially derived propositions." 20. Fine 1995, p.273 21. I am greatly indebted, as usual, to Crispin Wright for much encouragement and constructive criticism. Thanks are due, too, to Jim Edwards, Scott Meikle, Pat Shaw, Peter Sullivan and Nick Zangwill. References

Dummett, Michael 1959 'Wittgenstein's philosophy of mathematics' The Philosophical Review 68 324-48, reprinted in Dummett's Truth and Other Enigmas, Duckworth, London 1978. Fine, Kit 1994 'Essence and modality' in James Tomberlin ed Philosophical Perspectives 8, Logic and Language, 1-16. 1995 'Ontological dependence' Proceedings of the Aristotelian Society, 269-89. Kripke, Saul 1980 Naming and necessity Blackwell, Oxford. McFetridge, Ian 1990 Logical Necessity and other essays, Aristotelian Society Series vol 11. Quine, W.V.O. 1966 'Truth by convention' originally published in O.H.Lee ed. Philosophical Essays for A.N. Whitehead New York, Longmans 1936, reprinted in Quine's Ways of Paradox Random House, New York 1966. Wittgenstein, L. 1921 Tractatus Logico-Philosophicus. English translation 1961 by D.F Pears & B.F.McGuinness. Routledge & Kegan Paul, London and New York. Wright, Crispin 1980 Wittgenstein on the Foundations of Mathematics Harvard University Press, Cambridge, Mass.