Knowledge of Possibility and of Necessity

I*—The Presidential Address KNOWLEDGE OF POSSIBILITY AND OF NECESSITY by Bob Hale I investigate two asymmetrical approaches to knowledge of absolute possibility and of necessity—one which treats knowledge of possibility as more fundamental, the other according epistemological priority to necessity. Two necessary conditions for the success of an asymmetrical approach are proposed. I argue that a possibility-based approach seems unable to meet my second condition, but that on certain assumptions—including, pivotally, the assumption that logical and conceptual necessities, while absolute, do not exhaust the class of absolute necessities—a necessity-based approach may be able to do so.

ABSTRACT

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he Problem of Modal Knowledge. How can we come to know, or at least arrive at reasonable beliefs about,1 what is possible and what is necessary? Kant famously remarked2 that we may learn from experience what is the case, but not what must be. He might have added that experience—roughly, sense-perception and introspection, together with what we can infer from their deliverances—leaves us almost equally in the dark about what might be. Not quite, of course, since wherever experience teaches us that p, we may safely reason, ab esse ad posse, that it is possible that p. But the interesting question concerns knowledge of unrealised possibilities (or at least knowledge of possibilities not known to be realised). It is precisely because possibilities may go unrealised that experience cannot teach us what must be so. Experience may inform us that p, but to know that it is not just true, but necessary that p, we need to know that there is no (unrealised) possibility that not-p. Whilst philosophical discussion of modality has often given greater prominence to the 1. This alternative is always to be understood, even when, for brevity, I suppress it, as should the caveat that any reasonable beliefs we may form are likely to be fallible. 2. Kant (1963), p. 43. *Meeting of the Aristotelian Society, held in Senate House, University of London, on Monday, 14th October, 2002 at 4.15 p.m.

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nature and basis of necessity and our knowledge of it than to corresponding questions about possibility, it seems clear that the central problems, at least, concern both modalities. Kant’s observation draws attention to one important aspect of the problem, but it can be just as well raised in other ways. In terms of possible worlds, both 䊐p and, except in special cases, 䉫p require p’s truth at other worlds (all, or at least one) besides the actual world. If we follow David Lewis in adopting a fullbloodedly realist attitude to possible worlds, there is an obvious and familiar difficulty to be confronted: how, given our isolation from兾inability to inspect other (merely possible) worlds, can we know these truth-conditions to be met? But the problem—while it may be aggravated by taking the truth-conditions of modal propositions to receive their most fundamental formulation in these terms—doesn’t depend on the adoption of possible world semantics, realistically construed. Even without talk of worlds,3 it is clear that knowledge of necessity and possibility goes beyond knowledge of what is actually the case—we can’t verify 䊐p or (except in special cases) 䉫p (just) by finding out what is (actually) the case. We need to know, in case of 䊐p, that p would have been true no matter how different things might have been in other respects—the problem is to see how we can be in a position to know a kind of generalised strong counterfactual: ∀q(q 䊐→ p), or in the case of 䉫p, a corresponding (existentially) generalised weak counterfactual ∃q(q 䉫→ p), equivalent4 to ™∀q(q 䊐→ ™p). Besides taking the problem of modal knowledge to be as much a problem about possibility as about necessity, I shall further take it to concern, primarily anyway, our knowledge of absolute—as contrasted with (merely) relatiûe—necessity and possibility. The rough idea in calling a kind of necessity—physical necessity, say—relative is straightforward enough: what is physically necessary is what is required by the laws of physics, i.e. what logically must be so, if there is to be no violation of physical law, and what is physically possible is what can be so, without violation of any physical law, i.e. what is logically consistent with 3. —which may, of course, be construed as involving only moderate realism in Robert Stalnaker’s sense, as opposed to Lewis’s uncompromising version. 4. Defining the weak counterfactual A 䉫→ B (If it were兾had been that A, it might be兾have been that B), with David Lewis ((1973), p. 2), as ™(A 䊐→ ™B).

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physical law.5 The contrast here is roughly that between being necessary conditionally on the truth of certain propositions—or necessary relative to a certain set of (true) propositions—and being unconditionally necessary, or necessary without qualification. If logical truths are necessary, they are presumably unconditionally so, and thus absolutely necessary. Being logical consequences of the empty set of premisses, and so of its union with any other, they will also be necessary relative to any chosen set of propositions and thus relatively necessary in a great many different ways or senses6 —the interesting contrast is thus between absolute and merely relative. In terms of the standard apparatus of possible worlds, absolute necessity requires truth at all worlds without exception, relative necessity demands only truth at a subset of worlds, and merely relative necessity consists in truth throughout a proper subset. If, as I do, we prefer to explain the distinction without relying on that apparatus, I think we can do so as follows. Let Φ be some set of true propositions—then we can define a type of necessity, φ-necessity, by: It is φ-necessary that p iff it is a logical consequence of Φ that p; and it is φ-possible that p iff it is logically consistent with Φ that p. φ-necessity and φ-possibility, so explained, are clearly relative notions—to be φ-necessary is to be necessary relative to Φ, and to be φ-possible is to be possible relative to Φ. To a first approximation, φ-necessity is merely relative iff there is a kind of possibility, ψ-possibility, such that there are some φ-necessary propositions whose negations are ψ-possible; equivalently, φ-necessity is absolute if and only if there is no kind of possibility, ψ-possibility, such that for some p, it is φnecessary that p but ψ-possible that ™p. These explanations will not quite do as they stand, because modal idioms are often used 5. The example is not, of course, entirely uncontroversial. Kripke remarks ((1980), p. 99): ‘Physical necessity might turn out to be necessity in the highest degree. At least for this sort of example, it might be that when something’s physically necessary, it always is necessary tout court.’ The examples of which Kripke speaks here are theoretical identifications such as ‘Heat is motion of molecules’ and ‘Light is a stream of photons.’ Later, speaking of examples like gold being the element with atomic number 79, Kripke envisages that ‘such statements representing scientific discoveries ... [may not be] contingent truths but necessary truths in the strictest possible sense’ (p. 125). Kripke is careful to restrict the scope of these remarks. I do not think he says anything to warrant interpreting him as holding that physical necessities are always and invariably absolute. 6. On standard assumptions, 2ℵ0 —nearly all of them completely uninteresting.

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to express epistemic rather than alethic modalities. When we say, for example, that Jones must have missed his train, we are not claiming that Jones’s failure to catch it was somehow a matter of physical—much less metaphysical or logical—necessity, but roughly, that his missing it is the only conclusion we can reasonably draw from the evidence available to us. And if we add that he may have caught a later one than he originally planned to take, we are not likely to be drawing attention to the mere logical (or even physical) possibility of his having done so; what we mean is roughly that his having done so isn’t ruled out by what we know. But while epistemic possibility in anything like the sense illustrated7 remains in play as an interpretation for ψ-possibility, our suggested explanations are bound to fail of their purpose. Even in the case of absolute necessities (e.g. on some views, truths of logic and mathematics) knowable a priori—and even more obviously in the case of absolute but a posteriori necessities of the kind brought into prominence by Kripke and Putnam8 — there seems to be no clear reason why the negation of an absolutely necessity should not be, in our sense, epistemically possible. For all we know, for example, there is some very large even number which cannot be expressed as the sum of two primes; but this ought not to preclude Goldbach’s Conjecture from being, if true, absolutely necessarily so. More generally, if the contrast between absolute and merely relative kinds of necessity is to be non-empty on both sides, we must qualify our proposed explanation: φnecessity is absolute iff there is no non-epistemic kind of possibility, ψ-possibility, such that for some p, it is φ-necessary that p but ψ-possible that ™p.9 7. There are, arguably, stronger and weaker ways of understanding epistemic possibility, as expressed by such ordinary locutions as ‘For all we know, p’ and ‘That p is not ruled out by what we know.’ In addition, there is the rather different notion of epistemic possibility suggested by Kripke ((1980), Lecture 3, p. 141ff.) when he discusses the (apparent) possibility of water’s turning out to be something other than H2O, etc. 8. These are often, following Kripke’s own example, called ‘metaphysical’ necessities, and I shall do so myself. But I shall—disregarding what is perhaps a quite widespread practice—restrict my application of the term to a posteriori necessities of the KripkePutnam variety, in contrast with both strictly logical and conceptual necessities. 9. This complication is discussed a little more fully in Hale (1999), pp. 24–5. Note that on at least one way of construing it, and assuming our knowledge extends beyond just logic, epistemic possibility and necessity qualify as merely relative in this sense.

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If φ-necessity and φ-possibility are merely relative in this sense, they raise no extra epistemological problem, over an above that raised by absolute modalities, at least if we agree that logical necessity is absolute10 —knowledge that φ-necessarily p is knowledge that p is a logical consequence of Φ, i.e. that it is logically impossible that every member of Φ should be true but p false, etc.

II Asymmetrical Approaches to the Problem. An account of how we may come to know兾justifiably believe that something is necessarily so is not as such, and does not automatically lead to, an account of how we may know anything not to be so, and likewise for possibility. But if necessity and possibility are interdefinable in the usual way—or at least if we have the equivalences →™䉫™p, 䉫p ← → ™䊐™p—then it appears that we don’t 䊐p ← need four separate accounts of how we know about possibility and about the absence of necessity and about necessity and about the absence of possibility. Still, we do seem to be left with two questions: How do we know, for given p, that 䊐p兾™䉫™p? How do we know, for given p, that 䉫p兾™䊐™p? even if we don’t face four. It is consistent with acknowledging this much that the most fruitful approach to our problem should accord priority to one of these questions over the other—treating knowledge of necessity, say, as more fundamental than knowledge of possibility, or vice versa. To put the idea roughly and suggestively, we might think of possibility as more revealing characterised as just absence of necessity, so that knowledge of possibilities is primarily knowledge of the absence of any relevant necessities—or oppositely, we may view necessity as just absence of possibility, and knowledge of necessity as primarily knowledge of the absence of any relevant possibility. This suggests a distinction between two broadly opposed asymmetrical approaches to our problem—necessity-based 10. This assumption is defended in Hale (1996).

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approaches, which treat knowledge of necessities as more fundamental, and possibility-based approaches, which accord priority to knowledge of possibilities. In an asymmetrical approach, one of the modalities is taken as dominant in the sense, roughly, that our most basic modal knowledge is taken to be of truths in whose expression the dominant modal operator is principal (i.e. has widest scope), and the other is recessiûe in the sense, again roughly, that knowledge of modal truths in whose expression that operator is principal is essentially a matter of well-founded belief that there are no conflicting dominant modal truths. Does an asymmetrical approach have any prospect of success? And if so, are there grounds to favour one of the two such approaches over the other?—or, more generally, reasons to view our knowledge of possibility, say, as in any way more basic than our knowledge of necessity, or vice versa? To give these questions some more definite shape and focus, I shall begin by proposing two necessary conditions that an asymmetric approach must meet.

III Two Necessary Conditions. One necessary condition emerges from consideration of an obvious objection to the asymmetrical strategy. According to that strategy, beliefs about the recessive modality are to be justified by appeal to the fact that we have found no dominant modal truths which rule out the recessive modal claim up for assessment. For example, on a necessitybased version of the approach, each particular possibility claim is to be justified by appeal to the fact that we know of no necessity which rules it out. The obvious objection is that this simply and grotesquely conflates lack of grounds to believe it impossible that p with grounds to believe that it is possible—isn’t that just a special case of the obviously bad move from: we haûe no reason to belieûe ™p to: we haûe reason to belieûe p?11 An asymmetric theorist can scarcely deny that the latter shift is bad, so she must dispute the invidious comparison. But it is at least not obvious that she cannot do so. She may begin by 11. This kind of objection is brought by Stephen Yablo against taking conceivability as grounds for belief in possibility, when ‘conceivable that p’ is interpreted as ‘believable that p’ or as ‘believable that p is possible’. See Yablo (1993), pp. 8, 20.

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observing that there are cases in which what looks superficially like this bad shift is defensible—cases in which our failure to find evidence for a proposition does constitute a good ground for believing its contradictory. For example, our failure, after searching carefully for evidence of the burglar’s having made his entry through the French windows—signs of the windows having been forced, footprints in the adjacent flowerbed, etc.—and finding none, we may justifiably (albeit fallibly) conclude that he didn’t get in that way. The general thought is that whilst mere lack of evidence for ™p never, in and by itself, constitutes reason to believe p, it can do so, in the context of a well-directed and thorough search for evidence in p’s favour. If, applying this to the modal case, the asymmetric theorist is to rebut the objection, she must make out that in following the route she proposes, we need not be simply passing, gratuitously, from mere lack of knowledge of any relevant countervailing necessities, but may have looked responsibly for them and failed to find any. This requires that we can give decent operational sense to the idea of a well-directed and thorough search for necessities relevant to the assessment of a given possibility claim (or, of course, for possibilities relevant to a given necessity claim, in case of a possibility-based asymmetric approach). I shall return later to the question whether this necessary condition—hereafter the First Condition—can be met. First, I want to introduce a second necessary condition. Given that an asymmetrical approach will treat claims about its dominant modality as basic, and as, in effect, a key part of the essential background against which claims about its recessive modality are assessed, it may seem to be a further necessary condition for its successful implementation that there should be a way of coming to know dominant modal truths which neither involves nor presupposes knowledge of any recessive modal truths. Something like this is surely correct. But the condition as stated is open to a stronger and a weaker interpretation. Should it be taken as requiring that there be a way of coming to know dominant modal truths by means of which we can gain knowledge of any such truth, independently of any knowledge of recessive modal truths? Or should it be taken to require only that there be a such a way of coming to know some dominant modal truths? The stronger condition is too strong, but the weaker condition, while indeed necessary, is needlessly weak. The stronger

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condition is too strong because it overlooks the possibility that modal knowledge may be structured—in the sense, roughly, that once certain dominant modality claims are secured, this enables the assessment of some recessive claims which are in turn required for the assessment of further dominant claims, which may in their turn bear upon the assessment of yet further recessive claims. With this possibility in view, we can formulate a necessary condition which is weaker than the strong condition but stronger than the weak one. This requires—my Second Condition—that there be a base class of dominant modal truths which meet two conditions: (i) they can be known without reliance upon any recessive modality claims, and (ii) they are collectively strong enough to support the superstructure of modal knowledge to be erected over them. In what follows, I explore the prospects for an asymmetrical approach in the light of these conditions, beginning with the second. First, however, I want to make explicit an assumption— perhaps quite widely accepted, at least amongst those who believe in absolute necessities, but certainly not uncontroversial—which will play an important roˆ le. This is that broadly logical necessities (which comprise, in my usage, just narrowly or strictly logical together with analytic or conceptual necessities) are properly included within the class of absolute necessities. Strictly, of course, there are two assumptions here: first, that broadly logical necessities are indeed absolute,12 and second, that they do not exhaust the class of absolute necessities. Many who accept the second assumption will do so because they take metaphysical necessities13 —e.g. Hesperus is Phosphorus, Water is H2O, etc.—to be absolute but not broadly logical. Although I think the acknowledgement of different kinds of absolute necessity raises some difficulties,14 I am inclined to agree with this view.

12. See note 11. 13. See note 9. 14. Since, if one thinks primarily in terms of possible worlds, absolute necessities are those which hold at all worlds, metaphysical and broadly logical necessities can’t be distinguished by reference to the worlds through which they hold. One must instead think of the distinction as epistemological, or perhaps as having to do with different sources of necessity. For discussion of a quite different sort of difficulty, see Hale (1996), Sections 4–7.

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IV Asymmetrical Approaches and the Second Condition. As applied to a necessity-based approach, my second condition requires the existence of a class of absolute necessities which can be known without reliance upon any knowledge of possibilities and which is rich enough to serve as a basis for all other modal knowledge. Under the assumption that broadly logical necessities are absolute, it is plausible that a necessity-based approach meets the first part of this condition, and so at least clears the first hurdle. In very many cases in which we know it to be broadly logically necessary that p, our knowledge will be inferential—we know that it is necessary that p because we have inferred that p from some further premiss q which we know to be necessary by steps which are (known to be) necessarily truth-preserving. In such cases, mastery of the concepts involved will not suffice for knowledge that it is necessary that p unless that mastery is taken to include a capacity to perform or ratify the requisite inferential steps. And even if some degree of inferential competence is required for mastery of logical concepts, it is implausible to hold that this would suffice for recognition of all broadly logical but consequential necessities. But that does not matter for present purposes, since the second condition does not require that all knowledge of absolute necessities is independent of knowledge of possibilities.15 It would suffice to meet the first part of our second condition that knowledge of some broadly logical necessities demands no more than mastery of the concepts involved. I claim that there is a base class of necessities meeting that condition. This comprises necessities which constitute or reflect ways in which certain ingredient concepts are fixed. As an example, we may take the necessity that if a conjunction is true each of its conjuncts is so. Since the concept of conjunction is the concept of a function which takes the value truth only if both its arguments are true, someone who possesses that concept is in a position directly to see that a conjunction cannot be true without each of its conjuncts being so—i.e. that any compound proposition that is true despite the falsehood of one or its immediate 15. For the same reason, it is unnecessary to maintain—although it might anyway be argued—that our knowledge of logical consequence relations need not rest upon any knowledge of possibilities.

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constituents cannot be a conjunction. Examples of this sort stand, it seems to me, in significant contrast with others in which recognition of the necessity of a truth calls for a more or less substantial train of reasoning. A thinker’s need to be convinced, by rehearsal of the obvious reasoning, that the square of an odd number must itself be odd (not to mention more recondite cases), does not eo ipso raise any doubt about her grasp of the concepts of integer, oddness and square. But someone’s affecting to think that something might be a true conjunction with a false conjunct would immediately put it in doubt that it was conjunction she was thinking about. It is a further question whether a necessity-based approach can meet the second condition in full. I shall find it convenient to defer that question pro tem, as I think it best approached in the light of my discussion of the first condition. So I want instead to turn now to the possibility-based approach, and consider whether it can do at least as well as its competitor vis-a`-vis the second condition. Obviously there is some knowledge of possibility which is independent of any knowledge of necessity—ab esse ad posse16 —but equally obviously, this is insufficient for a possibility-based approach, which requires knowledge of unrealised possibilities. Hume notoriously claimed it to be ... an establish’d maxim in metaphysics, That whateûer the mind clearly conceiûes, includes the idea of possible existence, or in other words, that nothing we imagine is absolutely impossible.17

Hume’s principle18 has been thought clearly incorrect or at least open to decisive objection. Even so, it seems to me that it, or something close to it, has to be upheld if a possibility-based approach is to meet even the first part of the second condition. I shall take it as obvious without argument that we may put aside any interpretation of conceiving or imagining in terms of having visual imagery, if only because there are few, if any, questions of 16. As already noted—see my opening remarks. 17. Hume (1888), p. 32. 18. The label has, of course, is widely used—perhaps somewhat inappropriately—to refer to a quite different principle concerning identity of cardinal numbers. Yablo (1993) defends a version of Hume’s principle, as so called here. I regret that I have not found space to discuss this here.

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possibility to which what we (can) visualise is even relevant. A better reading of Hume’s principle has it that we may infer that it is possible that p if we can imagine or conceive of a situation in which it would be true that p, where we can do that if we can describe or represent such a situation without logical inconsistency or conceptual incoherence. I shall say that if this is so, it is conceiûable1 that p.19 An immediate ground for pessimism about the prospects of a possibility-based approach using conceivability1 arises from our assumption that broadly logical necessity is properly included within absolute necessity. It is a corollary of this assumption that the class of absolute possibilities is properly included within the class of logical or conceptual possibilities, and hence that whilst broadly logical possibility is a necessary condition for absolute possibility, it is not sufficient. This does not entail that there is no knowledge of absolute possibilities which does not depend upon any knowledge of absolute necessities. But it does, I think, mean that even if mastery of relevant concepts suffices not only for knowledge of some broadly logical necessities but also for knowledge of some broadly logical possibilities, it is never by itself sufficient for knowledge of any absolute possibilities. There is, for example,—and it seems to me that we can know that— no purely logical or conceptual obstacle to the supposition that there might exist talking donkeys, or blue dahlias. But whilst the first, but perhaps not the second, of these is—and can, plausibly, be known to be—not merely a broadly logical but also an absolute possibility, our grounds to think it broadly logically possible do not entitle us to the stronger conclusion that it is absolutely so. Purely logical and conceptual considerations fail to ensure the non-existence of talking donkeys; but such considerations likewise fail to exclude the existence of blue dahlias,20 or, to take more widely discussed examples, of water that isn’t H2O, gold that isn’t an element or doesn’t have atomic number 79, and the 19. Note that conceivability1 is no mere defeasible ground for absolute possibility, but strictly implies it. But while our reasons to take something to be conceivable1 will be a priori, they will in general—in my view anyway—fallible. 20. I don’t know whether this is absolutely possible—horticulturalists tell me that ‘there is no such thing as a blue dahlia’. I think they think they know that attempts to produce one aren’t just unlikely to succeed, but that they are bound to fail. If so, the example is of mild interest, as going against Hume’s principle understood in terms of visual imaginability, as well as in terms of conceivability1.

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like. If we think that it is, nevertheless, metaphysically (and so absolutely) necessary that water is H2O, so that our inability to rule out the existence of water that isn’t H2O on purely logical or conceptual grounds doesn’t justify us in taking that to be an absolute possibility, we ought equally to acknowledge that our inability similarly to rule out the existence of talking donkeys doesn’t by itself entitle us to claim that this is an absolute possibility.21 In short, the difficulty is that we need a sense of ‘conceivable’ in which (i) its being conceivable that p constitutes adequate grounds for taking it to be absolutely possible that p, but in which (ii) it can be recognised that it is conceivable that p without reliance on any assumptions about absolute necessity. But p’s conceivability1 does not give adequate grounds to think it is more than broadly logically possible, and so fails condition (i).22 Can we do better? One well-known line of theorising23 we can. Instead of taking the impossibility of water’s being other than H2O to show that what is conceivable may be impossible, we should deny that it really is conceivable that water should be something other than H2O. Seemingly successful attempts so to conceive misfire—not in the way that any attempt to conceive of cousins without shared grandparents misfires, as a result of some analytic or conceptual connection, but—because, for reasons having to do with rigidity of reference, it cannot be water one thinks of, if one thinks of something other than H2O. ‘Water’ is used rigidly to designate a certain substance—the substance which actually fills our lakes and rivers, flows from our domestic taps, etc. ‘H2O’ is likewise so used—rigidly designates the substance composed of hydrogen and oxygen as indicated. Since the substance which actually fills our lakes, etc., is H2O, one cannot, if one conceives of a transparent, colourless and nearly tasteless liquid that is not so composed, be thinking of water. 21. To stress—I am not claiming that we cannot be justified in taking this and similar things to be absolutely possible; nor am I claiming that there are no broadly logical— specifically conceptual—considerations that bear on whether it is absolutely possible. 22. Conceivability in this sense does, of course, provide adequate grounds for— because it entails—broadly logical possibility. Obviously we can be mistaken in taking something to be so conceivable. 23. Developed originally, of course, by Saul Kripke (in Kripke (1980), Lecture 3). I shall not consider how far the view, as I present it, is entirely faithful to Kripke’s text, or corresponds accurately to what he intended.

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While whether or not certain expressions are rigid may24 be reckoned a feature of their use which competent speakers can, as such and at least on reflection, recognise, and so may be reckoned as an aspect of our competence with the concepts— say, those of water and H2O—we use them to express, that does not, of course, have the result that it is after all conceptually, and so broadly logically, necessary that they are co-extensive (e.g. that it is broadly logically necessary that water is H2O), or that it is not after all conceivable1 that water isn’t H2O. We do, that is, have a different and more inclusive notion—inconceivability2 — which does not imply inconceivability1 (but is, most naturally at least, understood as implied by it). How exactly to characterise this notion, and the corresponding notion of conceivability2, is a matter of some difficulty. Our example suggests a sufficient condition: it will be inconceivable2 that p if (i) the statement that p employs rigid general terms φ and ψ and asserts their divergence in extension while (ii) φ and ψ coincide in extension. But this will not25 be a necessary condition unless all cases in which rigidity results in inconceivability2 are of, or can be somehow reduced to, this kind.26 Since I have no good proposal to offer, I shall make the simplifying assumption, for the purposes of the immediately following discussion, that this is so. So: it is inconceivable2 that p iff either it is inconceivable1 that p or conditions (i) and (ii) above hold; and conceivable2 that p iff it is not inconceivable2 that p. Inconceivability2 is not in general something we can recognise a priori. Of course, it will be recognisable a priori when it results from inconceivability1, if that is so recognisable (albeit fallibly). In other 24. Yablo ((1993), p. 3) claims that to know that ‘Alexander’s teacher’ is not rigid we must establish that it is possible that Aristotle should not have taught Alexander, but this seems to me wrong. It is perhaps suggested by Kripke’s characterisation of a rigid designator as an expression which designates the same object in all possible worlds, but this characterisation is potentially misleading—a less misleading one, which seems to me to accord with Kripke’s intentions as reflected in other things he says on the matter, has it that a designator is rigid if it is used on the understanding that it designates in modal contexts whatever it designates outside such contexts. On this account, the fact that there is no possible world at which ‘the first odd prime’ is not uniquely satisfied by 3 does not suffice to make the description a rigid designator. 25. I.e. when disjoined with a condition to the effect that the statement that p is inconceivable1. 26. Our case involves general stuff- or substance-terms—other cases include, of course, singular terms (e.g. HesperusGPhosphorus) and general sortal terms (Tigers are animals).

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cases, where inconceivability2 results from the joint satisfaction of conditions (i) and (ii), empirical grounds will (normally27) be required for thinking that φ and ψ are co-extensive. It is worth observing that the situation as regards conceivability2 differs in this respect. When we are concerned with propositions whose truth is conceivable1, their conceivability2 turns upon satisfaction of the disjunctive condition: not-(i) or not-(ii). And for this we can have a priori grounds, since we can have such grounds for the first disjunct. If we are indeed not using φ and ψ rigidly, that is something we can tell simply by reflection on how we intend them. The upshot is that, without even enquiring whether φ and ψ diverge in extension, we can tell that it is conceivable2 that they diverge. Does conceivability2 fit the possibility-based theorist’s bill? On the assumption, which I shall not challenge, that conceivability2 suffices for (i.e. strictly implies) absolute possibility,28 it may seem well-equipped to do so. For can’t we have grounds—admittedly fallible, but that is no objection—for thinking something to be conceivable2, and so possible, which do not require knowledge of any absolute necessities? Well, perhaps we can. But I am not so sure, and would like to air one reason for doubt—not, I should emphasize, a reason to doubt that beliefs about possibility can be based on grounds for conceivability2, but for doubt that such grounds are suitably independent of knowledge of necessities. We have so far taken it for granted that conceivability1 is at the service of a possibilitybased approach. Since conceivability2 requires conceivability1, the assumption is crucial. But is it true? I am not sure that one can be justified in taking it to be conceivable1 that p without some assurance that potentially relevant broadly logical necessities—those relating to concepts involved in p—do not ensure that ™p. And to have any such assurance, it seems, one needs to 27. Normally, because at least on some views, coincidence in extension of certain terms—e.g. mathematical ones—may be determinable a priori, but not solely on the basis of logical or conceptual considerations. 28. Although I shan’t challenge it, I don’t think it—or the equivalent assumption that absolute impossibility implies inconceivability2 —is obviously correct. It isn’t necessary to assume that all absolute necessities that aren’t broadly logical have their source in, or are to be explained in terms of, facts about rigidity. That might well be denied by some kinds of essentialist. What does have to be assumed is that such necessities are always reflected in the facts about rigidity which ensure satisifaction of conditions (i) and (ii) (or some refinement of them, if one drops my simplifying assumption). That is what I don’t find obvious, although it seems plausible.

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know what the potentially relevant necessities are—those necessities which are directly reflected in the requirements for the application of the concepts involved. A possibility-based theorist might counter that there is, really, no asymmetry with necessity here—that it could just as well be claimed that to be justified in taking it to be necessary that p, one needs assurance that no relevant possibilities go against one’s claim, and so needs to know what the relevant possibilities are. I have already indicated why I do not think this is always so. But it seems to me that the counter is anyway unconvincing. Whereas there is a reasonably well-circumscribed class of broadly logical necessities relevant to the assessment of a claim about conceivability1 —those necessities, appreciation of which goes along with mastery of the concepts involved—there is no similarly well-circumscribed class of possibilities relevant to the assessment of a claim about broadly logical necessity. Of course, there will be indefinitely many prima facie broadly logical possibilities, involving some or all of the concepts featured in the given necessity-claim, which competence with those concepts does not enable us to rule out. But these will not be germane to the assessment of that necessity-claim. And of course, to justify the claim that it is broadly logically necessary that p, we must justify the claim that it is not (perhaps despite some appearance to the contrary) broadly logically possible that ™p—for these are the same claim. My point is that there is no well-defined class of genuine (broadly logical) possibilities we need to review.

V Recessiûe Modal Beliefs. If the central arguments of the last two sections are good, they give some reason to think that if an asymmetrical approach can work at all, it will be one which accords epistemological priority to necessity rather than possibility. Although I am well short of confident that they are irresistible, I shall focus, in returning to the questions deferred in Sections III and IV, on a necessity-based approach. First, then, can such an approach meet my first necessary condition? As we saw, if a necessity-based theorist is to answer the charge that her account of the basis of our beliefs about possibility simply confuses having grounds to believe a proposition with lacking grounds to disbelieve it, she needs to give decent sense to the idea of a wellconducted search for countervailing necessities. And secondly, can my second condition be met in full?

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If my suggestion about the structure of our modal knowledge or beliefs is roughly correct, our first question29 concerns the composition of the class of absolute necessities which constitutes the basis of that structure. That class will comprise all absolute necessities which are basic in the sense that their necessity is not to be viewed as transmitted from and so consequential upon that of other absolute necessities—so if we can make sense of a wellconducted search through it for necessities which block a given candidate possibility-claim, then a necessity-based approach would seem to be in business. To a first approximation, it would seem that, provided this base class is finite, it should be possible to search through it for necessities potentially relevant to a given prima facie possibility—that is, necessities which directly or, typically, indirectly rule it out. To be sure, there will be plenty of scope for things to go wrong, since we may overlook relevant basic necessities altogether, fail to spot their relevant logical consequences, or be mistaken about them. But a procedure does not have to be effective30 or algorithmic in order for its implementation with negative result to provide what are admitted to be defeasible grounds for possibility claims. There is, however, a more serious complication to be faced, arising from my pivotal assumption that the class of absolute necessities is not exhausted by broadly logical or conceptual necessities, and its corollary that not all broadly logical possibilities will be genuine absolute possibilities. Our question about the composition of the base class and the complication just noted are closely related. The base class will certainly include some basic logical necessities, whose status as such derives from their constitutive role vis-a`-vis basic logical concepts. I assume it will also include a further range of conceptual necessities whose status as such similarly derives from their constitutive role vis-a`-vis their ingredient non-logical concepts. But—in line with our assumption—there will be a host of metaphysical necessities which, in contrast with necessities of the first two kinds, are knowable only a posteriori, and which are not logical consequences of just broadly logical necessities. 29. And, as we shall soon see, our second. 30. I.e. in the strict sense of being guaranteed, if properly implemented, to yield the correct answer after a finite number of steps.

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If the class of absolute necessities comprises necessities of just these kinds, however, the complication noted may be tractable— at least if we accept Kripke’s plausible suggestion about the way a posteriori knowledge of such necessities as that water is H2O, etc., may be acquired. As is well known, Kripke’s proposal is that our knowledge that, say, water is necessarily H2O, is gained inferentially, by detachment from the conditional: water is H2O → 䊐 water is H2O Since our knowledge of the minor premise is a posteriori, so is our resulting knowledge of the consequent. Whence our knowledge of the major premise? Kripke says we know it ‘by a priori philosophical analysis.’ 31 I don’t think Kripke means that it’s straightforwardly analytic (assuming some things are!) in the way that it’s analytic that vixens are female, etc. But it may be held that the conditional is knowable a priori because it is a conceptual truth of sorts,32 even if the reasons for this are less straightforward than in simple cases like the vixens and perhaps involve considerations of a kind that are conceptual only in a quite broad sense. Roughly, this is because it follows from a more general truth of the same sort, to the effect that: ∀C (water has chemical composition or nature C → 䊐 water has C) 31. Kripke(1993), p. 180. 32. This phrase, together with the qualification immediately following it, is designed to avoid engagement with an issue which I cannot take on here. I think Kripke’s conditional, and the generalised versions of it which follow in the text, could be held to be analytic—albeit perhaps less obviously so—in essentially the same sense as ‘Vixens are female’, etc. But what Kripke says provides little, if any, support for such interpretation, and is consistent with the view that the conditional is guaranteed true by considerations concerning rigidity of reference, rather than anything to do with sense or meaning, and so isn’t a conceptual truth in anything like the way in which ‘Vixens are female’ is. On this latter view, our ability to know the conditional a priori would, presumably, be explained by observing that we can know by reflection on our linguistic intentions that we are using the ingredient terms rigidly, with the result that, if they co-refer, there can be no counterfactual situation in which their references diverge (because their references, in any counterfactual situation, remain just as they are). On this view, my generalised conditionals would still be true, provided the bound variables are taken as holding place for rigid terms. Probably this is more widely accepted, both as what Kripke meant and as what is true. I don’t think the other view is obviously incorrect or indefensible. However, I don’t need to resolve that issue for present purposes, as long as it is accepted that the conditionals in question are knowable a priori, and that they are themselves absolutely necessary.

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and this in turn holds because it follows from a yet more general principle about substances: ∀S ∀C(substance S has chemical composition or nature C → 䊐 S has C) A necessity-based theorist can claim—plausibly, and in my view rightly—that Kripke’s water-H2O conditional is itself necessary, and absolutely so, and that the same goes not only for my generalisations of it, but for other general conditional principles similarly corresponding to a posteriori necessities of identity, etc. I see no reason why these necessary general principles should not be reckoned to fall within his base class. If so, then a crucial question—crucial, anyway, for the necessity-based theory—is whether, to the extent that the class of absolute necessities exceeds that of broadly logical ones, it comprises just necessities of this sort. An affirmative answer, if we could assure ourselves of its correctness, would give grounds for optimism that such an approach can both meet my first condition and meet my second condition in full. If absolute necessities are limited to necessities of these two kinds, there may be a manageable base class—comprising non-consequential broadly logical ncessities together with necessary general conditional principles of the kind just illustrated—knowledge of which suffices—in principle, and in conjunction, where appropriate, with empirical investigation—for responsible appraisal of all other claims about absolute necessity and possibility.33 33. As emphasized previously, there will be plenty of scope for error. In particular, and in addition to the sources of possible error already noted, it is not clear how we could gain assurance that we have reckoned with all necessary general principles underlying a posteriori metaphysical necessities. A very similar issue about completeness arises on the ‘principle-based’ approach developed in recent work by Christopher Peacocke (Peacocke (1997), (1999)—see especially (1999), p. 158). As Peacocke notes, some assurance of completeness is needed, if his approach is to yield a reductive account of modality. It is also required, of course, that the approach should involve no implicit reliance on modal notions. Peacocke does not commit himself to the feasibility of a reductive account, but to the extent that his approach seeks to give truth-conditions of modal propositions in terms of ‘admissible assignments’ regulated by what he calls ‘Principles of Possibility’, it may be seen as possibility-based in my sense. In addition to the general doubt about possibility-based approaches aired in Section IV, I am sceptical about Peacocke’s account for a reason well brought out by Crispin Wright—essentially, that its capacity to underpin knowledge of necessity and possibility depends not only upon the correctness, but upon the necessity, of Peacocke’s equations of necessity with truth in all admissible assignments and of possibility with truth in some such assignment, and that the account cannot explain how their necessity is known. For details, see Wright (forthcoming). There is also a problem over whether the account can avoid endorsing the controversial Barcan principle that ∀x䊐Fx → 䊐∀xFx. Peacocke believes it can do so, claiming that he can

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Since I could not—even if I had a good answer to it—take on that large question here, let conclude with some brief, and partly cautionary, remarks. First, while that question has emerged through considering the feasibility of a necessity-based approach, it seems to me clearly important in its own right, at least for anyone who accepts that there are any absolute necessities at all. Second, I have been entirely concerned with whether an asymmetrical approach can meet my two necessary conditions. Even if the arguments I’ve advanced in support of a necessity-based approach, and against the competing approach through possibility, have some force, I could not claim that, as presented, they are decisive. And even if they are, as far as they go, good, it would be premature to conclude that an necessity-based approach is right. Quite apart from the unresolved crucial question, the conditions I discussed were put forward only as individually necessary—I have given no reason to think them jointly sufficient, i.e. for thinking that there are no other necessary conditions. So I’m afraid much work remains to be done, if there is anything at all of value in the overall approach I have been pursuing.34 Department of Philosophy Uniûersity of Glasgow G12 8QQ

‘acknowledge the possibility of objects which do not actually exist, provided these are constructed from the materials of the actual world’ (cf. (1999), p. 153—Peacocke explicitly claims only that an Actualist can do so, but he needs to make the claim himself). But again, I am sceptical, in part because I haven’t been able to find an interpretation of ‘constructed from the materials of the actual world’ which is clear, plausible and otherwise suitable for Peacocke’s purpose, and partly because I think it is clear that ‘are constructed’ has to be understood as ‘can be constructed’, which looks to bring in a problematic appeal to modality again. Obviously much fuller discussion of these issues is needed than I have space for here. But it is worth noting that a necessity-based approach faces no special difficulty over merely possible objects—claims such as that there might have been many more aardvarks than there actually are can be straightforwardly true, simply because no necessities impose an upper bound on the number of aardvarks. 34. I am grateful to John Benson, Agustin Rayo and Crispin Wright for helpful discussion of some of the ideas in this paper.

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REFERENCES Hale, Bob (1996) ‘Absolute Necessities’, in James Tomberlin (ed.) Philosophical Perspectiûes Vol 10, (Cambridge MA, Blackwell), pp. 93–117. Hale, Bob (1999) ‘On Some Arguments for the Necessity of Necessity’, Mind 108, pp. 23–52. Hume, David (1888) A Treatise of Human Nature, ed. Selby-Bigge (Oxford: Clarendon Press). Kant, Immanuel (1963) Critique of Pure Reason, trans. Norman Kemp-Smith (London: Macmillan). Kripke, Saul (1980), Naming and Necessity (Oxford: Basil Blackwell). Kripke, Saul (1993) ‘Identity and Necessity’, in A.W. Moore, ed. Meaning and Reference (Oxford: Oxford University Press), pp. 162–91. Lewis, David (1973) Counterfactuals (Oxford: Basil Blackwell). Peacocke, Christopher (1997) ‘Metaphysical Necessity: Understanding, Truth and Epistemology’ Mind 106, pp. 521–74. Peacocke, Christopher (1999) Being Known (Oxford: Clarendon Press). Wright, Crispin (forthcoming) ‘On Knowing What is Necessary: Three Limitations of Peacocke’s Account’, Philosophy and Phenomenological Research, LXIV, pp. 656–63. Yablo, Stephen (1993), ‘Is Conceivability a Guide to Possibility?’. Philosophy and Phenomenological Research, LIII, pp. 1–42.