Knowledge of Possibility and of Necessity

I*—The Presidential Address

KNOWLEDGE OF POSSIBILITYAND OF NECESSITY

by Bob Hale

ABSTRACT I investigate two asymmetrical approaches to knowledge of abso-lute possibility and of necessity—one which treats knowledge of possibility asmore fundamental, the other according epistemological priority to necessity.Two necessary conditions for the success of an asymmetrical approach are pro-posed. I argue that a possibility-based approach seems unable to meet my secondcondition, but that on certain assumptions—including, pivotally, the assump-tion that logical and conceptual necessities, while absolute, do not exhaust theclass of absolute necessities—a necessity-based approach may be able to do so.

I

The Problem of Modal Knowledge. How can we come to know,or at least arrive at reasonable beliefs about,1 what is poss-

ible and what is necessary? Kant famously remarked2 that wemay learn from experience what is the case, but not what must be.He might have added that experience—roughly, sense-perceptionand introspection, together with what we can infer from theirdeliverances—leaves us almost equally in the dark about whatmight be. Not quite, of course, since wherever experience teachesus that p, we may safely reason, ab esse ad posse, that it is poss-ible that p. But the interesting question concerns knowledge ofunrealised possibilities (or at least knowledge of possibilities notknown to be realised). It is precisely because possibilities maygo unrealised that experience cannot teach us what must be so.Experience may inform us that p, but to know that it is not justtrue, but necessary that p, we need to know that there is no(unrealised) possibility that not-p. Whilst philosophical dis-cussion 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 tocorresponding questions about possibility, it seems clear that thecentral problems, at least, concern both modalities.

Kant’s observation draws attention to one important aspect ofthe problem, but it can be just as well raised in other ways. Interms of possible worlds, both �p and, except in special cases,�p require p’s truth at other worlds (all, or at least one) besidesthe actual world. If we follow David Lewis in adopting a full-bloodedly realist attitude to possible worlds, there is an obviousand familiar difficulty to be confronted: how, given our isolationfrom�inability to inspect other (merely possible) worlds, can weknow these truth-conditions to be met? But the problem—whileit may be aggravated by taking the truth-conditions of modalpropositions to receive their most fundamental formulation inthese terms—doesn’t depend on the adoption of possible worldsemantics, realistically construed. Even without talk of worlds,3

it is clear that knowledge of necessity and possibility goes beyondknowledge of what is actually the case—we can’t verify �p or(except in special cases) �p (just) by finding out what is (actually)the case. We need to know, in case of �p, that p would havebeen true no matter how different things might have been inother respects—the problem is to see how we can be in a positionto know a kind of generalised strong counterfactual: ∀q(q �→ p),or in the case of �p, a corresponding (existentially) generalisedweak counterfactual ∃q(q �→ p), equivalent4 to ™∀q(q �→™p).

Besides taking the problem of modal knowledge to be as mucha problem about possibility as about necessity, I shall furthertake it to concern, primarily anyway, our knowledge of abso-lute—as contrasted with (merely) relatiûe—necessity and pos-sibility. The rough idea in calling a kind of necessity—physicalnecessity, say—relative is straightforward enough: what is physi-cally 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 physicallaw, and what is physically possible is what can be so, withoutviolation of any physical law, i.e. what is logically consistent with

3. —which may, of course, be construed as involving only moderate realism in Rob-ert Stalnaker’s sense, as opposed to Lewis’s uncompromising version.

4. Defining the weak counterfactual A �→B (If it were�had been that A, it mightbe�have been that B), with David Lewis ((1973), p. 2), as ™(A �→™B).

KNOWLEDGE OF POSSIBILITY AND OF NECESSITY 3

physical law.5 The contrast here is roughly that between beingnecessary conditionally on the truth of certain propositions—ornecessary relative to a certain set of (true) propositions—andbeing unconditionally necessary, or necessary without qualifi-cation. If logical truths are necessary, they are presumablyunconditionally so, and thus absolutely necessary. Being logicalconsequences of the empty set of premisses, and so of its unionwith any other, they will also be necessary relative to any chosenset of propositions and thus relatively necessary in a great manydifferent ways or senses6—the interesting contrast is thusbetween absolute and merely relative.

In terms of the standard apparatus of possible worlds, absolutenecessity requires truth at all worlds without exception, relativenecessity demands only truth at a subset of worlds, and merelyrelative necessity consists in truth throughout a proper subset. If,as I do, we prefer to explain the distinction without relying onthat apparatus, I think we can do so as follows. Let Φ be someset 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 conse-quence of Φ that p; and it is φ-possible that p iff it is logicallyconsistent with Φ that p. φ-necessity and φ-possibility, soexplained, are clearly relative notions—to be φ-necessary is to benecessary relative to Φ, and to be φ-possible is to be possiblerelative to Φ. To a first approximation, φ-necessity is merely rela-tive iff there is a kind of possibility, ψ-possibility, such that thereare some φ-necessary propositions whose negations are ψ-poss-ible; equivalently, φ-necessity is absolute if and only if there is nokind of possibility, ψ-possibility, such that for some p, it is φ-necessary that p but ψ-possible that ™p. These explanations willnot 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 leastfor this sort of example, it might be that when something’s physically necessary, italways is necessary tout court.’ The examples of which Kripke speaks here are theor-etical identifications such as ‘Heat is motion of molecules’ and ‘Light is a stream ofphotons.’ Later, speaking of examples like gold being the element with atomic num-ber 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 hesays anything to warrant interpreting him as holding that physical necessities arealways 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 notclaiming that Jones’s failure to catch it was somehow a matterof physical—much less metaphysical or logical—necessity, butroughly, that his missing it is the only conclusion we can reason-ably draw from the evidence available to us. And if we add thathe may have caught a later one than he originally planned totake, we are not likely to be drawing attention to the mere logical(or even physical) possibility of his having done so; what wemean is roughly that his having done so isn’t ruled out by whatwe know. But while epistemic possibility in anything like thesense illustrated7 remains in play as an interpretation for ψ-pos-sibility, our suggested explanations are bound to fail of their pur-pose. Even in the case of absolute necessities (e.g. on some views,truths of logic and mathematics) knowable a priori—and evenmore obviously in the case of absolute but a posteriori necessitiesof the kind brought into prominence by Kripke and Putnam8—there seems to be no clear reason why the negation of an absol-utely necessity should not be, in our sense, epistemically possible.For all we know, for example, there is some very large even num-ber which cannot be expressed as the sum of two primes; but thisought not to preclude Goldbach’s Conjecture from being, if true,absolutely necessarily so. More generally, if the contrast betweenabsolute and merely relative kinds of necessity is to be non-emptyon both sides, we must qualify our proposed explanation: φ-necessity is absolute iff there is no non-epistemic kind of possibil-ity, ψ-possibility, such that for some p, it is φ-necessary that pbut ψ-possible that ™p.9

7. There are, arguably, stronger and weaker ways of understanding epistemic pos-sibility, as expressed by such ordinary locutions as ‘For all we know, p’ and ‘That pis not ruled out by what we know.’ In addition, there is the rather different notionof epistemic possibility suggested by Kripke ((1980), Lecture 3, p. 141ff.) when hediscusses the (apparent) possibility of water’s turning out to be something other thanH2O, 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 widespreadpractice—restrict my application of the term to a posteriori necessities of the Kripke-Putnam 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. Notethat on at least one way of construing it, and assuming our knowledge extendsbeyond just logic, epistemic possibility and necessity qualify as merely relative in thissense.


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

II

Asymmetrical Approaches to the Problem. An account of howwe may come to know�justifiably believe that something is neces-sarily so is not as such, and does not automatically lead to, anaccount of how we may know anything not to be so, and likewisefor possibility. But if necessity and possibility are interdefinablein the usual way—or at least if we have the equivalences�p→←™�™p, �p→←™�™p—then it appears that we don’tneed four separate accounts of how we know about possibilityand about the absence of necessity and about necessity and aboutthe absence of possibility. Still, we do seem to be left with twoquestions:

How do we know, for given p, that �p�™�™p?How do we know, for given p, that �p�™�™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 oneof these questions over the other—treating knowledge of necess-ity, say, as more fundamental than knowledge of possibility, orvice versa. To put the idea roughly and suggestively, we mightthink of possibility as more revealing characterised as justabsence of necessity, so that knowledge of possibilities is primar-ily knowledge of the absence of any relevant necessities—oroppositely, we may view necessity as just absence of possibility,and knowledge of necessity as primarily knowledge of theabsence of any relevant possibility.

This suggests a distinction between two broadly opposedasymmetrical 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 funda-mental, and possibility-based approaches, which accord priorityto knowledge of possibilities. In an asymmetrical approach, oneof the modalities is taken as dominant in the sense, roughly, thatour most basic modal knowledge is taken to be of truths in whoseexpression the dominant modal operator is principal (i.e. haswidest scope), and the other is recessiûe in the sense, againroughly, that knowledge of modal truths in whose expressionthat operator is principal is essentially a matter of well-foundedbelief 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 suchapproaches over the other?—or, more generally, reasons to viewour knowledge of possibility, say, as in any way more basic thanour knowledge of necessity, or vice versa? To give these questionssome more definite shape and focus, I shall begin by proposingtwo necessary conditions that an asymmetric approach mustmeet.

III

Two Necessary Conditions. One necessary condition emergesfrom consideration of an obvious objection to the asymmetricalstrategy. According to that strategy, beliefs about the recessivemodality are to be justified by appeal to the fact that we havefound no dominant modal truths which rule out the recessivemodal claim up for assessment. For example, on a necessity-based version of the approach, each particular possibility claimis to be justified by appeal to the fact that we know of no necess-ity which rules it out. The obvious objection is that this simplyand grotesquely conflates lack of grounds to believe it impossiblethat p with grounds to believe that it is possible—isn’t that justa special case of the obviously bad move from: we haûe no reasonto belieûe ™p to: we haûe reason to belieûe p?11

An asymmetric theorist can scarcely deny that the latter shiftis bad, so she must dispute the invidious comparison. But it is atleast not obvious that she cannot do so. She may begin by

11. This kind of objection is brought by Stephen Yablo against taking conceivabilityas grounds for belief in possibility, when ‘conceivable that p’ is interpreted as ‘believ-able that p’ or as ‘believable that p is possible’. See Yablo (1993), pp. 8, 20.


observing that there are cases in which what looks superficiallylike this bad shift is defensible—cases in which our failure tofind evidence for a proposition does constitute a good groundfor believing its contradictory. For example, our failure, aftersearching carefully for evidence of the burglar’s having made hisentry through the French windows—signs of the windows havingbeen forced, footprints in the adjacent flowerbed, etc.—and find-ing none, we may justifiably (albeit fallibly) conclude that hedidn’t get in that way. The general thought is that whilst merelack of evidence for ™p never, in and by itself, constitutes reasonto believe p, it can do so, in the context of a well-directed andthorough search for evidence in p’s favour. If, applying this tothe modal case, the asymmetric theorist is to rebut the objection,she must make out that in following the route she proposes, weneed not be simply passing, gratuitously, from mere lack ofknowledge of any relevant countervailing necessities, but mayhave looked responsibly for them and failed to find any. Thisrequires that we can give decent operational sense to the idea ofa well-directed and thorough search for necessities relevant tothe assessment of a given possibility claim (or, of course, for pos-sibilities relevant to a given necessity claim, in case of a possibil-ity-based asymmetric approach). I shall return later to thequestion whether this necessary condition—hereafter the FirstCondition—can be met. First, I want to introduce a second neces-sary condition.

Given that an asymmetrical approach will treat claims aboutits dominant modality as basic, and as, in effect, a key part ofthe essential background against which claims about its recessivemodality are assessed, it may seem to be a further necessary con-dition for its successful implementation that there should be away of coming to know dominant modal truths which neitherinvolves nor presupposes knowledge of any recessive modaltruths. Something like this is surely correct. But the condition asstated is open to a stronger and a weaker interpretation. Shouldit be taken as requiring that there be a way of coming to knowdominant modal truths by means of which we can gain knowl-edge of any such truth, independently of any knowledge ofrecessive modal truths? Or should it be taken to require only thatthere be a such a way of coming to know some dominant modaltruths? The stronger condition is too strong, but the weaker con-dition, while indeed necessary, is needlessly weak. The stronger

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condition is too strong because it overlooks the possibility thatmodal knowledge may be structured—in the sense, roughly, thatonce certain dominant modality claims are secured, this enablesthe assessment of some recessive claims which are in turnrequired for the assessment of further dominant claims, whichmay in their turn bear upon the assessment of yet further recess-ive claims. With this possibility in view, we can formulate anecessary condition which is weaker than the strong conditionbut stronger than the weak one. This requires—my Second Con-dition—that there be a base class of dominant modal truthswhich meet two conditions: (i) they can be known withoutreliance upon any recessive modality claims, and (ii) they arecollectively strong enough to support the superstructure of modalknowledge to be erected over them.

In what follows, I explore the prospects for an asymmetricalapproach in the light of these conditions, beginning with thesecond. First, however, I want to make explicit an assumption—perhaps quite widely accepted, at least amongst those whobelieve in absolute necessities, but certainly not uncontro-versial—which will play an important role. This is that broadlylogical necessities (which comprise, in my usage, just narrowly orstrictly 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, thatbroadly logical necessities are indeed absolute,12 and second, thatthey do not exhaust the class of absolute necessities. Many whoaccept the second assumption will do so because they take meta-physical necessities13—e.g. Hesperus is Phosphorus, Water isH2O, etc.—to be absolute but not broadly logical. Although Ithink the acknowledgement of different kinds of absolute necess-ity 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 arethose which hold at all worlds, metaphysical and broadly logical necessities can’t bedistinguished by reference to the worlds through which they hold. One must insteadthink of the distinction as epistemological, or perhaps as having to do with differentsources of necessity. For discussion of a quite different sort of difficulty, see Hale(1996), Sections 4–7.


IV

Asymmetrical Approaches and the Second Condition. As appliedto a necessity-based approach, my second condition requires theexistence of a class of absolute necessities which can be knownwithout reliance upon any knowledge of possibilities and whichis rich enough to serve as a basis for all other modal knowledge.Under the assumption that broadly logical necessities are abso-lute, it is plausible that a necessity-based approach meets the firstpart of this condition, and so at least clears the first hurdle.

In very many cases in which we know it to be broadly logicallynecessary that p, our knowledge will be inferential—we knowthat it is necessary that p because we have inferred that p fromsome further premiss q which we know to be necessary by stepswhich are (known to be) necessarily truth-preserving. In suchcases, mastery of the concepts involved will not suffice for knowl-edge that it is necessary that p unless that mastery is taken toinclude a capacity to perform or ratify the requisite inferentialsteps. And even if some degree of inferential competence isrequired for mastery of logical concepts, it is implausible to holdthat this would suffice for recognition of all broadly logical butconsequential necessities. But that does not matter for presentpurposes, since the second condition does not require that allknowledge of absolute necessities is independent of knowledgeof possibilities.15 It would suffice to meet the first part of oursecond condition that knowledge of some broadly logical necessi-ties demands no more than mastery of the concepts involved. Iclaim that there is a base class of necessities meeting that con-dition. This comprises necessities which constitute or reflect waysin which certain ingredient concepts are fixed. As an example, wemay take the necessity that if a conjunction is true each of itsconjuncts is so. Since the concept of conjunction is the conceptof a function which takes the value truth only if both its argu-ments are true, someone who possesses that concept is in a posi-tion directly to see that a conjunction cannot be true withouteach of its conjuncts being so—i.e. that any compound prop-osition that is true despite the falsehood of one or its immediate

15. For the same reason, it is unnecessary to maintain—although it might anywaybe argued—that our knowledge of logical consequence relations need not rest uponany knowledge of possibilities.

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constituents cannot be a conjunction. Examples of this sortstand, it seems to me, in significant contrast with others in whichrecognition of the necessity of a truth calls for a more or lesssubstantial train of reasoning. A thinker’s need to be convinced,by rehearsal of the obvious reasoning, that the square of an oddnumber must itself be odd (not to mention more recondite cases),does not eo ipso raise any doubt about her grasp of the conceptsof integer, oddness and square. But someone’s affecting to thinkthat something might be a true conjunction with a false conjunctwould immediately put it in doubt that it was conjunction shewas thinking about.

It is a further question whether a necessity-based approach canmeet the second condition in full. I shall find it convenient todefer that question pro tem, as I think it best approached in thelight of my discussion of the first condition. So I want instead toturn now to the possibility-based approach, and considerwhether it can do at least as well as its competitor vis-a-vis thesecond condition.

Obviously there is some knowledge of possibility which is inde-pendent of any knowledge of necessity—ab esse ad posse16—butequally obviously, this is insufficient for a possibility-basedapproach, which requires knowledge of unrealised possibilities.Hume notoriously claimed it to be

... an establish’d maxim in metaphysics, That whateûer the mindclearly conceiûes, includes the idea of possible existence, or in otherwords, that nothing we imagine is absolutely impossible.17

Hume’s principle18 has been thought clearly incorrect or at leastopen to decisive objection. Even so, it seems to me that it, orsomething close to it, has to be upheld if a possibility-basedapproach is to meet even the first part of the second condition.I shall take it as obvious without argument that we may put asideany interpretation of conceiving or imagining in terms of havingvisual 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—torefer 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 havenot found space to discuss this here.


possibility to which what we (can) visualise is even relevant. Abetter reading of Hume’s principle has it that we may infer thatit is possible that p if we can imagine or conceive of a situationin which it would be true that p, where we can do that if we candescribe or represent such a situation without logical inconsis-tency or conceptual incoherence. I shall say that if this is so, it isconceiûable1 that p.19

An immediate ground for pessimism about the prospects of apossibility-based approach using conceivability1 arises from ourassumption that broadly logical necessity is properly includedwithin absolute necessity. It is a corollary of this assumption thatthe class of absolute possibilities is properly included within theclass of logical or conceptual possibilities, and hence that whilstbroadly logical possibility is a necessary condition for absolutepossibility, it is not sufficient. This does not entail that there isno knowledge of absolute possibilities which does not dependupon any knowledge of absolute necessities. But it does, I think,mean that even if mastery of relevant concepts suffices not onlyfor knowledge of some broadly logical necessities but also forknowledge of some broadly logical possibilities, it is never byitself sufficient for knowledge of any absolute possibilities. Thereis, for example,—and it seems to me that we can know that—no purely logical or conceptual obstacle to the supposition thatthere might exist talking donkeys, or blue dahlias. But whilst thefirst, but perhaps not the second, of these is—and can, plausibly,be known to be—not merely a broadly logical but also an abso-lute possibility, our grounds to think it broadly logically possibledo not entitle us to the stronger conclusion that it is absolutelyso. Purely logical and conceptual considerations fail to ensurethe non-existence of talking donkeys; but such considerationslikewise fail to exclude the existence of blue dahlias,20 or, to takemore widely discussed examples, of water that isn’t H2O, goldthat 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 willbe 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 attemptsto 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 termsof visual imaginability, as well as in terms of conceivability1.

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like. If we think that it is, nevertheless, metaphysically (and soabsolutely) necessary that water is H2O, so that our inability torule out the existence of water that isn’t H2O on purely logicalor conceptual grounds doesn’t justify us in taking that to be anabsolute possibility, we ought equally to acknowledge that ourinability similarly to rule out the existence of talking donkeysdoesn’t by itself entitle us to claim that this is an absolutepossibility.21 In short, the difficulty is that we need a sense of‘conceivable’ in which (i) its being conceivable that p constitutesadequate grounds for taking it to be absolutely possible that p,but in which (ii) it can be recognised that it is conceivable that pwithout reliance on any assumptions about absolute necessity.But p’s conceivability1 does not give adequate grounds to thinkit 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 thanH2O to show that what is conceivable may be impossible, weshould deny that it really is conceivable that water should besomething other than H2O. Seemingly successful attempts so toconceive misfire—not in the way that any attempt to conceive ofcousins without shared grandparents misfires, as a result of someanalytic or conceptual connection, but—because, for reasonshaving to do with rigidity of reference, it cannot be water onethinks of, if one thinks of something other than H2O. ‘Water’ isused rigidly to designate a certain substance—the substancewhich actually fills our lakes and rivers, flows from our domestictaps, etc. ‘H2O’ is likewise so used—it rigidly designates the sub-stance composed of hydrogen and oxygen as indicated. Since thesubstance which actually fills our lakes, etc., is H2O, one cannot,if one conceives of a transparent, colourless and nearly tastelessliquid 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 similarthings 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 tak-ing something to be so conceivable.

23. Developed originally, of course, by Saul Kripke (in Kripke (1980), Lecture 3). Ishall not consider how far the view, as I present it, is entirely faithful to Kripke’stext, or corresponds accurately to what he intended.


While whether or not certain expressions are rigid may24 bereckoned a feature of their use which competent speakers can, assuch and at least on reflection, recognise, and so may bereckoned as an aspect of our competence with the concepts—say, those of water and H2O—we use them to express, that doesnot, 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 thatit 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 atleast, understood as implied by it). How exactly to characterisethis notion, and the corresponding notion of conceivability2, isa matter of some difficulty. Our example suggests a sufficientcondition: it will be inconceivable2 that p if (i) the statement thatp employs rigid general terms φ and ψ and asserts their diver-gence in extension while (ii) φ and ψ coincide in extension. Butthis will not25 be a necessary condition unless all cases in whichrigidity results in inconceivability2 are of, or can be somehowreduced to, this kind.26 Since I have no good proposal to offer,I shall make the simplifying assumption, for the purposes of theimmediately following discussion, that this is so. So: it isinconceivable2 that p iff either it is inconceivable1 that p or con-ditions (i) and (ii) above hold; and conceivable2 that p iff it is notinconceivable2 that p.

Inconceivability2 is not in general something we can recognise apriori. Of course, it will be recognisable a priori when it results frominconceivability1, if that is so recognisable (albeit fallibly). In other

24. Yablo ((1993), p. 3) claims that to know that ‘Alexander’s teacher’ is not rigidwe 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 ofa rigid designator as an expression which designates the same object in all possibleworlds, 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 hesays on the matter, has it that a designator is rigid if it is used on the understandingthat it designates in modal contexts whatever it designates outside such contexts. Onthis account, the fact that there is no possible world at which ‘the first odd prime’ isnot 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 isinconceivable1.

26. Our case involves general stuff- or substance-terms—other cases include, ofcourse, singular terms (e.g. HesperusGPhosphorus) and general sortal terms (Tigersare animals).

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cases, where inconceivability2 results from the joint satisfaction ofconditions (i) and (ii), empirical grounds will (normally27) berequired for thinking that φ and ψ are co-extensive. It is worthobserving that the situation as regards conceivability2 differs inthis respect. When we are concerned with propositions whose truthis conceivable1, their conceivability2 turns upon satisfaction of thedisjunctive condition: not-(i) or not-(ii). And for this we can havea priori grounds, since we can have such grounds for the firstdisjunct. If we are indeed not using φ and ψ rigidly, that is some-thing we can tell simply by reflection on how we intend them. Theupshot is that, without even enquiring whether φ and ψ diverge inextension, we can tell that it is conceivable2 that they diverge.

Does conceivability2 fit the possibility-based theorist’s bill? Onthe assumption, which I shall not challenge, that conceivability2suffices for (i.e. strictly implies) absolute possibility,28 it mayseem well-equipped to do so. For can’t we have grounds—admit-tedly fallible, but that is no objection—for thinking somethingto be conceivable2, and so possible, which do not require knowl-edge of any absolute necessities?

Well, perhaps we can. But I am not so sure, and would like toair one reason for doubt—not, I should emphasize, a reason todoubt that beliefs about possibility can be based on grounds forconceivability2, but for doubt that such grounds are suitablyindependent of knowledge of necessities. We have so far taken itfor granted that conceivability1 is at the service of a possibility-based approach. Since conceivability2 requires conceivability1,the assumption is crucial. But is it true? I am not sure that onecan be justified in taking it to be conceivable1 that p withoutsome assurance that potentially relevant broadly logical necessi-ties—those relating to concepts involved in p—do not ensurethat ™p. And to have any such assurance, it seems, one needs to

27. Normally, because at least on some views, coincidence in extension of certainterms—e.g. mathematical ones—may be determinable a priori, but not solely on thebasis of logical or conceptual considerations.

28. Although I shan’t challenge it, I don’t think it—or the equivalent assumptionthat absolute impossibility implies inconceivability2—is obviously correct. It isn’tnecessary to assume that all absolute necessities that aren’t broadly logical have theirsource in, or are to be explained in terms of, facts about rigidity. That might well bedenied by some kinds of essentialist. What does have to be assumed is that suchnecessities are always reflected in the facts about rigidity which ensure satisifactionof conditions (i) and (ii) (or some refinement of them, if one drops my simplifyingassumption). That is what I don’t find obvious, although it seems plausible.


know what the potentially relevant necessities are—those necessi-ties which are directly reflected in the requirements for the appli-cation of the concepts involved. A possibility-based theoristmight counter that there is, really, no asymmetry with necessityhere—that it could just as well be claimed that to be justified intaking it to be necessary that p, one needs assurance that norelevant possibilities go against one’s claim, and so needs toknow what the relevant possibilities are. I have already indicatedwhy I do not think this is always so. But it seems to me that thecounter is anyway unconvincing. Whereas there is a reasonablywell-circumscribed class of broadly logical necessities relevant tothe assessment of a claim about conceivability1—those necessi-ties, appreciation of which goes along with mastery of the con-cepts involved—there is no similarly well-circumscribed class ofpossibilities relevant to the assessment of a claim about broadlylogical necessity. Of course, there will be indefinitely many primafacie broadly logical possibilities, involving some or all of theconcepts featured in the given necessity-claim, which competencewith those concepts does not enable us to rule out. But these willnot be germane to the assessment of that necessity-claim. And ofcourse, to justify the claim that it is broadly logically necessarythat p, we must justify the claim that it is not (perhaps despitesome appearance to the contrary) broadly logically possible that™p—for these are the same claim. My point is that there is nowell-defined class of genuine (broadly logical) possibilities weneed to review.

VRecessiûe Modal Beliefs. If the central arguments of the last twosections are good, they give some reason to think that if an asym-metrical approach can work at all, it will be one which accordsepistemological 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 SectionsIII and IV, on a necessity-based approach. First, then, can suchan approach meet my first necessary condition? As we saw, if anecessity-based theorist is to answer the charge that her accountof the basis of our beliefs about possibility simply confuses hav-ing grounds to believe a proposition with lacking grounds to dis-believe it, she needs to give decent sense to the idea of a well-conducted search for countervailing necessities. And secondly,can my second condition be met in full?

BOB HALE16

If my suggestion about the structure of our modal knowledgeor beliefs is roughly correct, our first question29 concerns thecomposition of the class of absolute necessities which constitutesthe basis of that structure. That class will comprise all absolutenecessities which are basic in the sense that their necessity is notto be viewed as transmitted from and so consequential upon thatof other absolute necessities—so if we can make sense of a well-conducted search through it for necessities which block a givencandidate possibility-claim, then a necessity-based approachwould seem to be in business. To a first approximation, it wouldseem that, provided this base class is finite, it should be possibleto search through it for necessities potentially relevant to a givenprima facie possibility—that is, necessities which directly or,typically, indirectly rule it out. To be sure, there will be plentyof scope for things to go wrong, since we may overlook relevantbasic necessities altogether, fail to spot their relevant logicalconsequences, or be mistaken about them. But a procedure doesnot have to be effective30 or algorithmic in order for itsimplementation with negative result to provide what are admit-ted to be defeasible grounds for possibility claims. There is, how-ever, a more serious complication to be faced, arising from mypivotal assumption that the class of absolute necessities is notexhausted by broadly logical or conceptual necessities, and itscorollary that not all broadly logical possibilities will be genuineabsolute possibilities.

Our question about the composition of the base class and thecomplication just noted are closely related. The base class willcertainly include some basic logical necessities, whose status assuch derives from their constitutive role vis-a-vis basic logicalconcepts. I assume it will also include a further range of concep-tual necessities whose status as such similarly derives from theirconstitutive role vis-a-vis their ingredient non-logical concepts.But—in line with our assumption—there will be a host of meta-physical necessities which, in contrast with necessities of the firsttwo kinds, are knowable only a posteriori, and which are notlogical 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 thecorrect answer after a finite number of steps.


If the class of absolute necessities comprises necessities of justthese kinds, however, the complication noted may be tractable—at least if we accept Kripke’s plausible suggestion about the waya posteriori knowledge of such necessities as that water is H2O,etc., may be acquired. As is well known, Kripke’s proposal isthat our knowledge that, say, water is necessarily H2O, is gainedinferentially, by detachment from the conditional:

water is H2O→� water is H2O

Since our knowledge of the minor premise is a posteriori, so isour resulting knowledge of the consequent. Whence our knowl-edge of the major premise? Kripke says we know it ‘by a prioriphilosophical analysis’.31 I don’t think Kripke means that it’sstraightforwardly analytic (assuming some things are!) in the waythat it’s analytic that vixens are female, etc. But it may be heldthat the conditional is knowable a priori because it is a concep-tual truth of sorts,32 even if the reasons for this are less straight-forward than in simple cases like the vixens and perhaps involveconsiderations of a kind that are conceptual only in a quite broadsense. Roughly, this is because it follows from a more generaltruth of the same sort, to the effect that:

∀C (water has chemical composition or natureC→� water has C)

31. Kripke(1993), p. 180.

32. This phrase, together with the qualification immediately following it, is designedto avoid engagement with an issue which I cannot take on here. I think Kripke’sconditional, and the generalised versions of it which follow in the text, could be heldto 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 suchinterpretation, and is consistent with the view that the conditional is guaranteed trueby considerations concerning rigidity of reference, rather than anything to do withsense 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 prioriwould, presumably, be explained by observing that we can know by reflection on ourlinguistic intentions that we are using the ingredient terms rigidly, with the resultthat, if they co-refer, there can be no counterfactual situation in which their referencesdiverge (because their references, in any counterfactual situation, remain just as theyare). On this view, my generalised conditionals would still be true, provided thebound variables are taken as holding place for rigid terms. Probably this is morewidely accepted, both as what Kripke meant and as what is true. I don’t think theother view is obviously incorrect or indefensible. However, I don’t need to resolvethat issue for present purposes, as long as it is accepted that the conditionals inquestion are knowable a priori, and that they are themselves absolutely necessary.

BOB HALE18

and this in turn holds because it follows from a yet more generalprinciple about substances:

∀S∀C(substance S has chemical composition or natureC→� S has C)

A necessity-based theorist can claim—plausibly, and in myview rightly—that Kripke’s water-H2O conditional is itself neces-sary, and absolutely so, and that the same goes not only for mygeneralisations of it, but for other general conditional principlessimilarly corresponding to a posteriori necessities of identity, etc.I see no reason why these necessary general principles should notbe reckoned to fall within his base class. If so, then a crucialquestion—crucial, anyway, for the necessity-based theory—iswhether, to the extent that the class of absolute necessitiesexceeds that of broadly logical ones, it comprises just necessitiesof this sort. An affirmative answer, if we could assure ourselvesof its correctness, would give grounds for optimism that such anapproach can both meet my first condition and meet my secondcondition in full. If absolute necessities are limited to necessitiesof these two kinds, there may be a manageable base class—com-prising non-consequential broadly logical ncessities together withnecessary general conditional principles of the kind just illus-trated—knowledge of which suffices—in principle, and in con-junction, where appropriate, with empirical investigation—forresponsible appraisal of all other claims about absolute necessityand 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 wecould gain assurance that we have reckoned with all necessary general principlesunderlying a posteriori metaphysical necessities. A very similar issue about complete-ness arises on the ‘principle-based’ approach developed in recent work by ChristopherPeacocke (Peacocke (1997), (1999)—see especially (1999), p. 158). As Peacocke notes,some assurance of completeness is needed, if his approach is to yield a reductiveaccount of modality. It is also required, of course, that the approach should involveno implicit reliance on modal notions. Peacocke does not commit himself to thefeasibility of a reductive account, but to the extent that his approach seeks to givetruth-conditions of modal propositions in terms of ‘admissible assignments’ regulatedby what he calls ‘Principles of Possibility’, it may be seen as possibility-based in mysense. In addition to the general doubt about possibility-based approaches aired inSection IV, I am sceptical about Peacocke’s account for a reason well brought outby Crispin Wright—essentially, that its capacity to underpin knowledge of necessityand possibility depends not only upon the correctness, but upon the necessity, ofPeacocke’s equations of necessity with truth in all admissible assignments and ofpossibility with truth in some such assignment, and that the account cannot explainhow their necessity is known. For details, see Wright (forthcoming). There is also aproblem over whether the account can avoid endorsing the controversial Barcan prin-ciple that ∀x�Fx → �∀xFx. Peacocke believes it can do so, claiming that he can


Since I could not—even if I had a good answer to it—take onthat large question here, let conclude with some brief, and partlycautionary, remarks. First, while that question has emergedthrough considering the feasibility of a necessity-based approach,it seems to me clearly important in its own right, at least foranyone who accepts that there are any absolute necessities at all.Second, I have been entirely concerned with whether an asym-metrical approach can meet my two necessary conditions. Evenif the arguments I’ve advanced in support of a necessity-basedapproach, and against the competing approach through possibil-ity, have some force, I could not claim that, as presented, theyare decisive. And even if they are, as far as they go, good, itwould be premature to conclude that an necessity-basedapproach is right. Quite apart from the unresolved crucial ques-tion, the conditions I discussed were put forward only as indi-vidually necessary—I have given no reason to think them jointlysufficient, i.e. for thinking that there are no other necessary con-ditions. So I’m afraid much work remains to be done, if thereis anything at all of value in the overall approach I have beenpursuing.34

Department of PhilosophyUniûersity of GlasgowG12 8QQ

‘acknowledge the possibility of objects which do not actually exist, provided theseare constructed from the materials of the actual world’ (cf. (1999), p. 153—Peacockeexplicitly claims only that an Actualist can do so, but he needs to make the claimhimself). But again, I am sceptical, in part because I haven’t been able to find aninterpretation of ‘constructed from the materials of the actual world’ which is clear,plausible and otherwise suitable for Peacocke’s purpose, and partly because I thinkit is clear that ‘are constructed’ has to be understood as ‘can be constructed’, whichlooks to bring in a problematic appeal to modality again. Obviously much fullerdiscussion of these issues is needed than I have space for here. But it is worth notingthat a necessity-based approach faces no special difficulty over merely possibleobjects—claims such as that there might have been many more aardvarks than thereactually are can be straightforwardly true, simply because no necessities impose anupper bound on the number of aardvarks.

34. I am grateful to John Benson, Agustin Rayo and Crispin Wright for helpfuldiscussion of some of the ideas in this paper.

BOB HALE20

REFERENCES

Hale, Bob (1996) ‘Absolute Necessities’, in James Tomberlin (ed.) PhilosophicalPerspectiûes Vol 10, (Cambridge MA, Blackwell), pp. 93–117.

Hale, Bob (1999) ‘On Some Arguments for the Necessity of Necessity’, Mind108, 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 Limit-

ations of Peacocke’s Account’, Philosophy and Phenomenological Research,LXIV, pp. 656–63.

Yablo, Stephen (1993), ‘Is Conceivability a Guide to Possibility?’, Philosophyand Phenomenological Research, LIII, pp. 1–42.

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Knowledge of Possibility and of Necessity