TIMOTHY WILLIAMSON

WHAT MAKES IT A HEAP?

ABSTRACT. On the epistemic view of vagueness, a vague expression has sharp boundaries whose location speakers of the language cannot recognize. The paper argues that one of the deepest sources of resistance to the epistemic view is the idea that all truths are cognitively accessible from truths in a language for natural science, conceived as precise, in a sense explained. The implications of the epistemic view for issues about the relations between vague predicates and scientific predicates are investigated.

.

On the epistemic view of vagueness, a vague expression has sharp bound- aries whose location speakers of the language cannot recognize. This is not to deny that vagueness exists; it is to assert that its underlying nature is epistemic. The view may go back to the great Stoic logician Chrysippus. 1 It permits the application of classical logic and bivalent truth-conditional semantics in their full strength to vague languages. Nevertheless, many philosophers are willing to forego these advantages and reject the epis- temic view, because they believe that it outrages compelling intuitions.

This paper does not attempt to rehearse a full-scale defence of the view, still less to survey all the alternatives. That has been done elsewhere; it involves an investigation of the general cognitive principles which explain our ignorance in borderline cases. 2 A more limited project will be under- taken here. It will be argued that one of the deepest sources of resistance to the epistemic view is the idea that all truths are cognitively accessible in a sense explained below from truths in a language for natural science, conceived as precise. Once the idea is stated clearly, its questionable nature becomes apparent. Vagueness has too often been studied as though from the standpoint o f a privileged perfectly precise language. Such an external standpoint on vagueness is arguably impossible, not even fully imagin- able. One has to study the phenomenon from within. Doing that is just what many formal theories of vagueness attempt to avoid.

Section 2 makes the connection between resistance to the epistemic view and the language of natural science. Section 3 investigates the impli- cations of the epistemic view for issues about the relations between vague predicates and scientific predicates.

Erkenntnis 44: 327-339, 1996. (~) 1996 Kluwer Academic Publishers. Printed in the Netherlands.

3 2 8 TIMOTHY WlLLIAMSON

.

The epistemic view of vagueness strikes many philosophers as simply outrageous. The elaborate arguments against it which they sometimes go on to construct seem to be rationalizations of a more basic feeling. To say this is not to deny that the arguments require a serious answer, but the present concern is the feeling itself. It is often expressed like this:

The epistemic view permits this situation: h is a heap, although h is a borderline case of a heap. But what makes h a heap?

The suggestion is that h is a heap only if something makes it a heap, and that the epistemic theorist can offer nothing to make it a heap. Alternatively, the feeling may be expressed metalinguistically, like this:

The epistemic view permits this situation: the word 'heap' applies to h, although h is a borderline case of the applica- tion of 'heap'. But what makes 'heap' apply to h?

The suggestion is that 'heap' applies to h only if something makes it apply, and that the epistemic theorist can offer nothing to make it apply.

The two objections are. not mere rephrasings of each other. A descrip- tion of the use of the word 'heap' would be directly relevant to the latter objection but not to the former, for whether h is a heap depends on its physical properties, not on the use of a word. Nevertheless, the two objec- tions spring from the same source, and raise similar issues. For simplicity, the focus will be on the first objection. After all, if it is unobjectionable for h to be a heap, then it is equally unobjectionable for the word 'heap' to apply to h, given that the word 'heap' applies to all heaps.

Suppose that the result of subtracting one grain from h is a non-heap h - , which is also a borderline case. It may further be asked what the relevant difference is between h and h - , which makes the difference between a heap and a non-heap. Presumably, if X makes h a heap, and Y makes h - a non-heap, then the relevant difference is the difference between X and Y. Thus the epistemic view can satisfactorily answer the question 'What makes the relevant difference between h and h - ? ' if and only if it can satisfactorily answer the questions 'What makes h a heap?' and 'What makes h - a non-heap?'. Moreover, the question 'What makes h - a non- heap?' raises the same theoretical issues as does the question 'What makes h a heap?'. It is therefore sufficient to focus on the latter question as an objection to the epistemic view.

The most austere reply to the objector is that what makes h a heap is simply the fact that h is a heap. In answer to the semantic objection,

WHAT MAKES IT A HEAP? 329

the corresponding reply is that what makes the word 'heap' apply to h is simply the fact that 'heap' does apply to h. An equally austere but slightly more structured reply is also possible: what makes 'heap' apply to h is the combination of the semantic fact that the condition for 'heap' to apply to something is that it be a heap with the non-semantic fact that h is a heap. These replies are likely to leave the objector baffled yet quite unsatisfied. They have somehow met the letter of the objection but missed its spirit. What, then, is the spirit of the objection? The objector is likely to complain 'It cannot be just a brute fact that h is a heap'; what does this mean and why should it be true? Questions of the form 'What makes it the case that P? ' or 'In virtue of what is it the case that P? ' or 'In what does its being the case that P consist?' should never be allowed to go unchallenged. They demand the replacement o f ' P ' by some other statement 'Q' that is somehow prior to ' P ' , without explaining what kind of priority is in question or why we should expect to be able to find anything prior to ,p,.3

The best way to elicit the objector's tacit constraints on an acceptable answer to the rhetorical question is to ask it in a non-borderline case. Let h* be clearly a heap. what makes h* a heap? The austere reply is again that it is simply the fact that h* is a heap. The objector should find this reply unsatisfying, too, for if it is legitimate here, then it is equally legitimate in the borderline case. Of course, the objector is likely to accept that h* is a heap (since h* is clearly a heap), and unlikely to accept that h is a heap (since h is a borderline case). However, to conclude from this that the austere reply is question-begging in the borderline case would be to mistake the dialectical position. The objector is trying to articulate a devastating objection to the epistemic view, to render the view untenable by asking a question that proponents of the view feel the need but lack the ability to answer on their own terms. They willingly grant that some object h is a heap but not clearly a heap (of course, when we see h we cannot recognize it as such an object). Since the challenge is to reply on that hypothesis, they are entitled to assume that h is a heap. The objector seeks more than the austere account of what makes something a heap, whether it is a clear or borderline case of a heap. The difference is that the objector assumes that a richer account is possible only in the clear case. what kind of rich account does the objector envisage for the clear case? what makes h* a heap?

The objector may want to say something like this: what makes h* a heap is the arrangement of its constituent grains. Having exactly the microphysical properties that h* actually has is a sufficient (although not necessary) condition for being a heap; h* is a heap in virtue of those more fundamental facts. More generally, the truths of everyday life supervene

3 3 0 TIMOTHY WILLIAMSON

on those of natural science; situations cannot differ in the former without differing in the latter too. 4 In Sellars' terms, the manifest image supervenes on the scientific image.

The problem for the objector is that the premise of the objection is now quite consistent with the epistemic view of vagueness. For the epistemic theorist can grant that situations cannot differ in everyday ways without differing in scientific ways, too; in the borderline case, what makes h a heap is the arrangement of its constituent grains. Thus having exactly the microphysical properties that h actually has is a sufficient (although not necessary) condition for being a heap; h is a heap in virtue of those more fundamental facts. The epistemic theorist is not forced to treat it as a brute fact that h is a heap. Similarly, if the objector were to use everyday terms and say that what makes h* a heap is the fact that h* has enough grains, then the epistemic theorist would reply that what makes h a heap is the fact that h has enough grains.

Trying a slightly different line, the objector might say that in the clear case what makes h* a heap is speakers' agreement in applying the word 'heap' to it. After all, there is no agreement in applying the word to h, for it is a borderline case. This line runs an obvious risk of confusing use and mention, h* could easily have been a heap even if speakers had not agreed in applying the word 'heap' to it, for they could easily not have used that word in that sense, while h* retained the same microphysical properties. To avoid this mistake, the objector might specify that what makes z a heap in a possible situation s is not agreement in s in applying the word 'heap' to things with the microphysical properties of z in s but agreement in the actual situation in applying the word to things with the microphysical properties o f z in s. 5

As it stands, the new formulation is still too crude, for agreement in applying a word to a thing does not generally guarantee the correctness of that application. Whole societies can sometimes be mistaken. It takes only the mildest dose of realism to recognize that there can be agreement without truth, and truth without agreement. 6 That h* is a heap does not follow from the assumption that speakers in the actual situation agree in applying the word 'heap' to h*, unless one adds the dubious further premise that this is not the kind of matter agreement about which can rest on a mistake. Indeed, it would be uncharitable to interpret the objector as attempting to reduce the semantics of a word to its use, non-semantically described, for the reducibility claim is too contentious to ground an effective objection to the epistemic view. What the objector may more reasonably assume is that semantic truths supervene on non-semantic truths about use: situations cannot differ in the former without differing in the latter too (given a

WHAT MAKES IT A HEAP.'? 331

fixed contribution from the environment). Supervenience does not entail reducibility. At least, there is no entailment when the reducing formula is required to be of finite length; the objector implicitly required a finite answer by asking the epistemic theorist to say what makes h a heap. Given such a supervenience thesis, the total use of 'heap' with respect to h* in the actual situation is sufficient for h* to be a heap. 7

For the sake of argument, set aside the suspicion that the new view irrelevantly introduces linguistic facts into an account of the wholly non- linguistic fact that h* is a heap. The more pressing problem is that the premise of the objection is now quite consistent with the epistemic view of vagueness. For the epistemic theorist may grant that situations cannot differ semantically without differing in non-semantic facts about use, too (given a fixed contribution from the environment); in the borderline case, the total use of 'heap' with respect to h in the actual situation is sufficient for h to be a heap.

Thus the dialectical position does not greatly depend on whether the objector appeals to the microphysical properties of h* or to the use of the word 'heap' in saying what makes h* a heap. The epistemic theorist can accept the relevant supervenience thesis and apply it to the borderline case. In both cases, the crucial feature of the underlying truths is that they are conceived as precise (whether perfectly precise or just precise in relevant respects is an important question that will be left open here). 8 Once a precise sufficient condition has been given for something to be a heap, the issues raised by the further question 'What makes it meet that condition?' are not directly relevant to the problem of vagueness. Thus the pertinent supervenience thesis is that vague truths supervene on precise truths: situations cannot differ vaguely without differing precisely, too. Call this claim simply 'supervenience'. Although the epistemic view does not entail supervenience, the two are quite consistent. It will be simplest to assume that the epistemic theorist accepts supervenience.

Supervenience attributes a coarse-grained metaphysical completeness to precision. It implies that any vague truth is a metaphysically necessary consequence of some set of precise truths. The epistemic theorist and the objector agree on the metaphysical completeness of precision. What they disagree on is its fine-grained cognitive completeness. In the clear case, the conclusion that h* is a heap is cognitively accessible from some precise truth about h*, in the sense that, given knowledge of the latter, we are in a position to know that h* is a heap. Although the precise truth does not itself contain the concept of a heap, we can recognize it as entailing that h* is a heap by using our grasp of that concept. In the borderline case, by contrast, the conclusion that h is a heap is not cognitively accessible

332 TIMOTHY WILLIAMSON

from any precise truth about h. Given knowledge of such a truth, we are not in a position to know that h is a heap. The objector regards a vague statement as true only if cognitively accessible from a precise truth; the vague statement passes this test only in a clear case. The epistemic theorist regards vague statements as cognitively autonomous, and as such capable of truth in both cases. Belief in the cognitive completeness of precision is outraged by the epistemic view of vagueness.

It is worth noting what the cognitive completeness of precision does not imply. First, it does not imply that all precise truths are knowable, or that all vague ones are. It says that if we knew all precise truths then we should be in a position to know all vague ones, too, but it offers no guarantee of the antecedent, and as a result none of the consequent either. For all it says, there may be precise unknowable truths of physics. Second, the cognitive completeness of precision does not imply that all vague statements are analytically equivalent to precise ones (could they be vague if they were?). It says that every vague truth is a corollary of a precise one, as 'Most As are Bs' is a corollary of '97% of all As are Bs'. It does not suggest that the vague truth entails the precise truth, as it clearly does not in that example. 'Most As are Bs' need not be analytically equivalent to any precise statement. Only in a weak sense is it claimed that whatever can be said can be said clearly. Third, it is not even implied that vague knowledge is always derivable from precise knowledge. If a vague truth is known, then it is implied that there is a precise truth knowledge of which would have put one in a position to know the vague truth, but it is not implied that knowledge of that precise truth is possible.

In spite of what it does not imply, the claim that precision is cognitively complete should be suspect. For it does imply that all truths can be told in language of a privileged kind: precise language. The vague truths can be read off the precise ones. In this sense, the claim embodies a sort o f reductionism. Such reductionism undoubtedly has its attractions. But consider some of the vague words that we use to talk about what matters to us: 'cause'; 'reason'; ' say'; 'know'; 'desire'; 'pain'; 'person'; ' society'; 'environment'. Any truth formulated in such terms can be read off precise truths, if precision is cognitively complete. The objector may concede that some concepts permit us to formulate cognitively inaccessible truths, but must deny that vague concepts are an autonomous source of cognitive inaccessibility. Any inaccessible vague truth is a mere corollary of an inaccessible precise truth. This is clearly a contentious view. One may suspect it of being at bottom a form of scientism, on which all questions can be replaced without significant loss by questions of natural science. Be

WHAT MAKES IT A HEAP? 333

that as it may, the epistemic theorist can deny the cognitive completeness of precision without absurdity.

It would be hard to show that the claim of cognitive completeness lies behind all objections to the epistemic view of vagueness. But if you are inclined to object to the view by asking 'What makes h a heap?' or 'What makes "heap" apply to h?' in a borderline case, then you should ask yourself what you would accept as an answer to such a question in a clear case, and why you think that it must have such an answer there.

.

On the epistemic view of vagueness, vague predicates stand for properties that each thing has or lacks. The cognitive incompleteness of precision raises questions about these properties. What is their metaphysical status? Can they be identified with robust natural scientific properties? If so, what stops us from locating tfleir cut-offpoints? If not, are they somehow mind- dependent? These questions will be addressed on the basis of the epistemic view. As a convenient over-simplification, the language of natural science will be assumed to be unified and precise. This false assumption is harmless here, for it in no way helps the epistemic theorist.

The same property can be presented in different ways. The property of being part of Hesperus is the property of being part of Phosphorus. For definiteness, it will be assumed that the property of being F is the property of being G if and only if it is necessary that everything is F if and only if it is G. 9 The necessity here is to be understood as metaphysical necessity: what could not have been otherwise. Thus to ask whether the property of being a heap is a natural scientific property is to ask whether 'It is necessary that everything is a heap if and only if it is G' is true on some substitution for 'G' in the language of natural science.

Some vague predicates do stand for natural scientific properties. Some molecules contain many atoms; others do not. If the molecule x contains at least as many atoms in a possible situation s as the molecule y does in a possible situation t, and y contains many atoms in t, then x contains many atoms in s. l~ Let n be the least number i such that for some molecule y and situation t, y contains exactly i atoms in t and y contains many atoms in t. The definition of 'n ' assumes the least number principle, that every non-empty set of natural numbers has a least member, but that principle is unproblematic on the epistemic view of vagueness, whose consequences are being explored here. It follows that for every molecule x and situation s, x contains many atoms in s if and only if x contains at least n atoms in s. Thus the property of containing many atoms is the

3 3 4 TIMOTHY WILLIAMSON

property of containing at least n atoms. Since 'contains at least n atoms' is a predicate of natural science (where 'n ' goes proxy for a numeral for n, not the definite description used to define 'n'), the vague predicate 'contains many atoms' stands for a property of natural science.

We may know exactly how many atoms a molecule contains, yet be unable to find out whether it contains many atoms, because unable to find out which natural number n is. One might say therefore that we are unable to find out which property the property of containing many atoms is. However, one should not put much explanatory weight on that description, for the 'know which' locution is notoriously slippery. It works best in the context o f a system of canonical terms for the items in question, one for each item. The numerals are canonical names for the natural numbers, for example. In contrast, we lack a system of canonical predicates standing for all properties. The distinction between predicates that acquaint us with the properties for which they stand and predicates that do not is unclear. An adequately explicit account will avoid the 'know which' construction. What needs to be said is that the predicate 'contains at least n atoms' is more firmly embedded in our physical theory than the predicate 'contains many atoms'; the former has richer inferential connections with other expressions, making it easier for us to know its extension. We cannot measure the application of vague concepts because we cannot calibrate our measuring instruments to them; we lack the initial connections between our theory of the instruments and the concepts whose application is to be measured. 11

We do know some property identities expressed by means of an every- day and a scientific predicate; for example, we know that the property of being some gold is the property of being some of the element with atomic number 79. It might therefore be asked why no such knowledge is avail- able for vague predicates. To answer the question, one must first identify the conditions for such knowledge to be possible, and then explain how they can fail to be met. The following account is certainly too simple, but indicates at least the general lines of the epistemic theorist's answer.

Let ' F ' be an apparently vague pre-scientific predicate that we use in talking about some field of interest. We then develop a scientific taxonomy of the properties relevant to that field, which we think of as natural prop- erties. Each property is expressed by a canonical predicate in the language of the taxonomy; these predicates form a set F. We develop workable tests for the application of predicates in F. Suppose, indeed, that each predicate 'G ' in F is perfectly precise, so that it is necessary both that everything that is G is clearly G and that everything that is not G is clearly not G (the use

WHAT MAKES IT A HEAP? 3 3 5

of 'G ' outside quotation marks should be read as subject to the condition that 'G ' means G).

Call a predicate 'G' F-eligible just in case it is necessary both that everything that is clearly F is G and that everything that is clearly not F is not G. The two occurrences of 'clearly' here refer to what is clear prior to any theoretical identification of the property of being F. F-eligibility is a necessary but not sufficient condition for necessary coextensiveness with 'F'.12 It is necessary, for being clearly the case entails being the case: necessarily, everything that is clearly F is F and everything that is clearly not F is not F. Moreover, if a predicate 'G' in F is not F-eligible then a clear counter-example to the supposed coextensiveness of ' F ' and 'G' is possible, for it is possible that something is either both clearly F and not G (and therefore clearly not G, because 'G' is precise) or both clearly not F and G (and therefore clearly G, because 'G' is precise). Thus if 'G ' in F is not F-eligible, vagueness does not prevent us from recognizing that the property of being F is not the property of being G. In the other direction, F-eligibility is not sufficient for necessary coextensiveness with 'F ' . Prior to any theoretical identification of the property for which it stands, ' F ' appears vague: it is possible for something to be neither clearly F nor clearly not F; the mere F-eligibility of 'G' does not require such a thing to be F if and only if it is G.

Now suppose that the predicate 'G' in F is F-eligible, and that no other predicate in F is. By the remarks just made, 'G' is the only predicate in F that stands a chance of being necessarily coextensive with 'F ' . Moreover, with luck, most of the things that we have agreed in applying the predicate ' F ' to will be G, because most of them will be clearly F , and most of the things that we have agreed in applying the predicate 'not F ' to will be not G, because most of them will be clearly not F. Thus, if our scientific taxonomy of the field is a good one, the only natural property underlying our use of the predicate ' F ' as a serious candidate for its reference is that marked out by the predicate 'G'. If ' F ' functions like a natural kind term in the way described by Putnam and Kripke, then it applies to just what has that property with respect to any possible world. There are no special obstacles to its so functioning. We may assume that, either by our intention or (more likely) by default, it does so. On this basis, we are defeasibly warranted in asserting that the property of being F is the property of being G. By relying on that assertion, we can clear up the apparent vagueness in 'F ' .

The property of being F could be theoretically identified because there was exactly one F-eligible predicate in 1-', the set of scientific predicates for the relevant properties. But nothing in our pre-theoretic use of ' F '

3 3 6 TIMOTHY WlLLIAMSON

gave us an a priori guarantee that F would contain exactly one F-eligible predicate. If it had contained none, or more than one, no such theoretical identification would have been possible. We should therefore have been unable to clear up the apparent vagueness in 'F ' ; it would have been deep vagueness.

How could F fail to contain exactly one F-eligible predicate? If it contains more than one, they must be mutually compatible. For all F - eligible predicates apply to whatever is clearly F, and it may be assumed that something is clearly F. If the predicates in F partition the field at each level of generality into mutually exclusive and jointly exhaustive sets, then two predicates in F are compatible only if one is at a higher level of generality than the other; it will not be easy for predicates at different levels of generality both to be F-eligible. However, there is no general reason to expect the natural properties to be mutually incompatible; in an example below, they are not. In the other case, no predicate in F is F-eligible. Thus for any predicate 'G' in F, if 'G' is general enough necessarily to apply to everything that is clearly F, then it is so general that it can apply to something that is clearly not F. This outcome is quite feasible, indeed common. Every natural property diverges clearly from the property of being F at some point.

Even if ' F ' is necessarily coextensive with a scientific predicate, there need not be exactly one F-eligible predicate in F. Suppose, for example, that ' F ' , the vague predicate 'contains many atoms', is necessarily coex- tensive with the F-eligible scientific predicate "Gn', 'contains at least n atoms' (assume for the sake of the example that if any F-eligible predicate is in F, 'Gn' is). Since ' F ' is vague, for some number m it is unclear whether molecules that contain exactly m atoms contain many atoms. It follows that the predicates "Gin" ('contains at least m atoms') and 'Gm+l' ('contains at least m + 1 atoms') are both F-eligible. For let a molecule x contain exactly i atoms; i fx is clearly F, then i > m, so x is both Gm and Gin+l; i f x is clearly not F , then i < ra, so x is neither Gm nor Gm+l. If 'Gn' is in F, then presumably "Gin' and 'Gm+l ' are too, and 1-' contains more than one F-eligible predicate. If 'Gn' is not in F, then presumably "Gin' and 'Gm+l ' are not either, and F contains no F-eligible predicate. Either way, F does not contain exactly one F-eligible predicate. Thus the standard way of identifying a property as presented by an everyday pred- icate with a property as presented by a scientific predicate is blocked. We are in no position to discover that the property of containing many atoms is the property of containing at least n atoms.

Many vague predicates may fail to be necessarily coextensive with any (finite) scientific predicate at all, even unknowably. Call the properties that

WHAT MAKES IT A HEAP? 3 3 7

such predicates stand for unscientific properties. The property of being mountainous may be an example. 13 Perhaps creatures whose sensibilities differed from ours would never talk or think about the unscientific proper- ties that we talk and think about. Should we then say that those properties depend on our sensibilities? In the counterfactual sense, they need not do so. If something is mountainous, it would still have been mountainous even if our sensibilities had been different, provided that the difference in our sensibilities made no measurable difference to the lie of the land. So much is obvious on the basis of a reflective understanding of the word 'mountainous'. That we might have used the word differently is a quite separate matter. If unscientific properties do depend on our sensibilities in some deeper sense, it is quite unclear what it is. One may suspect the idea that unscientific properties must be mind-dependent of a kind of scientism. Although such a suspicion is no substitute for an argument, it will make one hesitate to join the search for an etiolated sense in which unscientific properties are mind-dependent. Equally, one will doubt that the properties, unscientific or not, for which vague predicates stand are mind-dependent in any interesting sense. The present sketchy remarks are a challenge to explain such a sense.

The epistemic theorist takes ordinary vague terms at face value, insisting that we can understand them theoretically only by engaging in the practice of using them. From this perspective, borderline cases confront us with unknowable mind-independent truths. There is a popular interpretation or misinterpretation of the later Wittgenstein that may be summed up in the slogans 'Practices must be viewed from within' and 'Nothing is hidden'. The phenomenon of vagueness suggests that those two slogans cannot both be right. ~4

NOTES

On the Stoic treatment of Sorites paradoxes see Barnes 1982, Bumyeat 1982, Williamson 1994: 12-27. 2 See Williamson 1992 and 1994. Earlier defences of the epistemic view include Cargile 1969, 1979, Campbell 1974, Scheffler 1979, Sorensen 1988: 217-252. 3 Quotation marks will be used as comer quotes where appropriate. 4 The most controversial aspects of the supervenience thesis in the text do not directly impinge on this paper: for example, the supervenience of the mental on the physical. The argument is also not very sensitive to the technical details of the supervenience claim, with respect to which there are many options. For more on the supervenience claim see Williamson 1992: 152-153, 1994: 202-204. 5 A similar move is often made in discussions of secondary qualities; the idea goes back to Davies and Humberstone 1980. 6 The point is forcefully argued in the context of vagueness in Putnam 1983: 281-283.

3 3 8 TIMOTHY WILLIAMSON

7 For more on the supervenience of semantic truths see Williamson 1992: 154-155, 1994: 205-207, Putnam 1994: 290. 8 That scientific predicates are not perfectly precise is argued in Williamson 1994:169--170; see also Putnam 1983: 277. 9 Sober 1982 argues against the view of property identity in the text. If he is right, causal factors may play a more central role. The arguments in the text could be adapted accord- ingly. 10 The possible objection that the standard for what counts as many in a situation s should depend on what is normal in s probably confuses the context of evaluation with the context of utterance. Even if the principle in the text does over-simplify, it does .so harmlessly. 11 See Williamson 1994:216--247 for an explanation on the basis of independently con- firmed epistemic principles of our inability to judge the application of vague predicates directly in borderline cases. 12 'Necessary coextensiveness' should be taken in the sense of the account of property identity, i.e. the predicates ' F ' and 'G ' are used, not mentioned, in the formula to which the necessity operator is applied. J3 For a brief account of vague properties based on the epistemic view see Williamson 1994: 267-268. 14 An earlier version of this paper was presented to a conference on the philosophy of Hilary Putnam at Utrecht in September 1994, where Bob Hale, Charles Travis, Crispin Wright and, especially, Putnam himself made helpful comments. So did a referee. Thanks also to John Campbell for stimulating discussions of the issues.

REFERENCES

Barnes, J.: 1982, 'Medicine, Experience and Logic', in J. Barnes, J. Brunschwig, M. E Burnyeat, and M. Schofield (eds.), Science and Speculation, Cambridge, Cambridge University Press.

Burnyeat, M. F.: 1982, 'Gods and Heaps', in M. Schofield and M. C. Nussbaum (eds.), Language and Logos, Cambridge, Cambridge University Press.

Campbell, R. 1974. 'The Sorites Paradox', Philosophical Studies 26, 175-191. Cargile, J.: 1969, 'The Sorites Paradox', British Journal for the Philosophy of Science 20,

193-202. Cargile, J.: 1979, Paradoxes: A Study in Form and Predication, Cambridge, Cambridge

University Press. Davies, M. K., and Humberstone, I. L.: 1980, 'Two Notions of Necessity', Philosophical

Studies 38, 1-30. Putnam, H.: 1983, 'Vagueness and Alternative Logic', Erkenntnis 19, (1983): 297-314, as

reprinted in his Realism and Reason: Philosophical Papers Vol. 3, Cambridge, Cambridge University Press.

Putnam, H.: 1994, 'Realism without Absolutes', International Journal of Philosophical Studies 1 (1993): 179-192, as reprinted in his Words and Life, J. Conant, (ed.), Harvard University Press, Cambridge, Mass.

Scheffler, I.: 1979, Beyond the Letter, Routledge and Kegan Paul, London. Sober, E.: 1982, 'Why Logically Equivalent Predicates may Pick Out Different Properties',

American Philosophical Quarterly 19, 183-189. Sorensen, R. A.: 1988, Blindspots, Clarendon Press, Oxford.

WHAT MAKES IT A HEAP? 339

Williamson, T.: 1992, 'Vagueness and Ignorance', Aristotelian Society suppl. 66, 145-162. Williamson, T.: 1994, Vagueness, Routledge, London.

Manuscript submitted October 21, 1994 Final version received December 4, 1995

Department of Philosophy University of Edinburgh Edinburgh EH8 9JX U.K.