Distinctions Without a Difference

The Southern Journal of Philosophy (1994) Vol. XXXIII, Supplement

Distinctions Without a Difference

Vann McGee Brian McLaughlin Rutgers University

Imagine a sequence of 10,000 exactly similar vats of red dye. One drop of yellow dye is stirred into the second vat, two drops into the third vat, three drops in the fourth vat, and so on, until 9,999 drops of yellow dye a re meticulously stirred into the 10,000th vat. Then, 10,000 exactly similar white ce- ramic tiles a re lowered into the vats and held at the same depth for the same length of time. The dyed tiles are then set out in a row to dry. Visual inspection reveals no difference at all in color between any tile and its immediate neighbors, yet the first tile is unmistakably red, the last tile unmistakably orange.

Let us say tha t there is no visible difference in color between two tiles if and only if no normal human visual observer could ascertain by visual inspection any difference in how the tiles look in color when viewed from the same angle and dis- tance, and under the same conditions of illumination, no matter how carefully and thoroughly she inspected them, and no matter how well trained she were in discerning fine color distinctions and in applying color concepts. We are using the number 10,000 as proxy for some number large enough so that there is no visible difference in color between adjacent tiles. Inasmuch as there are no other differences-they are alike in shape, size, texture, and luster-one can perceive no difference at all between adjacent tiles, other than their location.

The following premise, which we will call a sorites premise, is intuitively obvious:

If there is no visible difference in color between two tiles, then if one tile looks red to you (a normal human visual per- ceiver in normal circumstances of visual observation), the other does as well.

From this premise, together with the unmistakably t rue premise ‘The first tile looks red to you’ and 9,999 auxiliary premises of the form ‘There is no visible difference in color between the nth tile and the (n+l)st,’ we derive the unmistakably false conclusion, ‘The 10,000th tile looks red to you’.

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McGee and McLaughlin

LOGIC OF THE SORITES ARGUMENT

We have a n argument t h a t leads from what seem t o be obviously correct premises to an obviously false conclusion. Our options are either t o accept the conclusion, to reject one or more of the premises, or to deny the validity of the argument.

The trouble with denying the validity of the argument is that the argument requires so little by way of logical resources. The only rules it requires a re universal specification and modus ponens.’

One might credibly contend tha t , even though universal specification is a valid rule of inference, i t is not being correctly applied here, because the sorites premise should not be understood as a strict universal generalization, but rather as a generalization tha t admits exceptions, like ‘Calves have one head.’2 The sorites paradox arises from excessive generalization. Regarded as an approximate generalization, the sorites premise is quite correct, and we are right to accept it. But we go too far when we treat the premise as if it were strictly uni- versally valid. This seems to us a reasonable response. I t ex- plains why the strictly universal version of the sorites premise is intuitively so credible-nearly all the instances of the premise are true-yet it does not saddle us with the premise’s outlandish consequences. But this is surely not the end of the matter. One wants to know where these exceptions to the sorites premise come from. For this purpose, we need a response to the version of the paradox in which the sorites premise is explicitly formulated as a strict universal generalization.

We could reformulate the argument so t h a t i t avoided universal specification by replacing the sorites premise by 9,999 of its instances, so that the only rule required was modus ponens; but the gain in conceptual economy is only apparent. Our intuitive reasoning in the sorites paradox makes use of universal specification. We do not accept the 9,999 premises because we have entertained each of them separately and smiled upon them; we accept them because we see how to derive them from the universal generalization. The formulation that avoids universal specification merely hides part of our intuitive reasoning outside the formalization.

The version of the paradox which replaces the sorites premise by 9,999 of its instances invites a n interesting response. Perhaps each of the 9,999 premises, while extremely credible, has subjective probability slightly less than one (or would have subjective probability slightly less than one for a fully rational agent). Because there are so many premises, i t is perfectly possible for each of the premises to have probability near one even though the conclusion has probability near zero.

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The sorites paradox is revealed as a n elaborately disguised form of the lottery para do^.^ This seems to us a promising response, and we will come back to it; but, for now, let us focus our attention on the version of the argument that starts from a strict universal generalization.

As we noted, there is no denying that our sorites argument is valid. So, we have no choice but to accept the conclusion or to reject one or more premises. No one, as far as we know, is willing to force upon you the conclusion that the last tile looks red to you. So we must reject a premise.

There are those4 who would deny tha t the first tile looks red to you, the premise we precipitously acclaimed ‘unmistakably true’. One thing can be redder than another, the story goes, but nothing can be truly and purely red. Indeed, nothing can even genuinely look red. People mouth the words ‘That looks red to me’, with sincere conviction, but they do so only because they lack a proper philosophical understanding of the meaning of color terms.

This is, as far as we know, a consistent position. However, to adopt it, not only with respect t o color terms but all the other terms for which versions of the sorites argument can be run, would do such violence t o the way we talk and the way we think, both colloquially and scientifically, that we should surely only accept it as a last resort. We are not yet so desper- ate.

The premises to the effect that there is no visible difference in color between a tile and its immediate neighbors are true in virtue of the way we carefully constructed the sequence of tiles. With enough tiles, these premises must surely be true.

The only premise left is the sorites premise. Now the sorites premise is not analytic. Given the way we character- ized ‘no visible difference in color’, there is no inconsistency in supposing that there is no visible difference in color between the tiles, yet one looks red and the other fails to look red. This would imply a difference in how the tiles look in color tha t cannot be ascertained by direct visual inspection. It is conceiv- able that such a difference should be detected by some means other than visual inspection. For example, it might happen tha t a systematic pattern in your behavioral response when forced to classify tiles as ‘red’ or ‘not red’ indicates that tiles tinted in dye batch A look different to you from tiles tinted in dye batch B, even when the difference cannot be detected by any visual inspection, however careful.

While not analytic, the sorites premise is intuitively nearly irresistible. Still, it seems, we must reject it.

Rejection is a slippery fish. One might suppose that the sorites premise, though not strictly true, nonetheless has some privileged status which enables it to serve as a fundamental guide to our thinking.5 But it is hard to see how this guide can

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do its job. If the sorites principle is not to be conceptually in- ert, it must be allowed to play some sort of inferential role. To be sure, we could restrict i t from entering into the full range of classical inferences, but the bad consequences of the sorites premise can be obtained in even the weakest logics. Kleene’s weak three-valued logic6 or Prawitz’s minimal logic7 will suffice to enable us to derive from the sorites premise the blatantly false conclusion t h a t the last tile looks red. We can “accept” the sorites premise only if we restrict it from playing any inferential role at all, keeping i t on display like an an- tique chair with a velvet rope across the seat.

We cannot, in any practically useful sense, accept the sorites premise. Should we, therefore, deny it, that is, should we assert its negation? Classical logic tells us we must. When a given premise, taken together with other premises that are unimpeachable, leads us to a n absurd conclusion, classical logic tells us t o reject t he premise. But many have found themselves so unsettled by the prospect of denying the sorites premise that they have instead been willing to abandon classical logic. They have been willing to do so even though what they get in re turn is not the r ight to accept t he sorites premise, but merely the privilege of avoiding its denial.

It is hard to see how to settle the issue of whether classical logic should be retained, for it is hard to find a starting point. Once people agree on the rules of inference, they can hope to settle their disputes by argumentation; but what principles can they use to determine the correct rules of inference? An observation of Quine8 is worth repeating: we cannot hope to determine the correct rules of inference by a semantic investigation, trying to determine which rules are truth-preserving, because the same question ‘What are the legitimate rules of inference?’ is going to recur as we t ry to develop the metatheory. The meaning of the logical connectives cannot be given by the metatheory, because the very same connectives are employed in the metalanguage. Instead, what determines what the connectives mean are the inferences in which we employ them; the rules of inference implicitly define the connective^.^ People who employ different rules of inference mean different things by the connectives; so they are inevitably talking at cross purposes.

Lacking a direct argument, we resort to pragmatic con- cerns, invoking the glories of modern mathematics and the wonders of modern science, and asking whether i t is worth giving all this up merely to avoid denying the sorites premise. We would not get to accept the sorites premise, mind you; the premise leads incorrigibly to absurdity. All we purchase for the price of giving up classical mathematics is the right to remain silent about the premise.1° It is not worth the cost. The sorites premise has absurd consequences, so we deny it.

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When we deny the sorites premise, we assert a negated universal sentence. Are we willing also to assert the classically equivalent existentially quantified negation, tha t is, are we willing to countenance the inference from -(Vn)P to (3x)-P? If we prefer the alternative formulation of the argument which uses many instances of the sorites premise in place of the universal statement, the question we need to ask is this: reductio ad absurdum permits us to infer the negation of the conjunction of 9,999 instances of the sorites premise. Are we willing to go further, to make de Morgan’s inference from

Here again, opinions vary. According to the classical perspective, i t is in virtue of the meaning of the existential quantifier that (3 n) P is logically equivalent to - (V n) - P ; and i t is in virtue of the meaning of the disjunction sign t h a t ( P v Q) is logically equivalent to - ( - P A - Q). So when we deny the sorites premise, we are required to accept the conclusion that there exists an n less than 10,000 such that the nth tile looks red to you but the (n+l)st tile does not. We are not, however, required to suppose that it is possible to specify such an n. In classical logic, ‘there exists’ does not mean ‘it is possible to specify’ or even ‘it is possible for God to specify’, since, in classical logic, existence claims are nonconstructive.” To be sure, i t is possible and even useful to have a quantifier that means ‘it is possible to specify’, but there is precious little reason to suppose that the constructive quantifier is the only sort of existential quantification we ought to be allowed or that the constructive quantifier is the existential quantifier of Eng- lish. l2

As before, we cannot hope to settle by direct argument the conflict between classical and intuitionist readings of the existential quantifier, for lack of common ground; s o again we must resort to pragmatic considerations. For the price of giving up classical logic and classical mathematics, we get to avoid offending a few metaphysical intuitions. The price, we want to say, is not worth it. We propose to maintain classical logic and to follow the consequences of our denial of the sorites premise wherever they may lead us. We hope to convince you tha t these consequences are not so unpleasant as one might fear.

Pleasant or not, consequences of the t rea tment of the sorites argument as a reductio proof of the denial of the sorites premise are ubiquitous, for a version of the argument can be run with virtually any vague term in place of ‘looks red to you’; and virtually every term of English is to some extent, either actually or counterfactually, vague. Familiar versions take as their sorites premises the following statements:

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If we pluck a single hair from the head of a man who is not bald, he will still not be bald.

Giving a single penny to a poor person will leave the person poor.

Removing a single grain from a heap of sand will leave a heap of sand.

Though plausible, these premises must be rejected, so that we are led to the following results:

There is a number n such that , if we pluck n hairs from Harry’s head, he will still not be bald, but if we excise (n + 1) hairs, he will be.13

There is a number n such that , if we give Nell n cents, she will still be poor, whereas, if we give her ( n + 1) cents, she will no longer be poor.

There is a number n such that, if we remove n grains, but not if we remove ( n + l), we shall st i l l have a heap of sand.

If the only problem with these results were tha t they were counterintuitive, we could end the discussion abruptly. Our ad- vice would be simply, “Develop better intuitions.” After all, one of the principal purposes of philosophical inquiry is to enable us to replace intuitions that are familiar, comforting, and false by correct intuitions.

The problem is, however, more serious than merely a n offended intuition. The thesis tha t , for some n, the n th tile looks red t o you but the (n + 1)st does not is thought to entail that the phrase ‘looks red to you’ has a sharply defined extension. Similarly, we are forced to conclude tha t the adjective ‘bald’ has precise conditions of application, that the adjective ‘poor’ has sharp boundaries, and that there is an exact distinction between those collections of grains of sand that constitute heaps and those that do not. But these conclusions are prepos- terous. To accept them is t o deny tha t there is vagueness, thereby denying one of the most pervasive and prominent features of human l ang~age . ’~

11. TRUTH, DEFINITE TRUTH, AND PLUTH

How do we get the inference from the thesis that, for some n, the nth tile looks red to you but the (n+l)s t does not to the

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metatheoretical conclusion that the concept expressed by the phrase ‘looks red to you’ has a sharp boundary? This, it seems to us, is the crux of the sorites problem. That there is such an n is forced upon us by a straightforward inference from unimpeachable premises. But the metatheoretical conclusion had better not be forced upon us, for it is blatantly false. What, then, is the argument t h a t leads to th i s unhappy metatheoretical conclusion? An argument is surely called for, for the premise says something strictly in the object language, something about t i les and how they look to you, whereas the conclusion says something about words and concepts. To get from the object language premise to the met a1 a ng uage conclusion, met a t he ore t i c a1 principles a re needed. What principles are they? We have never seen the principles spelled out. Attention has been so focused on the sorites premise tha t scarcely any notice has been taken of what we would regard as the central fallacy in the sorites reasoning, namely, the inference from the denial of the sorites premise to the conclusion tha t our words and concepts a re precise.

Indeed, we suspect t h a t the principal reason why the statement that, for some n, the nth tile looks red to you and the (n + 1)st tile does not is so profoundly offensive to our intuit ions is t h a t we have fallaciously supposed t h a t the s ta tement implies t h a t the phrase ‘looks red to you’ has sharply defined conditions of application. We would like to offer a diagnosis of what goes wrong in the fallacious reasoning, bu t anything we can say must necessarily be quite tentative, because we have never seen the argument spelled out. We have to conjecture what the argument must be before we attempt to say what is wrong with it.

Before pursuing the diagnostic problem, however, we would like to begin a very faint sketch of a semantics of vague terms. Our semantics will make use of an adverbial operator ‘definitely’ to mark the distinction between clear and borderline applications of vague terms. We shall speak of clear-cut cases for the application of the predicate ‘bald’ as either ‘definitely bald’ or ‘definitely not bald’, whereas the borderline cases will be ‘neither definitely bald nor definitely not bald’.

‘Definitely’ is sometimes used as a term of emphasis, like ‘very’ or ‘really’, and i t sometimes has an epistemic use, according to which ‘Definitely P’ means roughly ‘I am sure that P’. Here, however, we want t o use the word ‘definitely’ in a semantic sense, according to which to say that an object a is definitely a n F means t h a t the thoughts and practices of speakers of the language determine conditions of application for the word F, and the facts about a determine tha t these conditions are met. (Note added after the Spindel conference:

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perhaps ‘determinately’ would be a better word, since ‘definitely’ has so many uses, but we realized this too late to make the change.9

The thoughts, experiences, and practices of the speakers of a language determine the conditions of application of its predicates, and a predicate definitely applies to an object jus t in case the facts about the object determine that these conditions are met. There are a number of possibilities for what the conditions of application might be. One possibility is that satisfying a predicate F is a matter of meeting certain criteria. Borderline cases arise in situations in which an object satisfies some but not all of the criteria, and also in cases in which the criteria themselves are vague. Another possibility is that satisfying a predicate F i s a mat ter of resembling a certain prototype. Things tha t closely resemble the prototype are definitely F , whereas things that scarcely resemble the prototype are definitely not F. Borderline cases arise in situations where the resemblance is weak. Whatever the correct theory of predication is, it will surely allow for borderline cases; a borderline case is an object that is neither definitely F nor definitely not F.

A sufficient condition for an object t o definitely satisfy a complex open sentence is given by the fact that the set of open sentences definitely satisfied by a n individual is closed under logical consequence. Thus, Harry is definitely either a bald man or a nonbald man, even though Harry is not definitely a bald man and he is not definitely a nonbald man, because Harry is definitely a man, and ‘Either x is a bald man or x is a nonbald man’ follows logically from ‘x is a man’. On the other hand, not every complex open sentence definitely satisfied by Harry follows logically from some simple open sentence he satisfies. Thus, Harry definitely satisfies ‘If x is a bald man, then x does not have much hair growing on top of his head’, not because of any peculiarities of Harry but because the general principle ‘A bald man does not have much hair growing top of his head’ follows from the meaning of the word ‘bald’. Again, Tarmin may have enough canine characteristics to ensure that she is definitely either a dog or a wolf, without being either definitely a dog or definitely a wolf.

We do not intend to attempt even the beginning of a general psycholinguistic account of when and why an individual definitely satisfies an open sentence; we expect tha t such a n account would be exceedingly complex. Here we are going to take for granted a naive intuitive distinction between clear-cut and borderline applications of open sentences and we are going to use this distinction to describe some of the logical behavior of the ‘definitely’ operator.16

Once we have the ‘definitely’ operator, we have the notion of definite truth: ‘Harry is bald’ is definitely true if Harry is definitely bald, definitely false if Harry is definitely not bald, and

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unsettled if Harry is a borderline case. When a sentence is definitely true, our thoughts and practices in using the language have established truth conditions for the sentence, and the nonlinguistic facts have determined that these conditions are met.”

Whereas we have not been able to be very definite about the meaning of ‘definite’, we can be quite specific about what the semantic theory we are outlining needs to say about truth. The theory must provide the (TI-sentences, that is, it must imply the sentences that follow the paradigm:

‘Snow is white’ is true if and only if snow is white.

The theory might tell us more about t ruth than we get from the (Tbsentences. For example, rather than taking the ( T h e n - tences as axioms, it might derive them from a Tarski-style theory of reference and satisfaction; but the theory must at least give us the (Tbsentences.

A couple of caveats are required. First, on account of the liar paradox, any theory which entails the (TI-sentences is going to be inconsistent with the basic laws of syntax. We believe that there are deep connections between the liar paradox and the paradoxes of vagueness, but we shall not explore them here.18 Here we shall set liar-type paradoxes aside by scrupu- lously observing an object language/metalanguage distinction.

Second, even within the object language the (Tksentences give crazy results, on account of indexicals and other context dependent features. We surely do not want this:

‘I am hungry’ is true if and only if I am hungry.

The sentence-that is, the sentence type-‘I am hungry’ will not have a truth-value. What we want instead is something like this:

An utterance of ‘I am hungry’ is t rue in English if and only if the speaker is hungry at the time of utterance.

We need to be especially attentive to contextual features when we are talking about vague terms, since the range of application of a vague predicate is nearly always heavily context-dependent. What s tandards to employ in determining the applicability of the adjective ‘hungry’ will depend upon the conversational context, so to assess an utterance of ‘I am hungry’, we must specify not only a speaker and a time but also a context of evaluation.

While the general importance of sensitivity to contextual dependencies cannot be gainsaid, it turns out that, for the specific limited purposes of the present paper, it will do no real

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harm if we quietly suppress contextual parameters, treating the sentences we discuss as if they were eternal ~entences.’~ If we explicitly listed the contextual parameters, we would have to carry them around throughout the discussion, even though they wouldn’t be doing any work. It is a nuisance to carry them around, so we leave them out. In particular, ‘true’ will ordinarily mean ‘true in English’ (or, in tight places, ‘true in my idiolect of English’), though sometimes it will be used instead to refer to sentences of a formal language we use to formalize a fragment of English.

Having the adverb ‘definitely’ and the adjectives ‘true’ and ‘false’-we define falsity as the truth of the negation-we com- bine them to get a classification of sentences as definitely true, definitely false, and unsettled. The most striking formal feature of the use of the phrase ‘definitely true’ is that a disjunction can be definitely true without any disjunct being definitely true. The disjunction ‘Harry is bald or Harry is not bald’ is definitely true-it is, after all, a truth of logic-but neither disjunct is definitely true; that neither disjunct is definitely true is what makes Harry a borderline case of ‘bald’. This failure of ‘definitely true’ t o distribute over disjunctions is the most dramatic difference between the logical behavior of ‘true’ and that of ‘definitely true’. A disjunction is true if and only if one or more disjuncts are true. The same does not hold for definite truth. A disjunction is definitely true if one or more disjuncts are definitely true, but not conversely.

An existential sentence can be definitely true without any instance being definitely true; this again distinguishes ‘definitely true’ from ‘true.’20 ‘ (3 x ) ( ( x = 0 A Harry is bald) v ( x = 1 Harry is not bald))’ is definitely true, but none of its instances is definitely true.

The failure of the existential quantifier to commute with the ‘definitely’ operator is crucial to understanding the sorites paradox. The sentences ‘The first tile looks red t o you’ and ‘The 10,000th tile does not look red to you’ are both definitely true, as are the axioms of Robinson’s R.21 Since the set of definite truths is closed under logical consequence, it follows that i t is definitely true that, for some n, the nth tile looks red to you and the ( n + 1)st tile does not.22 I t does not follow that there is an n such that it is definitely true that the nth tile looks red t o you and i t is definitely true that the ( n + 1)st tile does not. If there were such an n, then it would be definitely true that the nth tile looks red and definitely false that the ( n + 1)st tile looks red, and so we would have a sharp partition between the tiles that look red to you and those that do not. It is the failure of the ‘definitely’ operator to commute with the existential quantifier that blocks the fallacious inference from the definite truth of the existence claim to the absurd conclusion that there is a sharp partition.

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The definite truth of a compound sentence need not be com- positionally grounded (in the manner described in Kripke’s fa- mous paper on the l iar paradox23) in the definite t ru th or falsity of its atomic components. A disjunction can be definitely true without any disjunct being definitely true, an existential sentence can be definitely true without any instance being definitely true, and a biconditional can be definitely true without either component being either definitely true or definitely false. This is not to say that a sentence can be definitely true for no reason at all-for a sentence to be definitely true, there must be something we do, think, or say which, together with the nonlinguistic facts, makes it true-but the grounding relation is more complicated than simply compound sentences being grounded in atomic facts.24 In particular, what makes the existential claim that there is an n such that the nth tile looks red though its immediate neighbor does not definitely t rue is tha t the claim is a logical consequence of definitely true premises. What ensures that the set of definite truths is closed under logical consequence is that the rules of inference a re constitutive of the meaning of the connectives and quantifiers.

We can diagnose the sorites fallacy as coming from a failure clearly to distinguish definite t ruth from truth. We see, quite correctly, that:

I t is true that, for some n c 10,000, the nth tile looks red and the (n + 1)st tile does not look red.

Using the (Tbsentences and simple arithmetic, we conclude:

For some n c 10,000, it is true that the nth tile looks red and false that the (n + 1)st tile looks red.

If we fail to make a clear distinction between definite t ru th and truth, referring to both of them as ‘truth’, we are apt to misread this claim as this:

For some n c 10,000, it is definitely true that the nth tile looks red and definitely false that the (n + 1)st tile looks red,

which asserts, absurdly, that there is a sharp partition. This diagnosis takes for granted that we have the ( T h e n -

tences at our disposal. At first glance, this assumption seems harmless enough. What could be more na tura l to assume about truth than the (TI-sentences? On closer inspection, however, we see that, when we are dealing with vague terms, the (TI-sentences have consequences that are quite puzzling. One of the (T)-sentences is this:

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‘Harry is bald’ is true if and only if Harry is bald,

Another of the (TI-sentences is this:

‘Harry is not bald’ is true if and only if Harry is not bald.

Since we are treating falsity as truth of the negation, this gives us what we may call an (FI-sentence:

‘Harry is bald’ is false if and only if Harry is not bald.

Together these imply, in classical logic or even in weak three- valued logic:

‘Harry is bald‘ is either true or false.

But, one wants to say, precisely what marks Harry as a borderline case of ‘bald’ is that ‘Harry is bald’ is neither true nor false.

The truth and falsity conditions for the sentence ‘Harry is bald’ are established by the thoughts and practices of speakers of the language; what else could do it? If the facts about Harry’s head are such as to ensure that the truth conditions are met, then the sentence is true. If the facts about Harry’s head are such as to ensure that the falsity conditions are met, the sentence is false. But if our thoughts and practices in using the word ‘bald‘ leave it unsettled whether ‘bald’ should apply to someone like Harry, then surely the sentence is neither true nor false. The thoughts and practices of the speakers of the language, together with the facts about Harry’s head, do not determine a truth-value for the sentence, and there do not appear to be any other factors that could even be relevant.

In thinking this way, we are employing what Tarski calls a “semantic” conception of truth, according to which truth is based upon “connexions between the expressions of a language and the objects and states of affairs referred to by those expression^."^^ The semantic conception is a natural and deeply ingrained part of our ordinary way of thinking about truth, but it comes into conflict with an equally natural and equally deeply ingrained part of our ordinary thinking, namely the (Tbsentences.

We are witnessing a conflict between two deeply held principles about truth. The disquotation principle tell us that any adequate understanding of truth ought to give us the (TI- sentences and the (FI-sentences. The correspondence pr in - cipZez6 tells us that the truth conditions for a sentence are established by the thoughts and practices of the speakers of the language, and tha t a sentence is true only if the nonlinguistic factsz7 determine that these conditions are met.

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In other words, the correspondence principle tells us that, in order for a sentence to be true, it must be what we have been referring to in this paper as ‘definitely true’. Acceptance of the correspondence principle does not commit you to any particular account of what truth conditions are or how they are established. I t only requires the general thesis that, if a sentence is true, there is something speakers of the language do, say, or think tha t , together with the nonlinguistic facts, makes it true.

Both principles are natural, even obvious, but they come into conflict. The disquotation principle implies that ‘Harry is bald’ is either true or false, whereas the correspondence principle tells us tha t it cannot be either. The only way consistently to hold on to both principles would be to deny the phenomenon of vagueness.

Formally, the conflict manifests itself as a dispute about the permissible ways to fill in the blanks in the (Tbschema:

1 ’ is true if and only if

Clearly, not every English sentence can be plugged in. Ques- tions, for instance, cannot. Sentences containing indexicals cannot either, although one presumes tha t they could be in- serted into a suitably modified schema. Whether sentences containing nondenoting singular terms can be plugged in is a matter of debate. If you side with Russell,28 you will think they can be, whereas if you side with S t r a w s ~ n , ~ ~ you will only countenance a conditional version of the schema, thus:

If the present king of France exists, then ‘The present king of France is bald‘ is true if and only if the present king of France is bald.

For us, the relevant question is not whether the present king of France is bald but whether Harry is. When can sentences containing vague terms be plugged into the (TI-schema? One extreme, the disquotation principle, tells us that vagueness is no impediment to substitution into the (TI-schema, whereas the correspondence principle would urge us to restrict the principle somehow. The opposite extreme would be to say that a sentence cannot be substituted into the schema if any of its terms are vague, but this obviously goes too far. Nearly every term in the language is to some extent vague, so such a restriction reduces the schema to the vanishing point. To restrict the schema by permitting us to substi tute a sentence only when we know that the sentence is not about a borderline case would impair the usefulness of the notion of t ru th nearly as badly, for we seldom can be sure a sentence is not about a borderline case unless we already know its truth-value.

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A more promising strategy is to adopt a conditional version of the schema, something along the following lines:

If rP1 expresses a proposition, then rP7 is true if and only if P.

How credible this schema is depends on how we understand ‘proposition’. According to some philosophical uses, a sentence expresses a proposition if, given the rules of the language and the conversational situation, it could be used to make a meaningful assertion; or it could be used to express a thought, belief, or desire;30 or it is capable of having a t r ~ t h - v a l u e . ~ ~ So understood, borderline attributions of vague terms express proposi- tions, since they can be meaningfully asserted, they can express thoughts, beliefs, and desires, and they are capable of having truth-values (even if, as a matter of fact, they have none). If the condition is going to allow borderline attributions to be exempt from the disquotation principle, it must be understood in such a way t h a t a sentence is taken to express a proposition only if it, in fact, has a truth value. If we make this restriction explicit, we get the following conditional version of the (T)-schema:

If rP1 is either true or false, then rP1 is true if and only if P,

and likewise this conditional version of (F):

If rP1 is either true or false, then rP1 is false if and only if not-P.

Together, these two schemata are logically equivalent t o the left-to-right directions of (T) and (F):

If rP1 is true, then P. If rP1 is false, then not-P.

Since we are treating falsity as the t ruth of the negation, the left-to-right direction of (F) is an immediate consequence of the left-to-right direction of (T). The battleground of the conflict between the disquotation and correspondence principles is the question when we ought also t o accept the right-to-left direction of (T). Clearly we need to accept some instances of the right-to-left schema if the notion of truth is going to be of any use a t all; for the theory consisting of the left-to-right schema alone is consistent with the thesis tha t nothing is either true or false. How generous we ought to be in accepting right-to-left instances is the question in dispute.

The disquotation and the correspondence principles clash. So long as we admit that there is vagueness in the language,

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we cannot consistently maintain both. Facing up to the difficulty forthrightly, we have proposed to abandon the correspondence principle. Once we jettison the correspondence principle, we are left with the disquotational conception of a way of using the word ‘true’ whose fundamental governing postu- late is the disquotation principle.

We foresee an objection to the stance we have taken. What is left after we abandon the correspondence principle is too thin, it will be said, to count as truth. The idea that truth consists in a genuine connection between language and the world is so fundamental, both to our intuitive understanding of truth and to our theoretical usage of ‘true’ as a term of philosophical inquiry, tha t we cannot afford to relinquish it. Given the in- exorable conflict between the disquotation and correspondence principles, we must surrender the former, gett ing the correspondence conception of t ruth, which gives up the (T)- schema in the interest of maintaining a robustly semantical conception of truth.

The practical advantages of having a notion of t ruth that satisfies the (Tbsentences have been stressed by Q ~ i n e ~ ~ and others. Such a notion enables us to simulate operations of infinitary conjunction and disjunction by semantic ascent. Happily, these advantages will still be available to us, even if we adopt a correspondence conception. While we allow our usage of the word ‘true’ to be governed by the correspondence principle, we can introduce a new predicate ‘plue’ (an elision of ‘pleonastically true’), implicitly defined by the axiom schema:

(P) rP1 is plue if and only if P.

As Tarski such a n axiom schema will give us a conservative extension of whatever theory we hold.

It seems to us that the issue here is purely verbal. Both the disquotation principle and the correspondence principle are integral to our ordinary usage of the word ‘true’, and neither has any obvious claim to being more fundamental than the other. We have proposed the adoption of a disquotational conception, but we have no substantive disagreement with someone who prefers a correspondence conception. Things we shall say using the words ‘true’ and ‘definitely true’ the correspondence theorist will say using the words ‘plue’ and ‘true’; there is no genuine conflict. We shall continue to use ‘true’ and ‘definitely true’, but we have no quarrel at all with someone who uses ‘plue’ and ‘true’ instead.

Once we have the concepts of truth and definite truth sepa- rated, we can view the sorites reasoning as committing a fallacy of equivocation. If we take t ru th to be disquotational t ru th , the (T)-sentences, together with some rudimentary arithmetic give us this:

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If the first tile looks red t o you and the 10,OOOth tile does not, then there is an n such that i t is true that the nth tile looks red to you and false that the ( n + 1)st tile looks red to you.

If we take truth to be correspondence truth, we get this:

If there is an n such that it is true that the nth tile looks red to you and false that the (n + 1)st tile looks red to you, then the facts about our thoughts and practices and the facts about the way the tiles look to you pick out an n such that i t is determined that the nth tile looks red to you and its immediate neighbor does not.

Combining these two observations, one can derive the following absurd conclusion:

If the first tile looks red to you and the 10,000th tile does not, then the facts about our thoughts and practices and the facts about the way the tiles look to you pick out an n such that i t is determined that the nth tile looks red to you and its immediate neighbor does not.

But it is illegitimate to use the word ‘true’ in two disparate senses within a single argument.

Because it presupposes the distinction between truth and definite truth, this way of describing the sorites reasoning is m i ~ l e a d i n g , ~ ~ for it suggests that ordinary usage is ambiguous between two different uses of ‘true.’ We see no reason t o believe it is.36 Instead, there is a single notion of truth whose use is governed by conflicting principles. The correspondence principle and the disquotation principle are both integral components of our ordinary usage of ‘true’, but their joint application is what brings the sorites reasoning to its absurd conclusion. We are correct to believe the following, having derived it by a valid deduction from unimpeachable premises:

There is an n such that the nth tile looks red to you and the (n + 1)st tile does not look red to you.

From this we conclude, using the disquotation principle:

There is an n such that i t is true that the nth tile looks red to you and false that the (n + 1)st tile looks red to you.

Hence we derive, using the correspondence principle:

There is an n such that the thoughts and practices of the speakers of the language, together with the facts about

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the ways the tiles look to you, determine tha t the sentence ‘The n th t i le looks red to you’ is t rue but ‘The (n + 1)st tile looks red to you’ is false.

which tells us, falsely, that our thoughts and practices produce a sharp partition.

Given t h a t there is vagueness in the language, the two principles cannot both be maintained. One principle or the other must be given up. We have proposed giving up the correspondence principle, but one could, with equal justice, relinquish the disquotation principle, since neither principle is more fundamental than the other to our ordinary thinking about truth.

A paradox is, as Tarski tells us,37 a symptom of a disease. The disease of which the sorites paradox is a symptom is the conflict between the disquotation principle and the correspondence principle. As therapy,38 we have recommended replacing our old, conflicted notion of t ruth by two consistent notions. Pretherapeutically, we have only the one notion of truth, but within our ordinary, inconsistent notion of truth, there is material enough to permit the construction of two consistent notions. To call both of them ‘truth’ would invite fallacies of equivocation, so we call one ‘truth’ and the other ‘definite truth’ (although one might instead use ‘pluth’ and ‘truth’).

111. DISTINCTIONS WITHOUT A DIFFERENCE?

We come to the title question: are we required to make verbal distinctions in cases in which there are no relevant differences? Clearly we make distinctions among the tiles. We call some of them ‘red’, others ‘not red’. We say of some of them ‘That looks red to me’, of others ‘That doesn’t look red to me’. Clearly also, there are relevant differences among the tiles. Some are red, some are not red; some look red to you, others do not.

There are even relevant differences between adjacent tiles. Tile 4,013 has ever so slightly more yellow dye on it than tile 4,012. The difference is minute, so tiny i t could only be detected by a sophisticated spectrographic analysis, but it is a difference nonetheless, and a relevant one. In a particular run of a “forced-march sorites”3s sequence-a sequence in which you are led through the tiles one by one and forced to answer the question “Does that tile look red to you?”-there will be a first tile to which your verbal response is something other than a simple “Yes.” Suppose you balk at the 4,013th tile. Then, there is one particular si tuation in which, while a t tempting to classify the tiles in terms of your visual response to their

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color, you have responded differently t o the 4,013th tile from the way you responded to the 4,012th. Thus, there is a difference, tiny though it may be, between the 4,012th and 4,013th tile that is relevant to the question whether the tiles look red to you. Similarly, the difference you make to Nell’s fortune by giving her a penny is minuscule, but it is relevant to the question whether Nell is poor: she now has a teeny bit more money. Plucking a single hair from Harry’s head makes a minute but relevant difference to the question whether Harry is bald.

There are differences between adjacent tiles that are relevant to the question whether a tile is red or whether a tile looks red to you, but the differences are exquisitely small, much too small to justify distinguishing one from the other. We would expect to be justified in saying that the 4,012th tile looks red but the 4,013th does not only if i t is definitely true that the former looks red and the latter does

Whereas there is not any n such that we would be justified in saying that the nth tile looks red and the (n + 1)st tile does not, we are justified in making the existential claim that there is an n such tha t the nth tile looks red and the (n + 1)st tile does not. We are justified in asserting this because we have in- ferred it by careful deduction from secure premises.

Another existential sentence we are in a position to assert is that there is an n such that it is true that the nth tile looks red and false tha t the (n + 1)st tile looks red. We derive this from secure premises by means of the (Tbsentences. On a correspondence conception of truth, this would imply (modulo some small if elaborate qualifications) that the difference in appearance between the nth tile and the tile next to it is sufficient to permit a normal observer in normal circumstances to judge tha t the nth tile looks red and the (n + 1)st does not, a conclusion manifestly contrary to fact. But on a disquotational conception of truth-the only coherent conception that makes the (Tbsentences freely available-no such implication ensues.

Yet another existential judgment we are in a position to make is that there is some n such that the nth tile definitely looks red and the (n + 1)st tile does not definitely look red. This follows from the fact that some but not all the tiles definitely look red.

One existential claim we surely do not want t o make is this: there is a n n such tha t the nth tile definitely looks red and the (n + 1)st tile definitely does not look red. Such a claim would allege a justifiable distinction where there is no significant difference.

There is nothing odd about being able to assert an existential sentence without being in a position to assert any of its instances. Most of the time when this happens, i t is due to our ignorance. There is some fact or combination of facts of which

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we are unaware, knowledge of which would enable us to assert an instance. We know someone stole the cookies and we do not know who it was, but we could find out who it was by gather- ing suitable evidence.

In cases of semantic indeterminacy, our inability t o assert an instance would persist even if we had all the relevant facts. We know tha t there is a wealthiest poor person-that is, a poor person such that no one even slightly wealthier than that person is poor41-but we do not know who the wealthiest poor person is. We would not be able to identify the wealthiest poor person even if we had access to all the economic facts and unlimited patience and ability in sorting through those facts, for the economic facts do not determine an answer to the question ‘Who is the wealthiest poor person?’

Both ignorance and indeterminacy give us cases in which we know an existential sentence to be true without knowing any instance of it to be true, and in which we know a disjunction without knowing any disjunct. Indeterminacy gives us an excuse: the reason we do not know is that there is no fact of the matter there to be known.

Semantic indeterminacy is a timeless affair. The total history of the world, past, present, and future, does not determine an answer to the question ‘Who was the wealthiest poor person in the world on June 15, 1994?’

We can compare semantic indeterminacy with physical indeterminacy, with which i t shares a lot of logical features. In an indeterministic world, the physical facts up till now may well determine that an existential sentence is true without determining of any of its instances that it is true. If the outcome of the lottery is truly a matter of chance, then i t may well be the case that i t is now determined that someone will win the lottery, but it is not yet determined who the winner is going to be. A disjunction can be determined to be true without either disjunction being determined to be true. It is now determined tha t the coin will land either heads or tails, even if i t is not yet either determined tha t i t will land heads o r determined that it will land tails.

The difference is that what is physically determined is con- stantly changing; what is undetermined today may well be determined tomorrow. What is semantically determined can change too, because the meanings of our words can change; but what is physically determined can change even while the meanings of our words remains fixed. Today i t is physically undetermined who is going to win the lottery; tomorrow it will be determined who won, and what has made the difference will be new events, rather than new meanings. The laws of nature and the facts of history up to the present do not determine an answer to the question, ‘Who will win the lottery?’ but tomorrow’s facts will have settled on a winner. In cases of

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semantic indeterminacy, the whole history of the world will not suffice to fix an answer.

So there is an important difference between physical and semantical indeterminacy. If Chuck Flamish wins the lottery on October 15, then on October 14 the sentence ‘Chuck Flamish will win the lottery on October 15’ was definitely true, even though i t was not physically determinately true. But the similarity between the two kinds of indeterminacy is more important t han the difference. Both a re noncom- positional. An existential sentence can be definitely t rue or physically determinately true without any instance being definitely true or physically determinately true, and a disjunction can be definitely true or physically determinately true without either disjunct being definitely t rue or physically determinately true.

IV. SKETCH OF A MODEL THEORY

We would now like to give a t least a sketch of how a semantic theory of the type we have been outlining might be worked out model-theoretically. We do not really need the model theory to account for the sorites paradox. The solution to the sorites paradox is to resolve the conflict between the disquotation principle and the correspondence principle, which we propose to do by splitting our naive concept of t ru th into two notions, definite t ru th and t ruth. We do not need any model theory for that. Even so, until we have the rudiments of a semantic theory, we are not going to feel that we understand the logic of vague terms.

Fortunately, the tools we need are ready to hand. Some in- genious techniques developed by Bas van F r a a s ~ e n ~ ~ and Kit Fine43 will enable us t o utilize the methods of ordinary model theory t o get the beginning of a theory of definite truth. The account we shall give will only apply to a limited class of formalized languages, and, even for them, i t will be only a faint sketch; but at least it is a start.

First, some technicalities. We are only going to worry about languages for the predicate calculus, and we are only going to worry about what to do about vague predicates and vague proper names. In particular, we are not going to worry about vagueness in the quantifiers, such as might arise either because of some indeterminacy in what there is (such as we might get with quantum creation operators) or some indefi- niteness in the universe of discourse. Instead, we presume tha t our universe of discourse is, or can be treated as if it were, precisely determined. We take a first-order language L whose intended universe of discourse is a set A. We extend L by adding a new constant a for each member a of A, getting an

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enlarged language LA.44 An A-model is a function which assigns a member of A to each individual constant, assigning a-to a, and assigns a set of n-tuples from A to each n-place predicate, assigning I < a,a > : a E A) to ‘2. Truth in an A-model is given an explicit set-theoretic definition in the usual keep in mind that truth in an A-model is not the same as truth.

We have already noticed tha t the set of definite t ruths is closed under first-order consequence. Also, we may suppose tha t a universal generalization is definitely true if all i ts instances a re definitely true. This last assumption is not any deep reflection on the nature of definite truth, but rather an artificial consequence of the fact tha t we have chosen to re- s t r ic t our attention to si tuations in which there is no quantificational v a g u e n e s ~ . ~ ~ Taking these assumptions for granted, we have the following deep resul t of Henkin and ore^:^'

Assuming tha t L,is countable, a set r of sentences is closed under first-order consequence and the A-rule (the infinitary rule that permits you to infer (V x)Fx from the totality of the Fas) if and only if there is a class of A-models K such that r is equal t o the set of sentences true in every member of K.

This theorem provides our basic strategy in attempting to understand the logical structure of the set of definite truths: try to find and examine an appropriate class of A-models.

For countable languages, the strategy of trying to identify the set of definite truths as the sentences true in every member of some class of A-models presumes that the set of definite truths is closed under first-order consequence and the A-rule. What does the strategy presume about uncountable languages? To answer this , we introduce a new notion, derived from Tarski’s definition of logical con~equence .~~

According to Tarski, to see whether a sentence S is a logical consequence of a set of sentences r, first replace, in a uniform fashion,49 all the nonlogical constants that appear in r and in S by new variables of appropriate types; tha t is, individual constants are replaced by individual variables, unary predicates are replaced by unary second-order variables, and so on. S is a logical consequence of r if and only if every variable assignment (sequence assigning objects of appropriate types to the variables) which satisfies all the formulas gotten by making the substitutions into r also satisfies the formula gotten by making the substitutions into S. We certainly want to require the set of definite t ru ths to be closed under logical consequence. Taking account of the fact that we have a fixed universe of discourse and we have fixed the reference of the as , we can reasonably require a still stronger condition that incorpo-

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rates both closure under logical consequence and closure under the A-rule:

Definition. A sentence S is a Tarski A-consequence of a set of sentences r if and only if, if all the nonlogical con- s tants other than the as are uniformly replaced by new variables of appropriate type, we find that any variable assignment that satisfies all the formulas gotten by making the substitutions into r also satisfies the formula gotten by making the substitutions into S.

Theorem. The following are equivalent:

(i)

(ii)

r is closed under Tarski A-consequence.

There is a set of A-models K such that r is equal to the set of sentences true in every member of K.

There is a set A of sentences such that r consists of the sentences true in every A-model of A.

(iii)

If LA is countable, there is a fourth equivalent:

(iv) r is closed under first-order consequence and the A rule.

The proof (straightforward except for Part (iv), the Henkin- Orey theorem) is omitted.

As a example, take A to be an uncountable set containing the natural numbers, and take r to be the following theory:

All sentences Nb, for k a natural number.

All sentences 1 NG for a member of A that is not a natural number.

Let S be this sentence:

Intuitively, if all the members of r are definitely true, S must be definitely true. The fact that A is uncountable ensures that any function from A to the set of natural numbers will not be one-one. S is not derivable from r by first-order logic and the A-rule, but S is a Tarski A-consequence of r.

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The theorem suggests a plan: try to identify a set A of constraints such that the definite truths are the sentences true in every A-model of the constraints. Two sorts of constraint have been identified so far. There may well be others which are not yet suspected.

The first constraints ensure tha t our models get the easy cases right, successfully adjudicating the great mass of cases which are distant from the borderline. When we learn the language, we learn the conditions of application of its predicates; knowing how to apply the predicates is a par t of linguistic competence. While there are inevitably borderline cases, there are also clear cases tha t lie unmistakably on one side or another of the divide. Speakers of the language who are fully adept in the use of the predicate, who are appraised of all the relevant facts, and who suffer no cognitive impairment will be able to make the classification successfully, presumably, so tha t a speaker who is fully competent in the use of the one- place predicate R and who is in an epistemically ideal situation will classify a as a n R if a is definitely R , and she will classify a as non-R if a is definitely not R. (Saying what con- stitutes an epistemically ideal situation is notoriously difficult, as is figuring out what to do with predicates for which no observer can ever be epistemically ideally situated.) If a is a clear case of the predicate R, then our practice of applying the predicate as we do, together with the facts about a , will establish Re. as definitely true, and the sentence Ra will therefore be one of our constraints. Similarly, if a is clearly not B, 7 R a will be among the constraints. Likewise for n-place predicates.

There are two prominent theories of vagueness and proper names,6o so there a re two natural candidates for how t o account for proper names i n our semantics. Consider a geo- graphical entity with a fuzzy boundary, say Mt. Pisgah, and consider the many precisely delimited expanses of land in the vicinity of Mt. Pisgah. Since Mt. Pisgah has a fuzzy border, it will not be definitely identical to any of the sharply-defined land masses. Will it be definitely distinct from every sharp expanse? Here opinions vary.

One account, the ”vagueness in the world” viewpoint, has it that Mt. Pisgah is a vague object, distinct from every precisely bounded object. After we listed all the precisely bounded areas of land in the vicinity of Mt. Pisgah, we would find yet another entity, call it a , distinct from every precise expanse of terrain, such tha t the vague object a is identical to Mt. Pisgah and such that the sentence ‘g = Mt. Pisgah’ is in r.

The alternative story, the “vagueness in language” account, sees the source of the imprecision not in the ontological status of Mt. Pisgah but in the semantical s ta tus of ‘Mt. Pisgah.’ There are no imprecise objects, so the story goes, only precise objects imprecisely named. Once we have listed all the pre-

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cisely bounded expanses of land in the vicinity of Mt. Pisgah, there will not be anything further left over as a candidate for what Mt. Pisgah might be. The reason why there are various clods of Carolina clay for which there is no fact of the matter whether the clod is a part of Mt. Pisgah is that it is not precisely determined which of the precisely delimited expanses of land ‘Mt. Pisgah’ refers to. On this view, there will be no object a such tha t ‘a= Mt. Pisgah’ is in r. For every object b which is not a precisely limited land mass suitable as a candidate for the referent of ‘Mt. Pisgah’, the sentence ‘b# Mt. Pisgah’ will be in r.

Mt. Pisgah is definitely a mountain. On the vagueness-in- the-world account, there will be an object a such that the sentences ‘a is a mountain’ and ‘& = Mt. Pisgah’ are both among the constraints. On the vagueness-in-language account, there will be no such a. Yet, intuitively, Mt. Pisgah is a clear-cut case of a mountain, so we want to constrain our class of acceptable A-models so that, if the precisely delimited land mass a is the referent of ‘Mt. Pisgah’ in a model, u will also be in the extension of ‘mountain’ in that model. The simplest way to ensure this is simply to include the sentence ‘Mt. Pisgah is a mountain’ among the constraints. Also, if a , and u2 are slightly differing, precisely delimited land masses that are both candidates for the referent of ‘Mt. Pisgah’, we will not want both of them to be in the extension of ‘mountain’ in any of our A-models, since (presumably) ‘There is no mountain other than Mt. Pisgah in the immediate vicinity of Mt. Pisgah’ is definitely true. So the vagueness-in-language theorist will want to include the sentence ‘7 (a is a mountain A g2 is a mountain)’ among the constraint^.^^ +he semantic theory is perfectly ame- nable to both viewpoint^.^^

The second class of constraints a r e what Fine calls “penumbral connections,” which coordinate the ways in which our A-models adjudicate the borderline cases. If young Telemachus is a borderline case of ‘man’, there will be some models in which ‘Telemachus is a man’ is true and others in which i t is false. Telemachus is likewise a borderline case of ‘bachelor’, so t h a t there will be some models in which ‘Telemachus is a bachelor’ is t rue and others in which i t is false. But we want to rule out models in which ‘Telemachus is a bachelor’ is true but ‘Telemachus is a man’ is false, and likewise, Telemachus being definitely unmarried, to rule out models in which ‘Telemachus is a man’ is true and ‘Telemachus is a bachelor’ is false. We want to rule out such models because ‘Telemachus is a bachelor if and only if he is an unmarried man’, being analytic, should be definitely true. The simplest way to ensure this is to add the analytic generalization ‘An individual is a bachelor if and only if he is an unmarried man’ to our system of constraints. Similarly, if Belle and Nell are both

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borderline cases of ‘poor’, we shall have models in which ‘Belle is poor’ is true and models in which it is false, and we shall have models in which “ell is poor’ is true and models in which it is false. But, if Belle is definitely better off financially than Nell, we shall want to rule out models in which ‘Belle is poor’ is true and “ell is poor’ is false. The easiest way to do this is to add the conditional ‘If one person is better off financially than another, then, if the first is poor, the second is poor as well’ to our system of constraints.

The thoughts and practices of the speakers of the language, together with the nonlinguistic facts, pick out a set of sentences as definitely true. The constraints pick out a set of A- models. The hope is that these will exactly correspond, so that a sentence is picked out as definitely true if and only if it is true in every A-model of the constraints. Whereas a precise language has a single intended model, for vague languages there will typically be a great many A-models of the constraints. None of these models ought to be thought of as “intended models”, because i t is part of the intentions of the speakers that some of the terms should be used vaguely, yet each model assigns each of the terms a precise extension. The A-models of the constraints respect only part of our intentions, for they only have to pass a one-directional test. If M is such a model, then, whenever the thoughts and practices of the speakers of the language, together with the nonlinguistic facts, determine a sentence as true, that sentence should be true in M. But there is no converse requirement that, if a sentence is true in M, then the thoughts and practices of the speakers of the language, together with the nonlinguistic facts, determine that the sentence is true.

The idea that a sentence is definitely true if and only if it is true in every member of a certain class of models needs to be clearly distinguished from the thesis that a sentence of a vague language is definitely true if and only if it is true in all the fully precise languages that can be gotten by making the given language precise. This latter thesis seems to us insup- portable. It surely is not possible for us to devise a language which is fully precise,53 but, setting that problem aside, a more serious difficulty confronts us: we cannot get from English to a fully precise language merely by deciding what to say about some hard cases which English usage leaves unsettled. In- stead, getting a fully precise language will involve adopting rules that contradict the rules of English, for the meanings of certain English terms require that the terms be vague. ‘Bald’ and ‘looks red’ may well be such terms; ‘roughly six feet tall’ surely is. But there is no plausible reason to believe that examining a class of languages whose rules contradict the rules of English will cast any useful light upon the semantics of En- g l i ~ h . ~ ~ Luckily for us, the position we are developing here

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does not require looking at a lo t of different languages, but rather looking a t a lot of different models. The models we look a t are all models of the vague language whose semantics we are trying to describe, not models of some other languages gotten by precisifying that language.

Formally, ‘definitely true’ acts a lot like ‘necessary’ and the A-models of the constraints act a lot like possible worlds, but this is only a formal analogy.55 I t would be a mistake to think of the A-models of the constraints as possible worlds. Instead, each possible world is associated with a set of models, each of which conforms to the facts t h a t obtain in t h a t world. Whereas, in a fully precise language, there is one model that describes each world, in a vague language there will be many models that conform to the facts of the world in the sense that every sentence tha t is definitely t rue in the world is t rue in the model. We cannot really say tha t the models associated with the world describe the world, for the models will make t ru th assignments tha t are in no way justified by the facts that obtain in the world. Instead, we use the whole set of associated models to describe the world, so that a sentence is definitely true in the world if and only if i t is true in each of the associated models.

If the theory being constructed has a certain chewing-gum- and-baling-wire feel to it, this is to be explained by its origin. We are taking a well-developed and highly successful theory of the models of precise languages56 and retrofitting it to apply to vague languages. The resulting contraption seems to work rather well, in spite of occasional awkwardness.

We hasten preemptively to emphasize tha t what has been said so far, here and in the papers of van Fraassen and Fine, falls very far short of satisfactorily identifying the system of constraints. Still we would like to hope tha t it shows some promise as a first approximation.

However much we may hope to improve our understanding of what the constraints are, we cannot hope to get a fully precise characterization of t he se t of constraints. Such a characterization would engender a fully precise characterization of the set of definite truths. No such characterization is possible, for the boundary between the definite truths and the sentences that are not definitely true is not a sharp one. ‘The nth tile is red’ is definitely true if and only if the nth tile is definitely red, and there is no sharp partition between the tiles that are definitely red and those tha t are not definitely red. There is no n such tha t i t is definitely true that the nth tile is definitely red and that the (n + 1)st tile is not definitely red. ‘Definitely true’ is a vague notion.57

‘True’ is likewise vague. Because “‘Harry is bald’ is true if and only if Harry is bald” is definitely true-it is, on to the disquotational conception of truth, analytic-“‘Harry is bald’ is

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true” will be definitely true, definitely false, or unsettled according as ‘Harry is bald’ is definitely true, definitely false, or unsettled. ‘True’ inherits the vagueness of other vague terms, like ‘bald‘.

We would like to see how to formalize the metatheory, so as to reflect the fact that ‘true’ and ‘definitely true’ are vague terms. Since our metalanguage is going to be vague, we shall give the semantics of the metalanguage in just the way we give the semantics of the object language, by describing a system of constraints.

To get a theory of truth is remarkably easy; just adopt the (Tbsentences,

for P a sentence of LA, into the system of constraints. Just as penumbral constraints require the allowable models of the object language to treat ‘Telemachus is a man’ and ‘Telemachus is a bachelor’ as either both true o r both false, penumbral constraints require allowable models of the metalanguage to treat ‘Telemachus is a man’ and ‘12. ( ‘Telemachus is a ma n1 )’ as either both true or both false. A sentence P will be definitely true, definitely false, or unsettled according as IF ( ‘El is definitely true, definitely false, or unsettled. Thus ‘Tr’ will be a vague term, for it inherits the vagueness of every vague term of the object language.

The operator ‘Def’ will behave syntactically like ‘0’. If ‘BaZd’ symbolizes ‘bald’, ‘x is definitely bald’ will be symbol- ized ‘De@ald(x)’. ‘x is definitely true’ will be ‘DefTrbY. Thus, starting with an object language with no semantical terms, ‘12.’ will be a term of the metalanguage, and ‘DefTr’ will be a term of the metametalanguage.

Whereas giving a theory of truth is an easy formal exer- cise, giving a theory of definite truth will require serious linguistic labor. We need t o describe what distinguishes the clear-cut applications of vague terms from the borderline cases. We need to distinguish those high-level theoretical generalizations that should be regarded as partially constitutive of the meanings of their words, and so should be treated as penumbral constraints, from other deeply entrenched theoretical beliefs. A great deal of arduous effort will be required before we can satisfactorily describe the system of constraints. The ease with which we defined truth is deceptive. We were able to define truth without engaging in any serious linguistic labor because truth is, on the disquotational conception, such a singularly bloodless notion. When we get to substantive notions like definite truth, mere formal manipu- lations will no longer suffice; we have t o take a serious look at how the language works.

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Moreover, the metametatheory will demand a richer ontol- ogy than either the object theory or the metatheory. Namely, the universe of discourse of the metametatheory must include a sufficiently large fragment of the universe of set theory to permit a Tarski-style definition of truth in a model,

Once we have a description of the system of constraints, we can implicitly define ‘Def’ as follows: for each open sentence F(x) with one free variable in the metalanguage, the sentence:

(V‘x) ( D e F ( x ) t) [F(x)l is true in every A-model which satisfies the constraints)

will be included in the metametatheory. Similarly for closed sentences and for open sentences with more than one free variable.

We derive

for each sentence P of the object language. We also derive that the set of definite t ru ths is closed under first-order consequence and that a universal generalization is definitely true if and only if each of its instances is definitely true. However, to derive that every true sentence is true,

(Vx) ( D e m ( x ) + fi ( X I ) ,

will require some extra machinery, which we develop in an appendix.

Because our characterization of the system of constraints is sure to be vague, there will be many models of the metametatheory. I n any particular model of the metametatheory, there will be some particular set of sentences that the model assigns as the extension of the formalization of the word ‘constraint’. Hence, in any particular model of the metametatheory there will be a number n such t h a t ‘(DefRed (the nth tile) A 7 Defied (the (n + 1)st tile))’ is true in the model, and so ‘(32) (Defaed (the nth tile) A 7 Defaed (the (n+l)s t tile))’ is definitely true. But what that number might be will vary from model to model, since the extension of ‘constraint’ will vary from model to model; so there will be no number n such t h a t ‘(Defaed ( the n th t i le) A 7 DefRed ( the (n + 1)st tile))’ is definitely true. ‘Definitely red’ is not a sharp notion. We have not fallen into the trap of replacing a vague bifurcation ‘red’rnot red’ by a sharp tripartite division ‘definitely red‘rdefinitely not red’/ ‘unsettled’. As we ascend the hi- erarchy of metalanguages, we find vagueness all the way up.

One aspect of the formalism that deserves comment is i ts treatment of the identity relation.ss Because ‘=’ denotes the

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same relation in every A-model, identity will be a sharp relation, so that we have this:

(Vx) ( V y ) ( x = y + Def x = y ) (Vx) (Vy) ( x # y + Def x f y)

There can nonetheless be unsettled identity statements. Propo- nents of vagueness in language will have it t h a t ‘a = Mt. Pisgah’ is neither definitely true nor definitely false, but this is because ‘Mt. Pisgah’ is imprecise, not because ‘=’ is imprecise.

Some authorss9 have responded to the perplexities induced by such questions as ‘Which patient will survive the brain transplant?’ by supposing that identity is a vague concept. To accommodate these intuitions into the semantics, we have to let the extension of the identity sign vary from model to model. So we no longer t rea t ‘=’ as a logically privileged predicate with a fixed extension. We treat ‘=’ as an ordinary predicate, incorporating i ts special logical features into the system of constraints. Thus, the penumbral constraints are likely to include the reflexive law ‘(Vx) x = 2,’ as well as the following restricted versions of the law of the indiscernibility of identicals:

(VX) (Vy) ( X = y -+ (F ( x ) c) F (y))), for F an open sentence in which ‘Def’ does not appear.

(Vx) (Vy ) ( D e f x = y + (F ( x ) c) F (y))), for F arbitrary.

If we are to regard identity as vague, we would not want to adopt the unrestricted version of the law of indiscernibility of identicals, since an argument adapted from Gareth Evansso shows that the unrestricted version of the law entails this:

7 ( 3 x ) (3y) i t is definite tha t it is indeterminate whether X=E

which we may abbreviate

-(3x) (3) DefV x = y ,

employing VP as an abbreviation for (- Def P A 7 D e f - P) . The argument is as follows: take any a and b. ‘a = a’ is definitely true, being a law of logic; hence:

- V g = a .

The law of the indiscernibility of identicals gives us this:

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Consequently,

By contraposition,

V g = b + - g = b

This conclusion is a law of logic, hence definitely true:

Since ‘Def distributes over conditionals, we get:

Def V g = h + D e f - . g = b

On the other hand, the principle (Def P + P ) gives us this:

Def V a_ = b -+ V g = b,

and the definition of ‘V’ gives us this:

V g = b + 7Def 7 p = b

Hence, we derive:

7 Def V g = h

Since a and b were arbitrary, we conclude:

7 (3x1 (3y) Def V x = y ,

as required. If we assume the analogue of the Brouwerian principle

(P + 0 0 P ) , with ‘Def in place of ‘a’, we can derive the stronger conclusion:61

7 (3x) (3) v x = y ,

as follows: since a = a is a law of logic, we have this:

D e f a = a

The indiscernibility of identicals give us this:

g = b + ( D e f a = a t , D e f g = h ) .

Together these give us, by truth functional logic:

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p = h+ Def p = b

Contraposing,

Since we have proven this by logic, it is definitely true; hence:

Def (7 Def p = h + -, p = b)

Since ‘Def distributes over conditionals, we derive:

Def 7 Def p = h + Def -, a = h

The Brouwerian principle, together with the fact that Def P is definitely equivalent to Def .-) 7 P , gives us this:

Together, these two conditionals give us this:

which immediately implies, by the definition of ‘V’:

We already have (a = h + Def a_ = h), which implies, again by the definition of ‘V’:

Hence, by truth functional logic:

Since a and b were arbitrary, the desired conclusion follows. We do not know of any convincing reason either to accept

or to reject the thesis that the Brouwerian principle holds for ‘definitely’. By contrast, it is quite clear that we should reject the ‘definitely’ analogue of the 55 axiom (0 P + 0 0 P). The analogue of the 55 axiom is logically equivalent to this:

7 Def P + Def -, Def P

If we accepted this, then, since the S4 axiom (Op -+ Oop) is a theorem of $35, we would also have to accept the analogue of the S4 axiom:

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Def P + D e p e f P

Together these two principles tell us that, however indistinct P may be, ‘definitely P‘ is a sharp notion, a thesis we want vigor- ously to deny. But about the much weaker Brouwerian principle we have no firm opinion.

The perplexing problems about the nature of personhood raised by the brain transplant thought-experiments, and the equally difficult problems about the relation between essence and accident raised by other puzzles about persistence through change, cannot be simply dissolved by taking an appropriate stance on the logic of identity. The problems are still going to be there, whatever we say about identity. Instead of asking ‘Is the patient who awoke in Recovery Room A at noon identical to the patient wheeled into the operating room for brain surgery at 9 a.m.?’ we can ask ‘Did the patient who was wheeled into the operating room for brain surgery at 9 a.m. awake in Recovery Room A at noon?’ The two questions ask the same thing, even though in the second question the word ‘identical’ never appears. Although we do not expect tinkering with the logic of identity to solve the philosophical problems, we would like to be sure that our logical theory does not stand in the way of a philosophical solution. If our philosophical in- vestigations reveal the need for an imprecise notion of identity, we would like our logical theory to make such a notion available. So we are glad that an alternative treatment of the identity relation is available, even though in this paper we shall continue to treat ‘=’ as sharp.

The program we have been outlining is quite an ambitious one, and we are not altogether confident that it will succeed. The problem we anticipate is this: it may well be the case that the 5,800th tile and the 6’100th tile are both on the border between ‘red’ and ‘not red’, even though the former tile is visibly redder than the latter. In such a situation, the conditional ‘If the 6’100th tile is red, then the 5,800th tile is also red’ should count as definitely true. Arranging this is no problem. We simply introduce the generalization ‘If one thing is redder than another, then if the second is red, the first is red as well’ into our system of constraints. The difficulty we foresee is that we are only able to state a constraint which excludes models in which the 6,100th tile is counted as red but the 5,800th is not because the relation ‘redder than’ is expressible in English. Perhaps there is an analogous case in which there are some models that are intuitively unacceptable even though no constraints which effect their exclusion a re expressible in En- glish. We do not have an example of this phenomenon, but we worry that such a thing might happen.

We know from the theorem that , provided the set of definite truths is closed under Tarski A-consequence, there will be

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a class of models K such tha t the definite truths are the sentences true in every member of K, and there is a set A of constraints such that K consists of the A-models of A. The worry is that K will be an unnatural class, one we are unable to com- prehend or describe properly, so that we cannot characterize the set A satisfactorily. If this should happen, our hope is that i t will be possible to find some other, more natural set of A- models J such that the set of definite truths can also be char- acterized as the sentences true in every member of J, and that J can be satisfactorily described in terms of a system of constraints, if not in the original language, then in some extension of the original language. The following theorem shows that, in theory, i t is nearly always possible to describe a given class of A-models as the A-models of a system of constraints expressed in an extended language, though the proof gives no assurance at all that the constraints can be identified in practice:

Theorem. Provided the cardinality of A is at least as great as the cardinality of the set of sentences of L, for any class J of A-models of L, there will an extension of LA, gotten by adding new predicates, in which there is a set of sentences A such that an A-model of LA can be ex- panded to a model of A if and only if it is a member of J.

Proofi Since the cardinal number of A will be equal to the cardinal number of the set of sentences of LA, there will be a bijection which pairs an element rS1 of A with each sentence S of L ; this pairing can be entirely arbitrary. For each A-model M wiich is not in J, introduce a new predicate T,, and introduce an additional new predicate T. The theory A will consist of the following sentences:

All sentences T,(Q1), for M an A-model of LA that isn’t in J and S a sentence of L, that is true in M.

All conditionals (T (rS1) + S), for S a sentence of LA.

All sentences ( 3 x ) (T(z) A -T,(x)), for M an A-model of LA that is not in J.

It is easy to check that this works. There is nothing particularly new about the model theory

we have been developing; most of the ideas a re already present in Fine’s paper. The primary reason we nonetheless went through the effort is that we wanted to make it explicitly clear that our commitment to classical logic and the principle of bivalence does not entail any commitment either to a sharp distinction between red and nonred or to a sharp distinction

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between definitely red and not definitely red. On our account, we have vagueness in the object language, i n the metalanguage, in the metametalanguage, and so on, all the way up.

Turning the tables, we would like to raise an ad homines objection against those who develop a three-valued semantics for the object language by employing classical logic in the metalanguage. If, as these thinkers assert and we deny, the use of classical logic commits one to the existence of sharp distinctions, then their use of classical logic in the metatheory commits them to the existence of sharp distinctions in the metalanguage. To be sure, they do not have a sharp true/false dichotomy, but they do have a sharp true/false/neither tri- chotomy. But this sharp tripartite division strikes us as quite implausible. There is nothing we do, say, or think tha t pro- duces a sharp distinction between things in the extension of ‘red’, things in the antiextension, and things in the gap. Rather than an implausibly sharp three-part division, i t is better to have what we have proposed, a fuzzy two-part division. (We would not make the same complaint against the in- tuitionists, who typically employ an intuitionistic metalogic.)

We do not need the details of the model theory to see what has gone wrong in the sorites argument. What has gone wrong is the fallacious inference from ‘There is an n such the nth tile looks red and the ( n + 1)st tile does not’ to ‘There is a sharp distinction between the tiles tha t look red and those tha t do not’. This fallacious inference we have diagnosed as arising out of a conflict between the disquotation principle and the correspondence principle. In the model theory, we have at- tempted to go beyond the diagnostic problem to get the begin- nings of a positive account of the semantics of vague languages that gives each of the principles a place. But we do not need the positive account to defang the sorites monster.

V. DEGREES OF TRUTH

We would now like to apply the model theory to flesh out an idea we mentioned in Section I, the idea that the intuitive appeal of the sorites premise can largely be explained by the fact that every instance of the premise is very highly probable.

There is a class of acceptable A-models such tha t a sentence is definitely true if and only if it is true in every acceptable A-model. We therefore have a natural Boolean algebra which measures how nearly definitely true a given sentence is: take the Boolean value of a sentence to be the set of acceptable A-models in which it is true. The uni t element of the Boolean algebra is the set of acceptable A-models; a sentence

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has Boolean value 1 if and only if it is definitely true. The zero element of the Boolean algebra is the empty set; a sentence has Boolean value 0 if and only if it is definitely false. Other sentences assume Boolean values in between 0 and 1. The Boolean sum and product are union and intersection, respec- tively.

There is a clear sense in which ‘Tile 5,800 is red’ is more nearly definitely true that ‘Tile 6,100 is red‘, even though neither sentence is definitely true: there are acceptable A-models in which the former is true and the latter is false, but not vice versa. The Boolean algebra represents this fact by setting the Boolean value of ‘Tile 5,800 is red‘ greater than the value of ‘Tile 6,100 is red’.

We can go a step farther, if we like, by extending the Bool- ean algebra to a finitely additive probability measure on the set of acceptable models,62 then identifying the probability of a sentence with the measure of the set of sentences in which it is true. Thus, definitely true sentences will have probability one and definitely false sentences will have probability ‘Tile 5,800 is red’ will be assigned a numerical value intermediate between zero and one, and it will be greater than the number assigned to ‘Tile 6,100 is red’.

Developed by Dorothy E d g i n g t ~ n ~ ~ and R. M. Sainsbury,66 the probabilistic treatment expresses a clear intuition we have that, even though neither sentence ‘Tile 5,800 is red‘ and ‘Tile 6,100 is red’ is either definitely true or definitely false, the former sentence is truer or (better put) more nearly definitely true than the latter. It captures the spirit of the fuzzy logic approach invented by Lofti Zadeh,66 without the messy details. In particular, it does not require us to repudiate classical logic and classical mathematics, since it treats classical tautologies as definitely true, and it allows that, if P (classically) implies Q, the probability of Q is a t least as great as the probability of p.

Within any acceptable model, there is a critical point n such that the nth tile looks red to you but the (n + 1)st tile does not. On the other hand, for each number n, the propor- tion of the models (or, more accurately, the measure assigned to the set of models) in which n is such a critical point is close to zero, so that the probability that the nth tile looks red to you but the (n + 1)st does not is close to nil. This is helpful, we think, in explaining why the sorites premise, though demon- strably false, nonetheless has such intense intuitive appeal: each instance of the premise is almost surely true.

I t has been claimed tha t concepts are frequently built around prototypes.67 When an individual sufficiently resembles the prototype, the concept definitely applies. When an individual is sufficiently different from the prototype, the concept definitely does not apply. Intermediate cases arise when the

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resemblance is modest. One can think of the probability assigned to an attribution of the concept as measuring how far a given individual is removed from the prototype.

Our problem with the probabilistic approach is that we do not know where the numbers come from. The idea of using the probabilities to measure how closely an individual resembles the prototype is very attractive, but we do not yet see how to measure these degrees of similarity numerically; we do not see what scale to use.

For the simplest cases, we might use subjective probability as a start ing point. Thus, we could identify the probability that the nth tile looks red to you with your subjective probability that the nth tile is red, provided (and this is important) that your opinion about the color of the tile is based entirely and directly on how i t looks to you. Subjective probability must be understood here as a measure of the intensity of your inner feeling of confidence, since the usual method of measuring subjective probability, taking betting quotients, i s not available; in borderline cases, even God could not settle the bets. This takes care of the probability that a given tile looks red. To get the probability that a tile is red, we might try identifying the probability that the nth tile is red as the subjective probability that the tile is red of an ideal observer, examining it under ideal circumstances; but we see no reason to presume that all such observers would have the same subjective probability. So we have trouble even in the simplest cases. For more complicated cases and cases further removed from immediate sensory judgments, we do not know where to begin.

These difficulties do not really constitute an objection to the probabilistic program, which we regard a s extremely promising, but rather as an acknowledgment that a t this early stage in the program’s development, central foundational questions remain unanswered, so that we cannot yet feel confident that the program is going to succeed. Notice that, since we are allowing vagueness in the metalanguage, the use of the probability calculus need not commit us to a precise numerical measure. The metatheory can describe the probability function without specifying it precisely. So it is possible for the probabilistic account to be both useful and literally true, even if i t turns out to be impossible to specify exact values for the function.

That there are degrees intermediate between definite truth and definite falsity is striking, whether these degrees are measured numerically o r by a Boolean algebra. The measure we get from the Boolean algebra is compositional, so that, for example, the degree of a disjunction is the Boolean sum of the degrees of i ts disjuncts. In other words, we have a many-Val- ued logic, in the sense of Rosser and Turquette.68 The probabilistic approach is no longer compositional, in the sense tha t

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the probability of a compound sentence is not determined by the probabilities of its components. We do not think this is an objectionable feature of the probabilistic account. To be sure, the meaning of a compound sentences is determined by the meanings of its simple components and how those components are put together. To be sure, how nearly definitely true a sentence is is determined by the meaning of the sentence, together with the nonlinguistic facts. But i t would clearly be fallacious to conclude from these platitudes tha t how nearly definitely true a compound sentence is is determined by how nearly definitely true its simple components are.

A characteristic mark of vague terms is the appearance of degrees intermediate between definite applicability and definite inappl i~abi l i ty .~~ ‘Red’ is a vague term, so there are grada- tions of color intermediate between ‘definitely red’ and ‘definitely not red’. Since we want to t reat ‘true’ as a vague term, we are committed to the existence of degrees intermediate between definite truth and definite falsity. Our only reser- vation about the probabilistic account is tha t we are not yet certain how these degrees ought best to be measured.

APPENDIX: ANSWER TO A QUESTION OF SANFORD

Discussing the thesis that an English sentence is definitely true if and only if it is true in every precisification of English, David Sanford70 asks a telling question: why should the fact that a sentence is true in a lot of languages different from En- glish-different from i t in vitally important ways-have any bearing on the question whether the sentence is true in En- glish?

We reject the thesis tha t a sentence is definitely t rue in English if and only if i t is true in every language that could be gotten by making English fully precise. Instead, we propose that a sentence is definitely true in English if and only i t is true in every model of English. For us, no hypothetical languages different from English are involved. The models we are looking at are all models of English (though the relation between a vague language and i ts models is considerably less tau t than tha t between a precise language and its intended model). So for us Sanford’s question is a good deal less urgent than i t was initially. Still i t is a question worth asking: why does the fact that a sentence is true in every acceptable model assure us that the sentence is true?

In talking about models of English, we are exaggerating shamelessly, of course. What we have in hand is a sketch of a model theory for a class of very simple formalized languages

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that are intended t o simulate certain aspects of English semantics. The obstacles that stand in the way of getting from the models of these formalized languages to models of English are immense.

The question we really intend to try t o answer is a dumbed-down version of Sanford’s question. We are given a system of constraints for a language LA. We know what the sentences of the language L mean, because we know how t o translate them into English, and we know what the new constants we added t o L in forming LA refer to because we have stipulated that g refers t o a. LA is not an “interpreted language,” as that phrase is usually understood, because it does not have a unique, precisely identified intended model, but the language is meaningful nonetheless, and we can speak meaningfully about a sentence of the language being either true or false. We know that the constraints are all true; either they are clearly correct applications of a vague term, or else they are penumbral constraints, which are analytic. What we want to see is that it follows from this that every sentence true in every A-model of the constraints is likewise true.

The question is an analogue t o a question in modal logic: how do we know that a sentence true in every possible world is true? For modal logic, the answer is simple: a sentence true in every possible world is, a fortiori, true in the actual world, and a sentence true in the actual world is true. We cannot give the same answer here, because there is no model that is distinguished as the actual model.

We can give a mathematical demonstration that, if each of the constraints is true, then every sentence true in every A- model of the constraints will be true, but the demonstration will require a nontrivial set-theoretic assumption. The assumption is that the separation axiom schema holds even if the open sentence being substituted into the schema contains vague terms. The justification for this assumption is pragmatic: rejecting it does no good, inasmuch as the separation principle is not implicated in any of the known paradoxes about vagueness, and it is likely to do a great deal of harm. We employ the separation principle every time we count or measure, in order t o form the set of things being counted or measured; virtually every empirical quantity we wish to count or measure is, to some extent, vaguely defined; so to refuse to admit vague terms into the separation axioms would pretty nearly kill off applied mathematics.

Before we give the demonstration, let us quickly examine how set-theoretic notation behaves in the presence of vague terms. There are six billion or so human beings, and it is possible, in principle, for an assiduous census taker t o list them all as pl, p z , ..., so that it is definitely true that (V’x)(x is a human bc%$$?% = p1 v x = x z v ... v x = p6,000,000,000)).

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(We are pretending, for simplicity, that there are no borderline cases of ‘human being’.) There are roughly 26~000~000~000 sets of human beings, and, with enough time and grant money, we could list them all by enumerating their members, getting a list S,, S,, . . ., S,S~OOO~Ooo~Ooo.

and

(3! set R ) ( V y ) 0, E R t) y is bald)

are definitely true; the latter statement follows from the former by a separation axiom. Hence, the disjunction

{x: x is bald] = S,v {x: x is bald] = S, v ... v {x: x is bald] = S , 6 ~ O O O ~ ~ O * O O o

is definitely true; there is no room in the universe of set theory for any “fuzzy sets” of human beings in addition to S,, S,, ..., S,6.000~ooo~ooo. On the other hand, because ‘bald’ is a vague term, there is no number i such that

{x: x is bald) = Si

is definitely true. We may, if we like, introduce a notion of reference via a

Tarski-style definition. Such a definition will describe a “disquotational” rather than a “correspondence” notion of reference; to say that a name refers to an object will not imply any connection, intentional, causal, or otherwise, linking the name to the object.

(3! set R ) ( ‘ {x : x is bald)’ refers to R )

will be definitely true; so will this:

‘ ( x : x is bald]’ refers to S,v ‘ ( x : x is bald)’ refers to S, ... v ‘ { x : x is bald}’ refers to S~ooo~ooo~ooo.

But there is not any i such that

‘ {x : x is bald)’ refers to Si

is definitely true.

24 1


Assuming the requisite set theory, here is the argument: suppose each of the constraints is true and that S is true in every A-model that satisfies the constraints. I t is definitely true that there exists a unique A-model M that satisfies the following conditions:

M (‘Harry’) = Harry M (‘Mt. Pisgah’) = Mt. Pisgah M (‘bald’) = {x : x is bald] M (‘red’) = {x : x is red1

and so on, for every constant and predicate in the language. (Although, if we identified each of the A-models in some precise fashion, we would find no A-model of which i t was definitely true that i t satisfied these conditions.) An induction on the complexity of sentences gives us

rP1 is true in M t) rP1 is true

for each sentence P of LA. Since each of the constraints is true, each of the constraints is true in M , so that M is an A-model of the constraints. I t follows that S is true in M , and hence that S is true.

NOTES

’ There are a few, including one of the authors of this paper, who doubt the validity of modus ponens. (See Vann McGee, “A Counterex- ample to Modus Ponens,” Journal of Philosophy 82 (1988): 462-471). But even McGee allows that modus ponens is valid for conditionals that do not have conditionals as parts. So, to please McGee, the sorites premise could be reformulated thus:

If there is no visible difference in color between two tiles and one of the tiles looks red to you, the other looks red to you as well.

Now use universal specification, conjunction introduction, and modus ponens restricted to conditionals without conditional parts to complete the argument. Or, more simply, reformulate the argument in terms of material conditionals.

See Ernest W. Adams, “The Logic of ‘Almost All,”’ Journal of Philo- sophical Logic 2 (1974): 13-17.

See Henry Kyburg, “Probability, Rationality, and the Rule of De- tachment” in Yehoshua Bar-Hillel, ed., Logic, Methodology, and Philoso- phy of Science (Amsterdam: North-Holland, 19671, 301-310.

Peter Unger, “There Are No Ordinary Things,” Synthese 41 (1979): 117-154; and Samuel C. Wheeler, “On That Which Is Not,” Synthese 41

Jamie Tappenden (“The Liar and Sorites Paradoxes: Toward a Unified Treatment,” Journal of Philosophy 90 (1993): 551-577) regards the sorites premise as preanalytic, where a preanalytic statement has

(1979): 155-174.

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much the same status as has traditionally been accorded analytic statements, except tha t , whereas a n analytic sentence h a s to be t rue, a preanalytic sentence need only be nonfalse.

Stephen Cole Kleene, Introduction to Metamathematics (New York: American Elsevier, 19521, §54.

Dag Prawitz, Natural Deduction (Stockholm: Almqvust and Wiksell, 1965), 24.

“Truth by Convention” in 0. H. Lee, ed., Philosophical Essays for A. N. Whitehead (New York: Longmans, 1936), 90-124.

For classical logic, with the connectives “A and ”7” taken as primitive, we can give this observation a sharp form, as follows: an implication relation is a relation + between sets of sentences and sentences which meets the following conditions:

*

If P E r, then l- + P. If l-+P for e a c h P i n A and A + Q , then r + Q.

Our rules of inference are the following:

{E Ql + ( P A Q).

KP A Q N + Q. P, -Z‘\ + Q. If u {PI + Q and r u {-PI + Q, then I- + Q.

{(P &)I + Z?

To see t h a t these rules uniquely pin down the meaning of the connectives, suppose otherwise, so that there are at least two logically inequivalent possibilities for what is meant by the connectives. Disam- biguating, we can introduce symbols ‘A ’ and ‘yl’ for one of the candidates, ‘ A ~ ’ and (-I~’ for the other. Let PI be a sentence formed with ‘A ’ and ‘ll’ as its connectives, and let P, be the corresponding sentence with ‘A,’ and ‘12.’ Applying the rules of inference with both sets of connectives, we derive {P 1 + P, and {P,) + P,, so that the two candidates are logically equivalent atter all. Contradiction.

For a general development, applicable to a wide variety of classical and nonclassical logics, of the view that connectives are defined by their inferential role, see Arnold Koslow, A Structuralist Theory of Logic (New York: Cambridge University Press, 1992).

lo Many authors have wanted to relax their vow of silence a bit. While they are willing to neither assert nor deny the sorites premise, they have been willing to make the metatheoretical claim t h a t the sorites premise is neither true nor false. But we are unable to see how they got at this conclusion if not by the use of reductio ad absurdum in the metatheory.

If we countenance reductio ad absurdum in the metatheory, then in the metatheory we are forced to deny the sorites premise. If the object theory is silent about the premise, this only shows that the negation of the premise is a conclusion that can only be obtained at a higher level of reflection than the object theory affords.

l1 Anyone who wanted to uphold the validity of classical logic and si- multaneously to contend that, whenever an existential sentence is correctly asserted, it ought to be possible, at least in principle, to specify a witness would find herself committed to what seems to us a n outlandish position: that it is in principle possible to determine an answer to any

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well-formed ”Yes” or “NO” question. For each declarative sentence P , the classical logician is willing to assert the existential sentence

but she could only specify a witness to the existential claim if she could determine the truth value of P. Moreover, even someone who accepts the position that every ”Yes”/“No” question is potentially answerable must surely admit that this extreme epistemic optimism is not built into the meanings of the quantifiers.

If an existential claim is not to be understood as an offer to specify a witness, how is it to be understood? There are many possible answers, but the one that seems most likely is that the meaning of the quantifiers is given by the rules of inference. For the formal languages studied in the predicate calculus, it is easy to specify rules that accomplish this. Introduce universal quantification by adding to the rules of footnote 9 the following rules:

((Vx)Fx) + Fc. If + Fc and c doesn’t appear in the members of or in Fx, then r + ( V X ) FX.

(we assume an unlimited supply of constants), and introduce existential quantificatior by stipulating that ( 3 x ) Fx is to be interderivable with 1 (Vx) 1 F.;. The resulting system is categorical in the sense of footnote 9.

l2 Tappenden protests, posing the “objection from upper-case let- ters”: “When I say that there is a number n such that ... I mean that THERE IS a number n such that” (The Liar and Sorites Paradoxes, 564). While we agree with Tappenden that a natural first impulse, when confronted with an existential claim, is to demand to be shown a witness, we do not suppose that it is reasonable always to expect this first impulse to be satisfied. We can know a differential equation has a solution without having any idea how to calculate a solution, and, indeed, without a solution being calculable in principle. (See Marian B. Pour-El and Jonathan I. Richards, Computability in Analysis and Physics (Ber- lin: Springer-Verlag, 1989).) “There is a number n such that” means ”THERE IS a number n such that”; it doesn’t mean that one is able to produce such an n.

l3 One needs to state the sorites argument with some care if it is to be philosophically challenging. For example, the sorites argument will lead us to reject the premise “For any n, if every man with n hairs on his head is bald, every man with ( n + 1) hairs on his head is bald.” But this is a premise we should have rejected in any case, since whether a man is bald depends not only on the number of hairs on his head but also on their configuration. (See Linda Claire Burns, Vagueness (Dordrecht, Holland: Kluwer, 1991).) Again, the argument will lead us (by classical logic) to accept the conclusion “There is an n such that every man with n hairs is bald but not every man with ( n + 1) hairs is bald.” But this conclusion might hold for philosophically uninteresting reasons. It could be that the reason the conclusion holds is that for some number n, large enough to ensure unmistakably that no one with more than n hairs on his head is bald but small enough so that there is a

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man with more than n hairs, it so happens that there is no man with exactly n hairs on his head. So it is better to talk about a specific person, Harry say, than to talk in generalities.

Even when we restrict our attention to a single individual, we have to be careful. Consider the following reasoning:

Harry, who has 100,000 hairs, is not bald. If a man who isn’t bald had one fewer hair, he still would not be bald. Therefore (after many steps of argument), if Harry had no hair a t all, he still would not be bald.

The argument is not valid, or, a t least, i t is not valid according to Stalnaker’s semantics for conditionals (see Robert Stalnaker, “A Theory of Conditionals,” American Philosophical Quarterly Monograph Series (Oxford: Blackwell, 1968)); and, moreover, it would not be valid even if we treated the sorites premise as necessary. Indeed, the premises will not even suffice to enable us to derive the conclusion:

If Harry had two fewer hairs, he still wouldn’t be bald.

Let w be the nearest world to the actual world in which Harry has 99,999 hairs, let w 2 be the nearest world to w 1 in which Harry has 99,998 hairs, and let w be the nearest world to the actual world in which Harry has 99,99i hairs. I t is logically possible that “Harry is bald” should be true in w3 and false in w , giving us a model in which the conclusion is false in the actual world. If we reformulate the argument using material conditionals, the argument becomes valid, but uninteresting. Having a false antecedent, the conclusion is trivially true.

To avoid these difficulties, we prefer versions of the paradox in which it is assumed that all the intermediate cases we go through as we con- duct the sorites argument are actually and concretely available, either synchronically, like the tiles that are laid out in front of us all a t once, or sequentially, like poor Harry, whose hairs we pluck out one by one.

Sometimes the sorites paradox is understood as a problem about the vagueness of concepts or of Fregean senses, rather than about vagueness in natural language. We prefer to talk about vagueness in language, because words are a little more tangible, but we don’t expect that this matters in any important way.

l6 The remarks of Roy Sorenson, Timothy Williamson, and Crispin Wright have convinced us that ‘determinately’ would be a word less likely to cause confusion.

Timothy Williamson (Vagueness (London and New York: Routledge, 1994), 164) considers and rejects an account similar to the one being developed here, primarily on the basis that no one has ever given a satisfactory analysis of ‘definitely.’ Sad to say, he is right that no one has given a satisfactory analysis; we just hope that, in spite of this, the intuitive distinction between clear and border cases is comprehen- sible enough that it will not be wholly useless.

One can say that an object a definitely satisfies an open sentence if the facts about a are such as to ensure that the conditions of application of the open sentence are met; but this will only be helpful if we have a notion of ‘fact’ according to which, in the borderline cases, there is no

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fact of the matter whether the open sentence is satisfied. If we employ a pleonastic usage in talking about facts, so that ‘It is a fact that F“ means just the same thing as ‘P‘, we can say that the condition of application of the open sentence ‘x is bald’ is that 3c be bald, so that either Harry is bald, in which case the fact that Harry is bald ensures that the condition of application of ‘x is bald’ is met, or else Harry is not bald, in which case the fact that Harry is not bald ensures that the condition of application of ‘Harry is not bald’ is met. So if we try to apply our formula with the pleonastic notion of fact, we wind up with the unwanted conclusion that either Harry is definitely bald or definitely not bald.

l7 In saying that the nonlinguistic facts suffice, we are assuming that the sentence in question is not about language. More generally, it is the facts about whatever it is the sentence is about that determine the truth of a definitely true sentence.

l8 They are investigated in McGee’s Duth, Vagueness, and Paradox (Indianapolis: Hackett, 1991).

l9 See W. V. 0. Quine, Word and Object (Cambridge: MIT Press. 19601, 193.

2o More precisely, if every member of the universe of discourse has a name, then an existential sentence is true if and only if at least one of its substitution instances is true.

21 From Alfred Tarski, Andrzej Mostowski, and Raphael M. Robin- son, Undecidable Theories (Amsterdam: North Holland, 19531, 53.

22 It should be emphasized that Robinson’s R is all the arithmetical theory that is needed (in fact, we only need a little bit of R). If the proof used the principle that the natural numbers are well-ordered-There is an n such that the nth tile does not look red. Therefore there is a least number n such that the nth tile does not look red”-one might well object, as follows: “I agree that every nonempty, sharply defined set of natural numbers has a least element. But I do not agree that every nonempty fuzzy set of natural numbers has a least element; in particular, I do not agree that the set of numbers n such that the nth tile does- n’t look red has a least element. For to agree to that would be to say that the set had a sharp boundary.” To a proof that used well-ordering, this would be quite a n apt objection. But, in fact, since everything is bounded by 10,000, we do not require the principle that the natural numbers are well-ordered. Robinson’s R, a very weak theory which does not include either mathematical induction or the principle that the natural numbers are well-ordered, will suffice.

23 Saul A. Kripke, “Outline of a Theory of Truth,” Journal of Philoso- phy 72 (1972): 690-716.

More precisely, in Kripke’s theory, compound sentences a re grounded in the facts expressed by atomic and negated atomic sentences.

Alfred Tarski, ”The Establishment of a Scientific Semantics,” 401, in Tarski‘s Logic, Semantics, Metamathematics, 2nd ed. (Indianapolis: Hackett, 1983)) 401-408. This is a n English translation by J. H. Woodger of “Grundlegung der wissenschaftlichen Semantik,” Actes du Congr&s International o!e Philosophie Sckntifique 3 (1936): 1-8.

26 The correspondence principle is a weak and general thesis that stops far short of what are usually called “correspondence theories of truth,” though it is a step in the same direction. Although the correspondence principle does not allow a sentence to be true for no reason a t all, it leaves it almost completely open what that reason might be. It does

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not require the existence of states-of-affairs or other truthmakers, and it allows that the reasons for truth might vary drastically from sentence to sentence.

We would like to emphasize that, when we talk about the “thoughts and practices” by which speakers give their sentences their truth Val- ues, we intend to leave it a wide-open question by what mechanism our thoughts and practices accomplish this. In particular, we do not intend to limit the role which contingent causal features of the world that are independent of our activities can play in the mechanism by which truth conditions are established. We say nothing about the ratio of “external” to ”internal” factors.

Assuming the sentence is not about language. 98 Bertrand Russell, “On Denoting,” Mind 14 (1905): 479-493.

P. F. Strawson, “On Referring,” Mind 59 (1950): 320-344. Some allowance needs to be made here for sentences that express

thoughts too complex or too homble for anyone to entertain and for sentences that are too long or too uproarious for anyone to assert.

There are exotic problem cases to worry about. One could argue that ”The least large integer is even” expresses a proposition, even though there is no possible world in which it is either true or false.

The distinction between the disquotational and correspondence conceptions of truth is due to Hartry Field. See “The Deflationary Con- ception of Truth” in Graham MacDonald and Crispin Wright, eds., Fact, Science, and Morality (Oxford: Blackwell, 1986): 55-117. See also Paul Horwich, Duth (Oxford: Blackwell, 19901, especially , where a position on the paradoxes of vagueness is developed which is closely allied with that being advocated here, and Vann McGee, “A Semantic Conception of Truth?” Philosophical lbpics 21 (1993): 83-111.

ss W. V. 0. Quine, Philosophy of Logic, 2nd ed. (Cambridge: Harvard University Press, 19701, 10-13.

Alfred Tarski, “The Concept of Truth in Formalized Languages” in Logic, Semuntics, Metumathemutics, 256. This is the English translation by J. H. Woodger of “Der Warheitsbegriff in den formalisierten Sprachen,” Studia Logica 1 (1935): 261-405. Notice that Tarski’s proof relies essentially upon the use of classical logic. The (PI sentence,

[PI is plue if and only if P,

entails, in either strong or weak three-valued logic, the law of the excluded middle,

(P v -P),

and adjoining the law of the excluded middle to either strong or weak three-valued logic gives us full classical logic. So there is a sense in which the acceptance of the (PI-schema amounts to neither more nor less than the acceptance of classical logic.

We are grateful to Domenic Hyde’s insightful comments in this volume for helping us understand this.

36 Perhaps “true love” or “true north” involves a different notion of truth, but not one that’s relevant here.

87 Alfred Tarski, ” f iu th and Proof,” Scientific American 220 (6) (1969): 63-77.

98 The useful distinction between diagnostic and therapeutic aspects

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of an investigation of a paradox is due to Charles Chihara, “The Para- doxes: A Diagnostic Investigation,” Philosophical Review 88 (1979): 590- 618.

39 The term is Terence Horgan’s; see his “Robust Vagueness and the Forced-March Sorites Paradox,” Philosophical Perspectives 8 (1994):

40 One can come up with cases in which, even though a tile doesn’t look red to us, we are justified in saying that i t looks red to us, but such cases are quite exotic.

159-188.

41 There might be ties for wealthiest poor person, but we ignore this. 42 Bas van Fraassen, “Singular Terms, Truth Value Gaps, and Free

Logic,” Journal of Philosophy 63 (1966): 464-495. 43 Kit Fine, “Vagueness, Truth, and Logic,” Synthese 30 (1975): 265-

300. LA i s 9 mathematical abstraction, not a human language. The

various inscrutability of reference arguments of Quine (Chapter 2 of Word and Object) and others raise doubts whether it is humanly possible to establish a word-to-world relationship that unequivocally pairs a unique canonical name with each individual. But for the existence of the abstract language, all that is required is that there exist a mathematical function that pairs a member of A with whatever set is taken as its name. There is no presumption that the connection between a name and its bearer is uniquely determined by the thoughts and activities of the speakers of the language or, indeed, that there are any speakers of the language. So, within the mathematical language, the kind of “direct reference” which pairs p with a is unproblematic.

46 See C. C. Chang and H. J. Keisler, Model Theory (Amsterdam: North-Holland, 1973), g1.3.

46 The general thesis that a universal generalization is definitely true if all its instances are definitely true is an artifact of the formalism, an inconsequential consequence of the fact that we have fixed a universe of discourse and we have fmed names for all its members. The restricted thesis that an arithmetical generalization is definitely true if all its numerical instances are true is philosophically more interesting, for it marks a decisive formal contrast between definite truth and such epistemic notions as “would be accepted by an ideal observer in optimal epistemic circumstances.” An arithmetical generalization needn’t be accepted by an ideal observer in optimal circumstances just because all its instances are, but an arithmetical generalization has to be definitely true if all its instances are.

47 See Proposition 2.2.13 of Chang and Keisler. There A is taken to be the set of natural numbers, but it could just as well be any countable set. a Alfred Tarski, %her den Begriffder logischen Folgerung,” Actes du

Congrks International de Philosophie Scientifique 7 (1936): 1-11. En- glish translation by J. H. Woodger, “On the Concept of Logical Conse- quence” in Logic, Semantics, Metamathematics, 409-420.

48 That is, different occurrence of the same constant are replaced by the same variables, and occurrences of different constants are replaced by different variables.

6o See the papers by Michael Tye and R. M. Sainsbury in the present volume.

61 Rather than including each of the sentences ‘-(al is a mountain A &L is a mountain),’ for slightly differing mountainlike land

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masses a, and a2, separately among the constraints, we might include a general maxim to the effect that two distinct mountains cannot overlap very substantially among the penumbral constraints, discussed below. Cf. W. V. 0. Quine, “What Price Bivalence?” Journal of Philosophy 78 (1981): 90-95.

62 On the vagueness-in-the-world account, but not on its rival, we shall have:

(3) Def x = Mt. Pisgah,

which we can derive by the following argument

Def Mt. Pisgah = Mt. Pisgah. Therefore (3x) Def x = Mt. Pisgah.

How does the proponent of vagueness in language block the argument? The inference is formally analogous to the following modal inference:

0 (the author of Waverly exists -+ the author of Waverly wrote Waverly). Therefore (3) 0 (x exists + x wrote Waverly).

The modal inference is blocked by insisting that existential generalization is only permitted in modal contexts when the singular term in the premise is a rigid designator; ‘the author of Waverly’ is not a rigid designator.

‘Mt. Pisgah’ is a rigid designator, on the standard account of rigid designators, because it denotes the same individual in every possible world. Yet there is another sense in which it is not a rigid designator. It does not denote the same individual in every A-model of the constraints. The proponent of vagueness in language will block the unwelcome inference by insisting that a singular term can be used for existential generalizations within a “definitely” context only if it is a rigid designator in the alternative sense.

63 In trying to devise a fully precise language, we might attempt to replace the patently vague ‘looks red to me’ by the sharper ‘looks exactly the same to me in color a s the first tile.’ But this predicate, while sharper, still is not fully precise. There is a greatest n such that the nth tile looks exactly the same to you in color as the first tile, but there is no significant relevant difference between your visual response to that tile and your response to its immediate neighbors. So there is no n of which it is definitely true that the nth tile looks to you exactly the same in color as the first tile but the ( n + 1)st tile does not.

64 This objection is raised by David Sanford, “Competing Semantics of Vagueness: Many Values Versus Super-Truth,” Synthese 33 (1976): 195-210.

56 We are grateful to Jerry Fodor and Ernest LePore for helping us ~~

to understand-this point. 66 We do not suppose that it is possible for human beings to speak

fully precise languages, even in principle. The precise languages are a mathematical abstraction.

57 The vagueness of the system of constraints allows a useful fluidity. Vague predicates can be understood more or less strictly, in different

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settings, depending on our needs and purposes. 58 The papers by Jack Copeland and Michael Tye in this volume

were of great help to us in understanding the issues here. 59 For example, Peter Woodruff and Terence Parsons, “Indeterminacy

of Identity of Objects and Sets,” Philosophical Perspectives 8 (1994). 6o Gareth Evans, “Can There be Vague Objects?” Analysis 38 (1978):

208. 61 We cannot derive the stronger conclusion without the Brouwerian

principle, even if we allow the unrestricted application of the indiscernibility of identicals. We can show this by producing a modal system in which the principle of the indiscernibility of identicals holds without exception but the principle ‘7 (3) (3y)Vx = y’ is invalid. For this purpose, consider a modified Kripke semantics in which the “individuals” are functions assigning an object to each world (so an object is a world-slice of an individual), subject to the following constraint:

I f f and g are individuals with f ( w ) = g ( w ) and if u is accessible from w, then f l u ) = g(u).

As we make our way through an accessibility chain of worlds, individuals can merge, but they can never divide. Constants denote individuals. An n-tuple < f l , ..., f n > of individuals satisfies the predicate E in w if and only if the n-tuple of objects c f , (w) , ..., f,(w) > is in the extension in w of P . In particular, a = h is true in w if and only if a ( w ) = b(w) . On this semantics, the law of the indiscernibility of identicals holds without restriction, yet identity is indeterminate.

a = b + O a = b

is valid, but

is not; so we can have the conjunction:

V a = b A - a = b.

If our only identity axioms are the reflexive law a = a and the indiscernibility of identicals, we have the following results: of the fifteen normal modal systems formed using the axiom schemata D, T, B, 4, and 5 (see Brian Chellas, Modal Logic (Cambridge: Cambridge University Press, 1980), 94.3), the principle (a # b -+ D a # b ) is valid in the five systems that include Kl3 and invalid in the rest. 1 V a = b is valid in the nine systems that include either KB or K5 and invalid in the remaining six. 1 V a = b is valid in KT, KDB, KD4, KD5, KD45, KTB, KT4, and KT5 and invalid in K, KD, KB, K4, K5, K45, and KB4.

More precisely, we have a finitely additive measure on the field of sets of acceptable A-models that have the form

{acceptable A-models M: S is true in MI,

for some sentence S . 63 If the language is countable, we can arrange things so that a sen-

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tence has probability one if and only if it is definitely true. If the language is uncountable, it may not be possible to arrange this. See McGee, “We Turing Machines Aren’t Expected Utility Maximizers (Even Ide- ally),’’ Philosophical Studies 64 (1992): 115-123. a Dorothy Edgington, “Validity, Uncertainty, and Vagueness,”

Analysis 52 (1992): 193-204. 65 R. M. Sainsbury, Paradoxes (Cambridge: Cambridge University

Press, 19881, ch. 2. 66 Lofti Zadeh, “FUZZY Logic and Approximate Reasoning,” Synthese

30 (1975): 407-428. 67 See Eleanor Rosch, “On the Internal Structure of Perceptual and

Semantic Categories” in T. E. Moore, ed., Cognitive Development and the Acquisition of Language (New York: Academic Press, 1973).

BB John Barclay Rosser and Atwell R. Turquette, Many-Valued Logics (Amsterdam: North Holland, 1952). The fact that it is possible to repre- sent the degrees intermediate between definite truth and definite falsity by a system of many-valued logic does not indicate anything very special about definite truth and definite falsity, since it is possible to repre- sent virtually any system of implication by a many-valued logic; see Jerzy Tos and Roman Suszko, “Remarks on Sentential Logics,” Indagatzones Mathematicue 20 (1958): 117-183.

69 See Michael Tye’s paper in this volume. 70 Sanford, “Competing Semantics,” 195-210.

We would like to thank Ernest Adams, Jerry Fodor, Terence Horgan, Peter Klein, Ernest LePore, Barry Loewer, Graham Priest, and Michael “ye for their help. We would especially like to thank Dominic Hyde for his highly insightful comments.

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