VANN MCGEE and BRIAN P. MCLAUGHLIN
LOGICAL COMMITMENT AND SEMANTIC INDETERMINACY: A REPLY TO WILLIAMSON
Professor Williamson’s gracious and thoughtful reply to our review1 of Vagueness2 has clarified in a very useful way where he and we agree and disagree. Where we agree most enthusiastically is in the assessment that vagueness is largely – Williamson would say entirely – an epistemic phenomenon. Of course, everyone always knew that there were epistemic aspects to vagueness, but we daresay no one before Williamson realized just how much you could learn if you look at vagueness with epistemic concerns primarily in mind. Indeed, we would go so far as to say that Williamson has solved the sorites paradox. He hasn’t, of course, vindicated the sorites principles – sayings like “Anyone who owns just a penny more than a poor person is poor” – for those principles are beyond redemption. What Williamson has accomplished is the philosophical task that remains after we repudiate the principles, namely, to explain why, in spite of their preposterous consequences, the sorites principles have such enormous intuitive appeal. The explanation is that we cannot produce a counterexample, or even clearly imagine producing a counterexample. To apply a vague term reliably, we must leave a margin for error, so that people very close to the boundary of “poor” aren’t recognizable as poor, even if they are, in fact, poor. If we can recognize someone as poor, then a person with just a few pennies more is poor as well. So, although we can prove mathematically that there is a richest poor person, we can never hope to identify such a person by name. What we disagree about is meaning fixation. Allowing ourselves the simplifying fiction that whether a person is poor depends only on how much money the person possesses, Williamson contends that English usage determines a precise if unknown amount of money such that anyone with that amount of money or less satisfies “poor”, and anyone with even a penny more satisfies “not poor”. We don’t for a moment suppose that this doctrine is contradictory, but we nonetheless find it incredible that 1 Linguistics and Philosophy 21 (1998): 221–35. 2 London and New York: Routledge, 1994.
Linguistics and Philosophy 27: 123–136, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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our casual and careless practices establish a partition of such astonishing exactitude. The alternative is to suppose that English usage determines ranges of application for “poor” and for “not poor”, but it is not so incisive as to partition everyone into one category or the other; there are folk in the middle about whom usage is indecisive. The way we like to put it (the terminology is not obligatory), if Tom is in the first range, “Tom is poor” is determinately true. If in the second, the sentence is determinately false. Otherwise, the truth value of the sentence is undetermined. This threefold partition is itself vague. Williamson asks, reasonably enough, that we say explicitly what we mean by “determinately”, and we haven’t been able to do so. We anticipate that, if someday people have an adequate account of the procedures by which English words get their meanings, they will see that those procedures leave some difficult cases unsettled. That is our expectation, but no one has a satisfactory account of meaning fixation in English at present. The interim problem of trying to define “determinately” using the theoretical resources currently on hand is one on which we haven’t made much progress. Williamson’s complaint is entirely justified, but we ask, rather churlishly, if one cannot make a similar complaint against him? He confidently predicts that, if we do someday have an account of how natural-language expressions get their meanings, it will tell us that speakers’ practices establish an exactly delimited border for every meaningful predicate of the language. Doesn’t he owe us at least a sketch of an account of how this is done? We expressed the hope that someday a scientific understanding of meaning fixation would resolve the controversy between epistemist and semantic accounts of vagueness. Williamson is not nearly so optimistic. He thinks it not unlikely that the multifarious paths by which words get their meanings will never submit themselves to a unified scientific understanding, so that our dispute will never rise above the level of philosophical speculation. Certainly, we claim no powers of prognostication, but we would have thought that Williamson would have been embarrassed by the suggestion that the connections between linguistic usage and semantic values are too diverse and too imprecisely defined to allow significant generalizations. He himself advocates a principle of breathtaking generality, namely, that for every single meaningful predicate of the language, usage establishes a function that determines, for each possible context of utterance, an exact, exclusive, and exhaustive partition into things that satisfy the predicate and things that satisfy its negation.
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Rather than speculate further about the future of etymology, let us pursue a couple of rather technical points concerning Williamson’s objections to the possibility of maintaining a semantic theory of vagueness while upholding classical logic. According to Williamson, supervaluationists can’t retain classical logic, but proponents of classical logic needn’t fret. There is no need for supervaluations because there is no semantic indeterminacy; the indeterminacy of borderline cases is wholly epistemic. We’ll first examine his main objections to the claim that there is semantic indeterminacy. Then, we’ll address the issue of whether supervaluationists can retain classical logic.
1. S UPERVENIENCE AND S EMANTIC D ETERMINATION
There are two ways to express the thesis of semantic indeterminacy. The more prevalent is to deny bivalence, saying that, if Harry is a borderline case of “bald”, the sentence “Harry is bald” is neither true nor false. The other is to introduce a new operator, “determinately”, saying that, although “Harry is bald” is either true or false, it is neither determinately true nor determinately false.3 The appeal of the more complicated version is that it permits us (except in the vicinity of the paradoxes of self-reference) to hold onto the (T)- and (F)-sentences, sentences like the following, which together entail bivalence: “Harry is bald” is true if and only if Harry is bald. “Harry is bald” is false if and only if Harry is not bald. The merit of (T)- and (F)-sentences, in addition to their enormous intuitive appeal, is their ability to enable us to achieve some of the effects of infinitary logic by finite means. When we tell you “Everything the Pope says is true”, we are, in effect, conveying the conjunction of infinitely many sentences of the form, “If the Pope says ‘Flounders snore’, then flounders snore”.4 (In treating sentences as true or false, we are oversimplifying; if nothing else, we need context to tell us which Harry is salient. We’ll maintain the pretense for now, and come back to it presently.) Both approaches clash with our intuitions, but we think this can largely be explained in the way Williamson explained the counterintuitive outcome that there is a richest poor person. On the simpler approach, there 3 We ourselves prefer the latter. See “Distinctions Without a Difference”, Southern Journal of Philosophy 33 supp. (1995) (Spindel Conference volume for 1994): 203–52. 4 See Quine, Philosophy of Logic, 2nd ed. (Cambridge, Mass., and London: Harvard University Press, 1986), ch. 1. See also Volker Halbach, “Disquotationalism and Infinite Conjunctions”, Mind 108 (1999): 1–22.
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is a true sentence of the form “p, but ‘p’ isn’t true”, but we aren’t able to provide any such sentence. One or the other of “Harry is bald” and “Harry is not bald” is such a p, but we can’t say which. On the more complicated approach, we know that there is a sentence that is true but not determinately true, but we can’t provide such a sentence. According to Williamson, the second strategy is sure to fail, because the notions of determinate truth and determinate falsity collapse into truth and falsity.5 In making his case, he focuses on baldness, but the argument can be extended to other cases in which a vaguely defined attribute supervenes on a comparatively precisely defined attribute. Williamson begins with the simplifying assumption that whether a person is bald depends only on the number and configuration of the hairs on his head. It has become standard philosophical practice to cash out claims of metaphysical dependence in terms of supervenience, and this is what Williamson does, proposing the following supervenience principle: (SP)
For all N and C, necessarily, if one is bald and the number and configuration of one’s hairs are N and C respectively, then necessarily if the number and configuration of one’s hairs are N and C respectively then one is bald. (p. 116)
Assuming that our modal logic is S5 (a weaker modal logic would complicate the details, but it wouldn’t affect the substance of the argument), from (SP) we derive the analogous principle with “not bald” in place of “bald”. These two theses together entail: (1)
For every person a, number N, and configuration C, either it is necessary that, if a has N hairs in configuration C, a is bald, or it is necessary that, if a has N hairs in configuration C, a is not bald.
The next step of Williamson’s argument requires the following premise: (2)
To say that the number and configuration of a’s hairs determine that p is to say that necessarily if a has that number and configuration of hairs, then p. (p. 117)
5 Williamson speaks of definite truth rather than determinate truth, but the difference
is merely verbal. We prefer “determinate” because it is so natural to understand “definite” epistemically.
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This gives us: (3)
For any person a, number N, and configuration C, if a has N hairs in configuration C, then either the number and configuration of hairs on a’s head determine that a is bald or they determine that a is not bald.
Combining this with the innocuous: (5)
For any person a, there is number N and a configuration C that are the number and configuration of hairs on a’s head,
we derive: (6)
For any person a, either it is determined that a is bald or it is determined that a is not bald.
Wishing, as we do, to distinguish determinate baldness from baldness,6 we had best block the argument somewhere. The simplifying assumption that baldness depends only on number and configuration of hairs is harmless, and we have no quarrel with (1); we think of it as a penumbral constraint. We likewise have no complaint about the left-to-right direction – the “If number and configuration determine, then necessarily” direction – of (2). Our quarrel is with the right-to-left direction of (2). The supervenience principle (SP) precludes the possibility that any factor other than the number and configuration of a’s hairs could make a difference to the question whether a is bald, so it assures us that the number and configuration of a’s hair determines whether a is bald if anything does. But maybe nothing determines whether a is bald. The number-and-configuration-of-hairs facts determine the baldness facts, to the extent they are determined at all. The supervenience principle assures us that no other facts are relevant. But this leaves open the possibility that there should be no fact of the matter whether a is bald. Necessity is one thing, semantic determination is another; and neither implies the other.7 Modal statements can be semantically indeterminate. One who accepts the supervenience thesis will regard the following disjunction as determinately true “Either it is necessary that a person with 6 We treat “a is determinately bald” and “ ‘a is bald’ is determinately true” as
equivalent. 7 See McLaughlin “Supervenience, Vagueness, and Determination”, Philosophical Perspectives 11 (1997): 209–30.
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N hairs in C is bald or it is necessary that a person with N hairs in C is not bald”. But it may be indeterminate which disjunct holds, since a disjunction can be determinately true without either disjunct being so. To amplify the distinction between necessity and semantic determination, we’ll look at the possible-world semantics for modal logic within a supervaluationist context. The notions of an interpretation of a modal language and of what it is for a sentence to be true in a world under an interpretation are understood in the standard way. In particular, an interpretation assigns a set of n-tuples to each pair consisting of an n-place predicate and a world. Among the interpretations, those that respect, in an appropriate fashion, the meanings of the words of the language are distinguished as acceptable interpretations. To understand how the practices of a community of speakers single out the acceptable interpretations is the principal problem the supervaluationist program has to solve; it is too large a problem to confront here. But the leading idea is that, if a certain number and configuration of hairs are such that the factors that fix the meaning of “bald” establish that someone satisfies “bald” if he has that number and configurations of hairs, then in any acceptable model, for any world w, anyone with that number and configuration of hairs in w will be in the extension of “bald” in w. If a certain number and configuration of hairs are such that the factors that fix the meaning of “bald” establish that someone fails to satisfy “bald” if he has that number and configuration of hairs, then anyone with that number and configuration of hairs in w will be outside the extension of “bald” in w. If a certain number and configuration of hairs are not of either sort, then someone with that number and configuration of hairs in w will be in the extension of “bald” in w in some acceptable models, and outside the extension of “bald” in others. A sentence is necessarily true under an acceptable interpretation A just in case it is true in w under A, for every world w.8 A sentence is determinately true in a world w just in case it is true in w under A, for every acceptable interpretation A. A sentence is determinately true simpliciter if and only if it is true in the actual world under A, for every acceptable interpretation A. It is determinately true that a given sentence is necessarily true just in case the sentence is true in w under A, for every world w and every acceptable interpretation A. In general, to answer questions about necessity, we fix the interpretation of the language and we look at different ways the world might have been. To answer questions about determinate truth, we fix our attention on the actual world, and we look at 8 Setting aside vagueness that might afflict the notions of necessity and possibility,
we assume that the possible worlds are the same in all acceptable models. Recall we are employing S5, so that we don’t have to worry about an accessibility relation.
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different acceptable ways of interpreting the language.9 There is no reason at all to suppose that two such different variations should yield the same outcome.10 In going from (1) to (3), we inferred a conclusion of the form “Either the statement that p is determinately true or the statement that q is determinately true” from a premise of the form “Either it is necessary that p or it is necessary that q.” Such an inference is invalid, since it is perfectly possible that the acceptable ways of interpreting the language should fall into two nonempty categories, as follows: One category of acceptable interpretations makes p true in every world but makes q false in the actual world, whereas the other category makes q true in every world but makes p false in the actual world. In such a situation, (p ∨ q) will be determinately true, but neither p nor q will be determinately true. In our particular example, where p and q are the disjuncts of (1), the first category will consist of acceptable interpretations in which the threshold for the application of the word “bald” is set at N, C or lower, whereas the second category consists of acceptable models in which the threshold is set higher than N, C. Thus, the supervaluationist possible-world semantics yields the result that (1) is determinately true and (3) is false.
2. U TTERANCES AND D ISQUOTATION
Let us turn our attention now to the more prevalent method of describing semantic indeterminacy, according to which borderline attributions of vague terms are neither true nor false. In our review, we oversimplified badly by pretending that sentences are either true or false. That’s incorrect, of course. Utterances are true or false, and sentences are true or false in a context, but not true or false outright. The phrase “Utterance u means that p” is one of the most treacherous in philosophy, and we had hoped to avoid its perils. The simplifying pretense let us speak as if Williamson were committed to the (T)- and (F)-sentences, but his actual commitment is to a more subtle thesis we may call the utterance disquotation principle: If an utterance u means that p, then u is true if and only if p, 9 Cf. John Etchemendy, The Concept of Logical Consequence, 2nd printing (Stanford: CSLI Publications, 1999). 10 See our “The Lessons of the Many”, Philosophical Topics 28 (2000): 129–51, for a fuller discussion of the interplay between necessity and semantic determinacy in possibleworlds semantics.
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together with the analogous principle about falsity. Assuming, plausibly enough (though nothing in this neighborhood is entirely innocuous), that for every meaningful assertion u there is a p such that u means that p, we derive assertion bivalence: Every meaningful assertion is either true or false. On Williamson’s account, for an English assertion of the form “a is an F ” to be meaningful, there have to be features of English usage that, within the context of the utterance, pick out a unique individual as the referent of “a”, and there have to be other features of usage that, within the context, pick out an exact, exclusive, and exhaustive partition of individuals into those that satisfy “F ” and those that satisfy “not F ”. This is the aspect of Williamson’s doctrine that we find so incredible. It seems to us that the utterance disquotation principle will be difficult even for an epistemist to uphold in complete generality, although counterexamples will be far rarer for the epistemist than for the semantic indeterminist. It will not be impossible to maintain the fully general principle, but doing so will require the epistemist to suppose that everyday practices in making indirect speech reports are drastically in error. We shall consider two examples. The first involves fixing the sentence that is used in the utterance (along with its meaning) and varying the context in which the sentence is uttered. The second involves utterances made in the same context that use different sentences that (by ordinary standards) mean the same thing. For the first example, consider the adjective “shrewd”, whose conditions of application can vary drastically depending in the comparison class and on the specific mental abilities being evaluated. Within a political context, in which one was concerned with a politician’s ability to pitch an appeal to the voters so as to maximize partisan advantage, George W. Bush would be counted as shrewd by almost anyone. One would be less likely to make such a kind assessment in a context in which the politician’s ability to grasp the long-term implications of tax policy is at issue. Thus if Clarissa says “George W. Bush is shrewd” within a political conversation, she has said something true, whereas if Abigail says the same sentence in a discussion of economic planning, she has said something false. Or that, at least, is the way it seems to us; you may alter the example to suit your political views. Both Clarissa and Abigail have said that George W. Bush is shrewd. Because of the differences in context, they meant somewhat different things by the words they used, but each of them produced an utterance that meant that George W. Bush is shrewd. At least, that’s what we’ll say according
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to colloquial standards of how to report other people’s speech. What we’d ordinarily say is that both women said that George W. Bush is shrewd and they both meant what they said, but they had different things in mind when the said it. That both women said and meant that George W. Bush is shrewd is something we can’t say according to the utterance disquotation principle, if we want to allow that one of them uttered something true and the other something false. The principle has these instances If Clarissa’s utterance means that George W. Bush is shrewd, then Clarissa’s utterance is true if and only if George W. Bush is shrewd. If Abagail’s utterance means that George W. Bush is shrewd, then Abagail’s utterance is true if and only if George W. Bush is shrewd. These together imply: If Clarissa’s utterance means that George W. Bush is shrewd and Abagail’s utterance also means that George W. Bush is shrewd, then if either utterance is true, both are true. We can save the utterance disquotation principle from this counterexample by insisting that one or the other of the two utterances didn’t mean that George W. Bush is shrewd. Actually, since Clarissa’s political conversation and Abigail’s economic discussion had different concerns from our present philosophical context, we should probably say that neither woman’s utterance meant that George W. Bush is shrewd. Such a course would have severe practical disadvantages – context dependence is sufficiently prevalent in natural languages that the policy would appear to make it virtually impossible to report what anybody says outside the present context – but we don’t have any philosophical objection to it. For our second example, we shall consider two utterances that employ sentences in different languages that (at least by ordinary standards) have the same meaning. Let u1 be an utterance of the Spanish sentence “El paquete de la tía Merle pesa un poquito menos de diez libras”, and let u2 be an utterance in the same context of the Dutch sentence “Het pakket van tante Merle weegt een beetje minder dan tien pond”. Both utterances mean the same thing, namely, that Aunt Merle’s package weighs a little less than ten pounds. But we say that the utterances have the same meaning the way we say that two people have the same philosophical interests or the same color eyes. Two people never have exactly the same philosophical interests or eye color. The phrase “pesa un poquito menos de diez libra” is an expression of Spanish, and it means what it does because of the way speakers of Spanish use its constituent words. The phrase “weegt een beetje minder
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dan tien pond” is an expression of Dutch, and it means what it does because of the way speakers of Dutch use its constituent words. It would be nothing short of miraculous if the two historical developments ended up in exactly the same place. Barring a miracle, we can, by tinkering with the weight of Aunt Merle’s package, construct a hypothetical situation in which the package is within the extension of one phrase but outside the extension of the other, so that exactly one of the two utterances u1 and u2 is true. Because the biconditional, u1 is true if and only if u2 is true fails, one or the other of the following two instances of the disquotation principle must also fail: If u1 means that Aunt Merle’s package weighs a little less than ten pounds, then u1 is true if and only if Aunt Merle’s package weighs a little less than ten pounds. If u2 means that Aunt Merle’s package weighs a little less than ten pounds, then u2 is true if and Aunt Merle’s package weighs a little less than ten pounds. Thus we have a hypothetical counterexample to the utterance disquotation principle. Once again, we can block the counterexample by taking “means the same” to mean “means exactly the same (in the strictest sense of ‘exactly’)”, but the price would be to forbid all but homophonic translations.
3. S UPERVALUATIONS AND C ONDITIONAL P ROOF
It is obvious that one can accept classical logic, in the sense of being disposed to reason in ways classical logic approves of, without also accepting the metatheoretical principle that every meaningful statement is either true or false, because it is possible to accept classical logic without having any metatheoretical opinions whatever. Bas van Fraassen11 did something not at all obvious, namely, to sketch a plausible semantics that would justify a person in accepting classical logic while actively denying the principle of bivalence. On van Fraassen’s theory, all the classical validities are true, and every classical consequence of a true sentence is true, yet there are meaningful statements that are neither true nor false. 11 “Singular Terms, Truth Value Gaps, and Free Logic”, Journal of Philosophy 63
(1966): 464– 95.
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Williamson questions whether what van Fraassen’s semantics provides is really enough to count as full classical logic. To accept classical logic, Williamson says, one must be willing to reason in the ways classical logic recommends. Anyone who regards the patterns of argument found in, say, Euclid’s Elementsand Gauss’s Disquisitionesas illegitimate cannot in any useful sense be said to accept classical logic, even if, by some deviant pathway, he comes to regard all the classically appropriate sentences as true. While Williamson takes care to avoid engaging us in a futile debate over the definition of the phrase “accept classical logic”, he insists that conditional proof, reductio ad absurdum, and proof by cases occupy such a central place among the methods of reasoning employed both in classical mathematics and in the sciences that utilize classical mathematics that whatever reasons one might have for supposing that it is a good thing to accept classical logic would be forfeited by anyone who rejected these methods of proof. We completely agree. In order for the supervaluationist’s claim that she accepts classical logic to count as anything more than wordplay, she must commit herself to the employment of conditional proof, reductio ad absurdum, and proof by cases. Where we disagree with Williamson is in how, precisely, these three rules ought to be formulated. We’ll look at conditional proof, although similar considerations apply to the other two. As we understand it, what a commitment to classical logic demands is this: Conditional Proof (Inferential Version). If you assume p as a premise, and from this premise together with further premises that you accept you are able to conclude q by any combination of the rules of inference sanctioned by the classical predicate calculus, then you may discharge the assumption to derive the conclusion (p → q). The “rules of inference sanctioned by the classical predicate calculus” will vary slightly from one natural deduction system to another, but we may take them to include conditional proof, reductio ad absurdum, and proof by cases. Supervaluation theory establishes that conditional proof, in its inferential version, will never lead us to accept an untrue conclusion on the basis of true premises.
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Williamson proposes a more liberal12 version of the rule: Conditional Proof (Metatheoretical Version). If you assume p as a premise, and from this premise together with further premises that you accept you are able to conclude q by modes of inference you know to be truth preserving, then you may discharge the assumption to conclude (p → q). Williamson’s more liberal condition is implicit in Kit Fine’s 1975 paper,13 where he argues that a supervaluationist should regard an argument from p to T(p)14 as valid because it’s truth preserving, but she shouldn’t accept the conclusion (p → T(p)). One could respond to this argument simply by denying its presumption that (where “true” means “supertrue”), if a statement is true, then it’s true that it’s true. Williamson has constructed an ingenious example to the same effect that doesn’t require the presumption. The inference from (p∧ ∼ T(p)) to (p∧ ∼ p) is truth preserving, even for the supervaluationist who denies the maxim (T → TT), yet to accept the conditional ((p∧ ∼ T(p))) → (p∧ ∼ p)) is to repudiate truth-value gaps. Fine’s groundbreaking paper was the origin of the supervaluationist approach to vagueness, and it is a little impudent of us to say he’s got it wrong, even in the details. But that’s what we have to say nonetheless. The inferential version of conditional proof is all that’s needed for Euclid’s Elements and Gauss’s Disquisitiones. To extend the inferential version by allowing any old truth-preserving mode of inference to appear within conditional proofs is to go beyond the logic employed by Euclid and Gauss, to go beyond it in a way that’s mischievous as well as gratuitous. The metatheoretical version of conditional proof is justified by classical, bivalent semantics. To illustrate, suppose that the inference r p ∴q is truth preserving, so that, if r and p are both true, q is true as well. Allowing this new inference to appear within conditional proofs, we can 12 If an inference is sanctioned by classical logic, it’s known to be truth preserving, but
there are metatheoretical principles that aren’t a part of classical logic that are known to be truth preserving. 13 “Vagueness, Truth, and Logic”, Synthese 30 (1975): 265–300. Reprinted in Rosanna Keefe and Peter Smith, eds., Vagueness: A Reader (Cambridge, Mass., and London: MIT Press, 1996), pp. 119–50. The remarks on conditional proof are on p. 143 of Keefe and Smith. 14 “T(p)” formalizes the statement that p is true.
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derive (p → q) from r by assuming p as a new premise, deriving q by the new rule, and then discharging. This derived inference is again truth preserving, as we can establish by the following argument: Assume r is true. There are two cases: Case 1. p is true. Then, since the inference “r, p, ∴ q” is truth preserving, q must be true. Consequently, (p → q) must be true, since it has a true consequent. Case 2. p isn’t true. Then p is false, so (p → q) is true, because it has a false antecedent. This argument relies heavily upon bivalence. Once we allow truth-value gaps, we admit the possibility that the argument “r, p, ∴ q” is vacuously truth preserving, because, whereas r is true, neither p, q, nor (p → q) has a truth value. The form of conditional proof that we can all agree upon is this: if you assume p as a premise, and from this premise together with further premises that you accept you are able to conclude q by legitimate modes of inference, then you may discharge the assumption to conclude (p → q). The controversy is over what modes of inference to count as legitimate. To satisfy the needs of Euclid and Gauss, we need the legitimate mode of inference to include those sanctioned by the predicate calculus, so that an inference counts as legitimate if the conclusion of the inference is a logical consequence of its premises. If we go further and allow every truth-preserving mode of inference as legitimate, we reinstate bivalence, as Williamson’s argument shows. From a supervaluationist point of view, it’s hardly surprising that, if we try to characterize legitimate inferences in terms of truth-preservation, we get unsatisfactory results. The central hypothesis of supervaluation theory is that the familiar notions of truth and falsity do not suffice to describe and explain how people reason with vague terms. We need the further notions of truth and falsity in an acceptable model. The good inferences, according to supervaluation theory, are the ones that preserve truth in A, for each acceptable model A. Taking this as our standard of legitimacy, we obtain: Conditional Proof (Supervaluationist Metatheoretical Version). If you assume p as a premise, and from this premise together with further premises that you accept you are able to conclude q by modes of inference that are known to preserve truth in each acceptable model, then you may discharge the assumption to conclude (p → q).
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This version of conditional proof, employed either by itself or in combination with other modes of inference that preserve truth in acceptable models, invariably leads from (super)true premises to (super)true conclusions. Because modes of inference that preserve classical consequence also preserve truth in a model, the inferential version of conditional proof can be derived from the supervaluationist metatheoretical version. Neither the inference from p to T(p) nor the inference from (p∧ ∼ T(p)) to (p∧ ∼ p) preserve truth in an acceptable model, so the examples by which Fine and Williamson attempted to show that conditional proof enforces bivalence are thwarted. Similar considerations apply to the other modes of indirect reasoning. By taking the legitimate inferences – those that can be properly employed within both direct and indirect proofs – to be the ones that preserve truth in an acceptable model, conditional proof, reductio ad absurdum, and proof by cases are exonerated.15 Vann McGee Department of Linguistics and Philosophy Massachusetts Institute of Technology 77 Massachusetts Ave, Bldg. E39 Cambridge, MA 02139 USA E-mail:
[email protected] Brian P. McLaughlin Department of Philosophy Rutgers University 26 Nichol Ave New Brunswick, NJ 08901-1411 USA E-mail:
[email protected]
15 We are grateful to Delia Graff for very helpful comments.
