The Philosophy of Philosophy. Timothy Williamson

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СКАЧАТЬ and talk are not always about thought or talk. To judge by its overt compositional structure, the original question in particular is not about thought or talk. It is no metalinguistic or metaconceptual question. We have seen no reason to regard its overt structure as at all misleading in that respect. Our provisional conclusion must therefore be that the original question, although at least proto-philosophical, is not about thought or language in any distinctive sense. It does not support the linguistic or conceptual turn, interpreted as a conception of the subject matter of philosophy.

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      If the original question, read literally, had too obvious an answer, either positive or negative, that would give us reason to suspect that someone who uttered it had some other meaning in mind, to which the overt compositional structure of the question might be a poor guide. But competent speakers of English may find themselves quite unsure how to answer the question, read literally, so we have no such reason for interpreting it non-literally.

      It is useful to look at some proposals and arguments from the vagueness debate, for two reasons. First, they show why the original question is hard, when taken at face value. Second, they show how semantic considerations play a central role in the attempt to answer it, even though it is not itself a semantic question.

      The most straightforward reason for answering the original question positively is that “Mars was always either dry or not dry” is a logical truth, a generalization over instances of the law of excluded middle (A ∕ ¬A, “It is either so or not so”) for various times. In my view, that reasoning is sound. However, many think otherwise. They deny the validity of excluded middle for vague terms such as “dry.”

      The simplest way of opposing the law of excluded middle is to deny outright when Mars is a borderline case that it is either dry or not dry, and therefore to answer the original question in the negative. For instance, someone may hold that Mars was either dry or not dry at time t only if one can know (perhaps later) whether it was dry at t, given optimal conditions for answering the question (and no difference in the history of Mars): since one cannot know, even under such conditions, whether it is dry when the case is borderline, it is not either dry or not dry. One difficulty for this negative response to the original question is that it seems to imply that in a borderline case Mars is neither dry nor not dry: in other words, both not dry and not not dry. That is a contradiction, for “not not dry” is the negation of “not dry.”

Intuitionistic logic provides a subtler way to reject the law of excluded middle without denying any one of its instances. Intuitionists ground logic in states of increasing but incomplete information, rather than a once-for-all dichotomy of truth and falsity. They deny that anything can be both proved and refuted, but they do not assert that everything can be either proved or refuted. For intuitionists, the denial of an instance of excluded middle (¬(A ∕ ¬ ∼A), “It is not either so or not so”) entails a contradiction (¬A & ¬¬A, ‘It is both not so and not not so’), just as it does in classical logic, and contradictions are as bad for them as for anyone else. Thus they cannot assert that Mars was once not either dry or not dry (∃t ¬(Dry(m, t) ∕ ¬Dry(m, t))), for that would imply that a contradiction once obtained (∃t (¬Dry(m, t) & ¬¬Dry(m, t)), “Mars was once both not dry and not not dry”), which is intuitionistically inconsistent. However, although intuitionists insist that proving an existential claim in principle involves proving at least one instance of it, they allow that disproving a universal claim need not in principle involve disproving at least one instance of it. The claim that something lacks a property is intuitionistically stronger than the claim that not everything has that property. Thus one might assert that Mars was not always either dry or not dry (¬∀t (Dry(m, t) ∕ ¬Dry(m, t))), on the general grounds that there is no adequate procedure for sorting all the times into the two categories, without thereby committing oneself to the inconsistent existential assertion that it was once not either dry or not dry. Hilary Putnam once proposed the application of intuitionistic logic to the problem of vagueness for closely related reasons.⁶ Thus one might use intuitionistic logic to answer the original question in the negative.

      On closer inspection, this strategy looks less promising. For a paradigm borderline case is the worst case for the law of excluded middle (for a term such as ‘dry’ for which threats to the law other than from vagueness are irrelevant), in the sense that both proponents and opponents of the law can agree that it holds in a paradigm borderline case only if it holds universally. In symbols, if Mars was a paradigm borderline case at time τ: (Dry(m,τ) ∕ ¬Dry(m,τ)) → ∀ t Dry(m, t) ∕ ¬Dry(m, t)) (“If Mars was either dry or not dry at time τ, then Mars was always either dry or not dry”). But on this approach the law does not hold always hold in these cases (¬∀t (Dry(m,t)∕ ¬Dry(m, t)), “Mars was not always either dry or not dry”), from which intuitionistic logic allows us to deduce that it does not hold in the paradigm borderline case (¬ (Dry(m,τ) ∕ ¬Dry(m,τ)), “Mars was not either dry or not dry at”), which is a denial of a particular instance of the law, and therefore intuitionistically inconsistent (it entails ¬Dry(m,τ) & ¬¬Dry(m,τ), “Mars was both not dry and not not dry at τ”). Thus the intuitionistic denial of the universal generalization of excluded middle for a vague predicate forces one to deny that it has such paradigm borderline cases. The latter denial is hard to reconcile with experience: after all, the notion of a borderline case is usually explained by examples.

On closer inspection, this strategy looks less promising. For a paradigm borderline case is the worst case for the law of excluded middle (for a term such as ‘dry’ for which threats to the law other than from vagueness are irrelevant), in the sense that both proponents and opponents of the law can agree that it holds in a paradigm borderline case only if it holds universally. In symbols, if Mars was a paradigm borderline case at time τ: (Dry(m,τ) ∕ ¬Dry(m,τ)) → ∀ t Dry(m, t) ∕ ¬Dry(m, t)) (“If Mars was either dry or not dry at time τ, then Mars was always either dry or not dry”). But on this approach the law does not hold always hold in these cases (¬∀t (Dry(m,t)∕ ¬Dry(m, t)), “Mars was not always either dry or not dry”), from which intuitionistic logic allows us to deduce that it does not hold in the paradigm borderline case (¬(Dry(m,τ)∕ ¬ Dry(m,τ)), “Mars was not either dry or not dry at τ”), which is a denial of a particular instance of the law, and therefore intuitionistically inconsistent (it entails¬Dry(m,τ) &¬¬Dry(m,τ), “Mars was both not dry and not not dry at τ”). Thus the intuitionistic denial of the universal generalization of excluded middle for a vague predicate forces one to deny that it has such paradigm borderline cases. The latter denial is hard to reconcile with experience: after all, the notion of a borderline case is usually explained by examples.

The problems for the intuitionistic approach do not end there. One can show that the denial of the conjunction of any finite number of instances of the law of excluded middle is intuitionistically inconsistent.⁷ The denial of the universal generalization of the law over a finite domain is therefore intuitionistically false too. If time is infi-nitely divisible, the formula ∀t (Dry(m,t) ∕ ¬Dry(m,t)) generalizes the law over an infinite domain of moments of time, and its denial is intuitionistically consistent, but the possibility of infinitely divisible time is not crucial to the phenomena of vagueness. We could just as well have asked the original question about a long finite series of moments at one-second intervals; it would have been equally problematic. The classical sorites paradox depends on just such a finite series: a heap of sand consists of only finitely many grains, but when they are carefully removed one by one, we have no idea how to answer the question ‘When did there cease to be a heap?’ To deny that Mars was dry or not dry at each moment in the finite series is intuitionistically inconsistent. Thus intuitionistic logic provides a poor basis for a negative answer to the original question.

      Other theorists of vagueness СКАЧАТЬ