Читать книгу The Philosophy of Philosophy - Timothy Williamson - Страница 19

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 refuse to answer the original question either positively or negatively. They refuse to assert that Mars was always either dry or not dry; they also refuse to assert that it was not always either dry or not dry.

A simple version of this approach classifies vague sentences (relative to contexts) as true (T), false (F) or indefinite (I); borderline sentences are classified as indefinite. The generalized truth-tables of a three-valued logic are used to calculate which of these values to assign to a complex sentence in terms of the values assigned to its constituent sentences. The negation of A, ¬A, is true if A is false, false if A is true and indefinite if A is indefinite:

A	¬A
T	F
I	I
F	T

A conjunction A & B (“A and B”) is true if every conjunct is true; it is false if some conjunct is false; otherwise it is indefinite. A disjunction A∕ B (“Either A or B”) is true if some disjunct is true; it is false if every disjunct is false; otherwise it is indefinite:

A	B	A & B	A∕ B
T	T	T	T
T	I	I	T
T	F	F	T
I	T	I	T
I	I	I	I
I	F	F	I
F	T	F	T
F	I	F	I
F	F	F	F

A universal generalization is treated as if it were the conjunction of its instances, one for each member of the domain: it is true if every instance is true, false if some instance is false, and otherwise indefi-nite. An existential generalization is treated as if it were the disjunction of the instances: it is true if some instance is true, false if every instance is false, and otherwise indefinite. The three-valued tables generalize the familiar two-valued ones in the sense that one recovers the latter by deleting all lines with “I.”

Let us apply this three-valued approach to the original question. If Mars is definitely dry or definitely not dry at t (the time denoted by t), then Dry(m, t) is true or false, so the instance of excluded middle Dry(m, t)∕ ¬Dry(m, t) is true. But if Mars is neither definitely dry nor definitely not dry at t, then Dry(m, t) is indefinite, so ¬Dry(m, t) is indefinite too by the table for negation, so Dry(m, t)∕ ¬ Dry(m, t) is classified as indefinite by the table for disjunction. Since Mars was once a borderline case, the universal generalization ∀t (Dry(m, t) ∕ ¬Dry(m, t)) has a mixture of true and indefinite instances; hence it is classified as indefinite. Therefore its negation ¬∀t (Dry(m, t)∕ ¬ Dry(m, t)) is also indefinite. Thus three-valued theoreticians who wish to assert only truths neither assert ∀t (Dry(m, t) ∕ ¬Dry(m, t)) nor assert ¬∀t (Dry(m, t)∕ ¬ Dry(m, t)). They answer the original question neither positively nor negatively.

Three-valued logic replaces the classical dichotomy of truth and falsity by a three-way classification. Fuzzy logic goes further, replacing it by a continuum of degrees of truth between perfect truth and perfect falsity. According to proponents of fuzzy logic, vagueness should be understood in terms of this continuum of degrees of truth. For example, ‘It is dark’ may increase continuously in degree of truth as it gradually becomes dark. On the simplest version of the approach, degrees of truth are identified with real numbers in the interval from 0 to 1, with 1 as perfect truth and 0 as perfect falsity. The semantics of fuzzy logic provides rules for calculating the degree of truth of a complex sentence in terms of the degrees of truth of its constituent sentences. For example, the degrees of truth of a sentence and of its negation sum to exactly 1; the degree of truth of a disjunction is the maximum of the degrees of truth of its disjuncts; the degree of truth of a conjunction is the minimum of the degrees of truth of its conjuncts. For fuzzy logic, although the three-valued tables above are too coarse-grained to give complete information, they still give correct results if one classifies every sentence with an intermediate degree of truth, less than the maximum and more than the minimum, as indefinite.⁸ Thus the same reasoning as before shows that fuzzy logicians should answer the original question neither positively nor negatively.

Although three-valued and fuzzy logicians reject both the answer “Yes” and the answer “No” to the original question, they do not reject the question itself. What they reject is the restriction of possible answers to “Yes” and “No.” They require a third answer, “Indefi-nite,” when the queried sentence takes the value I. More formally, consider the three-valued table for the sentence operator Δ, read as “definitely” or “it is definite that”:

A	ΔA
T	T
I	F
F	F

Even for fuzzy logicians this table constitutes a complete semantics for Δ, since the only output values are T and F, which determine unique degrees of truth (1 and 0). A formula of the form ¬ΔA & ¬Δ¬ A (“It is neither definitely so nor definitely not so”) characterizes a borderline case, for it is true if A is indefinite and false otherwise. In response to the question A?, answering “Yes” is tantamount to asserting A, answering “No” is tantamount to asserting ¬A, and answering “Indefinite” is tantamount to asserting ¬ΔA & ¬Δ¬A. On the three-valued and fuzzy tables, exactly one of these three answers is true in any given case; in particular, the correct answer to the original question is “Indefinite.”

On the three-valued and fuzzy approaches, to answer “Indefinite” to the question “Is Mars dry?“ is to say something about Mars, just as it is if one answers “Yes” or “No.” It is not a metalinguistic response. For Δ is no more a metalinguistic operator ¬ than is. They have the same kind of semantics, given by a many-valued truth-table. Just as the negation ¬A is about whatever A is about, so are ΔA and ¬ΔA & ¬Δ¬A. Thus the answer “Indefinite” to the original question involves no semantic ascent to a metalinguistic or metaconceptual level. It remains at the level of discourse about Mars.

The three-valued and fuzzy approaches have many suspect features. For instance, they treat any sentence of the form ΔA as perfectly precise, because it always counts as true or false, never as indefinite, whatever the status of A; thus ΔΔA ∕ Δ¬Δ A (“It is definite whether it is definitely so”) is always true. This result does not fit the intended ∨ interpretation of Δ. For “Mars is definitely wet” is not perfectly precise. Just as no moment is clearly the last on which Mars was wet or the first on which it was not, so no moment is clearly the last on which it was definitely wet or the first on which it was not definitely wet. Just as it is sometimes unclear whether Mars is wet, so it is sometimes unclear whether it is definitely wet. This is one form of the notorious problem of higher-order vagueness: in other words, there are borderline cases of borderline cases, and borderline cases of borderline cases of borderline cases, and so on. The problem has never received an adequate treatment within the framework of threevalued or fuzzy logic; that it could is far from obvious.⁹

Some philosophers, often under the influence of the later Wittgenstein, deny the relevance of formal semantic theories to vague natural languages. They regard the attempt to give a systematic statement of the truth conditions of English sentences in terms of the meanings of their constituents as vain. For them, the formalization of “Mars was always either dry or not dry” as ∀t (Dry(m,t)∕ ¬ Dry(m,t)) is already a mistake. This attitude suggests a premature and slightly facile pessimism. No doubt formal semantics has not described any natural language with perfect accuracy; what has not been made plausible is that it provides no deep insights into natural languages. In particular, it has not been made plausible that the main semantic effects of vagueness are not susceptible to systematic formal analysis. In any case, for present purposes the claim that there can be no systematic theory of vagueness is just one more theory of vagueness, although – unless it is self-refuting – not a systematic one; it does not even answer the original question. Even if that theory were true, the other theories of vagueness, however false, would still exist, and would still have been accepted by some intelligent and linguistically competent speakers.

This is no place to resolve the debate between opposing theories of vagueness. The present point is just that different theories support contrary answers to the original question. All these theories have their believers. Any answer to the original question, positive, negative, or indefinite, is contentious. Of course, if everyone found their own answer obvious, but different people found different answers obvious, then we might suspect that they were interpreting the question in different ways, talking past each other. But that is not so: almost everyone who reflects on the original question finds it difficult and puzzling. Even when one has settled on an answer, one can see how intelligent and reasonable people could answer differently while understanding the meaning of the question in the same way. If it has an obvious answer, it is the answer “Yes” dictated by classical logic, but those of us who accept that answer can usually imagine or remember the frame of mind in which one is led to doubt it. Thus the original question, read literally, has no unproblematically obvious answer in any sense that would give us reason to suspect that someone who asked it had some other reading in mind.

Without recourse to non-literal readings, some theorists postulate ambiguity in the original question. For example, some three-valued logicians claim that “not” in English is ambiguous between the operators ¬(strong negation) and ¬Δ(weak negation): although ¬A and ¬ΔA have the same value if A is true or false, ¬ΔA is true while ¬ A is indefinite if A is indefinite. While A∕ ¬ A (“It is so or not so”) can be indefinite, A∕ ¬Δ A (“It is so or not definitely so”) is always true. On this view, the original question queries ∀t (Dry(m, t)∕ ¬ Dry(m, t)) on one reading, ∀t (Dry(m, t)∕ ¬ ΔDry(m, t)) on another; the latter is true (Mars was always either dry or not definitely dry) while the former is indefinite. Thus the correct answer to the original question depends on the reading of “not.” It is “Indefinite” if “not” is read as strong negation, “Yes” if “not” is read as weak negation. Although the threevalued logician’s reasoning here is undermined by higher-order vagueness, that is not the present issue.¹⁰

If ‘not’ were ambiguous in the way indicated, it would still not follow that the dispute over the original question is merely verbal. For even when we agree to consider it under the reading of ‘not’ as strong negation, which does not factorize in the manner of ¬Δ, we still find theories of vagueness in dispute over the correct answer. We have merely explained our terms in order to formulate more clearly a difficult question about Mars.

Still, it might be suggested, the dispute between different theories of vagueness is verbal in the sense that their rival semantics characterize different possible languages or conceptual schemes: our choice of which of them to speak or think would be pragmatic, based on considerations of usefulness rather than of truth. Quine defended a similar view of alternative logics (1970: 81–6).

To make sense of the pragmatic view, suppose that the original vague atomic sentences are classifiable both according to the bivalent scheme as true or false and according to the trivalent scheme as defi- nitely true, indefinite or definitely false, and that the truth-tables of each scheme define intelligible connectives, although the connective defined by a trivalent table should be distinguished from the similarlooking connective defined by the corresponding bivalent table. Definite truth implies truth, and definite falsity implies falsity, but indefiniteness does not discriminate between truth and falsity: although all borderline atomic sentences are indefinite, some are true and others false. As Mars dries, “Mars is dry” is first false and defi- nitely false, then false but indefinite, then true but indefinite, and finally true and definitely true. However, this attempted reconciliation of the contrasting theories does justice to neither side. For trivalent logicians, once we know that a sentence is indefinite, there is no further question of its truth or falsity to which we do not know the answer: the category of the indefinite was introduced in order not to postulate such a mystery. Similarly, for fuzzy logicians, once we know the intermediate degree of truth of a sentence, there is no further question of its truth or falsity to which we do not know the answer: intermediate degrees of truth were introduced in order not to postulate such a mystery. In formal terms, trivalent and fuzzy logics are undoubtedly less convenient than bivalent logic; the justification for introducing them was supposed to be the inapplicability of the bivalent scheme to vague sentences. If a bivalent vague language is a genuinely possible option, then the trivalent and fuzzy accounts of vagueness are mistaken. Conversely, from a bivalent perspective, the trivalent and fuzzy semantics do not fix possible meanings for the connectives, because they do not determine truth conditions for the resultant complex sentences: for example, the trivalent table for ¬ does not specify when ¬A is true in the bivalent sense. It would, therefore, be a fundamental misunderstanding of the issue at stake between theories of vagueness to conceive it as one of a pragmatic choice of language.

We already speak the language of the original question; we understand those words and how they are put together; we possess the concepts they express; we grasp what is being asked. That semantic knowledge may be necessary if we are to know the answer to the original question.¹¹ It is not sufficient, for it does not by itself put one in a position to arbitrate between conflicting theories of vagueness. For each of those theories has been endorsed by some competent speakers of English who fully grasp the question.

Competent speakers may of course fail to reflect adequately on their competence. Although the proponents of conflicting theories of vagueness presumably have reflected on their competence, their reflections may have contained mistakes. Perhaps reflection of sufficient length and depth on one’s competence would lead one to the correct answer to the original question. But the capacity for such more or less philosophical reflection is not a precondition of semantic competence. Philosophers should resist the professional temptation to require all speakers to be good at philosophy.

We can distinguish two levels of reflection, the logical and the metalogical. In response to the original question, logical reflection involves reasoning with terms of the kind in which the question is phrased; the aim is to reach a conclusion that answers the question. For example, one might conclude by classical logic that Mars was always either dry or not dry; one might conclude by fuzzy logic that it is indefinite whether it was always one or the other. The logical level is not purely mechanical. When the reasoning is complex, one needs skill to select from the many permissible applications of the rules one sequence that leads to an answer to the question. When the reasoning is informal, one needs good judgment to select only moves that really are permissible applications of the rules. But one is still thinking about whatever the question was about. One starts only at the metalogical level of reflection to think about the semantics of the logical connectives and other expressions one employed at the logical level. For example, at the metalogical level one may assert or deny that the sentence “Mars was always either dry or not dry” is a logical truth. The rules used at the logical level are articulated only at the metalogical level.

It must be possible to think logically without thinking metalogically, for otherwise by the same principle thinking metalogically would involve thinking metametalogically, and so ad infinitum: our thinking never goes all the way up such an infinite hierarchy. What can prompt ascent to the metalogical level are hard cases in which one feels unclear about the permissibility of a given move at the logical level. One’s mastery of the language and possession of concepts leave one quite uncertain how to go on. In the case of the original question, a salient line of classical reasoning leads to a positive answer: it persuades some competent speakers while leaving others unconvinced. Even to discuss the contentious reasoning we must semantically ascend. We cannot hope to resolve the dispute undogmatically if we never leave the lower level.