The Philosophy of Philosophy. Timothy Williamson

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СКАЧАТЬ 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 ¬ СКАЧАТЬ