By Zinov'ev A. A., Smirnov G. A., Sidorenko E. A., Fedina A. M., Bobrova L. A.
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Additional info for Foundations of the logical theory of scientific knowledge (Complex logic)
But if the semantics for a many-valued logic is described using a precise metalanguage, then sentences will always be assigned exact values, since sentences of the metalanguage ascribing degrees of truth will themselves be true or false simpliciter. The cluster of questions (6) has not always been directly confronted, but we outline several types of responses, which have been either defended or at least alluded to in the literature. Biting the bullet. First, we could accept that sentences do take unique exact (a)values, that there are sharp boundaries to the degree 1 sentences, that there is no higher-order vagueness, and that precise metalanguages are acceptable as they stand.
But suppose that "p" is true at some but not all points, so that "'p' is true" is true at no points. At those points where "p" is true, the conditional "if p, then 'p' is true" is false, since its antecedent is true and its consequent is false (compare the argument, p. 31, that A É DA is not always true). Consequently the biconditional (T) is also false at those points, and thus is not true simpliciter. g. Williamson pp. 26566; see also Machina pp. 17879). However, it remains the case that p entails and is entailed by Dp (the truth of p at the base-point guarantees the truth of Dp there, and vice versa).
Suppose additionally that the totality of these base-level facts fixes everything else. g. clouds or mountains) whose boundaries are left fuzzy. But what is true, false and left indeterminate about them would supervene on how things stand at the precise base level. Arguably, on this picture, any apparent ontic vagueness of the fuzzy-boundaried mountains etc. is merely superficial (compare the ontological framework of Lewis 1993). This suggests that to get a substantive thesis about ontic vagueness, we need to consider a contrasting picture of a world containing non-superficial vagueness, where the vagueness "goes all the way down".