Truth Values,Neither-true-nor-false,and Supervaluations |
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Authors: | Nuel Belnap |
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Institution: | (1) Department of Philosophy, University of Pittsburgh, Pittsburgh, PA 15260, USA |
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Abstract: | The first section (§1) of this essay defends reliance on truth values against those who, on nominalistic grounds, would uniformly
substitute a truth predicate. I rehearse some practical, Carnapian advantages of working with truth values in logic. In the
second section (§2), after introducing the key idea of auxiliary parameters (§2.1), I look at several cases in which logics
involve, as part of their semantics, an extra auxiliary parameter to which truth is relativized, a parameter that caters to
special kinds of sentences. In many cases, this facility is said to produce truth values for sentences that on the face of
it seem neither true nor false. Often enough, in this situation appeal is made to the method of supervaluations, which operate
by “quantifying out” auxiliary parameters, and thereby produce something like a truth value. Logics of this kind exhibit striking
differences. I first consider the role that Tarski gives to supervaluation in first order logic (§2.2), and then, after an
interlude that asks whether neither-true-nor-false is itself a truth value (§2.3), I consider sentences with non-denoting
terms (§2.4), vague sentences (§2.5), ambiguous sentences (§2.6), paradoxical sentences (§2.7), and future-tensed sentences
in indeterministic tense logic (§2.8). I conclude my survey with a look at alethic modal logic considered as a cousin (§2.9),
and finish with a few sentences of “advice to supervaluationists” (2.10), advice that is largely negative. The case for supervaluations
as a road to truth is strong only when the auxiliary parameter that is “quantified out” is in fact irrelevant to the sentences of interest—as
in Tarski’s definition of truth for classical logic. In all other cases, the best policy when reporting the results of supervaluation is to use only explicit
phrases such as “settled true” or “determinately true,” never dropping the qualification. |
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Keywords: | truth values neither-true-nor-false non-denoting terms vagueness paradoxes future tense |
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