What is a Truth Value And How Many Are There?

Abstract

Truth values are, properly understood, merely proxies for the various relations that can hold between language and the world. Once truth values are understood in this way, consideration of the Liar paradox and the revenge problem shows that our language is indefinitely extensible, as is the class of truth values that statements of our language can take -in short, there is a proper class of such truth values. As a result, important and unexpected connections emerge between the semantic paradoxes and the set-theoretic paradoxes.

Sorites, Curry and Suitable Models

“…It is an extension of the standard model of arithmetic. 6 2. It satisfies all of the instances of the T-schema and does not establish any order among semantically paradoxical sentences (it does not make any paradoxical sentence truer/falser than another).…”

“…Let us start with the first one. We want to consider the formal semantics as models, not only in the sense of them being useful interpretations (for instance, of show consistency), but in a quite stronger one, the same sense we find for instance in Goguen [10], Edgington [8] or Cook [6]. Just as a scale model of a building represents the building, our semantics are meant to represent those of natural language, and in particular, real numbers are used in place of verities, which are the semantic values that relate to English sentences.…”

Unifying the Philosophy of Truth

“…Because our approach relies primarily on such speech act distinctions, we thus believe the need to deal with determinateness operators in the object-language is less pressing than for other theories in which semantic values are primarily seen as ways of encoding the relation of a sentence with the world (see (Cook 2009) for such a conception). The main reason is that for us, strict and tolerant are primarily modes of assertion or acceptance; they qualify the force rather than the content of an assertion.…”

Possible predicates and actual properties

“…In addition, we need to guarantee that every property, and every predicate, has a satisfaction condition-hence the SatisfacFootnote 17 continued Of course, this is not all that surprising: satisfaction conditions (and our SC (... ) predicates holding of them) are semantic notions, and it is well-known that naïve formulation of the principles governing semantic notions can lead to paradox. Thus, a more careful and precise formulation of the principles governing satisfaction conditions would involve applying whatever prophylactic one prefers in the more well-known instances of this sort of phenomenon (i.e., plug in one's favored solution to the Liar paradox here: my own preference can be found in Cook 2007aCook , 2009). Given the straightforward nature of the arguments given below, however (and, in particular, the fact that they involve no diagonalization or other paradox-prone constructions), we can safely ignore this problem for the time being.…”