Foreword: Three-valued logics and their applications

摘要

Three-valued logics belong to a family of nonclassical logics that started to flourish in the 1920s and 1930s, following the work of Lukasiewicz (1920) and earlier insights coming from Frege and Peirce (see Fisch & Turquette, 1966; Frege, 1879, 1892). All of them were prompted by the idea that not all sentences need be true or false, but that some sentences can be indeterminate in truth value. In his pioneering paper, Lukasiewicz (1920) writes: 'Three-valued logic is a system of non-Aristotelian logic, since it assumes that in addition to true and false propositions there also are propositions that are neither true nor false, and hence, that there exists a third logical value'(Borkowski ed., 1970, p. 86). Lukasiewicz himself was led to introduce a third truth value in order to model the notion of possibility, and to formalise Aristotle's insight that contingent sentences about the future can be indeterminate. Before him, Frege had identified two other reasons to think of sentences as neither true nor false. The first concerns sentences in which a proper name in a sentence fails to refer to an existing individual, as in 'Odysseus was set ashore at Ithaca while sound asleep' (Frege, 1892, p. 27). What Frege points out is that 'anyone who seriously took the sentence to be true or false would ascribe to the name "Odysseus" a reference'. By contraposition, the failure of the name to have a reference should imply that the sentence fails to be either true or false. The second kind of case for which Frege thought sentences could be indeterminate concerns sentences involving vague predicates. For a predicate like 'heap of beans', Frege points out that the classical induction principle cannot be applied, 'on account of the indeterminateness of the notion "heap"' (Frege, 1879, p. 62). Frege did not propose a three-valued logic in relation to those observations, but his remarks find an echo in the supervaluationist system proposed by van Fraassen (1966) to account for the semantics of nonreferential singular terms, and used a decade later by others to account for vagueness (Fine, 1975; Kamp, 1976).