Dialogue Games as Foundation of Fuzzy Logics

Abstract

This chapter deals with fuzzy logic in Zadeh’s ‘narrow sense’ [27], pointing to t-norm based truth functional logics, where the truth values model ‘degrees of truth’, identified with reals from the unit interval [0,1]. In particular, we are interested in Łukasiewicz logic Ł and some of its variants, but also in Godel logic G, and in Product logic P. The literature on formal deduction systems for these and related many-valued logics is vast and even more has been written about their algebraic background. We refer to the monographs [23], [22], and [10] for more information and references. Most authors take the usefulness of these logics in the context of approximate reasoning, i.e., reasoning with vague and imprecise notions for granted. However the corresponding proof systems are hardly ever explicitly related to models of correct reasoning with vague information. In other words: the challenge to derive particular fuzzy logics from first principles about approximate reasoning is not addressed explicitly. The reference to general models of reasoning and to theories of vagueness – a prolific discourse in contemporary analytic philosophy – is only implicit, if not simply missing, in most presentations of inference systems for fuzzy logics. Some notable exceptions, where an explicit semantic foundation for particular fuzzy logics is aimed at, are: Ruspini’s similarity semantics [35]; voting semantics [25]; ‘re-randomizing semantics’ [24]; measurement-theoretic justifications [7]; the Ulam-Renyi game based interpretation of D. Mundici [29]; and ‘acceptability semantics’ of J. Paris [31]. As we have argued elsewhere [12], these formal semantics for various t-norm based logics (in particular Łukasiewicz logic) should be placed in the wider discourse about adequate theories of vagueness, a prolific subfield of analytic philosophy.