Handbook of Mathematical Fuzzy Logic

0372981 - ÚI 2012 RIV GB eng M - Monography Chapter
Cintula, Petr - Noguera, C.
A General Framework for Mathematical Fuzzy Logic. Chapter 2.
Handbook of Mathematical Fuzzy Logic. Vol. 1. London: College Publications, 2011 - (Cintula, P.; Hájek, P.; Noguera, C.), s. 103-207. Studies in Logic - Mathematical Logic and Foundations, 37. ISBN 978-1-84890-039-4
R&D Projects: GA ČR GEICC/08/E018; GA ČR GAP202/10/1826
Institutional research plan: CEZ:AV0Z10300504
Keywords : mathematical fuzzy logic * weakly implicative logics * abstract algebraic logic * non-classical logics * semilinear logics
Subject RIV: BA - General Mathematics

The aim of this chapter is to present a marriage of Mathematical Fuzzy Logic and (Abstract) Algebraic Logic in order to provide a general background for the rest of the handbook. We use the notions and techniques from the latter to create a new framework where we can develop in a natural way a particular technical notion corresponding to the intuition of fuzzy logics as the logics of chains. Our framework is the class of weakly implicative semilinear logics, roughly speaking logics with implication connective which are complete with respect to the class of linear ordered matrices. We choose the term `semilinear' instead of `fuzzy', because the term `fuzzy' is too heavily charged with many conflicting potential meanings. The chapter is structured as follows. In Section 1 we introduce the necessary notions from (Abstract) Algebraic Logic, the definition of weakly implicative logic and some refinements thereof and provide three increasingly stronger completeness theorems for them. Moreover, we present a very general notion of substructural logics as a particular family of weakly implicative logics, discuss their syntactical properties and deduction theorems, and we conclude with a rather general study of disjunction connectives. Section 2 presents and studies the main notion of this chapter: semilinearity. It characterizes semilinear logics in terms of properties of filters and properties of disjunctions, and gives methods to axiomatize semilinear logics. Section 3 studies first-order predicate systems built over weakly implicative semilinear logics. It gives axiomatizations, completeness theorems, and a general process of Skolemization. We conclude with Section 4 providing historical remarks to understand the genesis of the ideas and results presented in this chapter and many bibliographical references for further studies in related topics.
Permanent Link: http://hdl.handle.net/11104/0206164