MULTIPLE-VALUED LOGIC OF TIED IMPLICATIONS AND ITS USE IN FUZZY LOGIC

Abstract

The evolution of multiple-valued logics and fuzzy logic is motivated by the realization that traditional two-valued logical systems are inadequate for dealing with some uncertainties of the real world. This thesis presents a systematic treatment of deductive aspects and structures of the multiple-valued-logic component of fuzzy logic. The algebraic foundations of certain useful many-valued reasoning and their syntactical logical properties are studied. The main attention is oriented to the study of algebraic structures called "Tied Adjointness Algebras", in which conjunctions and implications are related by adjointness, and the implications are tied by commutative t-norms. The new achievement of this work is the development of a new complete syntax for quite a general multiple-valued logic, whose semantics is based on tied adjointness algebras. Such a formal system serves as a combined calculus for two, possibly different, types of uncertainty. We then extend that formal system to what we call Prelinear Tied Adjointness Algebras. The essentials of those algebras are introduced, employing the notions of filters and prime filters. A completeness theorem of this logic with respect to linearly ordered tied adjointness algebras is obtained. This is achieved through a representation of all prelinear tied adjointness algebras as subdirect products of linear tied adjointness algebras. We then develop predicate calculus corresponding to our proposjtional

calculus broadly analogous to the classical predicate logic. In particular, we deal only with two quantifiers, universal and existential, through a natural generalization of Tarskian semantics. Also, its completeness is established for semantics over linear tied adjointness algebras. Computational complexity of some particular cases of our propositional calculus has been studied, and positive results have been obtained. We end by investigating the use of that logic to analyze some typical patterns of approximate reasoning, through various patterns of logical deduction, namely, the compositional rule of inference, generalized modus ponens, and the inference modes in fuzzy control.