Study On Lattice-Valued Logic And Uncertain Reasoning Based On Linguistic Truth Values

In the recent forty years, since the theory of uncertain reasoning has been applied broadly in control systems, the logical foundation of it-non-classical logic have attracted a considerable deal of attention much more, In the late of 1970's, Pavelka etc. established elementarily fuzzy propositional logic, hereafter non-classical logics have been developed into an important research direction in Artificial Intelligence field. On the one hand, non-classical logics have been applied broadly in machine automatic prove, multi-agent system and program validation and so forth research fields. On the other hand, non-classical logics have also enriched and developed the theory of pure mathematics. Lattice-valued logic is a kind of important non-classical logic, it not only can characterizes the information with linearly ordered, but also the information with nonlinearly ordered, that is to say, the incomparable information. The main aim of this paper is to establish the lattice-valued propositional logic system with degree based on lattice implication algebras, which includes the semantic theory and syntactic theory, and further establish the theory and methods of uncertain reasoning with linguistic truth-valued, and set up the corresponding algorithms.In section one, the algebraic properties of lattice implication ordered semigroups induced by lattice implication algebras is obtained. We introduce two new concepts, i.e. lattice implication n-ordered semigroup and lattice implication p-ordered semigroup, prove that a lattice implication n-ordered semigroup is a residuated semigroup, and a lattice implication p-ordered semigroup is an arithmetic lattice-ordered semigroup. We also define the notion of lattice implication n-ordered semigroup homomorphism, and based on it, we characterize the algebraic properties of filters and sl ideals in lattice implication n-ordered semigroups and lattice implication p-ordered semigroups. At one time, we present several typical expansions of sl ideals in lattice implication n-ordered semigroups. It should be hopeful that these investigations can provide a kind of new train of thought investigations for further researching into the properties of lattice implication algebra and the theory of lattice-valued logic based on lattice implication algebras. Likewise we find out all subalgebras, filters and LI-ideals in linguistic truth-valued lattice implication algebra L18.In section two, the theory of generalized tautology is established in lattice-valued propositional system (?)P based on lattice implication algebras. We define several notions of L-type tautology, L-type contradiction andα-tautology etc., present several theorems about the relations among these generalized tautologies. We define the notion of satisfiability for L-fuzzy logic formulas set, based on it, define the semantic closure operation, and further define the notion of consistency of information and theory based on semantic closure operation. We also present several theorems about the compact property, logic compact property of semantic closure operation and the corresponding closure systems, establish the congruence relation on (?)P induced by certain given information and the corresponding quotient lattice implication algebra. Combining with the theory of fuzzy set and L-fuzzy set, we also obtained theorems about the properties of closure operation on P((?)p) induced by the semantic closure operation.In section three, we establish a kind of syntactic theory with some degree in the lattice-valued propositional logic system based on lattice implication algebras. We define form proof with some degree and syntactic closure operation, prove some theorems used often, and define the notion of provable equivalent relation, prove several important theorems about provable equivalent relations. We also investigate the consistency of the syntactic theory and the semantic theory established in above chapter, and establish generalized deduction theorem and completeness theorem. These results are useful to provide necessary academic preparation for establishing uncertain reasoning methods based on lattice-valued logic system.In section four, we characterized some properties of lattice-valued propositional logic system based on linguistic truth-valued lattice implication algebra L18, and then establish the theory and methods of uncertain reasoning with linguistic truth-valued, at one time, set up the corresponding reasoning algorithms. From the perspective of logic semantics and syntax, we also analyze the rationality of uncertain reasoning methods and algorithms which have been established ahead.