Syntactic Truth Degree Of Formulas And Constructive Triple-I Algorithm

Abstract fuzzy reasoning and fuzzy propositional logic are topics of general interest in many-valued logic. The present paper goes deep into the gradation of the basic concepts in propositional logic systems and the full implication triple-I algorithm in fuzzy reasoning. And some fruitful results are obtained. There are four main parts in this paper which are divided into three chapters. The first three parts are included in the second chapter while the last makes itself Chapter 3 alone. Syntax, semantics and quantitative logic theory in two logic systems, i.e., tow-valued propositional logic system L and fuzzy propositional logic system Luk, are introduced in Chapter 1 for preliminaries.This paper defines the syntactic truth degrees of formulas from the syntactical point of view, while [24] proposes the truth degrees of formulas from the point of view of semantics. In the first part, the concept of the syntactic truth degrees of formulas, its two equivalent depiction theorems and three examples are proposed in tow-valued propositional logic system L. It is pointed out that the similarity degree and pseudo-metric induced by syntactic truth degree possess respectively the basic properties of similarity degree and pseudo-metric introduced in [24]. Finally,the applications of syntactic truth degrees in reasoning are also discussed.Then, in part two, the concept of the syntactic truth degrees of formulas in fuzzy propositional logic system Luk is proposed similarly. Simultaneously two examples and three equivalent depictions of the syntactic truth degrees are given in Luk. It is illustrated by some examples that the depictions of syntactic truth degrees in L can't be achieved in Luk, and some main properties of the syntactic truth degrees in L are no longer available in Luk, although the definition of the syntactic truth degrees in Luk is similar to that in L. Finally, it is pointed out by three examples that the discussions about the syntactic truth degrees in Luk can't be developed plainly to L~*.The third part proposes in Luk the concept of lattice-valued truth degrees based on the second part by generalizing truth values from the MV-unit interval(a special MV-algebra) to a general MV-algebra. And the properties of lattice-valuedtruth degrees suggest that this generalization is natural and reasonable. Because of the equivalence between the two algebra systems, i.e. MV-algebra and lattice implication algebra, it's natural and easy to grade the truth of formulas in lattice-valued propositional logic whose basic algebra system is the lattice implication algebra.Full implication triple-I algorithm of fuzzy reasoning is studied further in the final part wherein Zadeh triple-I MP solution proposed in [2] is improved. Addi-tionallly this part proposes some necessary and sufficient conditions for Ro triple-I solution, Lukasiewicz triple-I solution, Zadeh triple-I MP solution and Godel triple-I solution possessing reductivity. And FMT(Fuzzy Modus ToUens) problem and its reductivity can be considered similarly.