Maximal Contractions In R₀Algebra And The Paradigm Representation Of Formula’s Truth Degree

In the software, it is a core problem in belief revision theory to solve all the maximal consistent subsets of a formula set about the collection of facts (i.e. maximal reduction) based on two-valued logic calculus. But in practical reasoning, since the human thinking pattern itself has some uncertainty and the experimental or observational data may also contain errors, the uncertainty reasoning has surpassed the scope that two-value logic can process. Because many-valued logic can effectively deal with uncertainty information encountered in practical reasoning and R0propositional logic is one of the most important many-valued logic, so it is one task of this paper to introduce the maximal reduction of software into both R0propositional logic and its corresponding algebra.On the other hand, in order to introduce the numerical calculation into mathematical logic so as to make it more flexible and extend its possible application range, professor Wang proposed the quantitative logic, the fundamental starting point of the quantitative logic lies in the degree of basic concepts, among these, the concept of truth degree of propositions introduced by grading the concept of tautology is the most basic one, other concepts, such as similarity degree, pseudo metric and so on, are all proposed by the truth degree of propositions. Therefore, how to calculate the truth degree of formulas is very important, so, another research task of this subject is to give a new and simple method for calculating the truth degree of formulas.The main results of this paper are as follows:1. The concepts of deductive element, maximal contraction as well as minimal subtraction are introduced in R0algebra and their important properties are studied. Moreover, an equivalent characterization of minimal subtraction of R0algebra is given and a sufficient and necessary condition that R0algebra has maximal contractions is obtained, which lays a preliminary foundation for the quantification of software in many-valued logic.2. Firstly, the concept of n-atom of formulas is introduced in classical propositional logic system and its important properties are studied. Secondly, the paradigm representation theorem of truth degree of formulas is given, i.e., the truth degree of a formula A is equal to the sum of the truth degree of its all n-atoms, which provides a new and simple method for calculating the truth degree of formulas. Finally, the specific disjunctive normal form which is logically equivalent to a formula A is obtained by using the paradigm representation theorem of formula’s truth degree.