Consistency Degrees Of Theories And Filters And Quotients Of Effect Algebras

From the point of modern logic, the logic developed from traditional logic to classical symbolic logic then to non-classical logic. Since the eighties of the 20th century, non-classical logic has obtained the basic position in the fields of computer science and artificial intelligence. Now, studies about fuzzy logic, quantum logic and the corresponding algebra systems are very active in non-classical logic. In this article, consistency degrees of theories in some systems of propositional fuzzy logic are referred to, so are filters and quotients of effect algebra.Whether a theory is consistent or not is one of crucial questions in any logical system. A theory is consistent or it is inconsistent, there is no intermediate situations, i.e., the concept of consistency of a theory is crisp rather than fuzzy. Naturally, it is worthy of concentrating on how to grade the consistency theories. Fortunately, the idea based on the concept of divergence degrees of theories can be used to solve this problem. We have known that every inconsistent theory must be fully divergent, i.e., its divergence degree is 1. However, are fully divergent theories to be inconsistent? For Lukasiewicz logical system, the answer has been given in [1, 2]. In the first chapter of this article, the question is discussed in logical systems of Godel, Product and L*, and a fully divergent and consistent theory is given there. In addition, a membership function is constructed to grade the consistency degrees of finite theories in Lukasiewicz logical system in [1]. Following that, the function is extended to general theories in Lukasiewicz and in two-valued logic systems by using the compactness of logical systems[3]. By introducing the concept of polar indexes, the first chapter defines a new reasonable concept of consistency degrees for general theories, finite or infinite, in basic logic systems(briefly, BL systems) including Lukasiewicz fuzzy logic system, Godel fuzzy logic system, and Product fuzzy logic system. The results obtained are also true in the R0-fuzzy logic system L* which is not included in BL systems, and the expression of the newly defined consistency degree ξ*(Γ) of theory Γ is much simpler than the expression ξ(Γ) given in [1, 3].Effect algebras were introduced as a mathematical model of quantum logic by American mathematicians Foulis and Bennett in 1994, which generalized ortho-...