L～* Predicate Logic And Newly Search For The R₀ Algebra

In the propositional logic, we regard simple propositions as the basic unit and we don't divide them anymore. We instead study complicated propositions which are made of simple propositions and connectives, such as their properties and logic relations. Let's consider the logic reasoning as follows:All even numbers can be divided by two.Ten is an even number.Then can be divided by two.There are no connectives in the premises and conclusions of this reasoning. There are not complicated but simple propositions. In the view of propositional logic, they are different simple propositions. They have the form of p, q, r and are apparantly not correct reasoning forms in propositional logic. Its accuracy can't be expressed in the propositional logic and is decided by the properties of pradicates and connectives. If we don't divide simple propositions into smaller parts and show the connections on the form and construction of promises and conclusions we could not grasp the forms and rules of the reasoning. Still, propositional logic can't include all the logic concepts.In the logic reasoning and even in our everyday life, such logic concepts as "must", "could", "all", "exist", "there is", "there is only one", "at least one", "at most one", " there is none", can always be met. Thus prepositional logic can only expressparts of the logic rules. My thesis built the theory of predicate logic on the basis of L* propositional logic.My thesis is divided into four parts. First part is preface. In the preface, I mainly in troduce the meaning of my work..The second part can be divided into three sections. § 2.1 took complete linearlyondered R0 algebra as semantics and built L* predicate logic by adding qualifiers and connectives to L* propositional logic. Meantime, in this section I brought up andproved some tautology. § 2.2 is about the syntactic of L* predicate logic. Axioms and inference rules are defined. Relative concepts such as proo, theory, conseqence, provability are brought up. Many basic proofs are given on the basis of axioms and inference rules. All these works get ready for the completeness of system L*. In the system L*, constants in [0,1] are not formulae, this to some extent confined the expressive ability of system L* as [0,1]-valued fuzzy propositions are always exist. Secondly, L* mainly discussed reasoning about tautology, this also confined the application of L*. Thus, § 2.3 adds truth constants as special formulae to L* predicate logic and axioms about truth constants. The truth degree and the provability of a formula are proposed. Its completeness is studied.R0 algebra is brought up in the need of system L* . Further study about R0 algebrawill make our knowledge about L* deeper. In third part, I discussed R0 algebra in two different aspect: (1) Defined Boole element in R0 algebra. Gave the properties of Boole element and distributive rules of Boole element about finite R0 algebra. Study the filter and the stone filter of L(M) are filters of M. But the filter of L(M) need not be the filter of M. (2) After the construction of fuzzy set theory, some scholars applied fuzzy theory to mathematical branch. Fuzzy logic, fuzzy topology ,fuzzy measure, fuzzy integral calculus were built. In § 3.2, we combine fuzzy mathematics with R0 algebra, then brought up the concepts of fuzzy R0 algebra and fuzzy filter, showed their properties. All these studies will enrich the content of R0 algebra.In the appendix I proved L* axioms applied in the text.