The Remaining Grid Depicts Several Logical Algebra And Propositional Logic System Conditional Truth Degree Of Comparison

Researching the logical problem by using the algebra is one of the attractive issues in recent years. Scholars use several different logical algebraic systems in different logical background. In terms of the non-classical logical system, different logics have set up the coordinate important logical algebraic systems, such as MV algebra, Ro algebra, FI-algebra, BL-algebra and BR0 algebra etc.MV algebra which is developed by the famous mathematician C.C. Chang aims at solving the completeness of Lukasiewicz logic which is a non-classical, many valued logic. In recent decades, researches on MV algebra are popular. In the reference [5] and [14], researchers give more systematic illustrations which are fruitful, like the analysis and comparison on the different algebraic relationships. Prof. Wang Guojun publishes three kinds of MV algebras in different forms which are equivalent in the reference [5], and proves the equivalent relationships between the lattice implication algebra, a kind of FI algebra and MV algebra.In the year 1997, Prof. Wang Guojun set up a fuzzy proposition mathematical calculation on L* and the relevant R0 algebra [3]. After it, some research findings appear ([16]-[22]). The comparison between R0 algebra and residuated lattice in the reference [4] which is based on the residuated lattice method and theory gives the equivalent definition which illustrates the essence of residuated lattice. Basing on this, Prof. Wu Hongbo extends the R0 algebra and L* system, and in the reference [13], gives the R0 algebra and proves the important essence of it. In the reference [6], he discusses the disorder indicative form of BR0. After the new definition of BR0 algebra, in the discussion, WBR0 algebra was discovered, and Prof. Wu develops the theory and essence of WBRo algebra.Reference [4] and [15] shows that the relations of previous kinds of algebras with the residuated lattice can be set up, so residuated lattice is the non classical logic, or the algebra structure, especially the fuzzy logical algebra. That is to say, residuated lattice is the ideal algebra frame in the fuzzy logic. In reference [7], research on residuated lattice is further discussed, and the concept of canonical residuated lattice is quoted, which gives the specific theorem of residuated lattic and normal residuated lattice. Finally, the independence and the relationship related algebra of the system contains the residuated lattice and normal residuated lattice is discussed.This paper is based on the previous theory and method, where MV algebra is further researched. Two specific theorems on MV algebra are concluded. In addition, in the orthomodular lattices, PMV algebra which is weaker than MV is set up and the relationship between orthomodular lattices and these algebraic systems is discovered. Besides, in this paper, we do the further research on WBR0 algebra and analyze the WBR0 algebra and residuated lattice; the conclusion that WBR0 shares the same value of regular residuated lattice is drawn. Based on this, two equivalent forms of values of WBR0 algebra are given in this relationship which simplifies the definition of WBR0 algebra.On the other hand, our research on fuzzy logic has made great progress in recent years. Prof. Wang Guojun has proposed the theory of the truth degree of formulas in fuzzy logic, binary logic and n-valued logic. In particular, the literature [2], which proposed the truth degree of formula for continuous-valued logic, has established the Integral Semantics and provides a framework for the approximate reasoning in continuous valued propositional logic. However, the establishments of these theories are all proposed without additional information. On this basis, literature [11] has proposed the concept of the conditional truth degree of formula A under informationΓ={p} and extends the concept of truth degree, but they are all built on the premise of limited and uniform assignments. For example, the concept of conditional truth degree in literature [12] is supported by the Generalized Deduction Theorem. So, we can also propose the concept of conditional truth degree in fuzzy propositional system Godel and L* and discuss their properties because there are corresponding deduction theorem in these systems.The order of the corresponding implication operatorâ†'and its accompanied triangle norm (?) in system Godel and L* is like this:â†'God≤â†'L*,(?)L*≤(?)God, This is because: that is to say, the order of implication operatorâ†'and triangle norm.(?) is opposite in these two systems. This paper contains four chapters, and the main content of each chapter is as follow:The first chapter introduces the relevant knowledge in the paper. It contains three parts:The first part introduces the concepts of several logic systems and residuated lattice. The second part introduces the relevant knowledge on the truth degree and deduction theorems in fuzzy logic system Godel and L*.The third part introduces the main work of this paper.In the second chapter, the equivalent relationships have been set up among MV algebra, R0 algebra, WBR0 algebra and the residuated lattice. More attention is paid to the WBR0 algebra, the conclusion that WBR0 algebra equals normal residuated lattice is drown when comparing WBR0 algebra with residuated lattice. Basing on this conclusion, two equal forms of WBR0 algebra is given to simplify the definision of WBR0 algebra with the help of this equal form.Further research on MV algebra is done in the third chapter. Two specific theories of MV algebra are gained based on the conclusions of MV algebra and a normal residuated lattice. Besides, the relationship between orthomodular lattice and these algebra is gained through setting up the PMV algebra in the orthomodular lattices.In the fourth chapter, a simple discussion on conditional truth degree in two fuzzy systems Godel and L*. Because the order of the implication operatorâ†'and the triangle norm (?) is opposite in these two systems, we try to gain the order of the conditional truth degree of the formulas which containsâ†'and (?). Of course, this part is just a preparation for the further result, which caculates just six formulas; there is more work to do to propose the final result.