The Deductive System Of The BL Propositional Logic System And Its Properties

The Propositional Logic System BL is a relatively complete system.The study on BL propositional logic system is an important direction of the mathematical logic.In order to reveal the characteristics of BL further,this research study it from several aspects.Firstly,the deduction system of BL is defined,the properties of the complete deduction system are investigate,and the relationship between the strong congruence and the deduction system in BL is discussed.Secondly,the concept of the deduction system set algebra is introduced and the corresponding properties is discussed,the definition of basis of a theory ? is introduced and the properties of? is discussed.Moreover,the properties of the quotient algebra generated by the deduction system of BL is investigated.Finally,the new proof of the completeness theorem in BL is given.The construction of chapters and the concrete contents of this paper are as follows:Chapter 1:Preliminaries.We give the basic concepts and correlation proper-ties of lattice,residual lattice,BL-algebras,BL propositional logic system and its semantic theory.Chapter 2:The deduction system of F(S)in BL system.At first,the concept of the deductive system is introduced in BL system,and the properties of the deduc-tion system and the complete deduction system are discussed.Finally,the strong congruence relation of BL is defined,and the one-to-one relationship between the strong congruence and the deduction system is discussed.Chapter 3:The deduction system set algebra of F(S).In the first place,the concept of the deduction system set algebra of BL is introduced.Then,the definition of basis of a theory ? is introduced and the properties of ? is discussed.Chapter 4:The completeness theorem of BL propositional logic system.Firstly,the properties of the quotient algebra generated by the deduction system of BL are investigated.Secondly,the embedding theorem of BL-Lindenbaum algebra is discussed.Finally,the new proof of the completeness theorem in BL is given.