On Truth Degree Theory In Axiomatic Extension System Of Goguen Propositional Logic

Axiomatic extension of n-valued Goguen propositional logic system is given by adding new connectors～and △,which is denoted as Π～,△.The definition of k truth degree,k similarity degree and k pseudo-metric between formulas are given in this system.Some properties such as MP rule and HS rule of k truth degree are proved.It is introduced that the k truth degree of formula keeps invariability when an exchange takes place between its atomic formulas and its negation.Subsequently,the calculation formula of k randomized truth degree,k randomized similarity degree and k randomized pseudo-metric between formulas are put forward in Π～,△.Some basic properties such as MP rule and HS rule of k randomized truth degree are demonstrated.Meanwhile,three different types of approximate reasoning patterns are introduced,and they are proved to be equivalent.Then,with induced function,the concepts of Γ-k truth degree,Γ-k similarity degree and Γ-k pseudo-metric between formulas in Π～,△ relative to local finite theory Γ under the k conjunction are given.Some basic theories such as intersection-union rule,MP rule and HS rule of Γ-k truth degree are obtained.Last,using the randomization method of valuation sets,the Γ-k randomized truth degree,Γ-k randomized similarity degree and Γ-k randomized pseudo-metric inΠ～,△ relative to local finite theory Γ under the k conjunction are put foward.Furthermore,the concepts of relative randomized divergence degree and the relative randomized consistency degree of theory r relative to the fixed theory Γ0 in Π～,△are introduced.The relationships between relative randomized divergence degree and relative randomized consistency degree are also established.