Gödel含意を用いた可能性と必然性について

抄録

By an analogy with Kripke model in modal logic, the possibility measure Π and the necessity measure N in possibility theory are interpreted as follows. The possibility measure Π is constructed by the minimum operator adopted as a conjunctive operator and the necessity measure N is constructed by Dienes implication adopted as an implicational operator.
In this paper, a necessity measure constructed by Gödel implication and some measures related to it are proposed in addition to the usual possibility measure Π and the usual necessity measure N. The necessity measure constructed by Gödel implication is needed for unifying the views of fuzzy decision making and fuzzy mathematical programming. The properties of these measures and relations among them are investigated. Since a necessity measure constructed by Gödel implication does not have the duality with respect to Π, it is called a pseudo-necessity measure Γ. Gödel implication does not satisfy the law of contraposition, whereas Dienes implication satisfies. Thus, the contraposition of Gödel implication generates another implication. The necessity measure constructed by this implication is not a dual measure of Π either, and is called a pseudo-necessity measure L. Furthermore, pseudo-possibility measures ∧ and V, which are dual measures of Γ and L respectively, can also be defined. As the result of investigation on the properties of these measures, it is clarified that Γ and L are the similar measures to N, and ∧ and V are the similar measures to Π. Comparing with the variations of the values of Γ_A(B) and L_A(B), the variation of the value of N_A(B) around 0.5 is more gradual, and likewise comparing with the variations of the values of ∧_A(B)and V_A(B), the variation of the value of Π_A(B) around 0.5 is also more gradual. Γ_A with respect to a fuzzy set A≠0 and L_A with respect to a fuzzy set A such that supx μ_A(x)=1 satisfy the axioms of necessity measure, and ∧_A with respect to a fuzzy set A≠0 and V_A with respect to a fuzzy set A such that supx μ_A(x)=1 satisfy the axioms of possibility measure. Namely, the existence of measures which satisfy the axioms of possibility measure or necessity measure except for Π_A and N_A with respect to a fuzzy set A such that supx μ_A(x)=1 is revealed.