Fuzzy Modal Logic Based On Non-uniform Probability

With the increasingly widespread application of modal logic in thinking science, computer science and other fields, to discuss fuzzy modal logic appears very significant. Many scholars use different methods to study fuzzy modal logic, such as the fuzzy definition of modal operator, the modal reduction, the intuitional fuzzy modal logic research, etc. Based on the basic modal logic system, this paper introduces the concept of fuzzy modal operator, then gives the new semantic of fuzzy modal operator by introducing non-even probability P into the domain W of the modal M=(W, R, V), proves the dual properties of the fuzzy modal operators, brings fuzzy modal operators into the modal logic and gives proper justice sets and reasoning rules, and gets a fuzzy modal logic system which is better than modal logic system in representing and reasoning ability. This paper also discusses the fuzzy modal logic assigned on Wn, gives the definition of generalized modal tautologies assigned on Wn, classifies the generalized modal tautology of Wn, proves that the reachable generalized tautology is not empty in types and kinds of mutual different theorem of generalized modal tautology, etc. These studies not only enable us to have a more profound understanding of generalized modal tautology, but also provide us new research method and new vitality into the study of fuzzy modal logic.As preliminaries, the first chapter introduces some simple knowledge of modal logic and probability.The second chapter based on the description of "a little likely","very likely" which often appears in daily life, puts forward two new modal words "□β" and "0a". By the new modal words, we give the semantic theory of the fuzzy modal phrase "α-possibility" and "P-certainty" and get some useful conclusions. By bringing the fuzzy modal operators into modal logic system, we get a fuzzy modal logic system which is better than basic modal logic system in representing and reasoning ability.In the third chapter, we discrete the assignment lattice [0,1] into Wn based on semantic of assignment V in modal logic system. Then we mainly discuss the concept of generalized modal tautology which satisfies V(φ,w)≥α, and classify the generalized modal tautology which assigned on Wn, and prove the mutually different classes theorem of generalized modal tautology. The modal tautology representation theorem of the generalized modal tautology is also given.The fourth chapter summarizes the main results of this paper, and arises some questions worth considering in the future.