Reasoning about Fuzzy Belief and Common Belief: With Emphasis on Incomparable Beliefs

We formalize reasoning about fuzzy belief and fuzzy common belief, especially incomparable beliefs, in multi-agent systems by using a logical system based on Fitting’s many-valued modal logic, where incomparable beliefs mean beliefs whose degrees are not totally ordered. Completeness and decidability results for the logic of fuzzy belief and common belief are established while implicitly exploiting the duality-theoretic perspective on Fitting’s logic that builds upon the author’s previous work. A conceptually novel feature is that incomparable beliefs and qualitative fuzziness can be formalized in the developed system, whereas they cannot be formalized in previously proposed systems for reasoning about fuzzy belief. We believe that belief degrees can ultimately be reduced to truth degrees, and we call this “the reduction thesis about belief degrees”, which is assumed in the present paper and motivates an axiom of our system. We finally argue that fuzzy reasoning sheds new light on old epistemic issues such as coordinated attack problem.