Reasoning about Fuzzy Belief and Common Belief: With Emphasis on Incomparable Beliefs
Yoshihiro Maruyama
In this paper 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. Kripke semantics for common belief operator is naturally extended to the many-valued case and thus the notion of fuzzy common belief can be expressed in the extended semantics. A Hilbert-style deductive system is then developed and shown to be sound and complete with respect to the extended semantics. Its decidability and finite model property are also shown. It is a novel feature of this work that the notion of incomparable beliefs can be formalized in the developed system, whereas they cannot be formalized in previously proposed systems for reasoning about fuzzy belief.