Comparing strengths of beliefs explicitly

Inspired by a similar use in provability logic, formulas p > B q and p ≥ B q are introduced in the existing logical framework for discussing beliefs to express that the strength of belief in p is greater than (or equal to) that in q. Besides its usefulness in studying the properties of the concept of greater strength of belief itself this explicit mention of the comparison in the logical language aids in defining several other concepts in a uniform way, namely, older and rather clear concepts like the operators for universality (the totality of possibilities considered by an agent), together with newer notions like plausibility (in the sense of ‘more plausible than not’) and disbelief. Relative expressive powers of the proposed logics are also discussed. A major role is played in our investigations by the relationship between the standard plausibility ordering of the worlds and the strength of belief ordering. If we try to define the strength of belief ordering in terms of the world plausibility ordering we get some undesirable consequences, so we have decided to keep the relation between the two orderings as light as possible to construct a system that allows for widely different interpretations. In fact, we start with considering these orderings to be independent of each other and towards the end we provide a discussion on their possible inter-relationship. Finally, we provide an extension of the framework to the multi-agent setting, and we discuss the possibilities of extending our system to a dynamic one.