Comparing Strengths of Beliefs Explicitly

Inspired by a similar use in provability logic, formulas pÂB 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. This explicit mention of the comparison in the logical language aids in defining several other concepts in a uniform way, viz. older and rather clear concepts like the operators for universality (which possibilities ought to be considered), together with newer notions like plausibility (in the sense of ‘more plausible than not’) and disbelief. Moreover, it assists in studying the properties of the concept of greater strength of belief itself. A heavy part 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. Finally, after a brief discussion on the multi-agent setting, we move on to talk about the dynamics the change of ordering under the influence of hard and soft information.