Inspired by a similar use in provability logic, formulas p > q
and p ≥ 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 enables one to apply the system
to situations which are outside of the range of standard doxastic logic.
Moreover it enables one to formalize the intuitive idea that one wants
to consider only those situations which one considers to be possible
(one believes in the occurrence of such situations more than in the
contradiction), and which are the ones that inhabit the usual models
of logics of knowledge and belief. It also aids in defining several less
common concepts in a uniform way, notions like plausibility (in the
sense of `more plausible than not’) and disbelief. Finally, it
assists in studying the properties of the concept of greater
strength of belief itself.
Our basic semantics uses an ordering of the sets of worlds (propositions)
of the model. A heavy part is played in our investigations by the
relationship between the standard plausibility ordering of the worlds and
our strength of belief ordering of the propositions. We discuss the
possibility of defining the one in terms of the other. The few words which
time will allow us by then will be devoted to dynamics – the change of the
ordering under the influence of hard and soft information.
The work is in cooperation with Sujata Ghosh.