Seminar Session on April 12

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.

Aumann proved in 1976 that if two agents with common priors have common
knowledge about their posteriors, then these posteriors must coincide.
Recently, Roy and Dégremont have formalized a qualitative version of this
theorem in the setting of epistemic plausibility models. In this talk, we
will formalize it in the setting of probabilistic Kripke models.

On the technical level, the (semantic) proof of the agreement theorem
turns out to be a straightforward adaptation of Aumann’s original proof,
thus establishing probabilistic Kripke models as a natural setting in
which to explore these issues. Furthermore, we will provide
characterization results for all of the assumptions needed in the theorem,
and look for a complete axiomatization that captures our semantic
reasoning.

On a more methodological level, we will emphasize the dynamics part as an
essential component of the agreement theorem, and even argue that Aumann’s
original theorem is static only because of his convoluted notion of model.
From this perspective, our work is a new illustration of van Benthem’s
claim that many classical static results are actually based on intuitions
involving a large dynamic component, and that these dynamic components can
(and should) be made explicit.