Probability Logic for Harsanyi Type Spaces

Probability logic has contributed to significant developments in belief typesfor game-theoretical economics. We present a new probability logic for HarsanyiType spaces, show its completeness, and prove both a de-nesting property and aunique extension theorem. We then prove that multi-agent interactiveepistemology has greater complexity than its single-agent counterpart byshowing that if the probability indices of the belief language are restrictedto a finite set of rationals and there are finitely many propositional letters,then the canonical space for probabilistic beliefs with one agent is finitewhile the canonical one with at least two agents has the cardinality of thecontinuum. Finally, we generalize the three notions of definability inmultimodal logics to logics of probabilistic belief and knowledge, namelyimplicit definability, reducibility, and explicit definability. We find thatS5-knowledge can be implicitly defined by probabilistic belief but not reducedto it and hence is not explicitly definable by probabilistic belief