Probability Logic for Harsanyi Type Spaces

TL;DR

This paper introduces a new probability logic for Harsanyi Type spaces, proves its completeness and properties, and explores the complexity of multi-agent belief systems compared to single-agent ones.

Contribution

It develops a novel probability logic for Harsanyi Type spaces, establishes key logical properties, and analyzes the complexity differences in multi-agent versus single-agent belief models.

Findings

The logic is complete and has a de-nesting property.

Multi-agent belief spaces are significantly more complex than single-agent spaces.

S5-knowledge is implicitly definable but not reducible or explicitly definable by probabilistic belief.

Abstract

Probability logic has contributed to significant developments in belief types for game-theoretical economics. We present a new probability logic for Harsanyi Type spaces, show its completeness, and prove both a de-nesting property and a unique extension theorem. We then prove that multi-agent interactive epistemology has greater complexity than its single-agent counterpart by showing that if the probability indices of the belief language are restricted to a finite set of rationals and there are finitely many propositional letters, then the canonical space for probabilistic beliefs with one agent is finite while the canonical one with at least two agents has the cardinality of the continuum. Finally, we generalize the three notions of definability in multimodal logics to logics of probabilistic belief and knowledge, namely implicit definability, reducibility, and explicit definability. We find that S5-knowledge can be implicitly defined by probabilistic belief but not reduced to it and hence is not explicitly definable by probabilistic belief.