Small Infinitary Epistemic Logics

We develop a series of small infinitary epistemic logics to study deductive inference involving intra/inter-personal beliefs/knowledge such as common knowledge, common beliefs, and infinite regress of beliefs. Specifically, propositional epistemic logics GL(L_{α}) are presented for ordinal α up to some given ordinal so that GL(L₀) is finitary KDⁿ with n agents and GL(L_α) (α≥1) allows conjunctions of certain countably infinite formulae. GL(L_α) is small in that the language is countable and can be constructive. The set of formulae L_α is increasing up to α = ω but stops at ω. We present Kripke-completeness for GL(L_α) for each α ≤ ω, which is proved using the Rasiowa-Sikorski lemma and Tanaka-Ono lemma. GL(L_α) has a sufficient expressive power to discuss intra/inter-personal beliefs with infinite lengths. As applications, we discuss the explicit definabilities of Axioms T (truthfulness), 4 (positive introspection), 5 (negative introspection), and of common knowledge in GL(L_α). Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL(L_α), α ≤ ω.