SMALL INFINITARY EPISTEMIC LOGICS
Tai‐Wei Hu, Mamoru Kaneko, NOBU-YUKI SUZUKI · The Review of Symbolic Logic · 2019
Abstract We develop a series of small infinitary epistemic logics to study deductive inference involving intra-/interpersonal beliefs/knowledge such as common knowledge, common beliefs, and infinite regress of beliefs. Specifically, propositional epistemic logics GL (Lα) are presented for ordinalαup to a givenαo(αo≥ω) so that GL(L0) is finitary KDnwithnagents and GL(Lα) (α≥ 1) allows conjunctions of certain countably infinite formulae. GL(Lα) is small in that the language is countable and can be constructive. The set of formulaeLαis increasing up toα=ωbut stops atωWe present Kripke-completeness for GL(Lα) for eachα ≤ ω,which is proved using the Rasiowa–Sikorski lemma and Tanaka–Ono lemma. GL(Lα) has a sufficient expressive power to discuss intra-/interpersonal beliefs with infinite lengths. As applications, we discuss the explicit definability of Axioms T (truthfulness), 4 (positive introspection), 5 (negative introspection), and of common knowledge in GL(Lα) Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL(Lα),α ≤ ω.