Beschreibung:
<jats:title>Abstract</jats:title><jats:p>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 (<jats:bold><jats:italic>L</jats:italic></jats:bold><jats:sub><jats:italic>α</jats:italic></jats:sub>) are presented for ordinal <jats:italic>α</jats:italic> up to a given <jats:sub><jats:italic>α</jats:italic></jats:sub><jats:sup><jats:italic>o</jats:italic></jats:sup> (<jats:sub><jats:italic>α</jats:italic></jats:sub><jats:sup><jats:italic>o</jats:italic></jats:sup> ≥ <jats:italic>ω</jats:italic>) so that GL(<jats:bold><jats:italic>L</jats:italic></jats:bold><jats:sub>0</jats:sub>) is finitary KD<jats:sup><jats:italic>n</jats:italic></jats:sup> with <jats:italic>n</jats:italic> agents and GL(<jats:bold>L</jats:bold><jats:sub><jats:italic>α</jats:italic></jats:sub>) (<jats:italic>α</jats:italic> ≥ 1) allows conjunctions of certain countably infinite formulae. GL(<jats:bold><jats:italic>L</jats:italic></jats:bold><jats:sub><jats:italic>α</jats:italic></jats:sub>) is small in that the language is countable and can be constructive. The set of formulae <jats:bold><jats:italic>L</jats:italic></jats:bold><jats:sub><jats:italic>α</jats:italic></jats:sub> is increasing up to <jats:italic>α</jats:italic> = <jats:italic>ω</jats:italic> but stops at <jats:italic>ω</jats:italic> We present Kripke-completeness for GL(<jats:bold><jats:italic>L</jats:italic></jats:bold><jats:sub><jats:italic>α</jats:italic></jats:sub>) for each <jats:italic>α ≤ ω,</jats:italic> which is proved using the Rasiowa–Sikorski lemma and Tanaka–Ono lemma. GL(<jats:bold><jats:italic>L</jats:italic></jats:bold><jats:sub><jats:italic>α</jats:italic></jats:sub>) 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(<jats:bold><jats:italic>L</jats:italic></jats:bold><jats:sub><jats:italic>α</jats:italic></jats:sub>) Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL(<jats:bold><jats:italic>L</jats:italic></jats:bold><jats:sub><jats:italic>α</jats:italic></jats:sub>), <jats:italic>α ≤ ω</jats:italic>.</jats:p>