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SMALL INFINITARY EPISTEMIC LOGICS

Published online by Cambridge University Press:  01 February 2019

TAI-WEI HU ,
MAMORU KANEKO  and
NOBU-YUKI SUZUKI
TAI-WEI HU*
Affiliation:
Department of Economics, University of Bristol
MAMORU KANEKO*
Affiliation:
Faculty of Political Science and Economics, Waseda University
NOBU-YUKI SUZUKI*
Affiliation:
Department of Mathematics, Faculty of Science, Shizuoka University
*
*DEPARTMENT OF ECONOMICS UNIVERSITY OF BRISTOL BRISTOL, UK E-mail: taiwei.hu@bristol.ac.ukURL: https://taiweihu.weebly.com/
†FACULTY OF POLITICAL SCIENCE AND ECONOMICS WASEDA UNIVERSITY TOKYO, JAPAN E-mail: mkanekoepi@waseda.jpURL: https://infoshako.sk.tsukuba.ac.jp/kaneko/
‡DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE SHIZUOKA UNIVERSITY SHIZUOKA, JAPAN E-mail: suzuki.nobuyuki@shizuoka.ac.jp
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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 ordinal α up to a given αo (αo ≥ ω) so that GL(L0) is finitary KDn with n agents 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 formulae Lα is increasing up to α = ω but stops at ω We present Kripke-completeness for GL(Lα) for each α ≤ 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.

Keywords

03B4203B4591A1091A26infinitary epistemic logiccompletenessRasiowa–Sikorski lemmaTanaka–Ono lemmacommon knowledgeexplicit definabilitygame theoryrationalizability
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 4 , December 2019 , pp. 702 - 735
Copyright
Copyright © Association for Symbolic Logic 2019 

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