SMALL INFINITARY EPISTEMIC LOGICS

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.

Scholastická logika „vědění“ I.

Fourteenth-century logic gave rise, among others, to the genre De scire et dubitare, which offered a unified framework for discussing different forms of epistemic sophisms by utilising the underlying systems of epistemic logic. One of the problems introduced in this context already by the founding father of this genre, William Heytesbury, was the so-called axiom of positive introspection, i.e., the principle that an agent who knows that something is the case, knows that she knows that it is the case. Owing to Heytesbury’s enormous popularity in the subsequent centuries, discussion of this problem became relatively widespread. This debate was addressed already in Boh’s seminal Epistemic Logic in the Later Middle Ages, which, despite its limitations acknowledged by its author, is a standard source. The present study elaborates on Boh by extending the corpus of his works (both in the sense of including new authors and of utilising manuscripts along with printed editions) and drawing new connections based on that. The core of the survey consists of an analysis of the positions of William Heytesbury and John Wyclif (both pertaining to the context of Merton College), their Italian reception by Peter of Mantua, and the “continental” reception of Heytesbury by John of Holland. The main goals of this study are to formalise the key arguments, which makes it possible to address the underlying systems of epistemic logic and their respective “strength”, and to articulate the conceptual background of those arguments and systems (the concepts of evidence, attention, and order of cognitive operations). The gist of the debate is, on one of the sides, an attempt to prove that it is impossible to doubt whether one knows that something is the case by employing whether the principles of positive introspection and of distribution of knowledge over implication, or the principles of positive and negative introspection combined.