Kripke completeness of some infinitary predicate modal logics is presented. More precisely, we prove that if a normal modal logic above is -persistent and universal, the infinitary and predicate extension of with BF and BF is Kripke complete, where BF and BF denote the formulas pi pi and x x, respectively. The results include the completeness of extensions of standard modal logics such as , and its extensions by the schemata T, B, 4, 5, D, and their combinations. The proof (...) is obtained by extending the correspondence between the representation of modal algebras and the completeness of propositional modal logic to infinite. (shrink)

Epistemic logic is the logic of knowledge and belief. It provides insight into the properties of individual knowers, has provided a means to model complicated scenarios involving groups of knowers and has improved our understanding of the dynamics of inquiry.

This paper investigates the epistemic assumptions that David Lewis makes in his account of social conventions. In particular, I focus on the assumption that the agents have common knowledge of the convention to which they are parties. While evolutionary analyses show that the common knowledge assumption is unnecessary in certain classes of games, Lewis’ original account (and, more recently, Cubitt and Sugden’s reconstruction) stresses the importance of including it in the definition of convention. I discuss arguments pro et contra to (...) argue that, although the assumption might be relevant to a descriptively adequate account of social convention, it is not required for its rational reconstruction. I then point out that Lewis’ account, properly speaking, is of common reason to believe, rather than of common knowledge, and argue that, in order to formalize aptly the distinction between reason to believe and belief, standard formal epistemic models need to be supplemented with so-called awareness structures. Finally, I stress that the notion of knowledge implicit in Lewis’ text involves interesting elements that cannot be captured in the standard propositional formalizations, but need the full expressive force of quantified epistemic logic. (shrink)

In order to capture the concept of common knowledge, various extensions of multi-modal epistemic logics, such as fixed-point ones and infinitary ones, have been proposed. Although we have now a good list of such proposed extensions, the relationships among them are still unclear. The purpose of this paper is to draw a map showing the relationships among them. In the propositional case, these extensions turn out to be all Kripke complete and can be comparable in a meaningful manner. F. Wolter (...) showed that the predicate extension of the Halpern-Moses fixed-point type common knowledge logic is Kripke incomplete. However, if we go further to an infinitary extension, Kripke completeness would be recovered. Thus there is some gap in the predicate case. In drawing the map, we focus on what is happening around the gap in the predicate case. The map enables us to better understand the common knowledge logics as a whole. (shrink)

If we express our knowledge in sentences, we will find that these sentences are linked in complex patterns governed by our observations and our inferences from these observations. These inferences are to a large extent driven by logical rules. We ask whether the structure logic imposes on our knowledge restricts what we forget and what we remember. The model is a two period S5 logic. In this logic, we propose a memory loss operator: the agent forgets a sentence pif and (...) only if he knows pat time 1 and he does not know pat time 2. Equipped with the operator, we prove theorems on the relation between knowledge and memory loss. The main results point to classes of formulas that an agent cannot forget, and classes of formulas he must forget. A desirable feature is that most results hold in the S4 logic. The results illustrate bounds to memory loss, and thus to bounded rationality. We apply the model to single-agent conventions: conventions made between an agent and himself. (shrink)

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 are presented for ordinal α up to a given αo so that GL is finitary KDn with n agents and GL allows conjunctions of certain countably infinite formulae. GL 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 for each α ≤ ω, which is proved using the Rasiowa–Sikorski lemma and Tanaka–Ono lemma. GL has a sufficient expressive power to discuss intra-/interpersonal beliefs with infinite lengths. As applications, we discuss the explicit definability of Axioms T, 4, 5, and of common knowledge in GL Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL, α ≤ ω. (shrink)

Predicate modal logics based on Kwith non-compact extra axioms are discussed and a sufficient condition for the model existence theorem is presented. We deal with various axioms in a general way by an algebraic method, instead of discussing concrete non-compact axioms one by one.

This paper provides a Genzten style formulation of the game logic framework GLm (0 m ), and proves the cut-elimination theorem for GLm. As its application, we prove the term existence theorem for GL used in Part I.

We show the faithful embedding of common knowledge logic CKL into game logic GL, that is, CKL is embedded into GL and GL is a conservative extension of the fragment obtained by this embedding. Then many results in GL are available in CKL, and vice versa. For example, an epistemic consideration of Nash equilibrium for a game with pure strategies in GL is carried over to CKL. Another important application is to obtain a Gentzen-style sequent calculus formulation of CKL and (...) its cut-elimination. The faithful embedding theorem is proved for the KD4-type propositional CKL and GL, but it holds for some variants of them. (shrink)

This note documents Aumann's reason for omitting the “empty shells” argument for the common prior assumption from the final version of “Correlated Equilibrium as an Expression of Bayesian Rationality.” It then continues to discuss the argument and concludes that rational entities cannot learn their own identity; if they do not know it a priori, they never will.