Multi-modal versions of propositional logics S5 or S4—commonly accepted as logics of knowledge—are capable of describing static states of knowledge but they do not reflect how the knowledge changes after communications among agents. In the present paper (part of broader research on logics of knowledge and communications) we define extensions of the logic S5 which can deal with public communications. The logics have natural semantics. We prove some completeness, decidability and interpretability results and formulate a general method that solves certain (...) kind of problems involving public communications—among them well known puzzles of Muddy Children and Mr. Sum & Mr. Product. As the paper gives a formal logical treatment of the operation of restriction of the universe of a Kripke model, it contributes also to investigations of semantics for modal logics. (shrink)

Closure spaces are generalizations of topological spaces, in which the Intersection of two open sets need not be open. The considered logic is related to closure spaces just as the standard logic S4 to topological ones. After describing basic properties of the logic we consider problems of representation of Lindenbaum algebras with some uncountable sets of infinite joins and meets, a notion of equality and a meaning of quantifiers. Results are extended onto the standard logic S4 and they are valid (...) also for some other standard modal logics. (shrink)

The first order modal logic of closure spaces belongs to the class of equationally definable standard modal logics . One can say, it satisfies no version of the deduction lemma. Nevertheless the Robinson and Beth theorems can be proved by means of an interpretation of modal theories in classical ones. LCS is described in [1], [2], [3], [4]. The logics obtained by adjoining axioms of quasi-equality or of equality to LCS are denoted by LCSQE and LCSE.