Advances in Modal Logic
Logic deals with the fundamental notions of truth and falsity. Modal logic arose from the philosophical study of “modes of truth” with the two most common modes being “necessarily true” and “possibly true”. Research in modal logic now spans philosophy, computer science, and mathematics, using techniques from relational structures, universal algebra, topology, and proof theory.
These proceedings record the papers presented at the 2018 conference on Advances in Modal Logic, a biennial conference series with the aim of reporting important new developments in pure and applied modal logic. The topics include decidability and complexity results, proof theory, model theory, interpolation, as well as other related problems in algebraic logic.
We prove completeness for some normal modal predicate logics in the standard Kripke semantics with expanding domains. We consider quantified versions of propositional logics with the axiom of density plus some others (transitivity, confluence). The method of proof modifies the technique developed for other cases (without density) by S. Ghilardi, G. Corsi and D. Skvorstov; but now we arrange the whole construction in a game-theoretic style.
It is well-known that every quantied modal logic complete with respect to a first-order definable class of Kripke frames is recursively enumerable. Numerous examples are also known of natural quantied modal logics complete with respect to a class of frames dened by an essentially second-order condition which are not recursively enumerable. It is not, however, known if these examples are instances of a pattern, i.e., whether every recursively enumerable, Kripke complete quantied modal logic can be characterized by a first-order definable class of frames. While the question remains open for normal logics, we show that, in the context of quasi-normal logics, this is not so, by exhibiting an example of a recursively enumerable, Kripke complete quasi-normal logic that is not complete with respect to any first-order definable class of (pointed) frames.