Search

We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of Π 1 -persistence and show that every weak positive modal logic is Π 1 -persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist's correspondence result.1 [ABSTRACT FROM AUTHOR]

The paper studies completeness and incompleteness of modal predicate logics in Kripke semantics, especially for logics of the form Q Λ , minimal predicate extensions of modal propositional logics. We show that Q Λ is incomplete for a continual family of logics Λ above K + □ (□ p → p) , in particular for well-known K5 and K45. On the other hand, in some cases we find completions of Q Λ ; they are obtained by adding a single extra axiom. Completeness proofs use canonical models, with some modifications, and the case of QK5 is the most interesting from the technical side. We also introduce the "boxing" operation for modal predicate logics and prove transfer results for Kripke and Kripke sheaf completeness with respect to this operation. [ABSTRACT FROM AUTHOR]

Justification logics are modal-like logics that provide a framework for reasoning about justifications. This paper introduces labeled sequent calculi for justification logics, as well as for combined modal-justification logics. Using a method due to Sara Negri, we internalize the Kripke-style semantics of justification and modal-justification logics, known as Fitting models, within the syntax of the sequent calculus to produce labeled sequent calculi. We show that all rules of these systems are invertible and the structural rules (weakening and contraction) and the cut rule are admissible. Soundness and completeness are established as well. The analyticity for some of our labeled sequent calculi are shown by proving that they enjoy the subformula, sublabel and subterm properties. We also present an analytic labeled sequent calculus for S4LPN based on Artemov–Fitting models. [ABSTRACT FROM AUTHOR]

We prove a single category-theoretic result encapsulating the notions of ultrafilters, ultrapower, ultraproduct, tensor product of ultrafilters, the Rudin–Kiesler partial ordering on ultrafilters, and Blass's category of ultrafilters UF. The result in its most basic form states that the category FC (S et , S et) of finite-coproduct-preserving endofunctors of S et is equivalent to the presheaf category [ UF , S et ]. Using this result, and some of its evident generalisations, we re-find in a natural manner the important model-theoretic realisation relation between n -types and n -tuples of model elements; and draw connections with Makkai and Lurie's work on conceptual completeness for first-order logic via ultracategories. As a further application of our main result, we use it to describe a first-order analogue of Jónsson and Tarski's canonical extension. Canonical extension is an algebraic formulation of the link between Lindenbaum–Tarski and Kripke semantics for intuitionistic and modal logic, and extending it to first-order logic has precedent in the topos of types construction studied by Joyal, Reyes, Makkai, Pitts, Coumans and others. Here, we study the closely related, but distinct, construction of the locally connected classifying topos of a first-order theory. The existence of this is known from work of Funk, but the description is inexplicit; ours, by contrast, is quite concrete. [ABSTRACT FROM AUTHOR]