In lecture 7, we have seen how axiomatics and semantics of modal logic fit together in soundness proofs and correspondence proofs. We have seen several examples of classes of Kripke frames that are characterized by formulas of propositional modal logic. These were several special cases. But we are looking for a general correspondence result. Can we find a full correspondence result? For any for...

We study the modal logicML of the countable random frame, which is contained in and ‘approximates’ the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of ML and show that it is not finitely axiomatizable. Then we describe the finite fra...

Despite half a century of fuzzy sets and fuzzy logic progress, as fuzzy sets address complex and uncertain information through the lens of human knowledge and subjectivity, more progress is needed in the semantics of fuzzy sets and in exploring the multi-modal aspect of fuzzy logic due to the different cognitive, emotional and behavioral angles of assessing truth. We lay here the foundations of...

7.1 Introduction This chapter is about decidability and complexity issues in modal logic; more specifically, we confine ourselves to satisfiability (and the complementary validity) problems. The satisfiability problem is the following: for a fixed class of models, to determine whether a given formula ϕ is satisfiable in some model of that class (a more precise definition will follow). The gener...

We present a fuzzy (or quantitative) version of the van Benthem theorem, which characterizes propositional modal logic as the bisimulation-invariant fragment of first-order logic. Specifically, we consider a first-order fuzzy predicate logic alongwith its modal fragment, and show that the fuzzy first-order formulas that are non-expansive w.r.t. the natural notion of bisimulation distance are ex...

We propose a notion of focusing for nested sequent calculi for modal logics which brings down the complexity of proof search to that of the corresponding sequent calculi. The resulting systems are amenable to specifications in linear logic. Examples include modal logic K, a simply dependent bimodal logic and the standard non-normal modal logics. As byproduct we obtain the first nested sequent c...

We briefly examine the modal formulae that can be derived in Multiplicative Additive Linear Logic (MALL) and some extensions by using Tarksi’s extensional modal operators. We also breifly compare this with a substructural form of the modal logic K.

The central idea of the paper is to look at both relation algebras (cf. [JT], [J]) and temporal logic of intervals ([vB1], [V]) from a viewpoint of twodimensional modal logic. Here we mean by a two-dimensional modal logic a system of modal logic in which (at least some of) the intended Kripke semantics is such that the set of possible worlds is (a subset of) a Cartesian product A×A for some set...