What is modal logic

Publisher Summary First-order modal logics are modal logics in which the underlying propositional logic is replaced by a first-order predicate logic. They pose some of the most difficult mathematical challenges. This chapter surveys basic first-order modal logics and examines recent attempts to find a general mathematical setting in which to analyze them. A number of logics that make use of constant domain, increasing domain, and varying domain semantics is discussed, and a first-order intensional logic and a first-order version of hybrid logic is presented. One criterion for selecting these logics is the availability of sound and complete proof procedures for them, typically axiom systems and/or tableau systems. The first-order modal logics are compared to fragments of sorted first-order logic through appropriate versions of the standard translation. Both positive and negative results concerning fragment decidability, Kripke completeness, and axiomatizability are reviewed. Modal hyperdoctrines are introduced as a unifying tool for analyzing the alternative semantics. These alternative semantics range from specific semantics for non-classical logics, to interpretations in well-established mathematical framework. The relationship between topological semantics and D. Lewis's counterpart semantics is investigated and an axiomatization is presented.

First Order Modal Logic

Day 5: Applications and Extensions The precise topics for the course will depend in part on the interests and background of the students attending the lectures. This document contains an introduction to the course including some pointers to relevant literature. The focus is on the basics of neighborhood semantics with only pointers to literature relating to the more advanced topics. The website for the course is staff.science.uva.nl/∼epacuit/nbhd esslli.html On this website you will find the lecture slides (and possibly course notes) updated daily. Two other sources will also be useful:

Neighborhood Semantics for Modal Logic

Publisher Summary This chapter discusses the semantic ideas underlying modern modal logic, and in particular, Kripke semantics—or relational semantics. It introduces the basic model theoretic constructions, explores links between modal logic and classical logic, both on models and on frames, and examines the extent to which the key semantic ideas transfer to richer modal logics and languages while maintaining a relatively low computational complexity. The basic modal languages and the graphs over which they are interpreted are discussed. The chapter also introduces the notion of bisimulation, based on which, modal logic as a fragment of first-order logic is characterized. The computability and computational complexity of modal logic is examined. The level of frames and the link between modal and classical logic are explored. Three alternatives to relational semantics––namely, algebraic semantics, neighborhood semantics, and topological semantics are also discussed.

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Modal logic: a semantic perspective

This is a survey of the origins of mathematical interpretations of modal logics, and their development over the last century or so. It focuses on the interconnections between algebraic semantics using Boolean algebras with operators and relational semantics using structures often called Kripke models. It reviews the ideas of a number of people who independently contributed to the emergence of relational semantics, and compares them with the work of Kripke. It concludes with an account of several applications of modal model theory to mathematics and theoretical computer science.

Mathematical modal logic: A view of its evolution

The paper studies propositional logics in a bimodal language, in which the first modality is interpreted as the local truth, and the second as the universal truth. The logic S4UC is introduced, which is finitely axiomatizable, has the f.m.p. and is determined by every connected separable metric space.

« Everywhere » and « here »

To describe semantics of a logical system one should define notions of a model and the truth in a model. A major part of classical first order model theory can be developed within the standard semantics, while alternative types of semantics (such as sheaves, forcing, polyadic algebras) play an auxiliary role.

Semantics of Non-Classical First Order Predicate Logics

This paper concerns modal logics appearing from the temporal ordering of domains in two-dimensional Minkowski spacetime. As R. Goldblatt has proved recently, the logic of the whole plane isS4.2. We consider closed or open convex polygons and closed or open domains bounded by simple differentiable curves; this leads to the logics:S4,S4.1,S4.2 orS4.1.2.

Modal logics of domains on the real plane.

Chronological Future Modality in Minkowski Spacetime.

Maximal Kripke-type semantics for modal and superintuitionistic predicate logics

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