What is modal logic

Logical formalisms for reasoning about relations between spatial regions play
a fundamental role in geographical information systems, spatial and constraint
databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's
modal logic of time intervals based on the Allen relations, we introduce a
family of modal logics equipped with eight modal operators that are interpreted
by the Egenhofer-Franzosa (or RCC8) relations between regions in topological
spaces such as the real plane. We investigate the expressive power and
computational complexity of logics obtained in this way. It turns out that our
modal logics have the same expressive power as the two-variable fragment of
first-order logic, but are exponentially less succinct. The complexity ranges
from (undecidable and) recursively enumerable to highly undecidable, where the
recursively enumerable logics are obtained by considering substructures of
structures induced by topological spaces. As our undecidability results also
capture logics based on the real line, they improve upon undecidability results
for interval temporal logics by Halpern and Shoham. We also analyze modal
logics based on the five RCC5 relations, with similar results regarding the
expressive power, but weaker results regarding the complexity.

Modal Logics of Topological Relations

A set theoretical assertion psi is forceable or possible, written lozenge psi, if psi holds in some forcing extension, and necessary, written square psi, if psi holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory S4.2.

The Modal Logic of Forcing

Publisher Summary This chapter focuses on logically possible worlds and counterpart semantics for modal logic. Originally conceived as the logic of necessity and possibility, philosophical roots of modal logic go back at least as far as Aristotle and the Stoic Diodorus Cronus. The most common semantics of modal logics introduce the notions of a world, a possibility, or a situation, and impose further structure by means of more or less complex relations, for example, by the notion of accessibility. Modal logic does not embody the commitment to possible worlds of any sort—rather the doctrines of modal realism and anti-realism are subject to considerable philosophical debate. The languages of modal predicate logic that is considered differs from the language of modal propositional logic as follows. First, there are neither propositional variables nor propositional constants. Second, first-order and second-order languages of modal predicate logic differ syntactically as well as with respect to the substitution principles assumed. This chapter discusses the notion of modal predicate logic, objects in counterpart frames, and the semantical impact of Haecceitism. Details of metaframes and status of the modal language are also presented in this chapter.

/pdf/logically-possible-worlds-and-counterpart-semantics-for-lbdqmee6v1.pdf

Logically Possible Worlds and Counterpart Semantics for Modal Logic

We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.

/pdf/sahlqvist-correspondence-for-modal-mu-calculus-mxkelswctg.pdf

Sahlqvist Correspondence for Modal mu-calculus

Chronological Future Modality in Minkowski Spacetime.

In this paper we prove that Japaridze's Polymodal Logic is PSPACE-decidable. To show this, we describe a decision proc edure for satisfiability on hereditarily ordered frames that can be applied to obtain upper complexity bounds for various modal logics.

/pdf/pspace-decidability-of-japaridze-s-polymodal-logic-7t6svo1em4.pdf

PSPACE-decidability of Japaridze's polymodal logic

On PSpace-decidability in transitive modal logic

Every world can see a Sahlqvist world.

Filtration is a standard tool for establishing the finite model property of modal logics. We consider logics and classes of frames that admit filtration, and identify some operations on them that preserve this property. In particular, the operation of adding the inverse or the transitive closure of a relation is shown to be safe in this sense.These results are then used to prove that every regular grammar logic with converse admits filtration. We present filtration constructions for right-linear and left-linear grammar logics. We also give a simple example of a grammar modal logic that is undecidable and hence does not admit filtration.

/pdf/filtration-safe-operations-on-frames-47y3lzy7bs.pdf

Filtration Safe Operations on Frames.

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