Categories and Modalities

Category theory provides various guiding principles for modal logic and its semantic modeling. In particular, Stone duality, or “syntax-semantics duality”, has been a prominent theme in semantics of modal logic since the early days of modern modal logic. This chapter focuses on duality and a few other categorical principles, and brings to light how they underlie a variety of concepts, constructions, and facts in philosophical applications as well as the model theory of modal logic. In the first half of the chapter, I review the syntax-semantics duality and illustrate some of its functions in Kripke semantics and topological semantics for propositional modal logic. In the second half, taking Kripke’s semantics for quantified modal logic and David Lewis’s counterpart theory as examples, I demonstrate how we can dissect and analyze assumptions behind different semantics for first-order modal logic from a structural and unifying perspective of category theory. (As an example, I give an analysis of the import of the converse Barcan formula that goes farther than just “increasing domains”.) It will be made clear that categorical principles play essential roles behind the interaction between logic, semantics, and ontology, and that category theory provides powerful methods that help us both mathematically and philosophically in the investigation of modal logic.

EVERYONE KNOWS THAT SOMEONE KNOWS: QUANTIFIERS OVER EPISTEMIC AGENTS

10.1017/s1755020318000497 ◽

2019 ◽

Vol 12 (2) ◽

pp. 255-270 ◽

Cited By ~ 1

Author(s):

PAVEL NAUMOV ◽

JIA TAO

Keyword(s):

Modal Logic ◽

Epistemic Logic ◽

Possible Worlds ◽

Completeness Theorem ◽

Kripke Semantics ◽

Possible Worlds Semantics ◽

Distributed Knowledge ◽

First Order ◽

Soundness And Completeness ◽

AbstractModal logic S5 is commonly viewed as an epistemic logic that captures the most basic properties of knowledge. Kripke proved a completeness theorem for the first-order modal logic S5 with respect to a possible worlds semantics. A multiagent version of the propositional S5 as well as a version of the propositional S5 that describes properties of distributed knowledge in multiagent systems has also been previously studied. This article proposes a version of S5-like epistemic logic of distributed knowledge with quantifiers ranging over the set of agents, and proves its soundness and completeness with respect to a Kripke semantics.

COMPLETENESS OF SECOND-ORDER PROPOSITIONAL S4 AND H IN TOPOLOGICAL SEMANTICS

10.1017/s1755020318000229 ◽

2018 ◽

Vol 11 (3) ◽

pp. 507-518

Author(s):

PHILIP KREMER

Keyword(s):

Modal Logic ◽

Intuitionistic Logic ◽

Second Order ◽

Topological Semantics ◽

First Order ◽

Propositional Modal Logic ◽

Propositional Quantifiers ◽

Modal Logic S4 ◽

Natural Way

AbstractWe add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionistic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical and intuitionistic logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.

QUANTIFIED MODAL LOGIC ON THE RATIONAL LINE

10.1017/s1755020314000021 ◽

2014 ◽

Vol 7 (3) ◽

pp. 439-454 ◽

Cited By ~ 4

Author(s):

PHILIP KREMER

Keyword(s):

Modal Logic ◽

Real Line ◽

Topological Spaces ◽

Topological Semantics ◽

Cantor Space ◽

Quantified Modal Logic ◽

Propositional Modal Logic ◽

The Family ◽

Modal Logic S4 ◽

Rational Line

AbstractIn the topological semantics for propositional modal logic, S4 is known to be complete for the class of all topological spaces, for the rational line, for Cantor space, and for the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete for the class of all topological spaces, and for the family of subspaces of the irrational line. The main result of the current paper is that QS4 is complete, indeed strongly complete, for the rational line.

Neighborhood-Sheaf Semantics for First-Order Modal Logic

Electronic Notes in Theoretical Computer Science ◽

10.1016/j.entcs.2011.10.011 ◽

2011 ◽

Vol 278 ◽

pp. 129-143 ◽

Cited By ~ 3

Author(s):

Kohei Kishida

Keyword(s):

Modal Logic ◽

First Order ◽

TOPOLOGY AND MODALITY: THE TOPOLOGICAL INTERPRETATION OF FIRST-ORDER MODAL LOGIC

10.1017/s1755020308080143 ◽

2008 ◽

Vol 1 (02) ◽

Cited By ~ 11

Author(s):

STEVE AWODEY ◽

KOHEI KISHIDA

Keyword(s):

Modal Logic ◽

First Order ◽

Topological Interpretation ◽

First-Order Modal Logic

10.1007/978-94-011-5292-1 ◽

1998 ◽

Cited By ~ 177

Author(s):

Melvin Fitting ◽

Richard L. Mendelsohn

Keyword(s):

Modal Logic ◽

First Order ◽

ERRATUM

10.1017/s175502030808026x ◽

2008 ◽

Vol 1 (3) ◽

pp. 393-393

Keyword(s):

Modal Logic ◽

Production Process ◽

Symbolic Logic ◽

First Order ◽

Topological Interpretation ◽

Final Equation ◽

Steve Awodey and Kohei Kishida (2008). Topology and Modality: The Topological Interpretation of First-Order Modal Logic. The Review of Symbolic Logic 1(2): 146-166.On page 148 of this article an error was introduced during the production process. The final equation in the displayed formula 8 lines from the bottom of the page should read,[0, 1) ≠ [0, 1]The publisher regrets this error.

First-order modal logic in the necessary framework of objects

Canadian Journal of Philosophy ◽

10.1080/00455091.2015.1132976 ◽

2016 ◽

Vol 46 (4-5) ◽

pp. 584-609 ◽

Cited By ~ 4

Author(s):

Peter Fritz

Keyword(s):

Modal Logic ◽

Model Theory ◽

Consequence Relations ◽

Possible World ◽

Logical Constants ◽

First Order ◽

Timothy Williamson ◽

Infinite Sets ◽

Infinite Set ◽

AbstractI consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson's argument.

On the expressive power of first-order modal logic with two-dimensional operators

Synthese ◽

10.1007/s11229-016-1172-3 ◽

2016 ◽

Vol 195 (10) ◽

pp. 4373-4417 ◽

Cited By ~ 2

Author(s):

Alexander W. Kocurek

Keyword(s):

Modal Logic ◽

Expressive Power ◽

Two Dimensional ◽

First Order ◽

Recursive enumerability and elementary frame definability in predicate modal logic

Journal of Logic and Computation ◽

10.1093/logcom/exz028 ◽

2019 ◽

Vol 30 (2) ◽

pp. 549-560 ◽

Cited By ~ 1

Author(s):

Mikhail Rybakov ◽

Dmitry Shkatov

Keyword(s):

Modal Logic ◽

Modal Logics ◽

The Other ◽

Kripke Semantics ◽

First Order ◽

Elementary Class ◽

Recursive Enumerability ◽

Kripke Frames ◽

Recursively Enumerable ◽

The Relationship

Abstract We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On one hand, it is well known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e. a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on ‘natural’ propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.