scholarly journals General Topos Semantics for Higher-Order Modal Logic

10.29007/nv5m ◽  
2018 ◽  
Author(s):  
Steve Awodey ◽  
Kohei Kishida ◽  
Hans-Christoph Kotzsch
Keyword(s):  
Modal Logic ◽  
Higher Order ◽  
Algebraic Generalization ◽  
Geometric Morphism

Topos-theoretic semantics for modal logic usually uses structures induced by a surjective geometric morphism between toposes. This talk develops an algebraic generalization of this framework. We take internal adjoints between certain internal frames within a topos, which provides semantics for (intuitionistic) higher-oder modal logic.

A relational modal logic for higher-order stateful ADTs

2010 ◽  
Vol 45 (1) ◽  
pp. 185-198 ◽  
Author(s):  
Derek Dreyer ◽  
Georg Neis ◽  
Andreas Rossberg ◽  
Lars Birkedal
Keyword(s):  
Modal Logic ◽  
Higher Order

Logical Tools

2021 ◽  
pp. 14-52
Author(s):  
Cian Dorr ◽  
John Hawthorne ◽  
Juhani Yli-Vakkuri
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
System P

This chapter presents the system of classical higher-order modal logic which will be employed throughout this book. Nothing more than a passing familiarity with classical first-order logic and standard systems of modal logic is presupposed. We offer some general remarks about the kind of commitment involved in endorsing this logic, and motivate some of its more non-standard features. We also discuss how talk about possible worlds can be represented within the system.

The Bounds of Possibility

2021 ◽  
Author(s):  
Cian Dorr ◽  
John Hawthorne ◽  
Juhani Yli-Vakkuri
Keyword(s):  
Modal Logic ◽  
Higher Order ◽  
Full Length ◽  
Entry Point ◽  
Material Objects ◽  
Gothic Novel ◽  
Fine Grained ◽  
Rigorous Treatment ◽  
Familiar Objects

This book didn’t have to consist of exactly the sentences that it in fact contains: any one of its sentences could have been very different. But it could not have consisted of an entirely different collection of sentences, such as to make it a gothic novel or a treatise on wine-tasting. Other familiar objects are similarly capable of being moderately different, but not radically different, in various respects. But there are puzzling arguments which threaten these apparently obvious judgments, exploiting the fact that an appropriate sequence of small differences can add up to a radical difference. This book presents the first full-length treatment of these puzzles, using them as an entry point to a broad range of metaphysical questions about possibility, necessity, and identity. It introduces tools of higher-order modal logic which enable a rigorous treatment of the puzzles, and develops a strategy for resolving them based on a plenitudinous ontology of material objects, which induces fine-grained variability in the reference of words like ‘book’.

scholarly journals An algebraic generalization of Kripke structures

2008 ◽  
Vol 145 (3) ◽  
pp. 549-577 ◽  
Author(s):  
SÉRGIO MARCELINO ◽  
PEDRO RESENDE
Keyword(s):  
Modal Logic ◽  
Operator Algebras ◽  
Inverse Semigroups ◽  
Kripke Semantics ◽  
Algebraic Semantics ◽  
Propositional Modal Logic ◽  
Kripke Structures ◽  
Algebraic Generalization ◽  
Foliated Manifolds ◽  
Topological Groupoids

AbstractThe Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.

scholarly journals THE DEVELOPMENT OF GÖDEL’S ONTOLOGICAL PROOF

2019 ◽  
pp. 1-19
Author(s):  
ANNIKA KANCKOS ◽  
TIM LETHEN
Keyword(s):  
Modal Logic ◽  
Higher Order ◽  
Second Order ◽  
Ontological Proof ◽  
Rigidity Condition

Abstract Gödel’s ontological proof is by now well known based on the 1970 version, written in Gödel’s own hand, and Scott’s version of the proof. In this article new manuscript sources found in Gödel’s Nachlass are presented. Three versions of Gödel’s ontological proof have been transcribed, and completed from context as true to Gödel’s notes as possible. The discussion in this article is based on these new sources and reveals Gödel’s early intentions of a liberal comprehension principle for the higher order modal logic, an explicit use of second-order Barcan schemas, as well as seemingly defining a rigidity condition for the system. None of these aspects occurs explicitly in the later 1970 version, and therefore they have long been in focus of the debate on Gödel’s ontological proof.

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