Improved Second-Order Quantifier Elimination in Modal Logic

2008 ◽  
pp. 375-388 ◽  
Author(s):  
Renate A. Schmidt
Keyword(s):  
Modal Logic ◽  
Quantifier Elimination ◽  
Second Order

Representability in second-order propositional poly-modal logic

2002 ◽  
Vol 67 (3) ◽  
pp. 1039-1054 ◽  
Author(s):  
G. Aldo Antonelli ◽  
Richmond H. Thomason
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Second Order ◽  
Order Logic ◽  
Multiple Modalities ◽  
Modal Operators ◽  
Second Order Systems ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Propositional System

AbstractA propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.In this paper we generalize this framework by allowing multiple modalities. While this does not affect the undecidability of K, B, T, K4 and S4, poly-modal second-order S5 is dramatically more expressive than its mono-modal counterpart. As an example, we establish the definability of the transitive closure of finitely many modal operators. We also take up the decidability issue, and, using a novel encoding of sets of unordered pairs by partitions of the leaves of certain graphs, we show that the second-order propositional logic of two S5 modalitities is also equivalent to full second-order logic.

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

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.

First-order definability in modal logic

1975 ◽  
Vol 40 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. I. Goldblatt
Keyword(s):  
Modal Logic ◽  
Binary Relation ◽  
Second Order ◽  
Order Conditions ◽  
Kripke Models ◽  
Order Condition ◽  
First Order ◽  
Axiom Systems ◽  
Set Of Points ◽  
First Order Definability

In the early days of the development of Kripke-style semantics for modal logic a great deal of effort was devoted to showing that particular axiom systems were characterised by a class of models describable by a first-order condition on a binary relation. For a time the approach seemed all encompassing, but recent work by Thomason [6] and Fine [2] has shown it to be somewhat limited—there are logics not determined by any class of Kripke models at all. In fact it now seems that modal logic is basically second-order in nature, in that any system may be analysed in terms of structures having a nominated class of second-order individuals (subsets) that serve as interpretations of propositional variables (cf. [7]). The question has thus arisen as to how much of modal logic can be handled in a first-order way, and precisely which modal sentences are determined by first-order conditions on their models. In this paper we present a model-theoretic characterisation of this class of sentences, and show that it does not include the much discussed LMp → MLp.Definition 1. A modal frame ℱ = 〈W, R〉 consists of a set W on which a binary relation R is defined. A valuation V on ℱ is a function that associates with each propositional variable p a subset V(p) of W (the set of points at which p is “true”).

scholarly journals Modal Structuralism and Theism

2017 ◽  
Author(s):  
Silvia Jonas
Keyword(s):  
Modal Logic ◽  
Religious Belief ◽  
Physical World ◽  
Second Order ◽  
Logical Possibility ◽  
Truth Values ◽  
Objective Truth ◽  
Metaphysical Priority

Drawing an analogy between modal structuralism about mathematics and theism, this chapter offers a structuralist account that implicitly defines theism in terms of three basic relations: logical and metaphysical priority, and epistemic superiority. On this view, statements like “God is omniscient” have a hypothetical and a categorical component. The hypothetical component provides a translation pattern according to which statements in theistic language are converted into statements of second-order modal logic. The categorical component asserts the logical possibility of the theism structure on the basis of uncontroversial facts about the physical world. This structuralist reading of theism preserves objective truth-values for theistic statements while remaining neutral on the question of ontology. Thus, it offers a way of understanding theism to which a naturalist cannot object, and it accommodates the fact that religious belief, for many theists, is an essentially relational matter.

Structuralism

2009 ◽  
Author(s):  
Geoffrey Hellman
Keyword(s):  
Modal Logic ◽  
Set Theory ◽  
Model Theory ◽  
Second Order ◽  
Mathematical Framework ◽  
Topos Theory ◽  
Mathematical Structuralism

The main types of mathematical structuralism that have been proposed and developed to the point of permitting systematic and instructive comparison are four: structuralism based on model theory, carried out formally in set theory (e.g., first- or second-order Zermelo–Fraenkel set theory), referred to as STS (for set-theoretic structuralism); the approach of philosophers such as Shapiro and Resnik of taking structures to be sui generis universals, patterns, or structures in an ante rem sense (explained in this article), referred to as SGS (for sui generis structuralism); an approach based on category and topos theory, proposed as an alternative to set theory as an overarching mathematical framework, referred to as CTS (for category-theoretic structuralism); and a kind of eliminative, quasi-nominalist structuralism employing modal logic, referred to as MS (for modal-structuralism). This article takes these up in turn, guided by few questions, with the aim of understanding their relative merits and the choices they present.

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