Pure extensions, proof rules, and hybrid axiomatics
type: Item
instance of: scholarly article
title: Pure extensions, proof rules, and hybrid axiomatics
zbMATH Open document ID: 1115.03009
author: N. E. Zubov
published in: Studia Logica
publication date: 2007-01-25T00:00:00Z
review text: Hybrid modal (formal) languages are modal (formal) languages having the expressive power to name possible worlds, asserting equalities between worlds, and describing accessibility relations. In this paper, the authors examine three different kinds of hybrid languages for propositional modal logic and one kind of hybrid language for first-order quantified modal logic; they also characterize semantic systems for all of these kinds of languages. These systems are possible worlds semantics in the fashion of Kripke's semantics, with additional clauses accommodating the syntactic peculiarities of the hybrid languages considered. For each kind of hybrid modal languages, axiomatic formal systems are presented by the authors. Some of these systems contain, among others, certain non-orthodox proof rules; these rules are widely discussed in the paper. Strong completeness theorems are proved for the two axiomatic systems formulated for the most expressive sort of the hybrid propositional logic languages (considered by the authors). For the other axiomatic systems, strong completeness theorems are rather proved for extensions of the systems by means of certain kinds of rules and/or axioms with respect to models satisfying the frame properties defined by those same rules and/or axioms. The completeness proofs are Henkin proofs and differ, in this way, from the standard modal approach by means of canonical models. It is important to point out that, as a result of the expressive power of the hybrid languages, the completeness results cover frame classes that are not definable in orthodox modal languages.
Mathematics Subject Classification ID: 03B45
zbMATH DE Number: 5118678
zbMATH Keywords: hybrid modal languages
zbMATH Keywords: hybrid modal logic
zbMATH Keywords: formal axiomatic systems
zbMATH Keywords: nominals
zbMATH Keywords: hybrid logic
zbMATH Keywords: completeness
zbMATH Keywords: possible worlds semantics
MaRDI profile type: Publication
full work available at URL: s11225-006-9009-6
OpenAlex ID: W1983240156
cites work: Hybrid logics: Characterization, interpolation and complexity
cites work: Repairing the interpolation theorem in quantified modal logic
cites work: Transfer Results for Hybrid Logic. Part I: The Case Without Satisfaction Operators
cites work: Internalizing labelled deduction
cites work: Arthur Prior and hybrid logic
cites work: Modal logic
cites work: Constructive interpolation in hybrid logic
cites work: Hybrid languages
cites work: Hybrid completeness
cites work: Hybrid languages and temporal logic
cites work: Natural Deduction for Hybrid Logic
cites work: An approach to tense logic1
cites work: Q4357061
cites work: The modal logic of inequality
cites work: Q5544276
cites work: Persistent and invariant formulas relative to theories of higher order
cites work: An incomplete logic containing S4
cites work: Modal logic with names
cites work: Hierarchies of modal and temporal logics with reference pointers
cites work: The price of universality
cites work: Epistemology Without Knowledge and Without Belief
cites work: The Morning Star Paradox
cites work: Q3241192
cites work: A completeness theorem in modal logic
cites work: Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi
cites work: Q5339289
cites work: Q4168590
cites work: A hierarchy of formulas in set theory
cites work: An essay in combinatory dynamic logic
cites work: Q5556395
cites work: Q4068699
cites work: Interpolation for extended modal languages
cites work: Expressivity of second order propositional modal logic
cites work: Modal languages for topology: expressivity and definability
cites work: Hybrid logics with Sahlqvist axioms
cites work: An incompleteness theorem in modal logic
DOI: 10.1007/S11225-006-9009-6
description: scientific article; zbMATH DE number 5118678