Fibred semantics and the weaving of logics. Part 1: Modal and intuitionistic logics

1996 ◽  
Vol 61 (4) ◽  
pp. 1057-1120 ◽  
Author(s):  
D. M. Gabbay
Keyword(s):  
Proof Theory ◽  
Finite Model ◽  
Finite Model Property ◽  
The Past ◽  
Combined Logics ◽  
Intermediate Logics ◽  
Combination Of Logics ◽  
Combining Logics ◽  
New System ◽  
Intuitionistic Logics

AbstractThis is Part 1 of a paper on fibred semantics and combination of logics. It aims to present a methodology for combining arbitrary logical systems Li, i ∈ I, to form a new system LI. The methodology ‘fibres’ the semantics i of Li into a semantics for LI, and ‘weaves’ the proof theory (axiomatics) of Li into a proof system of LI. There are various ways of doing this, we distinguish by different names such as ‘fibring’, ‘dovetailing’ etc, yielding different systems, denoted by etc. Once the logics are ‘weaved’, further ‘interaction’ axioms can be geometrically motivated and added, and then systematically studied. The methodology is general and is applied to modal and intuitionistic logics as well as to general algebraic logics. We obtain general results on bulk, in the sense that we develop standard combining techniques and refinements which can be applied to any family of initial logics to obtain further combined logics.The main results of this paper is a construction for combining arbitrary, (possibly not normal) modal or intermediate logics, each complete for a class of (not necessarily frame) Kripke models. We show transfer of recursive axiomatisability, decidability and finite model property.Some results on combining logics (normal modal extensions of K) have recently been introduced by Kracht and Wolter, Goranko and Passy and by Fine and Schurz as well as a multitude of special combined systems existing in the literature of the past 20–30 years. We hope our methodology will help organise the field systematically.

scholarly journals On the Decidability of Intuitionistic Tense Logic without Disjunction

2020 ◽  
Author(s):  
Fei Liang ◽  
Zhe Lin
Keyword(s):  
Proof Theory ◽  
Tense Logic ◽  
Heyting Algebras ◽  
Finite Model ◽  
Algebraic Proof ◽  
Model Property ◽  
Algebraic Models ◽  
Finite Axiomatizability ◽  
Finite Model Property ◽  
Modal Operators

Implicative semi-lattices (also known as Brouwerian semi-lattices) are a generalization of Heyting algebras, and have been already well studied both from a logical and an algebraic perspective. In this paper, we consider the variety ISt of the expansions of implicative semi-lattices with tense modal operators, which are algebraic models of the disjunction-free fragment of intuitionistic tense logic. Using methods from algebraic proof theory, we show that the logic of tense implicative semi-lattices has the finite model property. Combining with the finite axiomatizability of the logic, it follows that the logic is decidable.

scholarly journals CONDITIONAL BELIEFS: FROM NEIGHBOURHOOD SEMANTICS TO SEQUENT CALCULUS

2018 ◽  
Vol 11 (4) ◽  
pp. 736-779 ◽  
Author(s):  
MARIANNA GIRLANDO ◽  
SARA NEGRI ◽  
NICOLA OLIVETTI ◽  
VINCENT RISCH
Keyword(s):  
Decision Procedure ◽  
Proof Theory ◽  
Sequent Calculus ◽  
Constructive Proof ◽  
Finite Model ◽  
Finite Model Property ◽  
Semantic Completeness ◽  
Multi Agent ◽  
Labelled Sequent Calculus ◽  
Direct Correspondence

AbstractThe logic of Conditional Beliefs (CDL) has been introduced by Board, Baltag, and Smets to reason about knowledge and revisable beliefs in a multi-agent setting. In this article both the semantics and the proof theory for this logic are studied. First, a natural semantics forCDLis defined in terms of neighbourhood models, a multi-agent generalisation of Lewis’ spheres models, and it is shown that the axiomatization ofCDLis sound and complete with respect to this semantics. Second, it is shown that the neighbourhood semantics is equivalent to the original one defined in terms of plausibility models, by means of a direct correspondence between the two types of models. On the basis of neighbourhood semantics, a labelled sequent calculus forCDLis obtained. The calculus has strong proof-theoretic properties, in particular admissibility of contraction and cut, and it provides a decision procedure for the logic. Furthermore, its semantic completeness is used to obtain a constructive proof of the finite model property of the logic. Finally, it is shown that other doxastic operators can be easily captured within neighbourhood semantics. This fact provides further evidence of the naturalness of neighbourhood semantics for the analysis of epistemic/doxastic notions.

The finite embeddability property for noncommutative knotted extensions of RL

2015 ◽  
Vol 25 (03) ◽  
pp. 349-379 ◽  
Author(s):  
R. Cardona ◽  
N. Galatos
Keyword(s):  
Classical Logic ◽  
Proof Theory ◽  
Mathematical Linguistics ◽  
Residuated Lattices ◽  
Automata Theory ◽  
Finite Model ◽  
Algebraic Proof ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Embeddability Property

The finite embeddability property (FEP) for knotted extensions of residuated lattices holds under the assumption of commutativity, but fails in the general case. We identify weaker forms of the commutativity identity which ensure that the FEP holds. The results have applications outside of order algebra to non-classical logic, establishing the strong finite model property (SFMP) and the decidability for deductions, as well as to mathematical linguistics and automata theory, providing new conditions for recognizability of languages. Our proofs make use of residuated frames, developed in the context of algebraic proof theory.

Skolemization in intermediate logics with the finite model property

2016 ◽  
Vol 24 (3) ◽  
pp. 224-237 ◽  
Author(s):  
Matthias Baaz ◽  
Rosalie Iemhoff
Keyword(s):  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Intermediate Logics

A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property

1974 ◽  
Vol 39 (1) ◽  
pp. 67-78 ◽  
Author(s):  
D. M. Gabbay ◽  
D. H. J. De Jongh
Keyword(s):  
Intuitionistic Logic ◽  
Finite Model ◽  
Ordered Sets ◽  
Disjunction Property ◽  
University Of Illinois ◽  
Finite Model Property ◽  
Intermediate Logics ◽  
Heyting Arithmetic ◽  
Image Position ◽  
System 1

The intuitionistic propositional logic I has the following (disjunction) property.We are interested in extensions of the intuitionistic logic which are both decidable and have the disjunction property. Systems with the disjunction property are known, for example the Kreisel-Putnam system [1] which is I + (∼ϕ → (ψ ∨ α))→ ((∼ϕ→ψ) ∨ (∼ϕ→α)) and Scott's system I + ((∼ ∼ϕ→ϕ)→(ϕ ∨ ∼ϕ))→ (∼∼ϕ ∨ ∼ϕ). It was shown in [3c] that the first system has the finite-model property.In this note we shall construct a sequence of intermediate logics Dn with the following properties:These systems are presented both semantically and syntactically, using the remarkable correspondence between properties of partially ordered sets and axiom schemata of intuitionistic logic. This correspondence, apart from being interesting in itself (for giving geometric meaning to intuitionistic axioms), is also useful in giving independence proofs and obtaining proof theoretic results for intuitionistic systems (see for example, C. Smorynski, Thesis, University of Illinois, 1972, for independence and proof theoretic results in Heyting arithmetic).

Logics containing K4. Part I

1974 ◽  
Vol 39 (1) ◽  
pp. 31-42 ◽  
Author(s):  
Kit Fine
Keyword(s):  
Modal Logic ◽  
Recent Work ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Fourth Section ◽  
Negative Results ◽  
Intermediate Logics ◽  
Completeness Result ◽  
Background Material

There are two main lacunae in recent work on modal logic: a lack of general results and a lack of negative results. This or that logic is shown to have such and such a desirable property, but very little is known about the scope or bounds of the property. Thus there are numerous particular results on completeness, decidability, finite model property, compactness, etc., but very few general or negative results.In these papers I hope to help fill these lacunae. This first part contains a very general completeness result. Let In be the axiom that says there are at most n incomparable points related to a given point. Then the result is that any logic containing K4 and In is complete.The first three sections provide background material for the rest of the papers. The fourth section shows that certain models contain no infinite ascending chains, and the fifth section shows how certain elements can be dropped from the canonical model. The sixth section brings the previous results together to establish completeness, and the seventh and last section establishes compactness, though of a weak kind. All of the results apply to the corresponding intermediate logics.

Canonical rules

2009 ◽  
Vol 74 (4) ◽  
pp. 1171-1205 ◽  
Author(s):  
Emil Jeřábek
Keyword(s):  
Alternative Proof ◽  
Finite Model ◽  
Consequence Relations ◽  
Model Property ◽  
Finite Model Property ◽  
Admissible Rules ◽  
Superintuitionistic Logics ◽  
Canonical Formulas ◽  
Global Consequence

AbstractWe develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.

On the Modal μ-Calculus Over Finite Symmetric Graphs

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
Giovanna D’Agostino ◽  
Giacomo Lenzi
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Satisfiability Problem ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Symmetric Graphs ◽  
Trivial Consequence

AbstractIn this paper we consider the alternation hierarchy of the modal μ-calculus over finite symmetric graphs and show that in this class the hierarchy is infinite. The μ-calculus over the symmetric class does not enjoy the finite model property, hence this result is not a trivial consequence of the strictness of the hierarchy over symmetric graphs. We also find a lower bound and an upper bound for the satisfiability problem of the μ-calculus over finite symmetric graphs.

On simple, weak and strong models of propositional calculi

1973 ◽  
Vol 74 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Ronald Harrop
Keyword(s):  
Simple Model ◽  
Propositional Calculus ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Structure ◽  
Propositional Calculi ◽  
Simple Models

In this paper we will be concerned primarily with weak, strong and simple models of a propositional calculus, simple models being structures of a certain type in which all provable formulae of the calculus are valid. It is shown that the finite model property defined in terms of simple models holds for all calculi. This leads to a new proof of the fact that there is no general effective method for testing, given a finite structure and a calculus, whether or not the structure is a simple model of the calculus.

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