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.

An algebraic study of tense logics with linear time

1968 ◽  
Vol 33 (1) ◽  
pp. 27-38 ◽  
Author(s):  
R. A. Bull
Keyword(s):  
Linear Time ◽  
Propositional Calculus ◽  
Tense Logic ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Fourth Section ◽  
Classical Propositional Calculus ◽  
Axiom Systems ◽  
Time Linear

In [2] Prior puts forward a tense logic, GH1, which is intended to axiomatise tense logic with time linear and rational; he also contemplates the tense logic with time linear and real. The purpose of this paper is to give completeness proofs for three axiom systems, GH1, GHlr, GHli, with respect to tense logic with time linear and rational, real, and integral, respectively.1 In a fourth section I show that GH1 and GHlr have the finite model property, but that GHli lacks it.GH1 has the operators of the classical propositional calculus, together with operators P, H, F, G for ‘It has been the case that’, ‘It has always been the case that’, ‘It will be the case that’, ‘It will always be the case that’, respectively.

Finite Kripke models and predicate logics of provability

1990 ◽  
Vol 55 (3) ◽  
pp. 1090-1098 ◽  
Author(s):  
Sergei Artemov ◽  
Giorgie Dzhaparidze
Keyword(s):  
Modal Logic ◽  
Kripke Model ◽  
Finite Model ◽  
Kripke Models ◽  
Model Property ◽  
Logical Formula ◽  
Finite Model Property ◽  
Modal Predicate Logics ◽  
The One ◽  
Arithmetical Interpretation

AbstractThe paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic:If a closed modal predicate-logical formula R is not valid in some finite Kripke model, then there exists an arithmetical interpretation f such that PA ⊬ fR.This result implies the arithmetical completeness of arithmetically correct modal predicate logics with the finite model property (including the one-variable fragments of QGL and QS). The proof was obtained by adding “the predicate part” as a specific addition to the standard Solovay construction.

The finite model property in tense logic

1995 ◽  
Vol 60 (3) ◽  
pp. 757-774 ◽  
Author(s):  
Frank Wolter
Keyword(s):  
Modal Logics ◽  
Tense Logic ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Negative Results ◽  
Propositional Language ◽  
Positive Results

AbstractTense logics in the bimodal propositional language are investigated with respect to the Finite Model Property. In order to prove positive results techniques from investigations of modal logics above K4 are extended to tense logic. General negative results show the limits of the transfer.

Logics containing K4. Part II

1985 ◽  
Vol 50 (3) ◽  
pp. 619-651 ◽  
Author(s):  
Kit Fine
Keyword(s):  
General Result ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Completeness Result ◽  
Morphic Image ◽  
Elementary Condition

This paper establishes another very general completeness result for the logics within the field of K4. With each finite transitive frame ℭ we may associate a formula — Bℭ which validates just those frames ℑ in which ℭ is not in a certain sense embeddable (to be exact, ℭ is not the p-morphic image of any subframe of ℑ. By a subframe logic we mean the result of adding such formulas as axioms to K4. The general result is that each subframe logic has the finite model property.There are a continuum of subframe logics and they include many of the standard ones, such as T, S4, S4.3, S5 and G. It turns out that the subframe logics are exactly those complete for a condition that is closed under subframes (any subframe of a frame satisfying the condition also satisfies the condition). As a consequence, every logic complete for a condition closed under subframes has the finite model property.It is ascertained which of the subframe logics are compact. It turns out that the compact logics are just those whose axioms express an elementary condition. Tests are given for determining whether a given axiom expresses an elementary condition and for determining what it is in case it does.In one respect the present general completeness result differs from most of the others in the literature. The others have usually either been what one might call logic based or formula based. They have usually either been to the effect that all of the logics containing a given logic are complete or to the effect that all logics whose axioms come from a given syntactically characterized class of formulas are complete. The present result is, by contrast, what one might call frame based. The axioms of the logics to be proved complete are characterized most directly in terms of their connection with certain frames.

An incomplete decidable modal logic

1984 ◽  
Vol 49 (2) ◽  
pp. 520-527 ◽  
Author(s):  
M. J. Cresswell
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Order Theory ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
General Frame ◽  
Propositional Modal Logic ◽  
Modal Systems ◽  
Modal Propositional Logic

The most common way of proving decidability in propositional modal logic is to shew that the system in question has the finite model property. This is not however the only way. Gabbay in [4] proves the decidability of many modal systems using Rabin's result in [8] on the decidability of the second-order theory of successor functions. In particular [4, pp. 258-265] he is able to prove the decidability of a system which lacks the finite model property. Gabbay's system is however complete, in the sense of being characterized by a class of frames, and the question arises whether there is a decidable modal logic which is not complete. Since no incomplete modal logic has the finite model property [9, p. 33], any proof of decidability must employ some such method as Gabbay's. In this paper I use the Gabbay/Rabin technique to prove the decidability of a finitely axiomatized normal modal propositional logic which is not characterized by any class of frames. I am grateful to the referee for suggesting improvements in substance and presentation.The terminology I am using is standard in modal logic. By a frame is understood a pair 〈W, R〉 in which W is a class (of “possible worlds”) and R ⊆ W2. To avoid confusion in what follows, a frame will henceforth be referred to as a Kripke frame. By contrast, a general frame is a pair 〈, Π〉 in which is a Kripke frame and Π is a collection of subsets of W closed under the Boolean operations and satisfying the condition that if A is in Π then so is R−1 “A. A model on a frame (of either kind) is obtained by adding a function V which assigns sets of worlds to propositional variables. In the case of a general frame we require that V(p) ∈ Π.

scholarly journals Derivational modal logic of real line with difference modality

10.29007/7gcx ◽  
2018 ◽  
Author(s):  
Andrey Kudinov
Keyword(s):  
Modal Logic ◽  
Real Line ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Axiomatization

We study derivational modal logic of real line with difference modality and prove that it has finite model property but does not have finite axiomatization.

scholarly journals New New Modification of the Subformula Property for a Modal Logic

2020 ◽  
Author(s):  
Mitio Takano
Keyword(s):  
Modal Logic ◽  
Natural Extension ◽  
Modal Logics ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Additional Axiom ◽  
Subformula Property

A modified subformula property for the modal logic KD with the additional axiom $\Box\Diamond(A\vee B)\supset\Box\Diamond A\vee\Box\Diamond B$ is shown. A new modification of the notion of subformula is proposed for this purpose. This modification forms a natural extension of our former one on which modified subformula property for the modal logics K5, K5D and S4.2 has been shown (Bull Sect Logic 30:115--122, 2001 and 48:19--28, 2019). The finite model property as well as decidability for the logic follows from this.

A Modal Logic for Similarity Relations in Pawlak Knowledge Representation Systems

1991 ◽  
Vol 15 (1) ◽  
pp. 61-79
Author(s):  
Dimiter Vakarelov
Keyword(s):  
Modal Logic ◽  
Knowledge Representation ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
First Order ◽  
Similarity Relations ◽  
Representation Systems ◽  
Knowledge Representation Systems

One of the main results of the paper is a characterization of certain kind similarity relations in Pawlak knowledge representation systems by means of first order sentences. As an application we obtain a complete finite axiomatization of the corresponding poly modal logic, called in the paper MLSim. It is proved that MLSim possesses finite model property and is decidable.

scholarly journals STABLE CANONICAL RULES

2016 ◽  
Vol 81 (1) ◽  
pp. 284-315 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
NICK BEZHANISHVILI ◽  
ROSALIE IEMHOFF
Keyword(s):  
Modal Logic ◽  
Finite Model ◽  
Consequence Relations ◽  
Model Property ◽  
Consequence Relation ◽  
Finite Model Property ◽  
Normal Modal Logic

AbstractWe introduce stable canonical rules and prove that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. We apply these results to construct finite refutation patterns for modal formulas, and prove that each normal modal logic is axiomatizable by stable canonical rules. We also define stable multi-conclusion consequence relations and stable logics and prove that these systems have the finite model property. We conclude the paper with a number of examples of stable and nonstable systems, and show how to axiomatize them.

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
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