Relational dual tableau decision procedure for modal logic K

2011 ◽  
Vol 20 (4) ◽  
pp. 747-756 ◽  
Author(s):  
J. Golinska-Pilarek ◽  
E. Munoz-Velasco ◽  
A. Mora-Bonilla
Keyword(s):  
Modal Logic ◽  
Decision Procedure

scholarly journals A Note on the Issue of Cohesiveness in Canonical Models

2019 ◽  
Vol 29 (3) ◽  
pp. 331-348
Author(s):  
Matteo Pascucci
Keyword(s):  
Modal Logic ◽  
Decision Procedure ◽  
Canonical Model ◽  
Canonical Models

Abstract In their presentation of canonical models for normal systems of modal logic, Hughes and Cresswell observe that some of these models are based on a frame which can be also thought of as a collection of two or more isolated frames; they call such frames ‘non-cohesive’. The problem of checking whether the canonical model of a given system is cohesive is still rather unexplored and no general decision procedure is available. The main contribution of this article consists in introducing a method which is sufficient to show that canonical models of some relevant classes of normal monomodal and bimodal systems are always non-cohesive.

A cut-free Gentzen-type system for the modal logic S5

1980 ◽  
Vol 45 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Masahiko Sato
Keyword(s):  
Modal Logic ◽  
Natural Number ◽  
Decision Procedure ◽  
Type System ◽  
Point Of View ◽  
Usual Sense ◽  
Cut Elimination ◽  
Strict Sense ◽  
Theoretic Point ◽  
Subformula Property

The modal logic S5 has been formulated in Gentzen-style by several authors such as Ohnishi and Matsumoto [4], Kanger [2], Mints [3] and Sato [5]. The system by Ohnishi and Matsumoto is natural, but the cut-elimination theorem in it fails to hold. Kanger's system enjoys cut-elimination theorem, but, strictly speaking, it is not a Gentzen-type system since each formula in a sequent is indexed by a natural number. The system S5+ of Mints is also cut-free, and its cut-elimination theorem is proved constructively via the cut-elimination theorem of Gentzen's LK. However, one of his rules does not have the so-called subformula property, which is desirable from the proof-theoretic point of view. Our system in [5] also enjoys the cut-elimination theorem. However, it is also not a Gentzen-type system in the strict sense, since each sequent in this system consists of a pair of sequents in the usual sense.In the present paper, we give a Gentzen-type system for S5 and prove the cut-elimination theorem in a constructive way. A decision procedure for S5 can be obtained as a by-product.The author wishes to thank the referee for pointing out some errors in the first version of the paper as well as for his suggestions which improved the readability of the paper.

Correction to a paper on modal logic

1955 ◽  
Vol 20 (2) ◽  
pp. 150-150 ◽  
Author(s):  
Alan Ross Anderson
Keyword(s):  
Modal Logic ◽  
Decision Procedure ◽  
The Way

In a recent paper (here referred to as “IDP”) the writer outlined a decision procedure for Lewis's system S4 of modal logic. One of the clauses in definition 3.1 of IDP requires correction. Clause II of 3.1 (2) should read as follows.II. Some constituent of the form ◊β, of degree n1 ≤ n, has the value T in Row (i), and some constituents of the forms ◊δ1, … ◊δh, and ◊η1, …, ◊ηm, all have the value F in Row (i) (h ≥ 0, m ≥ 0, h+m ≥ 1), where β → (δ1 ∨ … ∨ δh ∨ ◊η1 ∨ … ∨ ◊ηm) is an (n1 − 1)-tautology of S4.This change is required in order to carry out the proof of Metathcorcm 3.19. In particular, the change guarantees the following. If the expression η of the second paragraph of 3.20 is of degree n − 1, then the antecedent λ of formula ζ on page 210 of IDP is also of degree n − 1; and consequently the formula ζ of 3.19 is of degree n (since ◊λ is a constituent of ζ). (If we fail to make the correction, then it might be the case that both ζ and (3) are of degree n − 1, in which case Row (i) would not satisfy clause II as originally stated, contrary to the claim at the end of 3.20.) The proofs for the remaining cases of 3.20 can then be carried out, using the revised clause II, in the way originally indicated.The proof of Metatheorem 3.2 requires only trivial corrections for case 2.

scholarly journals The interpretability logic of Peano arithmetic

1990 ◽  
Vol 55 (3) ◽  
pp. 1059-1089 ◽  
Author(s):  
Alessandro Berarducci
Keyword(s):  
Modal Logic ◽  
Modal Analysis ◽  
Set Theory ◽  
Decision Procedure ◽  
Peano Arithmetic ◽  
Relevant Modal Logic ◽  
Binary Operator ◽  
Base Theory ◽  
Interpretability Logic ◽  
Work Done

AbstractPA is Peano arithmetic. The formula InterpPA(α, β) is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, PrPA(x), to the case of the formalized interpretability relation InterpPA(x, y). The relevant modal logic, in addition to the usual provability operator ‘□’, has a binary operator ‘⊳’ to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem.

Aristotle's modal logic

1995 ◽  
Author(s):  
Richard Patterson
Keyword(s):  
Modal Logic

Modal Logic

1980 ◽  
Author(s):  
Brian F. Chellas
Keyword(s):  
Modal Logic

Modality and folk psychology

2019 ◽  
Vol 28 (1) ◽  
pp. 19-27
Author(s):  
Ja. O. Petik
Keyword(s):  
Modal Logic ◽  
Brain Activity ◽  
Formal System ◽  
Philosophy Of Logic ◽  
Psychological States ◽  
Formal Systems ◽  
Modern Psychology ◽  
Philosophical Interpretation ◽  
Important Direction

The connection of the modern psychology and formal systems remains an important direction of research. This paper is centered on philosophical problems surrounding relations between mental and logic. Main attention is given to philosophy of logic but certain ideas are introduced that can be incorporated into the practical philosophical logic. The definition and properties of basic modal logic and descending ones which are used in study of mental activity are in view. The defining role of philosophical interpretation of modality for the particular formal system used for research in the field of psychological states of agents is postulated. Different semantics of modal logic are studied. The hypothesis about the connection of research in cognitive psychology (semantics of brain activity) and formal systems connected to research of psychological states is stated.

A Decision Procedure for Bit-Vector Arithmetic

1998 ◽  
Author(s):  
Clark W. Barrett ◽  
David L. Dill ◽  
Jeremy R. Levitt
Keyword(s):  
Decision Procedure ◽  
Bit Vector
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