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.

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 Curves with canonical models on scrolls

2016 ◽  
Vol 27 (05) ◽  
pp. 1650045 ◽  
Author(s):  
Danielle Lara ◽  
Simone Marchesi ◽  
Renato Vidal Martins
Keyword(s):  
Singular Point ◽  
Complete Intersection ◽  
Canonical Model ◽  
Projective Curve ◽  
Canonical Models ◽  
Dimension Formula ◽  
Rational Normal Scroll ◽  
Monomial Curve

Let [Formula: see text] be an integral and projective curve whose canonical model [Formula: see text] lies on a rational normal scroll [Formula: see text] of dimension [Formula: see text]. We mainly study some properties on [Formula: see text], such as gonality and the kind of singularities, in the case where [Formula: see text] and [Formula: see text] is non-Gorenstein, and in the case where [Formula: see text], the scroll [Formula: see text] is smooth, and [Formula: see text] is a local complete intersection inside [Formula: see text]. We also prove that the canonical model of a rational monomial curve with just one singular point lies on a surface scroll if and only if the gonality of the curve is at most [Formula: see text], and that it lies on a threefold scroll if and only if the gonality is at most [Formula: see text].

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.

Topological facets of the logic of subset spaces (with emphasis on canonical models)

2016 ◽  
Vol 29 (7) ◽  
pp. 1099-1120 ◽  
Author(s):  
Bernhard Heinemann
Keyword(s):  
Modal Logic ◽  
Small Contribution ◽  
Topological Characterization ◽  
Canonical Models ◽  
Spatial Characteristics ◽  
Commutation Relations ◽  
Crucial Part

Abstract Among other things, this article makes a small contribution to the field of bi-topological modal logic, as it is examined herein to what extent Moss and Parikh's logic of subset spaces, LSS, is topologically relevant. For that purpose, several spatial characteristics are identified and proved to correspond with the axioms of this logic in the sense of topological definability first. It turns out that a certain bi-modal cover property plays a crucial part in doing so. Then, the question is raised whether these properties are valid on the canonical topo-model for LSS. Our investigation into this finally results in a topological characterization of that model, and it leads us to studying additional schemata in the same way, namely bi-modal commutation relations. As one of the questions opening our exposition asks for the nature of the topologies induced by the two LSS-modalities, we present a corresponding characterization in the concluding part of the article.

scholarly journals FIRST-ORDER POSSIBILITY MODELS AND FINITARY COMPLETENESS PROOFS

2019 ◽  
Vol 12 (4) ◽  
pp. 637-662
Author(s):  
MATTHEW HARRISON-TRAINOR
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Canonical Model ◽  
Model Construction ◽  
Completeness Proof ◽  
Quantified Modal Logic ◽  
First Order ◽  
Propositional Modal Logic ◽  
Computable Set ◽  
Order Case

AbstractThis article builds on Humberstone’s idea of defining models of propositional modal logic where total possible worlds are replaced by partial possibilities. We follow a suggestion of Humberstone by introducing possibility models for quantified modal logic. We show that a simple quantified modal logic is sound and complete for our semantics. Although Holliday showed that for many propositional modal logics, it is possible to give a completeness proof using a canonical model construction where every possibility consists of finitely many formulas, we show that this is impossible to do in the first-order case. However, one can still construct a canonical model where every possibility consists of a computable set of formulas and thus still of finitely much information.

scholarly journals Models for joint isometries

1996 ◽  
Vol 38 (2) ◽  
pp. 191-194 ◽  
Author(s):  
K. R. M. Attele ◽  
A. R. Lubin
Keyword(s):  
Hilbert Space ◽  
Unit Sphere ◽  
Canonical Model ◽  
Weighted Shifts ◽  
Canonical Models ◽  
Direct Sum ◽  
Commuting Contractions

An N-tuple ℐ= (T1…, TN) of commuting contractions on a Hilbert space H is said to be a joint isometry if for all x in H, or, equivalently, if Athavale in [1] characterized the joint isometries as subnormal N-tuples whose minimal normal extensions have joint spectra in the unit sphere S2N−X a geometric perspective of this is given in [4]. Subsequently, V. Müller and F.-H. Vasilescu proved that commuting N-tuples which are joint contractions, i.e. , can be represented as restrictions of certain weighted shifts direct sum a joint isometry. In this paper we adapt the canonical models of [3], and also construct a new canonical model, which completes the previous descriptions by showing joint isometries are indeed restrictions of specific multivariable weighted shifts [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.

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