The decidability of the Kreisel-Putnam system

1970 ◽  
Vol 35 (3) ◽  
pp. 431-437 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Intuitionistic Logic ◽  
Propositional Logic ◽  
Finite Model ◽  
Model Property ◽  
Intuitionistic Propositional Logic ◽  
Disjunction Property ◽  
Finite Model Property ◽  
Image Position

The intuitionistic propositional logic I has the following disjunction property This property does not characterize intuitionistic logic. For example Kreisel and Putnam [5] showed that the extension of I with the axiomhas the disjunction property. Another known system with this propery is due to Scott [5], and is obtained by adding to I the following axiom:In the present paper we shall prove, using methods originally introduced by Segerberg [10], that the Kreisel-Putnam logic is decidable. In fact we shall show that it has the finite model property, and since it is finitely axiomatizable, it is decidable by [4]. The decidability of Scott's system was proved by J. G. Anderson in his thesis in 1966.

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).

A consistent prepositional logic without any finite models

1985 ◽  
Vol 50 (1) ◽  
pp. 38-41 ◽  
Author(s):  
C. G. McKay
Keyword(s):  
Propositional Logic ◽  
Classical System ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Models ◽  
Positive Logic ◽  
Usual Manner ◽  
Constructive Criticism ◽  
Infinite Set

Some propositional logics (e.g. the classical system) can be characterized by a finite model, while others (e.g. Heyting's) which have the finite model property (FMP) can be characterized by an infinite set of finite models. Still others (e.g. certain extensions of Heyting's logic) which lack the FMP can only be characterized by a set of models, at least one of which is infinite. Yet all these logics admit finite models even though they may not be characterized by them. (For example, they all admit the 2-element Boolean algebra as a model in the sense that all their theorems are valid on that algebra when the propositional connectives are interpreted in the usual manner.) The object of the present paper is to give a (not too artificial) example of a propositional logic which is consistent and which admits only infinite models. It therefore lacks the FMP in a very strong sense. Such a propositional logic, I shall call hyperinfinite. The existence of hyperinfinite logics was already plausible from a result in abstract algebra which says that there are varieties of algebras of which the only finite element is the trivial algebra (see [3]).I wish to thank Professor A. S. Troelstra, Amsterdam, for comments on an early version of this paper. The constructive criticism of two anonymous referees has also been useful.The hyperinfinite propositional logic to be described is obtained from Positive Logic—the negative-free part of Heyting's logic—by adjoining certain axioms which govern the use of a unary modal connective.

Tense logic for discrete future time

1970 ◽  
Vol 35 (1) ◽  
pp. 105-118 ◽  
Author(s):  
Patrick Schindler
Keyword(s):  
Propositional Calculus ◽  
Tense Logic ◽  
Finite Model ◽  
Logical System ◽  
Model Property ◽  
Future Time ◽  
Finite Model Property ◽  
Complete Basis ◽  
Image Position ◽  
Classical Propositional Calculus

Prior has conjectured that the tense-logical system Gli obtained by adding to a complete basis for the classical propositional calculus the primitive symbol G, the definitionsDf. F: Fα = NGNαDf. L: Lα = KαGα,and the postulatesis complete for the logic of linear, infinite, transitive, discrete future time. In this paper it is demonstrated that that conjecture is correct and it is shown that Gli has the finite model property: see [4]. The techniques used are in part suggested by those used in Bull [2] and [3]:Gli can be shown to be complete for the logic of linear, infinite, transitive, discrete future time in the sense that every formula of Gli which is true of such time can be proved as a theorem of Gli. For this purpose the notion of truth needs to be formalized. This formalization is effected by the construction of a model for linear, infinite, transitive, discrete future time.

scholarly journals TOPOLOGICAL COMPLETENESS OF LOGICS ABOVE S4

2015 ◽  
Vol 80 (2) ◽  
pp. 520-566 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
DAVID GABELAIA ◽  
JOEL LUCERO-BRYAN
Keyword(s):  
Modal Logic ◽  
Binary Tree ◽  
Finite Model ◽  
Scott Topology ◽  
Model Property ◽  
Topological Semantics ◽  
Finite Model Property ◽  
Normal Modal Logic ◽  
General Frame ◽  
Image Position

AbstractIt is a celebrated result of McKinsey and Tarski [28] that S4 is the logic of the closure algebra Χ+ over any dense-in-itself separable metrizable space. In particular, S4 is the logic of the closure algebra over the reals R, the rationals Q, or the Cantor space C. By [5], each logic above S4 that has the finite model property is the logic of a subalgebra of Q+, as well as the logic of a subalgebra of C+. This is no longer true for R, and the main result of [5] states that each connected logic above S4 with the finite model property is the logic of a subalgebra of the closure algebra R+.In this paper we extend these results to all logics above S4. Namely, for a normal modal logic L, we prove that the following conditions are equivalent: (i) L is above S4, (ii) L is the logic of a subalgebra of Q+, (iii) L is the logic of a subalgebra of C+. We introduce the concept of a well-connected logic above S4 and prove that the following conditions are equivalent: (i) L is a well-connected logic, (ii) L is the logic of a subalgebra of the closure algebra $\xi _2^ + $ over the infinite binary tree, (iii) L is the logic of a subalgebra of the closure algebra ${\bf{L}}_2^ + $ over the infinite binary tree with limits equipped with the Scott topology. Finally, we prove that a logic L above S4 is connected iff L is the logic of a subalgebra of R+, and transfer our results to the setting of intermediate logics.Proving these general completeness results requires new tools. We introduce the countable general frame property (CGFP) and prove that each normal modal logic has the CGFP. We introduce general topological semantics for S4, which generalizes topological semantics the same way general frame semantics generalizes Kripke semantics. We prove that the categories of descriptive frames for S4 and descriptive spaces are isomorphic. It follows that every logic above S4 is complete with respect to the corresponding class of descriptive spaces. We provide several ways of realizing the infinite binary tree with limits, and prove that when equipped with the Scott topology, it is an interior image of both C and R. Finally, we introduce gluing of general spaces and prove that the space obtained by appropriate gluing involving certain quotients of L2 is an interior image of R.

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.

scholarly journals On density of truth of the intuitionistic logic in one variable

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Zofia Kostrzycka
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Propositional Logic ◽  
Propositional Variable ◽  
Intuitionistic Propositional Logic ◽  
International Audience

International audience In this paper we focus on the intuitionistic propositional logic with one propositional variable. More precisely we consider the standard fragment $\{ \to ,\vee ,\bot \}$ of this logic and compute the proportion of tautologies among all formulas. It turns out that this proportion is different from the analog one in the classical logic case.

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.

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