scholarly journals Uniform interpolation for propositional and modal team logics

2019 ◽  
Vol 29 (5) ◽  
pp. 785-802
Author(s):  
Giovanna D’Agostino
Keyword(s):  
Modal Logic ◽  
Interpolation Property ◽  
Classical Modal Logic ◽  
Uniform Interpolation

Abstract In this paper we consider modal team logic, a generalization of classical modal logic in which it is possible to describe dependence phenomena between data. We prove that most known fragments of full modal team logic allow the elimination of the so called ‘existential bisimulation quantifiers’, where the existence of a certain set is required only modulo bisimulation (i.e. not in the model itself but possibly in a bisimilar model). As a consequence, we prove that these fragments enjoy the uniform interpolation property.

UNIFORM INTERPOLATION IN SUBSTRUCTURAL LOGICS

2014 ◽  
Vol 7 (3) ◽  
pp. 455-483 ◽  
Author(s):  
MAJID ALIZADEH ◽  
FARZANEH DERAKHSHAN ◽  
HIROAKIRA ONO
Keyword(s):  
Propositional Logic ◽  
Predicate Logic ◽  
Interpolation Property ◽  
Substructural Logics ◽  
Substructural Logic ◽  
Sequent Calculi ◽  
Structural Rule ◽  
Modal Propositional Logic ◽  
Uniform Interpolation ◽  
Classical Predicate

AbstractUniform interpolation property of a given logic is a stronger form of Craig’s interpolation property where both pre-interpolant and post-interpolant always exist uniformly for any provable implication in the logic. It is known that there exist logics, e.g., modal propositional logic S4, which have Craig’s interpolation property but do not have uniform interpolation property. The situation is even worse for predicate logics, as classical predicate logic does not have uniform interpolation property as pointed out by L. Henkin.In this paper, uniform interpolation property of basic substructural logics is studied by applying the proof-theoretic method introduced by A. Pitts (Pitts, 1992). It is shown that uniform interpolation property holds even for their predicate extensions, as long as they can be formalized by sequent calculi without contraction rules. For instance, uniform interpolation property of full Lambek predicate calculus, i.e., the substructural logic without any structural rule, and of both linear and affine predicate logics without exponentials are proved.

scholarly journals First-Order Classical Modal Logic

Studia Logica ◽  
2006 ◽  
Vol 84 (2) ◽  
pp. 171-210 ◽  
Author(s):  
Horacio Arló-Costa ◽  
Eric Pacuit
Keyword(s):  
Modal Logic ◽  
First Order ◽  
Classical Modal Logic

scholarly journals Von Wright’s truth-logic and around

2013 ◽  
Vol 19 ◽  
pp. 39-50
Author(s):  
А.С. Карпенко
Keyword(s):  
Modal Logic ◽  
Interpolation Property ◽  
Tense Logic ◽  
Craig Interpolation ◽  
Craig Interpolation Property ◽  
Truth Logic

In this paper von Wright’s truth-logic T__ is considered. It seems that it is a De Morgan four-valued logic DM4 (or Belnap’s four-valued logic) with endomorphism e2. In connection with this many other issues are discussed: twin truth operators, a truth-logic with endomorphism g (or logic Tr), the lattice of extensions of DM4, modal logic V2, Craig interpolation property, von Wright–Segerberg’s tense logic W, and so on.

scholarly journals On a nonthesis of classical modal logic.

1974 ◽  
Vol 15 (3) ◽  
pp. 494-496
Author(s):  
Robert W. Murungi
Keyword(s):  
Modal Logic ◽  
Classical Modal Logic

scholarly journals Constructive interpolation in hybrid logic

2003 ◽  
Vol 68 (2) ◽  
pp. 463-480 ◽  
Author(s):  
Patrick Blackburn ◽  
Maarten Marx
Keyword(s):  
Modal Logic ◽  
Strong Evidence ◽  
Interpolation Property ◽  
Modal Logics ◽  
Hybrid Logic ◽  
Basic Algorithm ◽  
Technical Problems ◽  
First Order ◽  
Elementary Class ◽  
Interpolation Algorithms

AbstractCraig's interpolation lemma (if φ → ψ is valid, then φ → θ and θ → ψ are valid, for θ a formula constructed using only primitive symbols which occur both in φ and ψ) fails for many propositional and first order modal logics. The interpolation property is often regarded as a sign of well-matched syntax and semantics. Hybrid logicians claim that modal logic is missing important syntactic machinery, namely tools for referring to worlds, and that adding such machinery solves many technical problems. The paper presents strong evidence for this claim by defining interpolation algorithms for both propositional and first order hybrid logic. These algorithms produce interpolants for the hybrid logic of every elementary class of frames satisfying the property that a frame is in the class if and only if all its point-generated subframes are in the class. In addition, on the class of all frames, the basic algorithm is conservative: on purely modal input it computes interpolants in which the hybrid syntactic machinery does not occur.

scholarly journals Uniform interpolation and sequent calculi in modal logic

2018 ◽  
Vol 58 (1-2) ◽  
pp. 155-181 ◽  
Author(s):  
Rosalie Iemhoff
Keyword(s):  
Modal Logic ◽  
Sequent Calculi ◽  
Uniform Interpolation

Intuitionistic tense and modal logic

1986 ◽  
Vol 51 (1) ◽  
pp. 166-179 ◽  
Author(s):  
W. B. Ewald
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Tense Logic ◽  
List Type ◽  
Kripke Models ◽  
Classical Modal Logic

In this article we shall construct intuitionistic analogues to the main systems of classical tense logic. Since each classical modal logic can be gotten from some tense logic by one of the definitions(i) □ p ≡ p ∧ Gp ∧ Hp, ◇p ≡ p ∨ Fp ∨ Pp; or,(ii) □ p ≡ p ∧ Gp, ◇p = p ∨ Fp(see [5]), we shall find that our intuitionistic tense logics give us analogues to the classical modal logics as well.We shall not here discuss the philosophical issues raised by our logics. Readers interested in the intuitionistic view of time and modality should see [2] for a detailed discussion.In §2 we define the Kripke models for IKt, the intuitionistic analogue to Lemmon's system Kt. We then prove the completeness and decidability of this system (§§3–5). Finally, we extend our results to other sorts of tense logic and to modal logic.In the language of IKt, we have: sentence-letters p, q, r, etc.; the (intuitionistic) connectives ∧, ∨, →, ¬; and unary operators P (“it was the case”), F (it will be the case”), H (“it has always been the case”) and G (“it will always be the case”). Formulas are defined inductively: all sentence-letters are formulas; if X is a formula, so are ¬X, PX, FX, HX, and GX; if X and Y are formulas, so are X ∧ Y, X ∨ Y, and X → Y. We shall see that, in contrast to classical tense logic, F and P cannot be defined in terms of G and H.

scholarly journals NON-WELL-FOUNDED PROOFS FOR THE GRZEGORCZYK MODAL LOGIC

2020 ◽  
pp. 1-29
Author(s):  
YURY SAVATEEV ◽  
DANIYAR SHAMKANOV
Keyword(s):  
Modal Logic ◽  
Sequent Calculus ◽  
Interpolation Property ◽  
Cut Elimination

Abstract We present a sequent calculus for the Grzegorczyk modal logic $\mathsf {Grz}$ allowing cyclic and other non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs. As an application, we establish the Lyndon interpolation property for the logic $\mathsf {Grz}$ proof-theoretically.

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