Kripke semantics for the logic of problems and propositions

2020 ◽  
Vol 211 (5) ◽  
pp. 709-732
Author(s):  
A. A. Onoprienko
Keyword(s):  
Kripke Semantics

scholarly journals Kripke Semantics for Intersection Formulas

2021 ◽  
Vol 22 (3) ◽  
pp. 1-16
Author(s):  
Andrej Dudenhefner ◽  
Paweł Urzyczyn
Keyword(s):  
Kripke Semantics ◽  
Type Assignment ◽  
Soundness And Completeness

We propose a notion of the Kripke-style model for intersection logic. Using a game interpretation, we prove soundness and completeness of the proposed semantics. In other words, a formula is provable (a type is inhabited) if and only if it is forced in every model. As a by-product, we obtain another proof of normalization for the Barendregt–Coppo–Dezani intersection type assignment system.

scholarly journals Dynamic Łukasiewicz logic and its application to immune system

2021 ◽  
Author(s):  
Antonio Di Nola ◽  
Revaz Grigolia ◽  
Nunu Mitskevich ◽  
Gaetano Vitale
Keyword(s):  
Immune System ◽  
Scalar Multiplication ◽  
Kripke Semantics ◽  
Łukasiewicz Logic ◽  
Lukasiewicz Logic ◽  
Regular Algebras

AbstractIt is introduced an immune dynamic n-valued Łukasiewicz logic $$ID{\L }_n$$ I D Ł n on the base of n-valued Łukasiewicz logic $${\L }_n$$ Ł n and corresponding to it immune dynamic $$MV_n$$ M V n -algebra ($$IDL_n$$ I D L n -algebra), $$1< n < \omega $$ 1 < n < ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras $$(\mathcal {M}, \mathcal {R}, \Diamond )$$ ( M , R , ◊ ) that combine the varieties of $$MV_n$$ M V n -algebras $$\mathcal {M} = (M, \oplus , \odot , \sim , 0,1)$$ M = ( M , ⊕ , ⊙ , ∼ , 0 , 1 ) and regular algebras $$\mathcal {R} = (R,\cup , ;, ^*)$$ R = ( R , ∪ , ; , ∗ ) into a single finitely axiomatized variety resembling R-module with “scalar” multiplication $$\Diamond $$ ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic $$ID{\L }_n$$ I D Ł n with application in immune system.

Finitely generated free Heyting algebras

1986 ◽  
Vol 51 (1) ◽  
pp. 152-165 ◽  
Author(s):  
Fabio Bellissima
Keyword(s):  
Free Algebra ◽  
Kripke Semantics ◽  
Heyting Algebras ◽  
Free Algebras ◽  
Finitely Generated ◽  
Algebraic Properties

AbstractThe aim of this paper is to give, using the Kripke semantics for intuitionism, a representation of finitely generated free Heyting algebras. By means of the representation we determine in a constructive way some set of “special elements” of such algebras. Furthermore, we show that many algebraic properties which are satisfied by the free algebra on one generator are not satisfied by free algebras on more than one generator.

scholarly journals An ASP Approach to Generate Minimal Countermodels in Intuitionistic Propositional Logic

2019 ◽  
Author(s):  
Camillo Fiorentini
Keyword(s):  
Propositional Logic ◽  
Answer Set Programming ◽  
Kripke Model ◽  
Kripke Semantics ◽  
Minimal Models ◽  
Intuitionistic Propositional Logic ◽  
Answer Set

Intuitionistic Propositional Logic is complete w.r.t. Kripke semantics: if a formula is not intuitionistically valid, then there exists a finite Kripke model falsifying it. The problem of obtaining concise models has been scarcely investigated in the literature. We present a procedure to generate minimal models in the number of worlds relying on Answer Set Programming (ASP).

Tableaux for essence and contingency

2020 ◽  
Author(s):  
Giorgio Venturi ◽  
Pedro Teixeira Yago
Keyword(s):  
Kripke Semantics ◽  
Normal Operators ◽  
Soundness And Completeness

Abstract We offer tableaux systems for logics of essence and accident and logics of non-contingency, showing their soundness and completeness for Kripke semantics. We also show an interesting parallel between these logics based on the semantic insensitivity of the two non-normal operators by which these logics are expressed.

Relational semantics and a relational proof system for full Lambek calculus

1998 ◽  
Vol 63 (2) ◽  
pp. 623-637 ◽  
Author(s):  
Wendy MacCaull
Keyword(s):  
Proof Theory ◽  
Kripke Semantics ◽  
Substructural Logic ◽  
Lambek Calculus ◽  
Relational Semantics ◽  
Complete Proof ◽  
Proof Search ◽  
Proof Tree ◽  
Relational Logic ◽  
Completeness Theorems

AbstractIn this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation system analogous to that in [11]. Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska [14], [15], and Buszkowski and Orlowska [3], we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t(γ1 ● … ● γn ⊃ α)ευu, has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for t(γ1 ● … ● γn ⊃ α)ευu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ1 ● … ● γn ⊃ α)ευu and from this, build a RelKripke countermodel for γ1 ● … ● γn ⊃ α. These results allow us to add FL, the basic substructural logic, to the list of those logics of importance in computer science with a relational proof theory.

scholarly journals LOGICS FOR PROPOSITIONAL DETERMINACY AND INDEPENDENCE

2018 ◽  
Vol 11 (3) ◽  
pp. 470-506 ◽  
Author(s):  
VALENTIN GORANKO ◽  
ANTTI KUUSISTO
Keyword(s):  
Natural Language ◽  
Kripke Semantics ◽  
Dependence Logic ◽  
Team Semantics ◽  
Reasoning Tasks ◽  
Image Position ◽  
The Way

AbstractThis paper investigates formal logics for reasoning about determinacy and independence. Propositional Dependence Logic${\cal D}$and Propositional Independence Logic${\cal I}$are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics${{\cal L}_D}$and${{\cal L}_{\,I\,}}$, based on Kripke semantics, and propose them as alternatives for${\cal D}$and${\cal I}$, respectively. We analyse the relative expressive powers of these four logics and discuss the way these systems relate to natural language. We argue that${{\cal L}_D}$and${{\cal L}_{\,I\,}}$naturally resolve a range of interpretational problems that arise in${\cal D}$and${\cal I}$. We also obtain sound and complete axiomatizations for${{\cal L}_D}$and${{\cal L}_{\,I\,}}$.

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