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Problems of substitution and admissibility in the modal system Grz and in intuitionistic propositional calculus

V.V. Rybakov
1990 Annals of Pure and Applied Logic  
Questions connected with the admissibility of rules of inference and the solvability of the substitution problem for modal and intuitionistic logic are considered in an algebraic framework.  ...  Grz and for the intuitionistic calculus H is decidable, (c) intuitionistic propositional calculus H is decidable with respect to admissibility (a positive solution of Friedman's problem).  ...  Thus Admissibility and substitution problems The above mentioned problems wilI be solved first for the modal system Grz; after all they will carry over to intuitionistic propositional calculus H.  ... 
doi:10.1016/0168-0072(90)90055-7 fatcat:xbpyd36q6ndghgx6w44j4fzdnm

Page 2264 of Mathematical Reviews Vol. , Issue 87e [page]

1987 Mathematical Reviews  
V. (2-KRAS) 87e:03040 Bases of admissible rules of the modal system Grz and intuitionistic logic. (Russian) Mat. Sb. (N.S.) 128(170) (1985), no. 3, 321-338, 446.  ...  One of the logics is equivalent to the modal system G and the second one, de- noted by I“, is obtained by adding to the Heyting calculus a new unary operator A and the axioms p — Ap, (Ap — p) - p, Ap —  ... 

Page 30 of Mathematical Reviews Vol. , Issue 92a [page]

1992 Mathematical Reviews  
V. (2-KRAS) Problems of substitution and admissibility in the modal system Grz and in intuitionistic propositional calculus. Ann. Pure Appl. Logic 50 (1990), no. 1, 71-106.  ...  This is the latest in a series of papers that started in the middle 1980s, in which the author has obtained nice results about admis- sibility of rules and related matters in intuitionistic and modal propositional  ... 

A cut-free sequent system for Grzegorczyk logic, with an application to the Gödel–McKinsey–Tarski embedding

Roy Dyckhoff, Sara Negri
2013 Journal of Logic and Computation  
It is well-known that intuitionistic propositional logic Int may be faithfully embedded not just into the modal logic S4 but also into the provability logics GL and Grz of Gödel-Löb and Grzegorczyk, and  ...  This is simplified in that simple case by the use of labelled systems [11] ; we show in Section 6 how to attack the problem for the faithfulness of the embedding · 2 of Int into Grz.  ...  The complete sequent calculus permits a proof that the standard modal translation of intuitionistic logic is a faithful embedding into Grz.  ... 
doi:10.1093/logcom/ext036 fatcat:wxy26qn65reubasipfkzy2lmq4

Proof Complexity of Non-classical Logics [chapter]

Olaf Beyersdorff, Oliver Kutz
2012 Lecture Notes in Computer Science  
In these notes we give an introduction to this recent field of proof complexity of non-classical logics. We cover results from proof complexity of modal, intuitionistic, and non-monotonic logics.  ...  Traditionally, propositional proofs have been the main object of investigation in proof complexity.  ...  Acknowledgements We are very grateful to Rosalie Iemhoff and Emil Jeřábek for numerous helpful comments on an earlier version of these notes.  ... 
doi:10.1007/978-3-642-31485-8_1 fatcat:o2gh5kqzp5fpvb3fataonggkz4

On the complexity of the disjunction property in intuitionistic and modal logics

Mauro Ferrari, Camillo Fiorentini, Guido Fiorino
2005 ACM Transactions on Computational Logic  
In this article we study the complexity of disjunction property for intuitionistic logic, the modal logics S4, S4.1, Grzegorczyk logic, Gödel-Löb logic, and the intuitionistic counterpart of the modal  ...  For S4 we even prove the feasible interpolation theorem and we provide a lower bound for the length of proofs.  ...  PROPOSITIONAL INTUITIONISTIC LOGIC In this section we study the case of propositional intuitionistic logic Int.  ... 
doi:10.1145/1071596.1071598 fatcat:jwyvwet5hrhmxgtknx2ya3gq24

Unification in modal and description logics

F. Baader, S. Ghilardi
2010 Logic Journal of the IGPL  
In this article, we give a survey of the results on unification obtained in two closely related, yet different, application areas of unification: description logics and modal logics.  ...  Unification was originally introduced in automated deduction and term rewriting, but has recently also found applications in other fields.  ...  Recognizing admissible inferences is a hard task: the problem was solved positively for intuitionistic logic and for basic transitive modal systems by V.V. Rybakov (see [58, 60, 61] ).  ... 
doi:10.1093/jigpal/jzq008 fatcat:hrew2tewyrfopf2a5egnmhm23q

Frege systems for extensible modal logics

Emil Jeřábek
2006 Annals of Pure and Applied Logic  
Mints and Kojevnikov [11] have recently shown p-equivalence of Frege systems for the Intuitionistic Propositional Calculus (IPC ) in the standard language, building on a description of admissible rules  ...  By a well-known result of Cook and Reckhow [4, 12] , all Frege systems for the Classical Propositional Calculus (CPC ) are polynomially equivalent.  ...  Ghilardi [6, 7] found a characterization of admissible rules in terms of projective formulas, connecting admissibility to the unification problem for Heyting and modal algebras.  ... 
doi:10.1016/j.apal.2006.04.001 fatcat:mqjlbj4g7zbqbony6yobwfq7o4

Proof analysis in intermediate logics

Roy Dyckhoff, Sara Negri
2011 Archive for Mathematical Logic  
Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all  ...  Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the explanation of the rules.  ...  We recall (see Chapters 3 and 4 of [35] for detailed proofs) that all the rules of G3c are invertible and all the structural rules are admissible.  ... 
doi:10.1007/s00153-011-0254-7 fatcat:mfqotfqxivfi5igghlfzcclp6q

Admissible Rules of Modal Logics

Emil Jeřábek
2005 Journal of Logic and Computation  
We construct explicit bases of admissible rules for a representative class of normal modal logics (including the systems K4, GL, S4, Grz, and GL.3), by extending the methods of S. Ghilardi and R.  ...  We also investigate the notion of admissible multiple conclusion rules.  ...  A normal modal logic is a set L of modal formulas which is closed under substitution, contains all propositional tautologies and the schema (K) (ϕ → ψ) → ( ϕ → ψ), and is closed under the detachment and  ... 
doi:10.1093/logcom/exi029 fatcat:tksfg63qvrchvaplums5ea4fb4

Proofs and Countermodels in Non-Classical Logics

Sara Negri
2014 Logica Universalis  
Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart  ...  Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other.  ...  In a labelled system, a rule of substitution of labels is also often useful Γ → ∆ Γ(y/x) → ∆(y/x) Subst We have: Proposition 2.2.  ... 
doi:10.1007/s11787-014-0097-1 fatcat:yzxgrtllgrga5j3nsc4qm5hgoy

A bimodal perspective on possibility semantics

Johan van Benthem, Nick Bezhanishvili, Wesley H. Holliday
2016 Journal of Logic and Computation  
In this paper we develop a bimodal perspective on possibility semantics, a framework allowing partiality of states that provides an alternative modeling for classical propositional and modal logics [Humberstone  ...  In particular, we define a full and faithful translation of the basic modal logic K over possibility models into a bimodal logic of partial functions over partial orders, and we show how to modulate this  ...  Acknowledgements We wish to thank Guram Bezhanishvili and the three anonymous referees of this paper for helpful comments.  ... 
doi:10.1093/logcom/exw024 fatcat:paao4gu5efclhcvsybgrb7c5nq

Refutation systems in modal logic

Valentin Goranko
1994 Studia Logica: An International Journal for Symbolic Logic  
In section 2 sequential refutation systems are introduced for the classical propositional calculus and for some modal logics including K and KW.  ...  Luka- siewicz in [9], raising the general problem of the formal deduction of the non- valid statements of a given theory, suggested a complete refutation system for the non-valid classical propositions  ... 
doi:10.1007/bf01054714 fatcat:3bmczlgmozhv5bxhedwzhpb5v4

Master index to volumes 41-50

1990 Annals of Pure and Applied Logic  
) 41-63 Rubin, M., see Magidor, M. 46(1990) 41-63 Rybakov, V.V., Problems of substitution and admissibility in the modal system Grz and in intuitionistic propositional calculus 50 (1990) 71  ...  valued fields in weak sub- systems of second-order arithmetic Hella, L., Definability hierarchies of general quantifiers Henle, J., see Apter, A.  ... 
doi:10.1016/0168-0072(90)90061-6 fatcat:inapyhmvyffpzkhhdq4qje4ahi

Unification in lax logic

S. Ghilardi, G. Lenzi
2022 Journal of Algebraic Hyperstructures and Logical Algebras  
We prove that unification is finitary in this logic and supply algorithms for computing a basis of unifiers and for recognizing admissibility of inference rules, following analogous known results for intuitionistic  ...  In this paper, we focus on the intuitionistic propositional logic extended with a local operator [22] (also called nucleus [21]); such logic is commonly named lax logic after [9].  ...  Among logics with finitary unification types, we can mention intuitionistic logic [11] and the modal systems K4, S4, Grz, GL [12] ; however the modal system K lacks this property [19] .  ... 
doi:10.52547/hatef.jahla.3.1.6 fatcat:hgssa5i4fbbu5dmv54iycq4yiq
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