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Cut-free Gentzen calculus for multimodal CK

Michael Mendler, Stephan Scheele
2011 Information and Computation  
First, the fundamental nature of CK is discussed and compared with the intuitionistic modal logic IK which is traditionally taken to be the base line.  ...  This paper extends previous work on the modal logic CK as a reference system, both proof-theoretically and model-theoretically, for a correspondence theory of constructive modal logics.  ...  Acknowledgments We would like to thank the anonymous reviewers for their helpful comments to improve the presentation of this paper.  ... 
doi:10.1016/j.ic.2011.10.003 fatcat:m2xldxpbe5ajfhi64rpxxsvozi

A Tableau Calculus for Equilibrium Entailment [chapter]

David Pearce, Inmaculada P. de Guzmán, Agustín Valverde
2000 Lecture Notes in Computer Science  
A further tableau calculus is then presented for computing equilibrium entailment. This makes use of a new method for reducing the complexity of the tableau expansion rules, which we call signing-up.  ...  We apply tableau methods to the problem of computing entailment in the nonmonotonic system of equilibrium logic, a generalisation of the inference relation associated with the stable model and answer set  ...  (for atomic α) ∼α → ¬α These axioms are taken from the calculus of Vorob'ev [35, 36] , where N is treated as an extension of intuitionistic logic.  ... 
doi:10.1007/10722086_28 fatcat:ph4j6jn7d5ghnno2jjqqjtom6m

One-step Heyting Algebras and Hypersequent Calculi with the Bounded Proof Property

Nick Bezhanishvili, Silvio Ghilardi, Frederik Möllerström Lauridsen
2016 Journal of Logic and Computation  
Finally, using these criteria, we provide a number of examples of calculi for intermediate logics with and without the bounded proof property.  ...  We introduce a new weakly analytic subformula property (the bounded proof property) of hypersequent calculi for intermediate logics.  ...  If the class of finite conservative one-step algebras validating HC has the extension property then we have a one-step extension in form of a finite conservative one-step Heyting algebra H 1 = (D 1 , D  ... 
doi:10.1093/logcom/exw029 fatcat:hcyhyhwamzgpzbfpxkegszbjaa

The contribution of A.V. Kuznetsov to the theory of modal systems and structures

Alexei Y. Muravitsky
2008 Logic and Logical Philosophy  
In his research he focused mainly on semantic, i.e. algebraic, issues and lattices of extensions of particular modal logics, though his proof of the Full Conservativeness Theorem for the proof-intuitionistic  ...  We will outline the contributions of A. V. Kuznetsov to modal logic.  ...  , then, to the definition of Proof-Intuitionistic Logic, KM, and finally to his Full Conservativeness Theorem (see Theorem 17 below).  ... 
doi:10.12775/llp.2008.004 fatcat:ha3x7c2bwbf57fvn2nmkxwxwxu

A modal logic amalgam of classical and intuitionistic propositional logic

Steffen Lewitzka
2015 Journal of Logic and Computation  
Thus, the map φφ is an embedding of IPC into L, i.e. L contains a copy of IPC. Moreover, L is a conservative extension of classical propositional logic CPC.  ...  In this paper, we extend an intuitionistic version of modal logic S1+SP, introduced in our previous paper (S. Lewitzka, Algebraic semantics for a modal logic close to S1, J.  ...  That is, L contains a copy of IPC and behaves in a similar way as a conservative extension. Moreover, L is a conservative extension of classical propositional logic CPC.  ... 
doi:10.1093/logcom/exv048 fatcat:zwjhv2m4q5aajhfpwufexihogi

Epistemic extensions of combined classical and intuitionistic propositional logic

Steffen Lewitzka
2017 Logic Journal of the IGPL  
Logic L was introduced by Lewitzka [7] as a modal system that combines intuitionistic and classical logic: L is a conservative extension of CPC and it contains a copy of IPC via the embedding φφ.  ...  Finally, we show that a modification of our semantics yields algebraic models for the systems of Intuitionistic Epistemic Logic introduced in [1].  ...  L is a conservative modal extension of CPC and contains a copy of IPC via the embedding ϕ → ϕ. In this sense, L combines CPC and IPC.  ... 
doi:10.1093/jigpal/jzx004 fatcat:5rjivdztmzdljfhuc66xfag2gi

Page 8396 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
The sequent calculi are obtained by adding the axioms for these theories, which are given in the form of sequent calculus rules, to the logical calculus for intuitionistic propositional logic G3ip.  ...  The sequent calculi of extensions of the theory of excess (theory of positive linear order, positive lattices and Heyting algebra) are also considered.  ... 

Models for normal intuitionistic modal logics

Milan Božić, Kosta Došen
1984 Studia Logica: An International Journal for Symbolic Logic  
This paper lays the ground for an investigation of intuitionistio analogues of systems stronger than K. A brief survey is given of the existing literature on intuitionistic modal logic.  ...  Kripke-style models with two accessibility relations, one intuitionistio and the other modal, are given for analogues of the modal system K based on Heyting's propositional logic.  ...  It is also easy to show that ILK(> is a conservative, and hence ccnsistent~ extension of the Heyting propositional calculus in Z(> without (> by a procedure analogous r the procedure mentioned for IIK[  ... 
doi:10.1007/bf02429840 fatcat:mff6mewelrfxljnhysvrzhapte

Page 3988 of Mathematical Reviews Vol. , Issue 83j [page]

1983 Mathematical Reviews  
He obtains a conservative extension of the usual predicate calculus. {For the entire collection see MR 81a:03002.} G.  ...  A metrizable space X is said to be universal if every formula which cannot be derived in the intuitionistic predicate calculus has a counterexample in a sheaf over the complete Heyting algebra O( X) of  ... 

Page 2841 of Mathematical Reviews Vol. , Issue 87f [page]

1987 Mathematical Reviews  
The results are: (1) HATT’ is a conservative extension of HATT. (2) HATT' is consistent if and only if classical type theory is consistent (the proof uses a result of J.  ...  According to this theorem, if A is a theorem of the classical propositional calculus, then the double negation of A is a theorem of the intuitionistic propositional calculus.  ... 

Multi-type display calculus for dynamic epistemic logic

Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano, Vlasta Sikimić
2014 Journal of Logic and Computation  
The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design  ...  In the present paper, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus.  ...  The set L(m-IK) of the formulas A of the multi-modal version m-IK of Fischer Servi intuitionistic modal logic IK (cf.  ... 
doi:10.1093/logcom/exu068 fatcat:ulrfcu626rgpdmenrqv7alsa4y

A proof-theoretic study of bi-intuitionistic propositional sequent calculus

Luís Pinto, Tarmo Uustalu
2018 Journal of Logic and Computation  
Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication usually called "exclusion".  ...  A standard-style sequent calculus for this logic is easily obtained by extending multiple-conclusion sequent calculus for intuitionistic logic with exclusion rules dual to the implication rules (in particular  ...  This research was financed by Fundação para a Ciência e a Tecnologia (FCT) through project UID/MAT/00013/2013, by ERDF through the Estonian Centre of Excellence in Computer Science (EXCS), by the Estonian  ... 
doi:10.1093/logcom/exx044 fatcat:4msinmct6bhvhaqhbobujv6cqy

Page 2383 of Mathematical Reviews Vol. , Issue 92e [page]

1992 Mathematical Reviews  
(LOMI) 40 (1974), 101-109; MR 53 #2645], where he showed that a certain system HPCe of first-order Heyting predicate logic with Hilbert’s epsilon symbol is a conservative extension of the ordinary first  ...  Another result is that a system of first-order Heyting arithmetic with the epsilon symbol is a con- servative extension of ordinary first-order Heyting arithmetic.  ... 

Distributive Full Nonassociative Lambek Calculus with S4-Modalities Is Context-Free [chapter]

Zhe Lin
2012 Lecture Notes in Computer Science  
Z kolei BFNL jest PSPACEzupe lny, a jego modalne rozszerzenia sa PSPACE-trudne.  ...  Modalny nie laczny racunek Lambeka (NL♦) otrzymujemy wzbogacajac NL o modalności ♦, ↓ , spe lniajace prawo rezyduacji: ♦A ⇒ B wtedy i tylko wtedy, gdy A ⇒ ↓ B, co pozwala stosować pewne postulaty strukturalne  ...  So BFNL is a conservative extension of DFNL. The frame semantics corresponding to modal extensions of NL can be found in Kurtonina [1994] , which can be extended to DFNL and BFNL.  ... 
doi:10.1007/978-3-642-31262-5_11 fatcat:nzxxxs7nqnaxhbgclayh7oc7ti

Page 1428 of Mathematical Reviews Vol. 48, Issue 5 [page]

1974 Mathematical Reviews  
I. 8202 A new extension of S4. Notre Dame J. Formal Logic 14 (1973), 567-574. The author studies the system of modal logic obtained by adding O(MLIp)(CLMpLMLp) to the usual axiomatiza- tion of 84.  ...  The author gives a Hilbert-type axiomatization for the intuitionistic propositional calculus with equivalence as the only logical connective.  ... 
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