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Kripke semantics and proof systems for combining intuitionistic logic and classical logic

Chuck Liang, Dale Miller
2013 Annals of Pure and Applied Logic  
We combine intuitionistic logic and classical logic into a new, first-order logic called Polarized Intuitionistic Logic.  ...  The meaning of these polarities is defined model-theoretically by a Kripke-style semantics for the logic. Two proof systems are also formulated.  ...  The authors thank the anonymous reviewers for their detailed comments on an earlier draft of this paper. The authors also gratefully acknowledge the support of the ERC Advanced Grant ProofCert.  ... 
doi:10.1016/j.apal.2012.09.005 fatcat:hui7ezmjdjfrrpnvpozj7z3qta

Semantic Incompleteness of Hilbert System for a Combination of Classical and Intuitionistic Propositional Logic [article]

Masanobu Toyooka, Katsuhiko Sano
2022 arXiv   pre-print
This system is proposed as a proof theory for Kripke semantics for a combination of intuitionistic and classical propositional logic, which is obtained by adding the natural semantic clause of classical  ...  implication into intuitionistic Kripke semantics.  ...  This system was provided for a combination of intuitionistic and classical propositional logic.  ... 
arXiv:2207.07416v2 fatcat:67h5yudeqjejhakgqos2ud2hoi

Intuitionistic hybrid logic: Introduction and survey

Torben Braüner
2011 Information and Computation  
Intuitionistic hybrid logic is hybrid modal logic over an intuitionistic logic basis instead of a classical logical basis.  ...  In this short paper we introduce intuitionistic hybrid logic and we give a survey of work in the area.  ...  Acknowledgment The author wish to acknowledge the financial support received from The Danish Natural Science Research Council as funding for the project HYLOCORE (Hybrid Logic, Computation, and Reasoning  ... 
doi:10.1016/j.ic.2011.10.001 fatcat:rooysswgnfcqhgvido4vwi3qf4

Another Combination of Classical and Intuitionistic Conditionals

Satoru Niki, Hitoshi Omori
2022 Electronic Proceedings in Theoretical Computer Science  
Moreover, we discuss the interesting status of disjunction, and the possibility of combining classical logic and minimal logic. We also compare our system with the system of Caleiro and Ramos.  ...  To this end, we introduce an expansion of positive intuitionistic logic, both semantically and proof-theoretically, and establish soundness and strong completeness.  ...  Our system, however, is not the first attempt of combining classical positive logic and intuitionistic positive logic.  ... 
doi:10.4204/eptcs.358.13 fatcat:qppexa3djvapbksqd42gof2dpa

Ecumenical modal logic [article]

Sonia Marin, Luiz Carlos Pereira, Elaine Pimentel, Emerson Sales
2020 arXiv   pre-print
The discussion about how to put together Gentzen's systems for classical and intuitionistic logic in a single unified system is back in fashion.  ...  In Prawitz' system, the classical logician and the intuitionistic logician would share the universal quantifier, conjunction, negation, and the constant for the absurd, but they would each have their own  ...  Since part of the argumentation was given semantically, a notion of Kripke semantics for classical logic was stated, via the respective semantics for intuitionistic logic and the double negation interpretation  ... 
arXiv:2005.14325v1 fatcat:64xflibxz5a4jeyvjektutw4h4

Questions and Dependency in Intuitionistic Logic

Ivano Ciardelli, Rosalie Iemhoff, Fan Yang
2019 Notre Dame Journal of Formal Logic  
We propose an intuitionistic team semantics, where teams are embedded within intuitionistic Kripke models.  ...  These investigations have assumed classical logic as the background logic of statements, and added formulas expressing questions and dependencies to this classical core.  ...  Since our system includes a complete proof system for intuitionistic logic, and since truth for standard formulas coincides with truth in Kripke semantics, we know from the properties of the canonical  ... 
doi:10.1215/00294527-2019-0033 fatcat:blcfhlwwfbf5znuo7ywvvccxnu

Questions and dependency in intuitionistic logic [article]

Ivano Ciardelli, Rosalie Iemhoff, Fan Yang
2017 arXiv   pre-print
We propose an intuitionistic team semantics, where teams are embedded within intuitionistic Kripke models.  ...  These investigations have assumed classical logic as the background logic of statements, and added formulas expressing questions and dependencies to this classical core.  ...  Since our system includes a complete proof system for intuitionistic logic, and since truth for standard formulas coincides with truth in Kripke semantics, we know from the properties of the canonical  ... 
arXiv:1704.01866v1 fatcat:sfvvg4n3pranfdfftnjb75n2da

Relational and neighborhood semantics for intuitionistic modal logic

Kensuke Kojima
2012 Reports on Mathematical Logic  
We investigate semantics for an intuitionistic modal logic in which the "possibility" modality does not distribute over disjunction.  ...  Also, by considering degenerate cases of neighborhood and relational semantics, we demonstrate that a certain classical monotone modal logic has relational semantics, and can be embedded into a classical  ...  Relational and neighborhood semantics for intuitionistic modal logic are basically obtained by adding a preorder (taken from ordinary Kripke semantics for intuitionistic logic) to the corresponding classical  ... 
dblp:journals/rml/Kojima12 fatcat:uertyngchzf7vdvoeeajgphv5u

Structured Sequent Calculi for Combining Intuitionistic and Classical First-Order Logic [chapter]

Paqui Lucio
2000 Lecture Notes in Computer Science  
We define a sound and complete logic, called FO ⊃ , which extends classical first-order predicate logic with intuitionistic implication.  ...  With regard to proof systems for FO ⊃ , we follow Gentzen's approach of sequent calculi (cf. [8] ).  ...  In this paper, we introduce a logical system, called FO ⊃ , obtained by using a combination of classical and intuitionistic logic.  ... 
doi:10.1007/10720084_7 fatcat:tk2wid7vrfcktadqlyxld36rhq

Subtractive logic

Tristan Crolard
2001 Theoretical Computer Science  
, semantical and computational viewpoints.  ...  This paper is the ÿrst part of a work whose purpose is to investigate duality in some related frameworks (cartesian closed categories, lambda-calculi, intuitionistic and classical logics) from syntactic  ...  We give a direct proof of conservativity over intuitionistic logic using Kripke semantics.  ... 
doi:10.1016/s0304-3975(99)00124-3 fatcat:kxguirzxanaatl5y2c2ojppsty

Page 370 of Mathematical Reviews Vol. 53, Issue 2 [page]

1977 Mathematical Reviews  
The first four sections deal with the positive and the full propo- sitional logic, both intuitionist and classical.  ...  Strong semantical completeness for Kripke semantics is the name given to the theorem that S proves A if and only if every Kripke model of S satisfies A.  ... 

Combining First-Order Classical and Intuitionistic Logic

Masanobu Toyooka, Katsuhiko Sano
2022 Electronic Proceedings in Theoretical Computer Science  
This paper provides a multi-succedent sequent calculus G(FOC+J) for our combination of the first-order intuitionistic and classical logic.  ...  This paper studies a first-order expansion of a combination C+J of intuitionistic and classical propositional logic, which was studied by Humberstone (1979) and del Cerro and Herzig (1996), from a proof-theoretic  ...  Lucio [11] proved that FO ⊃ is sound and complete for the intended Kripke semantics. Other than C + J, there are many attempts to combine intuitionistic and classical logic.  ... 
doi:10.4204/eptcs.358.3 fatcat:ibhjvjkjlndxlfzp55ah3zt4lm

Sequent Calculus in the Topos of Trees [chapter]

Ranald Clouston, Rajeev Goré
2015 Lecture Notes in Computer Science  
In the computer science literature it has been most common to leave proof theory and search implicit and fix some concrete semantics; for example see Appel et al's Kripke semantics of stores [1] .  ...  We show that the semantics of the propositional fragment of this logic can be given by linear converse-well-founded intuitionistic Kripke frames, so this logic is a marriage of the intuitionistic modal  ...  Acknowledgments We gratefully acknowledge helpful discussions with Lars Birkedal, Stephané Demri, Tadeusz Litak, and Jimmy Thomson, and the comments of the reviewers of this and a previous unsuccessful  ... 
doi:10.1007/978-3-662-46678-0_9 fatcat:hxn3oyxyf5dvzij4abuepbdrke

A Kripke-style Semantics for the Intuitionistic Logic of Pragmatics ILP

G. Bellin
2003 Journal of Logic and Computation  
We give a Kripke-style semantics for the intuitionistic logic of pragmatics ILP and show completeness with respect to this semantics.  ...  Thus we have the finite model property and also a new proof that the cut rule is eliminable in ILP.  ...  Thanks for helpful comments and suggestions are due to Maria Emilia Maietti and Arnaud Fleury.  ... 
doi:10.1093/logcom/13.5.755 fatcat:elcqfphowncthbrhlz7icygld4

A Linear Logic Based Approach to Timed Petri Nets [chapter]

Norihiro Kamide
2008 Petri Net, Theory and Applications  
In order to handle both resource-sensitive and time-dependent properties of concurrent systems, combining linear logics with temporal operators has been desired, since the (classical) linear logic (as  ...  Kripke semantics Kripke model and soundness The following definition (except the existence of N) of the Kripke frame is the same as that for the (fragment of) intuitionistic linear logic (Kamide, 2003  ...  InTech Europe Although many other models of concurrent and distributed systems have been de-veloped since the introduction in 1964 Petri nets are still an essential model for concurrent systems with respect  ... 
doi:10.5772/5319 fatcat:gw25kquocnemtn3d5kfpenlqb4
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