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Natural Deduction for Full S5 Modal Logic with Weak Normalization

Ana Teresa Martins, Lília Ramalho Martins
2006 Electronical Notes in Theoretical Computer Science  
Weak normalization and subformula property is proved for full S5.  ...  We can mention cut-free Gentzen systems for S4/S5 [6, 7, 8, 1, 2, 16] , normalizable natural deduction systems for intuitionistic modal logics [12, 5] and for full classical S4 [3], but not for full classical  ...  A Natural Deduction System for Full S5 Modal Logic The first-order language L for S5 is considered as usual.  ... 
doi:10.1016/j.entcs.2005.05.027 fatcat:qibe3mraczgmfcza5t7ef76iaq

Full classical S5 in natural deduction with weak normalization

Ana Teresa Martins, Lilia Ramalho Martins
2008 Annals of Pure and Applied Logic  
Beyond weak normalization, we also prove the subformula property for full S5.  ...  We can mention many cut-free Gentzen systems for S4, S5 and K45/K45D, some normalizable natural deduction systems for intuitionistic modal logics and one more for full classical S4, but not for full classical  ...  deduction systems for intuitionistic modal logics [18, 4] and for full classical S4 [3] , but not for full classical S5.  ... 
doi:10.1016/j.apal.2007.11.007 fatcat:5cvg24y4h5g5phv4so4pq35eqy

Page 753 of Mathematical Reviews Vol. , Issue 99b [page]

1991 Mathematical Reviews  
A normal modal logic is a modal logic with the necessitation rule and the axiom K: O(A — B) — (OA — OB), a regular modal logic is one with the inference rule RR by which from A, + B we can infer the sequent  ...  (RS-KRASM; Krasnoyarsk) On admissible rules for the polymodal logic S5,,C. (Russian.  ... 

Page 2423 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews  
Logic Comput. 7 (1997), no. 6, 685-717; MR 98g:03049], we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes  ...  Basic modal logic shares several nice proper- ties with full predicate logic, namely, finite axiomatizability, Craig interpolation and Beth definability, as well as model-theoretic preservation results  ... 

Logics with Common Weak Completions

M. O. Galindo, J. A. Navarro Perez, J. R. Arrazola Ramirez, V. B. Macias
2006 Journal of Logic and Computation  
Two kinds of logics are mainly considered: paraconsistent logics and normal modal logics. For modal logics we use a translation proposed by Gelfond that identifies :oea with :a.  ...  We prove that, for normal logic programs, the result obtained by these weak completions is invariant with respect to a large family of logics.  ...  Classical deductions for normal programs In this section, we briefly summarize some of our early work [29] , where a natural deduction system that can be used to prove atoms, with respect to classical  ... 
doi:10.1093/logcom/exl013 fatcat:nlqsetzh55egrelp7thwzzumji

A Henkin-style completeness proof for the modal logic S5 [article]

Bruno Bentzen
2019 arXiv   pre-print
This paper presents a recent formalization of a Henkin-style completeness proof for the propositional modal logic S5 using the Lean theorem prover.  ...  The full source code is available online and has been typechecked with Lean 3.4.1.  ...  Modal logic We start with the syntax and semantics of the modal logic we have formalized, namely, S5, the strongest of the five systems of modal logic proposed by Lewis and Langford [10] .  ... 
arXiv:1910.01697v1 fatcat:h7ogr2oxczfohfifdf4c7rz7qy

Page 802 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews  
Positive modal logic, AML, is the restriction of the minimal normal modal logic K with the local consequence relation to the modal language with the connectives A, V,0,0, T, 1.  ...  Fitting [Proof methods for modal and intuitionistic logics, Reidel, Dordrecht, 1983; MR 84j:03036]. Weak connectedness is provided by the unique, non- branching v-rule.  ... 

The correctness and relevance of the modal ontological argument

Andrzej Biłat
2020 Synthese  
The paper points to some criteria of formal correctness, and distinguishes two types of relevance for these versions: strong and weak.  ...  AbstractThis paper deals with some metaphilosophical aspects of the modal ontological argument originating from Charles Hartshorne.  ...  . "[…] the modal logic of the argument is S5, the strongest modal system. This is not the case with every version of the modal argument.  ... 
doi:10.1007/s11229-020-02908-5 fatcat:4pb6rrwowfd7hnv6jgfrcocw2a

Notes on Nonmonotonic Autoepistemic Propositional Logic

Marek A. Suchenek
2011 Zeszyty Naukowe  
The paper also offers a proof that the consequence operation CnS5 of modal logic S5 is the maximal monotonic consequence operation satisfying CnS5(T)⊆MKA(T) for every modal theory T.  ...  It begins with a brief review of semantics of common patterns of nonmonotonic deduction arising from a lack of knowledge, including autoepistemic deduction, in terms of the fixed-point equation φ(T,E)=  ...  In particular, a natural candidate for characterization of Cn AE , Kripke's modal system S5, cannot be fully adequate because it is monotonic.  ... 
doi:10.26348/znwwsi.6.74 doaj:c6340de35be34085b8bb73eec7d6a7e1 fatcat:li3oy2wacfgffklqj7cpniycpy

Non-Boolean classical relevant logics II: Classicality through truth-constants

Tore Fjetland Øgaard
2021 Synthese  
One of the selection criteria related to expressive strength is having an "enthymematic" conditional for which a deduction theorem holds.  ...  I argue that this allows for a new interpretation of Anderson and Belnap's take on logical consequence, namely as committing them to pluralism about logical consequence.  ...  Acknowledgements I am very grateful to the participants of the Bergen Logic Group and the audience of the LanCog Workshop on Substructural Logics for constructive feedback.  ... 
doi:10.1007/s11229-021-03065-z fatcat:ret4qhgklvhyblco7gdolyclne

Sound and Complete Query Answering in Intensional P2P Data Integration [article]

Zoran Majkic
2011 arXiv   pre-print
equivalence relation for this logic and the weak deductive inference for it.  ...  In this paper we introduce an intensional FOL (First-order-logic) for P2P systems by fusing the Bealer's intensional algebraic FOL with the S5 possible-world semantics of the Montague, we define the intensional  ...  the weaker form of deduction that the omniscient deduction which uses Modus Ponens and Necessity rule (for normal modal logic) to derive all possible deductions.  ... 
arXiv:1103.0490v1 fatcat:3ejw4htgwjb6hbe3x5u6i2paby

Experimenting with Consistency [chapter]

Juliana Bueno-Soler, Walter Carnielli
2017 The Logical Legacy of Nikolai Vasiliev and Modern Logic  
Although this can be done with interesting consequences when starting from classical normal modal logics, some intriguing cases arise when starting from paraconsistent modalities and negations, as in the  ...  The paper also takes some first steps in exploring the philosophical significance of such logical tools, comparing them with the paraconsistent logics of the family of Logics of Formal Inconsistency (LFIs  ...  We are indebted to David Gilbert, Raymundo Morado, and Peter Verdee for discussions that helped to improve this paper.  ... 
doi:10.1007/978-3-319-66162-9_15 fatcat:q2tjqfrc5bcgjnukbhf3ad22ou

Page 3552 of Mathematical Reviews Vol. , Issue 83i [page]

1983 Mathematical Reviews  
normal form formula A* which is deductively equivalent to A, that is, A is a thesis of the logic if and only if A* is.  ...  Let L be a modal logic and let = be a set of modal formulas.  ... 

Algebraic semantics for a modal logic close to S1

Steffen Lewitzka
2014 Journal of Logic and Computation  
The modal systems S1--S3 were introduced by C. I. Lewis as logics for strict implication. While there are Kripke semantics for S2 and S3, there is no known natural semantics for S1.  ...  Suszko's non-Fregean logic adapted to the language of modal logic (we call these axioms the axioms of propositional identity).  ...  Conclusions We saw that replacing rule SPSE with the stronger principle SP in modal logic S1 results in a non-normal modal logic which has a natural algebraic semantics.  ... 
doi:10.1093/logcom/exu067 fatcat:evsrcc2lvjb4hopeeawn7nzvuq

Page 7595 of Mathematical Reviews Vol. , Issue 2000k [page]

2000 Mathematical Reviews  
modal logic.  ...  Summary: “Recently Ekman gave a derivation in natural deduction such that it either contains a substantial redundant part or else is not normal.  ... 
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