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Displaying similar documents to “Involutive Nonassociative Lambek Calculus: Sequent Systems and Complexity”

One-Sided Sequent Systems for Nonassociative Bilinear Logic: Cut Elimination and Complexity

Paweł Płaczek (2021)

Bulletin of the Section of Logic

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Bilinear Logic of Lambek amounts to Noncommutative MALL of Abrusci. Lambek proves the cut–elimination theorem for a one-sided (in fact, left-sided) sequent system for this logic. Here we prove an analogous result for the nonassociative version of this logic. Like Lambek, we consider a left-sided system, but the result also holds for its right-sided version, by a natural symmetry. The treatment of nonassociative sequent systems involves some subtleties, not appearing in associative logics....

An Alternative Natural Deduction for the Intuitionistic Propositional Logic

Mirjana Ilić (2016)

Bulletin of the Section of Logic

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A natural deduction system NI, for the full propositional intuitionistic logic, is proposed. The operational rules of NI are obtained by the translation from Gentzen’s calculus LJ and the normalization is proved, via translations from sequent calculus derivations to natural deduction derivations and back.

The Method of Socratic Proofs Meets Correspondence Analysis

Dorota Leszczyńska-Jasion, Yaroslav Petrukhin, Vasilyi Shangin (2019)

Bulletin of the Section of Logic

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The goal of this paper is to propose correspondence analysis as a technique for generating the so-called erotetic (i.e. pertaining to the logic of questions) calculi which constitute the method of Socratic proofs by Andrzej Wiśniewski. As we explain in the paper, in order to successfully design an erotetic calculus one needs invertible sequent-calculus-style rules. For this reason, the proposed correspondence analysis resulting in invertible rules can constitute a new foundation for...

Labeled Sequent Calculus for Orthologic

Tomoaki Kawano (2018)

Bulletin of the Section of Logic

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Orthologic (OL) is non-classical logic and has been studied as a part of quantumlogic. OL is based on an ortholattice and is also called minimal quantum logic. Sequent calculus is used as a tool for proof in logic and has been examinedfor several decades. Although there are many studies on sequent calculus forOL, these sequent calculi have some problems. In particular, they do not includeimplication connective and they are mostly incompatible with the cut-eliminationtheorem. In this...

An Investigation into Intuitionistic Logic with Identity

Szymon Chlebowski, Dorota Leszczyńska-Jasion (2019)

Bulletin of the Section of Logic

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We define Kripke semantics for propositional intuitionistic logic with Suszko’s identity (ISCI). We propose sequent calculus for ISCI along with cut-elimination theorem. We sketch a constructive interpretation of Suszko’s propositional identity connective.

A Modified Subformula Property for the Modal Logic S4.2

Mitio Takano (2019)

Bulletin of the Section of Logic

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The modal logic S4.2 is S4 with the additional axiom ◊□A ⊃ □◊A. In this article, the sequent calculus GS4.2 for this logic is presented, and by imposing an appropriate restriction on the application of the cut-rule, it is shown that, every GS4.2-provable sequent S has a GS4.2-proof such that every formula occurring in it is either a subformula of some formula in S, or the formula □¬□B or ¬□B, where □B occurs in the scope of some occurrence of □ in some formula of S. These are just the...

Axiomatization of a Basic Logic of Logical Bilattices

Mitio Takano (2016)

Bulletin of the Section of Logic

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A sequential axiomatization is given for the 16-valued logic that has been proposed by Shramko-Wansing (J Philos Logic 34:121–153, 2005) as a candidate for the basic logic of logical bilattices.

Grzegorczyk’s Logics. Part I

Taneli Huuskonen (2015)

Formalized Mathematics

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This article is the second in a series formalizing some results in my joint work with Prof. Joanna Golinska-Pilarek ([9] and [10]) concerning a logic proposed by Prof. Andrzej Grzegorczyk ([11]). This part presents the syntax and axioms of Grzegorczyk’s Logic of Descriptions (LD) as originally proposed by him, as well as some theorems not depending on any semantic constructions. There are both some clear similarities and fundamental differences between LD and the non-Fregean logics introduced...

New Modification of the Subformula Property for a Modal Logic

Mitio Takano (2020)

Bulletin of the Section of Logic

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A modified subformula property for the modal logic KD with the additionalaxiom □ ◊(A ∨ B) ⊃ □ ◊ A ∨ □ ◊B is shown. A new modification of the notion of subformula is proposed for this purpose. This modification forms a natural extension of our former one on which modified subformula property for the modal logics K5, K5D and S4.2 has been shown ([2] and [4]). The finite model property as well as decidability for the logic follows from this.

Deontic Paradoxes and Tableau System for Kalinowski’s Deontic Logic K1

Janusz Ciuciura (2017)

Bulletin of the Section of Logic

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In 1953, Jerzy Kalinowski published his paper on the logic of normative sentences. The paper is recognized as one of the first publications on the formal system of deontic logic. The aim of this paper is to present a tableau system for Kalinowski’s deontic logic and to discuss some of the topics related to the paradoxes of deontic logic.

Identity, equality, nameability and completeness. Part II

María Manzano, Manuel Crescencio Moreno (2018)

Bulletin of the Section of Logic

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This article is a continuation of our promenade along the winding roads of identity, equality, nameability and completeness. We continue looking for a place where all these concepts converge. We assume that identity is a binary relation between objects while equality is a symbolic relation between terms. Identity plays a central role in logic and we have looked at it from two different points of view. In one case, identity is a notion which has to be defined and, in the other case, identity...

Intuitionistic logic considered as an extension of classical logic : some critical remarks

Javier Legris, Jorge A. Molina (2001)

Philosophia Scientiae

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In this paper we analyze the consideration of intuitionistic logic as an extension of classical logic. This — at first sight surprising — point of view has been sustained explicitly by Jan Łukasiewicz on the basis of a mapping of classical propositional logic into intuitionistic propositional logic by Kurt Gödel in 1933. Simultaneously with Gödel, Gerhard Gentzen had proposed another mapping of Peano´s arithmetic into Heyting´s arithmetic. We shall discuss these mappings in connection...

Logics that are both paraconsistent and paracomplete

Newton C.A. da Costa (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The Author describes new systems of logic (called "nonalethic") which are both paraconsistent and paracomplete. These systems are connected with the logic of vagueness and with certain philosophical problems (e.g. with some aspects of Hegel's logic).

Useful Four-Valued Extension of the Temporal Logic KtT4

Vincent Degauquier (2018)

Bulletin of the Section of Logic

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The temporal logic KtT4 is the modal logic obtained from the minimal temporal logic Kt by requiring the accessibility relation to be reflexive (which corresponds to the axiom T) and transitive (which corresponds to the axiom 4). This article aims, firstly, at providing both a model-theoretic and a proof-theoretic characterisation of a four-valued extension of the temporal logic KtT4 and, secondly, at identifying some of the most useful properties of this extension in the context of partial...