Bulletin of the Section of Logic
https://czasopisma.uni.lodz.pl/bulletin
<div style="text-align: justify;"> <p>The <em>Bulletin of the Section of Logic</em> (<em>BSL</em>) is a quarterly peer-reviewed journal published with the support of the Lodz University Press. The <em>BSL</em> was founded in 1972 by Ryszard Wójcicki, Head of the Section of Logic of the Polish Academy of Sciences, then based in Wrocław, as a newsletter-journal designed for the exchange of scientific results among members of the Section with their national and international partners and colleagues. The first editor-in-chief of the <em>BSL</em> was Jan Zygmunt, who supervised the editorial process of the first six issues of the journal. In 1973 the role was taken over by Marek Tokarz. From 1975 to 2018 the journal was managed and edited by Grzegorz Malinowski. In 1992 the Department of Logic at the University of Łódź took over the publication from the Polish Academy of Sciences changing its policy into regular peer-reviewed journal. The aim of the <em>Bulletin</em> is to act as a forum for the prompt wide dissemination of original, significant results in logic through rapid publication. The <em>BSL</em> welcomes especially contributions dealing directly with logical calculi, their methodology, application and algebraic interpretations.</p> </div>Wydawnictwo Uniwersytetu Łódzkiegoen-USBulletin of the Section of Logic0138-0680Lifting Results for Finite Dimensions to the Transfinite in Systems of Varieties Using Ultraproducts
https://czasopisma.uni.lodz.pl/bulletin/article/view/14031
<p>We redefine a system of varieties definable by a schema of equations to include finite dimensions. Then we present a technique using <br />ultraproducts enabling one to lift results proved for every finite dimension to the transfinite. Let \(\bf Ord\) denote the class of all ordinals. Let \(\langle \mathbf{K}_{\alpha}: \alpha\in \bf Ord\rangle\) be a system of varieties definable by a schema. Given any ordinal \(\alpha\), we define an operator \(\mathsf{Nr}_{\alpha}\) that acts on \(\mathbf{K}_{\beta}\) for any \(\beta>\alpha\) giving an algebra in \(\mathbf{K}_{\alpha}\), as an abstraction of taking \(\alpha\)-neat reducts for cylindric algebras. We show that for any positive \(k\), and any infinite ordinal \(\alpha\) that \(\mathbf{S}\mathsf{Nr}_{\alpha}\mathbf{K}_{\alpha+k+1}\) cannot be axiomatized by a finite schema over \(\mathbf{S}\mathsf{Nr}_{\alpha}\mathbf{K}_{\alpha+k}\) given that the result is valid for all finite dimensions greater than some fixed finite ordinal. We apply our results to cylindric algebras and Halmos quasipolyadic algebras with equality. As an application to our algebraic result we obtain a strong incompleteness theorem (in the sense that validitities are not captured by finitary Hilbert style axiomatizations) for an algebraizable extension of \(L_{\omega,\omega}\).</p>Tarek Sayed Ahmed
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2024-02-122024-02-1253214515410.18778/0138-0680.2024.02SUP-Hesitant Fuzzy Interior Ideals in \(\Gamma\)-Semigroups
https://czasopisma.uni.lodz.pl/bulletin/article/view/14208
<p>In this paper, we defined the concept \(\mathcal{SUP}\)-hesitant fuzzy interior ideals in \(\Gamma\)-semigroups, which is generalized of hesitant fuzzy interior ideals in \(\Gamma\)-semigroups. Additionally, we study fundamental properties of \(\mathcal{SUP}\)-hesitant fuzzy interior ideals in \(\Gamma\)-semigroups. Finally, we investigate characterized properties of those.</p>Pannawit KhamrotThiti Gaketem
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2024-04-242024-04-2453215517110.18778/0138-0680.2024.09Some Logics in the Vicinity of Interpretability Logics
https://czasopisma.uni.lodz.pl/bulletin/article/view/16580
<p>In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic \(\mathbf{IL}\). We will introduce the logics \(\mathbf{BIL}\) and \(\mathbf{BIL}^{+}\) in the propositional language with a modal operator \(\square\) and a binary operator \(\Rightarrow\) such that \(\mathbf{BIL}\subseteq\mathbf{BIL}^{+}\subseteq\mathbf{IL}\). The logic \(\mathbf{BIL}\) is generated by the relational structures \(\left<X,R,N\right>\), called basic frames, where \(\left<X,R\right>\) is a Kripke frame and \(\left<X,N\right>\) is a neighborhood frame. We will prove that the logic \(\mathbf{BIL}^{+}\) is generated by the basic frames where the binary relation \(R\) is definable by the neighborhood relation \(N\) and, therefore, the neighborhood semantics is suitable to study the logic \(\mathbf{BIL}^{+}\) and its extensions. We shall also study some axiomatic extensions of \(\mathsf{\mathbf{BIL}}\) and we will prove that these extensions are sound and complete with respect to a certain classes of basic frames. Finally, we prove that the logic BIL+ and some of its extensions are complete respect with the class of neighborhood frames.</p>Sergio A. Celani
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2023-11-222023-11-2253217319310.18778/0138-0680.2023.26Fuzzy Sub-Equality Algebras Based on Fuzzy Points
https://czasopisma.uni.lodz.pl/bulletin/article/view/18350
<div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>In this paper, by using the notion of fuzzy points and equality algebras, the notions of fuzzy point equality algebra, equality-subalgebra, and ideal were established. Some characterizations of fuzzy subalgebras were provided by using such concepts. We defined the concepts of \((\in, \in)\) and \((\in, \in\! \vee \, {q})\)-fuzzy ideals of equality algebras, discussed some properties, and found some equivalent definitions of them. In addition, we investigated the relation between different kinds of \((\alpha,\beta)\)-fuzzy subalgebras and \((\alpha,\beta)\)-fuzzy ideals on equality algebras. Also, by using the notion of \((\in, \in)\)-fuzzy ideal, we defined two equivalence relations on equality algebras and we introduced an order on classes of \(X\), and we proved that the set of all classes of \(X\) by these order is a poset.</p> </div> </div> </div>Mona Aaly KologaniMohammad Mohseni TakalloYoung Bae JunRajab Ali Borzooei
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2023-12-182023-12-1853219522210.18778/0138-0680.2023.31A Syntactic Proof of the Decidability of First-Order Monadic Logic
https://czasopisma.uni.lodz.pl/bulletin/article/view/18448
<p>Decidability of monadic first-order classical logic was established by Löwenheim in 1915. The proof made use of a semantic argument and a purely syntactic proof has never been provided. In the present paper we introduce a syntactic proof of decidability of monadic first-order logic in innex normal form which exploits <strong>G3</strong>-style sequent calculi. In particular, we introduce a cut- and contraction-free calculus having a (complexity-optimal) terminating proof-search procedure. We also show that this logic can be faithfully embedded in the modal logic <strong>T</strong>.</p>Eugenio OrlandelliMatteo Tesi
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2024-02-092024-02-0953222324410.18778/0138-0680.2024.03Sequent Systems for Consequence Relations of Cyclic Linear Logics
https://czasopisma.uni.lodz.pl/bulletin/article/view/20419
<p>Linear Logic is a versatile framework with diverse applications in computer science and mathematics. One intriguing fragment of Linear Logic is Multiplicative-Additive Linear Logic (MALL), which forms the exponential-free component of the larger framework. Modifying MALL, researchers have explored weaker logics such as Noncommutative MALL (Bilinear Logic, BL) and Cyclic MALL (CyMALL) to investigate variations in commutativity. In this paper, we focus on Cyclic Nonassociative Bilinear Logic (CyNBL), a variant that combines noncommutativity and nonassociativity. We introduce a sequent system for CyNBL, which includes an auxiliary system for incorporating nonlogical axioms. Notably, we establish the cut elimination property for CyNBL. Moreover, we establish the strong conservativeness of CyNBL over Full Nonassociative Lambek Calculus (FNL) without additive constants. The paper highlights that all proofs are constructed using syntactic methods, ensuring their constructive nature. We provide insights into constructing cut-free proofs and establishing a logical relationship between CyNBL and FNL.</p>Paweł Płaczek
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2024-04-242024-04-2453224527410.18778/0138-0680.2024.06