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 Logic0138-0680Extended BCK-Ideal Based on Single-Valued Neutrosophic Hyper BCK-Ideals
https://czasopisma.uni.lodz.pl/bulletin/article/view/9134
<p>This paper introduces the concept of single-valued neutrosophic hyper \(BCK\)-subalgebras as a generalization and alternative of hyper \(BCK\)-algebras and on any given nonempty set constructs at least one single-valued neutrosophic hyper \(BCK\)-subalgebra and one a single-valued neutrosophic hyper \(BCK\)-ideal. In this study level subsets play the main role in the connection between singlevalued neutrosophic hyper \(BCK\)-subalgebras and hyper \(BCK\)-subalgebras and the connection between single-valued neutrosophic hyper \(BCK\)-ideals and hyper \(BCK\)-ideals. The congruence and (strongly) regular equivalence relations are the important tools for connecting hyperstructures and structures, so the major contribution of this study is to apply and introduce a (strongly) regular relation on hyper \(BCK\)-algebras and to investigate their categorical properties (quasi commutative diagram) via single-valued neutrosophic hyper \(BCK\)-ideals. Indeed, by using the single-valued neutrosophic hyper \(BCK\)-ideals, we define a congruence relation on (weak commutative) hyper \(BCK\)-algebras that under some conditions is strongly regular and the quotient of any (single-valued neutrosophic)hyper \(BCK\)-(sub)algebra via this relation is a (single-valued neutrosophic)(hyper \(BCK\)-subalgebra) \(BCK\)-(sub)algebra.</p>Mohammad Hamidi
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2023-08-102023-08-1052441144010.18778/0138-0680.2023.20Fundamental Relation on HvBE-Algebras
https://czasopisma.uni.lodz.pl/bulletin/article/view/13229
<p>In this paper, we are going to introduce a fundamental relation on \(H_{v}BE\)-algebra and investigate some of properties, also construct new \((H_{v})BE\)-algebras via this relation. We show that quotient of any \(H_{v}BE\)-algebra via a regular regulation is an \(H_{v}BE\)-algebra and this quotient, via any strongly relation is a \(BE\)-algebra. Furthermore, we investigate that under what conditions some relations on \(H_{v}BE\)-algebra are transitive relations.</p>Farzad IranmaneshMansour GhadiriArsham Borumand Saeid
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2023-08-092023-08-0952444145810.18778/0138-0680.2023.10Cut Elimination for Extended Sequent Calculi
https://czasopisma.uni.lodz.pl/bulletin/article/view/13263
<div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>We present a syntactical cut-elimination proof for an extended sequent calculus covering the classical modal logics in the \(\mathsf{K}\), \(\mathsf{D}\), \(\mathsf{T}\), \(\mathsf{K4}\), \(\mathsf{D4}\) and \(\mathsf{S4}\) spectrum. We design the systems uniformly since they all share the same set of rules. Different logics are obtained by “tuning” a single parameter, namely a constraint on the applicability of the cut rule and on the (left and right, respectively) rules for \(\Box\) and \(\Diamond\). Starting points for this research are 2-sequents and indexed-based calculi (sequents and tableaux). By extending and modifying existing proposals, we show how to achieve a syntactical proof of the cut-elimination theorem that is as close as possible to the one for first-order classical logic. In doing this, we implicitly show how small is the proof-theoretical distance between classical logic and the systems under consideration.</p> </div> </div> </div>Simone MartiniAndrea MasiniMargherita Zorzi
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2023-09-252023-09-2552445949510.18778/0138-0680.2023.22On Implicative and Positive Implicative GE Algebras
https://czasopisma.uni.lodz.pl/bulletin/article/view/14019
<p>GE algebras (generalized exchange algebras), transitive GE algebras (tGE algebras, for short) and aGE algebras (that is, GE algebras<br />verifying the antisymmetry) are a generalization of Hilbert algebras. Here some properties and characterizations of these algebras are investigated. Connections between GE algebras and other classes of algebras of logic are studied. The implicative and positive implicative properties are discussed. It is shown that the class of positive implicative GE algebras (resp. the class of implicative aGE algebras) coincides with the class of generalized Tarski algebras (resp. the class of Tarski algebras). It is proved that for any aGE algebra the property of implicativity is equivalent to the commutative property. Moreover, several examples to illustrate the results are given. Finally, the interrelationships between some classes of implicative and positive implicative algebras are presented.</p>Andrzej Walendziak
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2023-09-252023-09-2552449751510.18778/0138-0680.2023.21A Category of Ordered Algebras Equivalent to the Category of Multialgebras
https://czasopisma.uni.lodz.pl/bulletin/article/view/14375
<p>It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (\(\textit{CABA}\)s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \(\textbf{Set}\) and the category of \(\textit{CABA}\)s.</p> <p>We modify this result by taking multialgebras over a signature \(\Sigma\), specifically those whose non-deterministic operations cannot return the empty-set, to \(\textit{CABA}\)s with their zero element removed (which we call a \(\textit{bottomless Boolean algebra}\)) equipped with a structure of \(\Sigma\)-algebra compatible with its order (that we call \(\textit{ord-algebras}\)). Conversely, an ord-algebra over \(\Sigma\) is taken to its set of atomic elements equipped with a structure of multialgebra over \(\Sigma\). This leads to an equivalence between the category of \(\Sigma\)-multialgebras and the category of ord-algebras over \(\Sigma\).</p> <p>The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.</p>Marcelo Esteban ConiglioGuilherme Vicentin de Toledo
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2023-08-162023-08-1652451755010.18778/0138-0680.2023.23The Cardinal Squaring Principle and an Alternative Axiomatization of NFU
https://czasopisma.uni.lodz.pl/bulletin/article/view/14771
<p>In this paper, we rigorously prove the existence of type-level ordered pairs in Quine’s New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (<strong>NFU</strong> + <strong>Inf</strong> + <strong>AC</strong>). The proof uses the cardinal squaring principle; more precisely, its instance for the (infinite) universe (<strong>VCSP</strong>), which is a theorem of <strong>NFU</strong> + <strong>Inf</strong> + <strong>AC</strong>. Therefore, we have a justification for proposing a new axiomatic extension of <strong>NFU</strong>, in order to obtain type-level ordered pairs almost from the beginning. This axiomatic extension is <strong>NFU</strong> + <strong>Inf</strong> + <strong>AC</strong> + <strong>VCSP</strong>, which is equivalent to <strong>NFU</strong> + <strong>Inf</strong> + <strong>AC</strong>, but easier to reason about.</p>Tin AdlešićVedran Čačić
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2023-09-282023-09-2852455158110.18778/0138-0680.2023.25