Bulletin of the Section of Logic 2022-02-02T17:15:04+01:00 Andrzej Indrzejczak Open Journal Systems <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> On the Category of EQ-algebras 2022-02-02T17:15:04+01:00 Narges Akhlaghinia Mona Aaly Kologani Rajab Ali Borzooei Xiao Long Xin <p>In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculated coproduct and pushout in a special case. Also, we constructed a free EQ-algebra on a singleton.</p> 2021-01-20T00:00:00+01:00 Copyright (c) 2020 The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework) 2022-02-02T17:15:00+01:00 Damian E. Szmuc <p>We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined in terms of a \(p\)-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints.</p> 2021-05-27T00:00:00+02:00 Copyright (c) 2020 A Sound Interpretation of Leśniewski's Epsilon in Modal Logic KTB 2022-02-02T17:14:49+01:00 Takao Inoue <p>In this paper, we shall show that the following translation \(I^M\) from the propositional fragment \(\bf L_1\) of Leśniewski's ontology to modal logic \(\bf KTB\) is sound: for any formula \(\phi\) and \(\psi\) of \(\bf L_1\), it is defined as</p> <p>(M1) \(I^M(\phi \vee \psi) = I^M(\phi) \vee I^M(\psi)\),</p> <p>(M2) \(I^M(\neg \phi) = \neg I^M(\phi)\),</p> <p>(M3) \(I^M(\epsilon ab) = \Diamond p_a \supset p_a . \wedge . \Box p_a \supset \Box p_b .\wedge . \Diamond p_b \supset p_a\),</p> <p>where \(p_a\) and \(p_b\) are propositional variables corresponding to the name variables \(a\) and \(b\), respectively. In the last, we shall give some comments including some open problems and my conjectures.</p> 2021-11-09T00:00:00+01:00 Copyright (c) 2020 On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results 2022-02-02T17:14:56+01:00 Tarek Sayed Ahmed <p>Fix a finite ordinal \(n\geq 3\) and let \(\alpha\) be an arbitrary ordinal. Let \(\mathsf{CA}_n\) denote the class of cylindric algebras of dimension \(n\) and \(\sf RA\) denote the class of relation algebras. Let \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for the class of polyadic (equality) algebras of dimension \(\alpha\). We reprove that the class \(\mathsf{CRCA}_n\) of completely representable \(\mathsf{CA}_n\)s, and the class \(\sf CRRA\) of completely representable \(\mathsf{RA}\)s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \(\sf V\) between polyadic algebras of dimension \(n\) and diagonal free \(\mathsf{CA}_n\)s. We show that that the class of completely and strongly representable algebras in \(\sf V\) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \(\sf CRRA\) is not closed under \(\equiv_{\infty,\omega}\). In contrast, we show that given \(\alpha\geq \omega\), and an atomic \(\mathfrak{A}\in \mathsf{PEA}_{\alpha}\), then for any \(n&lt;\omega\), \(\mathfrak{Nr}_n\mathfrak{A}\) is a completely representable \(\mathsf{PEA}_n\). We show that for any \(\alpha\geq \omega\), the class of completely representable algebras in certain reducts of \(\mathsf{PA}_{\alpha}\)s, that happen to be varieties, is elementary. We show that for \(\alpha\geq \omega\), the the class of polyadic-cylindric algebras dimension \(\alpha\), introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension \(n\) and \(\mathsf{PA}_{\alpha}\)s for \(\alpha\) an infinite ordinal, proving negative results for the first and positive ones for the second.</p> 2021-07-21T00:00:00+02:00 Copyright (c) 2020 An Arithmetically Complete Predicate Modal Logic 2022-02-02T17:14:53+01:00 Yunge Hao George Tourlakis <p>This paper investigates a first-order extension of GL called \(\textup{ML}^3\). We outline briefly the history that led to \(\textup{ML}^3\), its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness (with respect to finite reverse well-founded Kripke models) is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that \(\textup{ML}^3\) is arithmetically complete. As expanded below, \(\textup{ML}^3\) is a first-order modal logic that along with its built-in ability to simulate general classical first-order provability―"\(\Box\)" simulating the the informal classical "\(\vdash\)"―is also arithmetically complete in the Solovay sense.</p> 2021-08-23T00:00:00+02:00 Copyright (c) 2020