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Deductive Systems in Traditional and Modern Logic • Urszula Wybraniec-Skardowska and Alex Citkin Deductive Systems in Traditional and Modern Logic Edited by Urszula Wybraniec-Skardowska and Alex Citkin Printed Edition of the Special Issue Published in Axioms www.mdpi.com/journal/axioms Deductive Systems in Traditional and Modern Logic Deductive Systems in Traditional and Modern Logic Editors Alex Citkin Urszula Wybraniec-Skardowska MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editors Alex Citkin Urszula Wybraniec-Skardowska Metropolitan Telecommunications Cardinal Stefan Wyszynski´ USA University in Warsaw, Department of Philosophy Poland Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Axioms (ISSN 2075-1680) (available at: http://www.mdpi.com/journal/axioms/special issues/deductive systems). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03943-358-2 (Pbk) ISBN 978-3-03943-359-9 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editors .............................................. vii Alex Citkin and Urszula Wybraniec-Skardowska Deductive Systems in Traditional and Modern Logic Reprinted from: Axioms 2020, 9, 108, doi:10.3390/axioms9030108 ................... 1 Piotr Kulicki Aristotle’s Syllogistic as a Deductive System Reprinted from: Axioms 2020, 9, 56, doi:10.3390/axioms9020056 ................... 5 Peter Simons Term Logic Reprinted from: Axioms 2020, 9, 18, doi:10.3390/axioms9010018 ................... 21 J.-Mart´ın Castro-Manzano Distribution Tableaux, Distribution Models Reprinted from: Axioms 2020, 9, 41, doi:10.3390/axioms9020041 ................... 31 Eugeniusz Wojciechowski The Zahl-Anzahl Distinction in Gottlob Frege: Arithmetic of Natural Numbers with Anzahl as a Primitive Term Reprinted from: Axioms 2020, 9, 6, doi:10.3390/axioms9010006 .................... 41 Valentin Goranko Hybrid Deduction–Refutation Systems Reprinted from: Axioms 2019, 8, 118, doi:10.3390/axioms8040118 ................... 49 Krystyna Mruczek-Nasieniewska and Marek Nasieniewski A Kotas-Style Characterisation of Minimal Discussive Logic Reprinted from: Axioms 2019, 8, 108, doi:10.3390/axioms8040108 ................... 69 Janusz Ciuciura A Note on Fernandez–Coniglio’s´ Hierarchy of Paraconsistent Systems Reprinted from: Axioms 2020, 9, 35, doi:10.3390/axioms9020035 ................... 87 Alex Citkin Deductive Systems with Multiple-Conclusion Rules and the Disjunction Property Reprinted from: Axioms 2019, 8, 100, doi:10.3390/axioms8030100 ................... 99 Dariusz Surowik Minimal Systems of Temporal Logic Reprinted from: Axioms 2020, 9, 67, doi:10.3390/axioms9020067 ...................123 Joanna Golinska-Pilarek ´ and Magdalena Welle Deduction in Non-Fregean Propositional Logic SCI Reprinted from: Axioms 2019, 8, 115, doi:10.3390/axioms8040115 ...................151 Sopo Pkhakadze and Hans Tompits Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic Reprinted from: Axioms 2020, 9, 84, doi:10.3390/axioms9030084 ...................171 v Henrique Antunes,Walter Carnielli, Andreas Kapsner and Abilio Rodrigues Kripke-Style Models for Logics of Evidence and Truth Reprinted from: Axioms 2020, 9, 100, doi:10.3390/axioms9030100 ...................201 Andrzej Malec Deontic Logics as Axiomatic Extensions of First-Order Predicate Logic: An Approach Inspired by Wolniewicz’s Formal Ontology of Situations Reprinted from: Axioms 2019, 8, 109, doi:10.3390/axioms8040109 ...................217 Dorota Leszczynska-Jasion ´ and Szymon Chlebowski Synthetic Tableaux with Unrestricted Cut for First-Order Theories Reprinted from: Axioms 2019, 8, 133, doi:10.3390/axioms8040133 ...................231 Urszula Wybraniec-Skardowska On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers Reprinted from: Axioms 2019, 8, 103, doi:10.3390/axioms8030103 ...................257 Jean-Pierre Descl´es and Anca Christine Pascu Logic of Typical and Atypical Instances of a Concept—A Mathematical Model Reprinted from: Axioms 2019, 8, 104, doi:10.3390/axioms8030104 ...................271 Alfredo Roque Freire Review of “The Significance of the New Logic” Willard Van Orman Quine. Edited and Translated by Walter Carnielli, Frederique Janssen-Lauret, and William Pickering. Cambridge University Press, Cambridge, UK, 2018, pp. 1–200. ISBN-10: 1107179025 ISBN-13: 978-1107179028 Reprinted from: Axioms 2019, 8, 64, doi:10.3390/axioms8020064 ...................285 vi About the Editors Alex Citkin spent 25 years of his career as a research fellow of Uzhgorod State University (Ukraine) studying non-classical and algebraic logics, as well as logical methods of pattern recognition. He was among the pioneers investigating the admissibility of inference rules and structural completeness of propositional logics. After relocating in 1994 to the United States, he was made CIO of Metropolitan Telecommunications (New York, USA). He has authored over 70 research papers in logic, universal algebra, group theory and informatics. Citkin is still actively involved in the research in propositional, modal and algebraic logics. Urszula Wybraniec-Skardowska is a retired professor of logic but is still actively working both scientifically and organizationally. She was named a Prof. of Humanities Science by the President of Poland in 1992. For many years she was a full professor of logic at Opole University and a co-founder and co-chairperson of the Group of Logic, Language and Information established there. Wybraniec-Skardowska is affiliated as a professor at Cardinal Stefan Wyszynski´ University in Warsaw. Her interdisciplinary research interests include: logic, philosophy, logic and philosophy of language, logical theory of communication, formal linguistics, information sciences and mathematics. She was a visiting professor at many outstanding universities. She is also a member of many Polish and international scientific associations, including: The Association for Symbolic Logic, The European Association for Logic, Language and Information, The Association of Logic and Philosophy of Science, Polish Society of Philosophy, Polish Society of Mathematics, Polish Association for Semiotic Studies. She is the author of about 130 publications. Her book “Theory of Language Syntax. Categorial Approach” was awarded for scientific research by the Polish Ministry of Education (1992). She is a recipient of many national awards. vii axioms Editorial Deductive Systems in Traditional and Modern Logic Alex Citkin 1,* and Urszula Wybraniec-Skardowska 2,* 1 Metropolitan Telecommunications, New York, NY 10041, USA 2 Institute of Philosophy, Cardinal Stefan Wyszynski University in Warsaw, Dewajtis 5, 01-815 Warsaw, Poland * Correspondence: [email protected] (A.C.); [email protected] (U.W.-S.) Received: 3 September 2020; Accepted: 9 September 2020; Published: 13 September 2020 Since its inception, logic has studied the acceptable rules of reasoning, the rules that allow us to pass from certain statements, serving as premises or assumptions, to a statement taken as a conclusion. The first kinds of such rules were distilled by Aristotle and are known as moduses. Stoics ramified Aristotle’s system, and for centuries, the syllogistic remained the main tool for logical deduction. With the birth of formal logic, new types of deduction emerged, and to support this new kind of inference, the deductive systems were used. Since then, the deductive systems have been at the heart of logical investigations. In one form or the other, they are used in all branches of logic. Contemporary understanding of science, as a theory of a high degree of exactness, requires treating it as a deductive theory (a deductive system). Generally speaking, such a theory (system) is a set of its sentential expressions which are derivable (deducible) from some expressions of the set selected as axioms, by means of deduction (inference) rules. The expressions obtained as a result of derivation from a given set of expressions are consequences; that is, they have a proof. The principal feature of deductive systems (theories) is the deducibility or provability of their theorems. From a very general point of view, there are two methods of deduction: (a) the axiomatic method (Hilbert style method) and (b) the natural deduction method (Ja´skowski–Słupecki–Borkowski, Gentzen or semantic tableaux). Method (b) leads to natural deduction systems, while the most often used method (a) leads to presentation (or characterization) of logical and mathematical theories as axiomatic deductive systems. Methods (a) and (b) are used in different scientific disciplines, such as physics, chemistry, sociology, philosophical and psychological sciences, information sciences, discursive sciences, computer science, and some technical sciences. Deductive sciences have not always been built explicitly
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