Studies in Universal Logic

Series Editor

Jean-Yves Béziau (Federal University of Rio de Janeiro and Brazilian Research Council, Rio de Janeiro, Brazil)

Editorial Board Members

Hajnal Andréka (Hungarian Academy of Sciences, Budapest, Hungary) Mark Burgin (University of California, Los Angeles, USA) Razvan˘ Diaconescu (Romanian Academy, Bucharest, Romania) Andreas Herzig (Centre National de la Recherche Scientifique, Toulouse, France) Arnold Koslow (City University of New York, USA) Jui-Lin Lee (National Formosa University, Huwei Township, Taiwan) Larissa Maksimova (Russian Academy of Sciences, Novosibirsk, Russia) Grzegorz Malinowski (University of Łód´z, ) Francesco Paoli (University of Cagliari, Italy) Darko Sarenac (Colorado State University, Fort Collins, USA) Peter Schröder-Heister (University Tübingen, Germany) Vladimir Vasyukov (Russian Academy of Sciences, Moscow, Russia)

This series is devoted to the universal approach to logic and the development of a general theory of logics. It covers topics such as global set-ups for fundamental theorems of logic and frameworks for the study of logics, in particular logical matrices, Kripke structures, combination of logics, categorical logic, abstract proof theory, consequence operators, and algebraic logic. It includes also books with historical and philosophical discussions about the nature and scope of logic. Three types of books will appear in the series: graduate textbooks, research monographs, and volumes with contributed papers.

More information about this series at http://www.springer.com/series/7391 Ángel Garrido • Urszula Wybraniec-Skardowska Editors

The Lvov- School. Past and Present Editors Ángel Garrido Urszula Wybraniec-Skardowska UNED Cardinal Stefan Wyszynski´ University Department of Fundamental in Warsaw Madrid, Spain Warsaw, Poland

ISSN 2297-0282 ISSN 2297-0290 (electronic) Studies in Universal Logic ISBN 978-3-319-65429-4 ISBN 978-3-319-65430-0 (eBook) https://doi.org/10.1007/978-3-319-65430-0

Library of Congress Control Number: 2018943846

Mathematics Subject Classification (2010): 01A72, 03-03

© Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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This book is published under the trade name Birkhäuser, www.birkhauser-science.com by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface

This book is about the Lvov-Warsaw School. I did not “order” this book, but I am very glad to have received this proposal and the book to be published in a book series I have created inspired by this school. The Lvov-Warsaw School is one of the most important schools in the history of logic but is still not very well known outside of Poland and a circle of aficionados. This book with contributions about the main members of this school and their works will certainly help to fill the gap, reinforcing the already existing attraction and promoting new interests for this school. I would like to thank the editors of this book, Urszula and Angel, for their considerable efforts to gather all the papers included in this big volume, as well as all the contributors of this book and Birkhäuser staff in Basel.

Editor-in-Chief Jean-Yves Beziau Studies in Universal Logic Rio de Janeiro, Brazil December 19, 2017

v Contents

Part I Twardowski’s School: The Period of Crystallization of LWS

Introduction. The School: Its Genesis, Development and Significance ..... 3 Urszula Wybraniec-Skardowska 1 A Short History and Influence of the School ...... 4 2 On the Structure and Contents of the Anthology ...... 13 References...... 13

Kazimierz Twardowski: A Great Teacher of Great Philosophers ...... 15 Anna Bro˙zek 1 Life...... 15 2 Personality ...... 16 3 Writings...... 17 4 Views...... 17 5 Philosophical School ...... 30 6 Conclusions...... 31 KazimierzTwardowski’sWorksMentionedinthePaper...... 32

Jan Łukasiewicz: A Creator of New Ideas in Logic and a Reinterpreter of Its History ...... 33 Jacek Jadacki 1 Personality ...... 33 2 Life...... 34 3 Works...... 35 4 Metaphysics ...... 35 5 Logic...... 37 6 Methodology ...... 42 7 Teacher ...... 44 JanŁukasiewicz’sWorksMentionedinthePaper...... 45

vii viii Contents

Kazimierz Ajdukiewicz: The Cognitive Role of Language ...... 47 Anna Jedynak 1 Life...... 47 2 MainPublications...... 48 3 ScientificActivity...... 48 4 DisciplesandContinuators...... 60 References...... 60

On Ajdukiewicz’s Project of the Semantic Theory of Knowledge ...... 65 Adam Olech 1 Introduction ...... 66 2 Presentation of the Meta-Epistemological Project of the Semantic TheoryofKnowledgeandAnalysisThereof...... 70 3 Conclusion...... 87 References...... 88

Categorial Grammars and Their Logics ...... 91 Wojciech Buszkowski 1 Introduction ...... 91 2 BasicCategorialGrammars...... 93 3 LambekCalculus...... 98 References...... 112

Material Implication and Conversational Implicature in Lvov-Warsaw School ...... 117 Rafal Urbaniak and Michał Tomasz Godziszewski 1 Ajdukiewicz’s Equivalence Argument ...... 117 2 Jackson’sArgument...... 118 3 Ajdukiewicz on Disjunction ...... 119 4 Ajdukiewicz vs. Quine on Assertibility ...... 121 5 Expressingvs.Stating...... 123 6 Ajdukiewicz on the Diagnostics of Improper Use ...... 124 7 Generalization, Weakening, Moore’s Paradox ...... 125 8 ApparentConnectionBetweenDisjuncts...... 125 9 Ajdukiewicz’s Diagnostics and Grice’s Cancellability ...... 127 10 Goła¸bonIndicativeConditionals...... 130 11 Słupecki’s Reply to Goła¸b...... 131 12 Ajdukiewicz’s Solution and Goła¸b’sProblem...... 131 References...... 132

Tadeusz Cze˙zowski ...... 133 Dariusz Łukasiewicz 1 Life...... 133 2 MainPapers...... 135 3 Views...... 135 4 Resonance...... 135 5 Pupils...... 135 6 Summary...... 136 Bibliography...... 136 Contents ix

Tadeusz Cze˙zowski’s Theory of Knowledge and Beliefs ...... 137 Dariusz Łukasiewicz 1 Introduction ...... 137 2 Modern Epistemological Individualism: Faith and Testimony inEpistemologicalFramework...... 138 3 Methodism, Particularism and Belief-Formation Process ...... 141 4 What Can Justify Beliefs: Basic Beliefs, Internalism and Externalism . . . . . 142 5Cze˙zowski’s Theory of Knowledge and Epistemological Individualism . . . . 143 6Cze˙zowski’s Internalism and Fallibilism ...... 147 7Cze˙zowski and Some Views on Belief-Formation Processes ...... 149 References...... 151

What Is Reasoning? ...... 153 Robert Kublikowski 1 Introduction ...... 153 2 The Definition of Reasoning ...... 155 3 The Conditions of Correctness of Reasoning ...... 157 4 Conclusions...... 162 References...... 162

Tadeusz Kotarbinski:´ Socrates of Warsaw ...... 165 Jacek Jadacki 1 Life...... 165 2 Writings...... 166 3 Views...... 166 4 Resonance...... 170 5 Pupils...... 173 6 Summary...... 173 Tadeusz Kotarbinski’sWorksMentionedinthePaper...... ´ 173

Agency in a Praxiological Approach ...... 175 Wojciech W. Gasparski 1 Introduction ...... 175 2 Agency in Tadeusz Kotarbinski’sApproach...... ´ 176 3 OtherApproaches...... 183 4 Conclusion...... 185 References...... 186

Zygmunt Zawirski: A Creator of New Ideas in Methodology of Science and Scientific Metaphysics ...... 189 Krzysztof Slezi´ nski´ 1 Life...... 189 2 Writings...... 191 3 Views...... 192 4 Resonance...... 196 5 Pupils...... 197 6 Summary...... 197 ZygmuntZawirski’sWorksMentionedinthePaper...... 198 x Contents

Zygmunt Zawirski’s Concept of Scientific Metaphysics ...... 201 Krzysztof Slezi´ nski´ 1 Introduction ...... 201 2 Relation Between Science and Metaphysics ...... 202 3 Synthesis of Human Knowledge and Metaphysics ...... 204 References...... 207

Stanisław Lesniewski:´ Original and Uncompromising Logical Genius ..... 209 Peter Simons 1 Life...... 209 2 PhasesofActivity...... 210 3 EarlyWritings...... 211 4 Antinomies,Classes,Parts...... 212 5 Formalization...... 213 6 ConsolidationandPublication...... 215 7 FinalYears...... 216 8 Metalogic...... 216 9 Teaching and Students ...... 217 10 Colleagues and Personality ...... 218 11 Legacy...... 218 Stanisław Lesniewski’sWorksMentionedinthePaper...... ´ 219

Izydora Dambska: ˛ The First Lady of the Twentieth-Century Polish Philosophy ...... 223 Anna Bro˙zek and Jacek Jadacki 1 Personality ...... 223 2 Life...... 224 3 Work...... 224 4 The Lvov-Warsaw School Versus Neopositivism and Linguistic Philosophy ...... 225 5 Metaphysics ...... 226 6 Semiotics...... 227 7 Methodology ...... 229 8 Axiology...... 231 9 Significance...... 233 Izydora Dambska’sWorksMentionedinthePaper...... ˛ 233

Maria Kokoszynska-Lutmanowa:´ The Great Polemist ...... 235 Filip Kawczynski´ 1 Life...... 235 2 MainPapers...... 236 3 Views...... 237 4 Pupils...... 239 5 Summary...... 240 References...... 240 Contents xi

Seweryna Łuszczewska-Romahnowa ...... 241 Roman Murawski and Jerzy Pogonowski 1 Life...... 241 2 MainWorks...... 242 3 Views...... 245 4 Resonance...... 245 5 Pupils...... 245 6 Summary...... 245 References...... 246

On Existential Dependence and Independence in the World of Thoughts and States of Affairs (with Reference to Eugenia Ginsberg-Blaustein’s and Roman Ingarden’s Analyses) ...... 249 Urszula M. Zegle˙ n´ 1 Introduction ...... 249 2 Ingarden’sDefinitionsandTheorems...... 251 3 Dependence in the Mental Sphere of Thoughts ...... 254 4 ConcludingRemarks...... 260 References...... 261

Józef I.M. Bochenski´ ...... 263 Korneliusz Policki 1 Life...... 263 2 MainResearchInterests...... 264 3 Bochenski’sViews...... ´ 265 4 Reactions...... 269 5 Criticism ...... 271 6 Summary...... 271

J.M. Bochenski’s´ Theory of Signs ...... 273 Korneliusz Policki 1 Introduction ...... 273 2 OntologicalAssumptions...... 273 3 PsychologicalAssumptions...... 274 4 TheConceptoftheSign...... 275 5 TheTheoryofSignsandAuthority...... 276 6 The Theory of Signs and ‘Philosophical Superstitions’ ...... 276 7 Conclusion...... 277 References...... 277

Jan Salamucha (1903–1944) ...... 279 Kordula Swi˛´ etorzecka 1 Beginnings...... 279 2 Education...... 279 3 Didactics...... 280 4 Priesthood ...... 281 5 Death...... 281 xii Contents

6 IntellectualFormation...... 282 SelectedPublications...... 283

Struve and Bieganski:´ Towards Modern Approach to Logic ...... 285 Roman Murawski 1 HenrykStruve...... 285 2 Władysław Bieganski...... ´ 290 3 Conclusion...... 295 References...... 296

Part II Warsaw School of Logic, Its Main Figures and Ideas: The Period of Prosperity

Łukasiewicz and His Followers in Many-Valued Logic ...... 301 Grzegorz Malinowski 1 Łukasiewicz’sWaytoMany-Valuedness...... 301 2 MatrixSemanticsforn-ValuedLogics...... 306 3 Post Logics and Słupecki’s Functional Completeness Programme ...... 309 4 AlgebraicInterpretationsofŁukasiewiczLogics...... 312 5 InSearchforInterpretationofŁukasiewiczLogics...... 315 6 On Applications and Influence of Łukasiewicz Work ...... 320 References...... 326

Tomorrow’s Sea-Battle and the Beginning of Temporal Logic ...... 329 Kazimierz Trz˛esicki 1 Introduction ...... 329 2 Language and Semantics ...... 331 3 BranchingTimeLogic...... 333 4 TemporalLogicofPossibleWorlds...... 334 5 Conclusion...... 335 References...... 336 Lesniewski´ and Mereology ...... 337 Peter Simons 1HowLesniewskiCametoMereology...... ´ 337 2Lesniewski’sUnderstandingofSetsorClasses...... ´ 339 3 Terminology...... 341 4 TheEarlySystem...... 341 5 Remarks About This System ...... 344 6 SymbolicReconstruction...... 345 7 LaterImprovements...... 347 8 Whitehead’sAlternativeMereology...... 349 9 ThePolishContinuation...... 351 10 TheHarvardVariant...... 352 11 SubsequentDevelopments...... 354 12 Prospects...... 358 References...... 358 Contents xiii

Alfred Tarski (1901–1983) ...... 361 Jan Wolenski´ Bibliography...... 370

Some Philosophical Aspects of Semantic Theory of Truth ...... 373 Jan Wolenski´ References...... 387

Tarski’s Influence on Computer Science ...... 391 Solomon Feferman References...... 402

The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper ...... 405 John Corcoran and José Miguel Sagüillo 1 Introduction ...... 406 2 Corcoran’s Awareness of the Issue in the 1960s and 1970s ...... 410 3 The Place of the Monistic-Pluralistic Distinction ...... 413 4 The Origin of the Modern form of the Monistic Framework ...... 417 5 ConcludingRemarks...... 419 References...... 421 : Auxiliary Notes on His Legacy ...... 425 Jan Zygmunt 1 Life...... 425 2 SetTheory...... 432 3 GeometryandMeasureTheory...... 436 4 DecidableandUndecidableTheories...... 439 5 SelectedWorksofTarski...... 448

Stanisław Jaskowski:´ Life and Work ...... 457 Andrzej Indrzejczak 1 Life...... 457 2 Works...... 459 3 Influence...... 462 References...... 463

Stanisław Jaskowski´ and Natural Deduction Systems ...... 465 Andrzej Indrzejczak 1 Introduction ...... 465 2 NaturalDeductioninGeneral...... 466 3Jaskowski’sResearchonND...... ´ 468 4 OtherApproachestoND...... 479 References...... 483 xiv Contents

Variations on Jaskowski’s´ Discursive Logic ...... 485 Barbara Dunin-K˛eplicz, Alina Powała, and Andrzej Szałas 1 Prelude...... 485 2Theme:Jaskowski’s´ Discursive Logic D2 ...... 487 3 Movement:BeliefBasesandBeliefStructures...... 488 4 Variations Part I: D4—A New Framework for Discursive Logics ...... 491 5 Variations Part II: Relation to Dialogue ...... 492 6 Variations Part III: Relation to Argumentation ...... 493 7 Coda...... 495 References...... 496

Czesław Lejewski: Propagator of Lvov-Warsaw Ideas Abroad ...... 499 Peter Simons 1 Life...... 499 2 MainWorks...... 500 3 Views...... 500 4 Influence, Teaching, Personality ...... 503 WorksbyCzesławLejewskiMentionedinthePaper...... 503 , Metric Spaces and Decompositions ...... 505 Robert Purdy and Jan Zygmunt 1 Introduction ...... 505 2 AShortLife...... 506 3 Metric Spaces ...... 520 4 Decomposition of Point Sets, and Their Equivalence by Decomposition . . . 529 5 Decompositions and Equivalence of Polygons in Elementary Geometry . . . . 534 6 Bibliography...... 537 7 Acknowledgments...... 547

Andrzej Mostowski: A Biographical Note ...... 551 Marcin Mostowski 1 Life...... 551 2 MainWorks...... 551 3 MainScientificAchievements...... 552 4 InfluenceofHisWorks...... 553 5 Students...... 553 6 LastWords...... 553 References...... 554

Foundations and Philosophy of Mathematics in Warsaw, the School of and Philosophy ...... 555 Marcin Mostowski 1 Introduction ...... 555 2 Some Mathematical Works in the Foundations ...... 556 3 Old School and Old Ideas ...... 562 4 The Old School and New Ideas: Computations ...... 563 5 What for Philosophy? ...... 563 6 WhoIsYourMaster?...... 564 References...... 564 Contents xv

Jerzy Słupecki (1904–1987) ...... 567 Jan Wolenski´ ReferencesandBibliography...... 572

Rejection in Łukasiewicz’s and Słupecki’s Sense ...... 575 Urszula Wybraniec-Skardowska 1 Introduction ...... 575 2 Reconstruction of Concepts of Rejection and Decidability: The Notions IntroducedandUsedbyŁukasiewicz...... 578 3 The Problem of Decidability of Aristotle’s Syllogistics, Set by Łukasiewicz,andItsSolutionGivenbySłupecki...... 579 4 Notions of Rejection in a Deductive System and Notions of Ł-Decidability ...... 585 5 More Important Findings Concerning Ł-Decidability of Deductive Systems...... 587 6 RejectionOperation...... 591 7 RejectionOperationasaPrimitiveNotion...... 593 References...... 594

Bolesław Sobocinski:´ The Ace of the Second Generation of the LWS ...... 599 Kordula Swi˛´ etorzecka 1 Life...... 600 2 NationalService...... 602 3 AcademicCareer...... 604 4 InterestsandAchievements...... 607 5 Sobocinski´ Personality ...... 609 6 Selected Publications by Bolesław Sobocinski...... ´ 610 References...... 612

Bolesław Sobocinski´ on Universals ...... 615 Kordula Swi˛´ etorzecka and Marek Porwolik 1Lesniewski’s´ Nominalist Argumentation in Sobocinski’s´ Version ...... 617 2 Sobocinski’sMetaconceptualism...... ´ 624 References...... 631

Many-Valued Logics in the Iberian Peninsula ...... 633 Angel Garrido 1 Introduction ...... 634 2 Many-Valued Logics and the Lvov-Warsaw School ...... 636 3 Reception of Many-valued Logics and Fuzzy Logic in the Iberian Peninsula...... 641 4 FinalNote...... 642 References...... 643

Ontology of Logic and Mathematics in Lvov-Warsaw School ...... 645 Roman Murawski 1 JanŁukasiewicz:(Neo)Platonism...... 646 2StanisławLesniewski:Nominalism...... ´ 648 3 AlfredTarski:Nominalism...... 650 xvi Contents

4 Tadeusz Kotarbinski:Reism...... ´ 652 5 LeonChwistek:Nominalism...... 655 6 Kazimierz Ajdukiewicz and Ontology of Mathematics ...... 656 7 Andrzej Mostowski: Nominalism, Reism, Constructivism ...... 657 8 Conclusion...... 659 References...... 660

Part III The War and Post-War Period

A View of Revival of Mathematical Logic in Warsaw, 1945–1975 ...... 665 Victor W. Marek 1 Introduction ...... 665 2 TheBeginnings...... 666 3 AndrzejMostowskiandHisStudents...... 667 4 HelenaRasiowaandHerGroup...... 668 5 AndrzejGrzegorczyk...... 669 6 Conclusions...... 671 References...... 671

Andrzej Mostowski and the Notion of a Model ...... 673 Wilfrid Hodges 1 TheEmergenceofaNewDiscipline...... 673 2 Mostowski’sWritingsinModelTheory...... 675 3 Background and Notation ...... 676 4 Some of Tarski’s Proposals ...... 677 5 ‘Theories’...... 679 6 ‘Models’...... 681 7 Thirty Years of Foundational Studies ...... 684 8 Robinson’sComplaint...... 685 9 Mostowski’sAttributionsinGeneral...... 687 10 Conclusion...... 688 References...... 689

All Quantifiers Versus the Quantifier All ...... 693 Stanisław Krajewski 1 TheUniversalQuantifierandItsDual...... 694 2 A Digression on the Range of the Familiar Quantifiers ...... 695 3ThePowerof∀ and ∃ ...... 696 4 GeneralizedQuantifiersinLogic...... 697 5 Characterizing Context-Independent Quantifiers ...... 699 6 Formalism-Free Definition of ∀-DefinableQuantifiers?...... 701 References...... 701

Helena Rasiowa (1917–1994) ...... 703 Andrzej Jankowski and Andrzej Skowron References...... 709 Contents xvii

Post Algebras in the Work of ...... 711 Ewa Orłowska 1 PlainSemi-PostAlgebras...... 712 2 PostAlgebrasofOrderm...... 715 3 Post Algebras of Order ω+ ...... 716 4 Post Algebras of Order ω + ω∗ ...... 717 References...... 719

Andrzej Grzegorczyk, a Logician Par Excellence ...... 723 Stanisław Krajewski 1 Life...... 723 2 LogicalAccomplishments...... 724 3 Views...... 726 4 Influence...... 727 References...... 728

A Mystery of Grzegorczyk’s Logic of Descriptions ...... 731 Joanna Golinska-Pilarek´ and Taneli Huuskonen 1 Introduction ...... 732 2 The Logic of Descriptions LD ...... 733 3 Properties of LD ...... 738 4 Extensions and Modifications of LD ...... 741 5 Conclusions...... 744 References...... 744

Roman Suszko: Logician and Philosopher ...... 747 Mieczysław Omyła 1 Life...... 747 2 MainPapers...... 748 3 Views...... 750 4 Influences...... 751 5 Summary...... 752

From Formal Theory of Knowledge to Non-Fregean Logic ...... 753 Mieczysław Omyła 1 DiachronicLogic...... 754 2 Non-FregeanLogic...... 757 References...... 762

Categories of First-Order Quantifiers ...... 763 Urszula Wybraniec-Skardowska 1 Introduction ...... 764 2 ProblemofQuantifiers...... 764 3 Some Intuitive Foundations of the Theory of Categorial Languages ...... 765 4 TheSolutionoftheProblemofQuantifiersof1st-Order...... 769 5 Conclusions...... 775 References...... 775 xviii Contents

The Lvov-Warsaw School: A True Mythology ...... 779 Jean-Yves Beziau 1 FromBahíaBlancatoWrocław...... 780 2 The Atopicity of the Lvov-Warsaw School ...... 787 3 The Future of the Lvov-Warsaw School ...... 798 References...... 812