Universal Logic and the Geography of Thought – Reflections on logical pluralism in the light of culture Tzu-Keng Fu Dissertation zur Erlangung des Grades eines Doktors der Ingenieurwissenschaften –Dr.-Ing.– Vorgelegt im Fachbereich 3 (Mathematik und Informatik) der Universit¨at Bremen Bremen, Dezember, 2012 PhD Committee Supervisor: Dr. Oliver Kutz (Spatial Cognition Re- search Center (SFB/TR 8), University of Bremen, Germany) External Referee : Prof. Dr. Jean-Yves B´eziau (Depart- ment of Philosophy, University of Rio de Janeiro, Brazil) Committee Member : Prof. Dr. Till Mossakowski (German re- search center for artificial intelligence (DFKI), Germany) Committee Member : Prof. Dr. Frieder Nake (Fachbereich 3, Faculty of Mathematics and Informatics, the University of Bremen, Germany) Committee Member : Dr. Mehul Bhatt (Cognitive Systems (CoSy) group, Faculty of Mathematics and Informatics, University of Bremen, Germany) Committee Member : Minqian Huang (Cognitive Systems (CoSy) group, Faculty of Mathematics and Informatics, University of Bremen, Germany) Sponsoring Institution The Chiang Ching-kuo Foundation for International Scholarly Exchange, Taipei, Taiwan. To my Parents. 獻給我的 父母。 Acknowledgment I owe many thanks to many people for their help and encouragement, without which I may never have finished this dissertation, and their constructive criticism, without which I would certainly have finished too soon. I would like to thank Dr. Oliver Kutz, Prof. Dr. Till Mossakowski, and Prof. Dr. Jean- Yves B´eziau for assisting me with this dissertation. Warmest and sincere thanks are also due to Prof. Dale Jacquette of the Philosophy Department, the University of Bern, Switzerland, and to Prof. Dr. Stephen Bloom of the Computer Science Department, Stevens Institute of Technology, Hoboken, New Jersey, USA, who provided me with valuable original typescripts on abstract logic. For their patience, I thank my family, and especially my parents, Mrs. Mei-Kuei Tseng and Mr. Yu-Lan Fu. I thank my wife Mrs. Jung-Ying Fang for her love and inspirational conversations on cross-cultural psychology. Fi- nally, I thank the Chiang Ching-kuo Foundation for International Scholarly Exchange for having financed my PhD research between 2008 and 2010. Tzu-Keng Fu Contents 1 Introduction 3 1.1 UniversalLogic .......................... 4 1.1.1 Bourbakism ........................ 5 1.1.2 LogicalStructuralism. 7 1.2 LogicTranslation ......................... 8 1.2.1 FromSub-logictoLogicTranslation. 9 1.2.2 TheSensesofLogicTranslation . 10 1.3 PluralisminLogic ........................ 12 1.4 TheGeographyofThought . 15 1.5 Publications............................ 16 2 On Universal Logic 19 2.1 Introduction............................ 19 2.2 AlgebraicStructuresandUniversalLogic . 20 2.2.1 The Relationship between Logic and Algebra . 21 2.2.2 Three Traditionsin AlgebraicLogic . 24 2.2.3 The Influence of the Development of Abstract Logic on UniversalLogic . 27 2.3 Neo-Bourbakism as a Conceptual Approach . 29 2.4 MainStrategiesandTechniques . 34 2.4.1 Strategies ......................... 35 2.4.2 MainTechniques . 37 2.4.3 Logical Two-Valued Semantics and Algebraic Valued Semantics ......................... 42 2.5 Conclusion............................. 44 3 LogicTranslationandParadox 47 3.1 Introduction............................ 47 1 3.2 TranslationParadox . 48 3.2.1 TheInclusionRelation . 48 3.2.2 TheTranslationRelation. 50 3.3 DevelopmentofLogicTranslation . 52 3.3.1 Consequence Relations and Logic Translation . 53 3.4 The Systematic Discussion of Logic Translations . 54 3.4.1 History: 1968–1933–1929–1925 . 55 3.4.2 History: 1988–1973–1971– . 57 3.4.3 History:2006–1990– . 59 3.5 LogicTranslation andtheDevianceofLogics. 61 3.5.1 From Expansion/Reduct Cases to Deviant Cases . 62 3.5.2 DevianceofLogics . 64 3.6 Conclusion............................. 66 4 Pluralism in Logic 69 4.1 Introduction............................ 69 4.2 Truth-PreservationandGTT . 70 4.2.1 TheClassicalcaseandGTT . 72 4.2.2 TheRelevantCaseandGTT . 74 4.2.3 TheConstructivecaseandGTT. 77 4.2.4 AUnifiedModelofNegation. 80 4.3 BRLP-Plurality . 83 4.3.1 Plurality.......................... 83 4.3.2 From a Strong Sense of Plurality to Constructing BRLP 85 4.4 TheScopeofMonisminBRLP . 87 4.4.1 FromBRLPtoMonisminLogic. 88 4.4.2 A Discussion of Priest’s BRLP-Challenge . 88 4.4.3 TheComplementofPlurality . 90 4.4.4 The Opposite-BRLP Version of Monism (i) . 91 4.4.5 The Opposite-BRLP Version of Monism (ii) . 94 4.4.6 The Opposite-BRLP Version of Monism (iii) . 95 4.4.7 FromBRLPtoRelativisminLogic? . 97 4.5 Conclusion.............................101 5 TheGeographyofCognitiveModes 103 5.1 Introduction............................103 5.2 TheModesofCognitiveProcessing . 105 5.3 CognitiveInstrumentalisminLogic . 106 5.3.1 TheCognitiveToolkit . .106 5.3.2 The Dialectical logic mode and ‘Principles’? . 108 5.4 Cross-Cultural and Interdisciplinary Notions oF Logic . .112 5.4.1 RevisionoFPluralisminLogic . 114 5.4.2 Some Cultural–psychobiological Perspectives on Mod- ificationoFCognition . .116 5.5 A Universal Logic System with Dialecticism . 119 5.5.1 From Fuzzy Logic to a Universal Logic System . 121 5.5.2 C-UniLogasaMeta-theory . 122 5.5.3 Dialecticism . .123 5.6 Conclusion.............................125 6 SummaryandFurtherResearch 127 6.1 Summary .............................127 6.2 FutureResearch. .136 AAChineseUniversalLogicSystem 167 A.1 Methodology ...........................168 A.2 FlexibleTruthValue . .168 A.3 Flexible Proposition Connectives Operation . 169 A.4 FlexibleQuantifiers . .172 A.5 Establishment oF Flexible Reasoning Models . 174 Abstract The aim of this dissertation is to provide an analysis for those involved and interested in the interdisciplinary study of logic, particularly Universal Logic. While continuing to remain aware of the importance of the central issues of logic, we hope that the factor of “culture” is also given serious consideration. Universal Logic provides a general theory of logic to study the most general and abstract properties of the various possible logics. As well as elucidating the basic knowledge and necessary definitions, we would especially like to ad- dress the problems of motivation concerning logical investigations in different cultures. First of all, I begin by considering Universal Logic as understood by Jean- Yves B´eziau, and examine the basic ideas underlying the Universal Logic proJect. The basic approach, as originally employed by Universal Logicians, is introduced, after which the relationship between algebras and logics at an abstract level is discussed, i.e., Universal Algebra and Universal Logic. Sec- ondly, I focus on a discussion of the “translation paradox”, which will enable readers to become more familiar with the new subJect of logical translation, and subsequently comprehensively summarize its development in the litera- ture. Besides helping readers to become more acquainted with the concept of logical translation, the discussion here will also attempt to formulate a new direction in support of logical pluralism as identified by Ruldof Carnap (1934), JC Beall and Greg Restall (2005), respectively. Thirdly, I provide a discussion of logical pluralism. Logical pluralism can be traced back to the principle of tolerance raised by Ruldof Carnap (1934), and readers will gain a comprehensive understanding of this concept from the discussion. More- over, an attempt will be made to clarify the real and important issues in the contemporary debate between pluralism and monism within the field of logic in general. Fourthly, I study the phenomena of cultural-difference as related to the geography of thought. Two general systems in the geogra- 1 2 phy of thought are distinguished, which we here call thought-analytic and thought-holistic. They are proposed to analyze and challenge the universality assumption regarding cognitive processes. People from different cultures and backgrounds have many differences in diverse areas, and these differences, if taken for granted, have proven particularly problematic in understanding logical thinking across cultures. Interestingly, the universality of cognitive processes has been challenged, especially by Richard Nisbett’s research in cul- tural psychology. With respect to these concepts, C-UniLog (appendix A) can also be considered in relation to empirical evidence obtained by Richard Nisbett et al. In the final stage of this dissertation, I will propose an in- terpretation of the concept of logical translation, i.e., translations between formal logical mode (as cognitive processes in the case of westerners) and dialectical logical mode (as cognitive processes in the case of Asians). From this, I will formulate a new interpretation of the principle of tolerance, as well as of logical pluralism. Chapter 1 Introduction Contents 1.1 Universal Logic . 4 1.2 Logic Translation . 8 1.3 Pluralism in Logic . 12 1.4 The Geography of Thought . 15 1.5 Publications . 16 Tarski’s account of the concept of logical consequence has gained great importance in modern mathematics, and moreover has founded the role of logic in scientific studies. Universal Logic as a general theory of logics is the view that has followed the most rudimentary stage of Tarski’s notion for logical consequence. The conceptual analysis of logic is what the Universal Logic project is about. There are many questions that one might ask about Tarski-oriented con- tributions. One – perhaps the most obvious – is why one might suppose the concepts of logical consequence to be