Logic in Reality Joseph E. Brenner Logic in Reality Joseph E. Brenner International Center for Transdisciplinary Research, Paris, France Postal address: P.O. Box 235 CH-1865 Les Diablerets Switzerland ISBN 978-1-4020-8374-7 e-ISBN 978-1-4020-0000-0 Library of Congress Control Number: 2008926590 © 2008 Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com CONTENTS Foreword……...............………………………………………………………….. xi Acknowledgments……………………………………………………………….. xv INTRODUCTION...............................................................................................xvii 1. REALITY AND LOGIC .............................................................................xvii 2. THE OBJECTIVE AND PLAN OF “LOGIC IN REALITY”..................... xix 1 LOGIC IN REALITY (LIR) AS A FORMAL LOGIC.................................. 1 1.1 INITIAL AXIOMATIZATION: THE FUNDAMENTAL POSTULATE.. 1 1.2 THE REAL AND REALITY ....................................................................... 6 1.2.1 Levels of Reality ................................................................................... 7 1.2.1.1 The Ontological Approach ............................................................ 8 1.2.1.2 The Physical Science Approach .................................................... 8 1.2.2 Contradiction, Counteraction or “Countervalence”.............................. 9 1.2.3 The Senses of Reduction..................................................................... 10 1.3 LIR VERSUS STANDARD LOGICS: DEDUCTION.............................. 11 1.3.1 Bivalence: Logical Truth, Logical Constants and Logical Consequence ................................................................................................ 12 1.3.2 Semantics ............................................................................................ 16 1.3.3 First-Order Predicate Logic (FOL) ..................................................... 19 1.4 NON-CLASSICAL LOGICS..................................................................... 20 1.4.1 Intuitionist, Paraconsistent and Paracomplete Logics ........................ 20 1.4.1.1 Intuitionist Logic and Paracompleteness..................................... 20 1.4.1.2 Paraconsistent Logic.................................................................... 21 1.4.2 Many-Valued and Fuzzy Logics......................................................... 22 1.4.3 Modal Logic........................................................................................ 24 1.4.4 Relevance and Natural Logics ............................................................ 26 1.4.4.1 Relevance (or Relevant) Logics .................................................. 26 1.4.4.2 Natural Logic............................................................................... 27 1.4.5 The Metalogic, Imaginary Logic and Empirical Logic of Vasiliyev.. 28 1.5 INDUCTIVE LOGIC AND ABDUCTIVE LOGIC: PROBABILITY ..... 30 1.5.1 Inference and Abductive Logic........................................................... 33 1.6 QUANTUM LOGIC................................................................................... 34 1.6.1 The Quantum Formalism of Aerts ...................................................... 35 1.7 THE FORMAL AXIOMATIZATION OF LIR......................................... 36 2 LIR AS A FORMAL SYSTEM....................................................................... 41 2.1 THE NON-CLASSICAL CALCULUS OF LIR: IMPLICATION............ 41 2.2 TRUTH VALUES, CONTRADICTION AND REALITY VALUES ...... 43 2.2.1 Other Theories of Truth ...................................................................... 46 v vi CONTENTS 2.2.2 Verificationism ................................................................................... 47 2.3 IMPLICATION BETWEEN THE LIMITS............................................... 49 2.3.1 The Transfinite.................................................................................... 50 2.3.2 Ortho-deductions................................................................................. 52 2.3.3 Para-deductions................................................................................... 53 2.3.4 Operational Aspects of the LIR Calculus: Logical Necessity ............ 54 2.4 CONJUNCTION AND DISJUNCTION ................................................... 56 2.4.1 A Classification Table for Connectives .............................................. 58 2.4.2 The Calculus of Events and Relations ................................................ 60 3 LIR AS A FORMAL ONTOLOGY ............................................................... 63 3.1 REALISM AND FORMAL ONTOLOGIES............................................. 63 3.2 THE LIR ONTOLOGICAL PREDICATES: DUALITY.......................... 65 3.3 THE DOMAIN OF ENTITIES: LEVELS OF REALITY......................... 66 3.3.1 Complexity Versus Levels of Reality................................................. 67 3.4 LIR AS AN INTERPRETED FORMAL SYSTEM .................................. 67 3.5 THREE CRITICAL CONCEPTS .............................................................. 68 3.5.1 Dynamisms.......................................................................................... 68 3.5.2 Processes ............................................................................................. 69 3.5.3 Properties ............................................................................................ 69 3.5.3.1 Intrinsic Versus Extrinsic Properties: Relations.......................... 72 3.6 SOME METALOGICAL CONSIDERATIONS ....................................... 73 3.7 THE LOGIC OF BEING............................................................................ 75 3.7.1 Abstract, Non-real, Non-existent and Non-spatio-temporal Objects.. 78 4 THE CATEGORIES OF LIR......................................................................... 81 4.1 THE DEFINITION AND FUNCTION OF ONTOLOGY ........................ 81 4.1.1 Formal and Material Categories.......................................................... 82 4.2 THE ESTABLISHMENT AND CONSTRUCTION OF CATEGORIES. 84 4.3 THE PHYSICS OF REALITY: THE FUNDAMENTAL DUALITIES ... 85 4.4 THE CATEGORY OF ENERGY .............................................................. 87 4.4.1 The Duality of Energy ........................................................................ 88 4.4.1.1 Intensity and Extensity ................................................................ 90 4.4.1.2 Identity and Diversity: Homogeneity and Heterogeneity ........... 92 4.4.1.3 Actuality and Potentiality............................................................ 93 4.4.1.4 Locality and Globality................................................................. 95 4.4.1.5 Energy Is Dual and Antagonistic ................................................ 96 4.4.2 The Fundamental Principle of LIR ..................................................... 97 4.5 THE CATEGORY AND SUB-CATEGORIES OF DYNAMIC OPPOSITION................................................................................................... 99 4.5.1 Non-separability................................................................................ 100 4.5.2 Is There a Mathematical Physics of LIR?......................................... 102 4.6 THE CATEGORY OF PROCESS: CHANGE ........................................ 104 4.6.1 Three Current Views of Process ....................................................... 104 CONTENTS vii 4.7 THE CATEGORY OF T-STATES .......................................................... 106 4.7.1 The Duality of Quantum Spin........................................................... 108 4.8 THE CATEGORIES OF SUBJECT, OBJECT AND SUBJECT-OBJECT........................................................................109 4.8.1 Self and Other: Self-Reference ......................................................... 112 4.9 LIR AS A FORMAL ONTOLOGY: NEO AND THE CATEGORY- AXIOM FIT.................................................................................................... 113 4.9.1 A Check-List of Principal Dynamic Relationships........................... 117 4.10 THE INTERPRETATION OF LIR........................................................ 119 5 THE CORE THESIS OF LIR: STRUCTURE AND EXPLANATION ... 123 5.1 THE CORE THESIS OF LIR................................................................... 124 5.2 A TWO-LEVEL FRAMEWORK OF RELATIONAL ANALYSIS....... 126 5.2.1 Mereology ......................................................................................... 128 5.2.2 Inter- and Intra-theoretic Relationships ............................................ 129 5.3 ONTOLOGY AND METAPHYSICS IN PARALLEL........................... 132 5.4 THE STRUCTURE OF REALITY IN LIR ............................................. 135 5.4.1 The Categorial Structure of Reality in LIR....................................... 136 5.4.1.1 Morphisms and Functors ........................................................... 136 5.4.1.2 Ontological Links ...................................................................... 139 5.4.2 The Structure