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Instantial Logic INSTANTIAL LOGIC An Investigation into Reasoning with Instances W.P.M. Meyer Viol ILLC Dissertation Series 1995-11 Instant ial Logic ILLC Dissertation Series 1995-11 language andcomputation For further information about ILLC-publications, please Contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-5256090 fax: +31-20-5255101 e-mail: [email protected] Instant ial Logic An Investigation into Reasoning with Instances Instantiéle Logica Een Onderzoek naar Redeneren met Voorbeelden (Met een Samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad Vandoctor aan de Universiteit Utrecht op gezag van de rector magnificus, Prof. dr. J .A. van Ginkel ingevolge het besluit van het college van decanen in het openbaar te verdedigen op dinsdag 20 juni 1995 des voormiddags te 10.30 uur door Wilfried Peter Marie Meyer Viol Promotoren: Prof. dr. D.J.N. van Eijck Prof. dr. J.F.A.K. van Benthem CIP—GEGEVENSKONINKLIJKE BIBLIOTHEEK, DEN HAAG Meyer Viol, Wilfried Peter Marie Instantial Logic: an investigation into reasoning with instances / Wilfried Peter Marie Meyer Viol. - Utrecht: LEd. - (ILLC dissertation series, 1995-11) Proefschrift Universiteit Utrecht. - Met lit. opg. ISBN 90-5434-O41-X NUGI 941 trefw.: logica/taalkunde ©1995, W.P.M. Meyer Viol, Amsterdam Contents Acknowledgments ix 1 What is Instantial Logic? 1 1.1 Introduction . 1 1.2 Overview of the Thesis . 3 2 A Brief History of Instantial Logic 6 2.1 Introduction . 6 2.2 Classical Predicate Logic . 7 2.2.1 Semantics . 7 2.2.2 Natural Deduction for Classical Predicate Logic . 11 2.2.3 Natural Deduction for Classical Generic Consequence 17 2.3 Intensional Epsilon Logic . 20 2.3.1 Background . 20 2.3.2 Language . 21 2.3.3 Semantics . 21 2.3.4 Generic Truth . 26 2.3.5 Logical Consequence . 27 2.3.6 Expressivity . 31 2.3.7 Proof Theory for the Intensional Epsilon Calculus 33 2.3.8 Natural Deduction for Intensional Epsilon Logic 37 2.4 Extensional Epsilon Logic . 41 2.4.1 The Extensionality Principle for Epsilon Terms . 41 2.4.2 Semantics . 42 2.4.3 Natural Deduction for Extensional Epsilon Logic . 43 2.5 Arbitrary Object Theory . 44 2.5.1 .Background . 44 Vi Contents 2.5.2 Semantics . 45 2.5.3 Natural Deduction with Arbitrary Objects 47 2.6 Comparison of Epsilon Logic and A0 Theory . 49 2.6.1 Semantic Comparison . 49 2.6.2 Proof Theoretic Comparison 53 2.7 Conclusion . 54 Intuitionistic Instantial Logic 55 3.1 Introduction . 55 3.2 Intuitionistic Predicate Logic . 55 3.2.1 Semantics . 55 3.2.2 Natural Deduction for Intuitionistic Predicate Logic . 58 3.3 Intensional Intuitionistic Epsilon Logic . 61 3.3.1 Plato’s Principle and the e—Rule 65 3.3.2 Interpretation of e-Terms 67 3.4 Intermediate Logics . 69 3.4.1 The Logic IPL+P_:.l . 70 3.4.2 The Logic IPL+P‘v’ . 79 3.4.3 The Logic IPL+P3+P\7' . 83 3.5 Conclusion . 87 3.6 Appendix . 88 Formula Dependencies 91 4.1 Dependence Management in Natural Deduction . 91 4.1.1 Dependence on Assumptions . 92 4.1.2 Dependence between Assumptions . 93 4.1.3 Classical Dependence Management . 97 4.1.4 Intuitionistic Dependence Management 101 4.2 Conservative Epsilon Extensions of IPL 102 4.3 Kripke Models for Epsilon Terms . 109 4.3.1 Semantic Strategies . 109 4.3.2 Partial Intuitionistic Epsilon Models . 111 4.3.3 Proof Calculus . 113 4.3.4 Completeness Proof 115 4.3.5 Additional Principles . 117 4.4 Conclusion . 118 4.5 Appendix . 119 Term Dependencies 126 5.1 Dependence as a Logical Parameter . 126 5.2 Dependence in Proofs 127 5.2.1 Sources . 127 5.2.2 Varieties of Dependence . 130 Contents vii 5.2.3 Proof Theoretic Formats . 131 5.2.4 Explicit Dependencies . 134 5.2.5 Substructural Variation 137 5.3 Epsilon Calculus as a Testing Ground . 5.4 Benchmark Problems . 142 5.5 Dependence Sensitive Prawitz Calculi . 144 5.5.1 Calculus 1: Local Restrictions on Rules . 145 5.5.2 Calculus II: Global Constraints on Proofs . 146 5.5.3 Discussion of Benchmarks . 146 5.6 Extended Dependence Language . 148 5.6.1 Quantifying in Dependence Structures . 148 5.6.2 An Explicit Language for Dependencies . 150 5.6.3 From First-Order Logic to Dependence Logic . 151 5.7 Extended Proof System . 153 5.7.1 Benchmarks Once More . 5.8 Possible Semantics . 5.8.1 Arbitrary Object Semantics . 160 5.9 Links to Linguistic Applications . 165 5.10 Conclusion 166 5.11 Appendixl . 5.12 Appendix II . 168 6 Epsilon Terms in Natural Language Analysis 170 6.1 Introduction . 170 6.2 Noun Phrases, Pronouns and e-Terms 172 6.3 Pronouns and Epsilon Terms . 174 6.3.1 Intersentential Donkey Pronouns . 176 6.3.2 Donkey Pronouns in Universal and Conditional Contexts 180 6.4 Truth-Conditional Semantics and Incrementality 184 6.5 Subject Predicate Form . 187 6.6 Harder Cases . 6.6.1 Bach-Peters Sentences . 189 6.6.2 Modal Subordination . 190 6.7 Plural E-type Pronouns . 191 6.8 Bare Plurals . 194 6.9 Generics Explained in Terms of Relevant Instances . 197 6.10 Extended 6-Calculi . 198 6.10.1 Standard Models . 199 6.10.2 Cumulative Models . 200 6.10.3 Preferential Models . 201 6.10.4 Monotonic Models . 202 6.10.5» Sensitive Generic Semantics . 203 viii Contents 6.11 Conclusion 204 Bibliography 205 Samenvatting 211 Curriculum Vitae 215 Acknowledgments It is rare that the initial cause of an event can be pinpointed as accurately as that of the origin of this thesis. In 1977 a young Johan van Benthem visited the department of philosophy of the University of Groningen to talk about his PhD thesis. I entered the lecture hall a dissatisfied student who read philosophy “wildly in all directions”. The conclusion of the lecture saw me a changed per­ son. For better or worse, my life took a different course that day and this thesis is one of the results. Both as my teacher and my supervisor Johan has been an inspiration to me. His good company and stubborn support, surviving even the most unrelenting resistance, leave me deeply indebted. Jan van Eijck and I go back an even longer way. We know each other from the time I was half-heartedly intending to graduate in ethics, and he was still active in faculty politics and sported a beard. Together with Johan and Ed Brinksma we formed a reading group of which I still have fond memories. A decade later he was instrumental in getting me to the OTS. Four years ago he handed me Kit Fine’s “Arbitrary Objects” with the admonition to read it. This thesis grew out of the study of that book. I want to thank him for putting up with my inadequacies as a PhD student, for his contributions to the content and form of the thesis, for our discussions about life and logic, and for his dinner parties. One must be considered fortunate to be presently a student of logic in the Netherlands. A more stimulating environment for a PhD student is hard to imagine. Of the people who formed my personal part of that environment I can only mention a few. Besides being great fun, my weekly meetings with Patrick Blackburn, Claire Gardent and Jennifer taught me a lot about linguistics. I thank Patrick for his inspiration, friendship and outstanding cooking. ix X Acknowledgments For significantly influencing the direction of the thesis I am indebted to J aap van der Does, Tim Fernando, Dov Gabbay, Ruth Kempson, Michiel van Lam­ balgen, Diana Santos, Anne Troelstra, Albert Visser, Fer-Jan de Vries. For contributions to the layout of this thesis beyond the call of duty I would like to thank Natasa Rakié and the amazing Maarten de Rijke. At the OTS, CWI and FWI it Was a pleasure to work, eat and party With Elena Marciori, Jan Rutten, Frank de Boer, Daniel Turi, Krzysztov Apt, Yde Venema, Kees Vermeulen, Marcus Kracht, Vera Stebletsova, Michael Moortgat, Annius Groenink, Natasha Alechina, Jan Jaspars, Dorit Ben-Shalom, Marianne Kalsbeek. Finally, I would like to thank my parents Peter Meyer Viol and Francoise Meyer Viol for showing me that love can have more than one face, and Natasa for giving it mine. Chapter 1 What is Instantial Logic? 1.1 Introduction In introductions to mathematics which take a logical perspective on their subject matter, students tend to be treated to warnings against using talk about ‘arbi­ trary objects’ in their proofs, the party line among logicians being that arbitrary object talk is dangerous for mental health, if not morally wrong then at least highly misleading, and that it should therefore at all costs be avoided. Doets [Doe94] constitutes a nice example of this attitude. Still, there is a wide gap between theory and practice. Arbitrary object talk abounds in mathematical discourse. Apparently, and maybe sadly, the warnings do not have much effect. This is how the Dutch engineer and mathematician Simon Stevin reasons about the center of gravity of a triangle. Theorem II. Proposition II The center of gravity of any triangle is in the line drawn from the vertex to the middle point of the opposite side. Supposition. Let ABC be a triangle of any form . Conclusion. Given therefore a triangle, we have found its center of gravity, as required. (Quoted from Struik [Str86, p. 189—191].) In informal mathematics, when we have shown of an ‘arbitrary triangle’ that its center of gravity is in the line drawn from the vertex to the middle point of the opposite side, we have established that this holds for all triangles.
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