The Intended Interpretation of the Intuitionistic First-Order Logical Operators
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THE INTENDED INTERPRETATION OF THE INTUITIONISTIC FIRST-ORDER LOGICAL OPERATORS PhD Dissertation by Enrique Gustavo FERNANDEZ-DIEZ-PICAZO The London School of Economics and Political Science (University of London) Year 1997 1 UMI Number: U613452 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U613452 Published by ProQuest LLC 2014. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 British Uljiarv/ o! Political and Econon tc Science ABSTRACT The present thesis is an investigation on an open problem in mathematical logic: the problem of devising an explanation of the meaning of the intuitionistic first-order logical operators, which is both mathematically rigorous and faithful to the interpretation intended by the intuitionistic mathematicians who invented and have been using them. This problem has been outstanding since the early thirties, when it was formulated and addressed for the first time. The thesis includes a historical, expository part, which focuses on the contributions of Kolmogorov, Heyting, Gentzen and Kreisel, and a long and detailed discussion of the various interpretations which have been proposed by these and other authors. Special attention is paid to the decidability of the proof relation and the introduction of Kreisel’s extra-clauses, to the various notions of ‘canonical proof' and to the attempt to reformulate the semantic definition in terms of proofs from premises. In this thesis I include a conclusive argument to the effect that if one wants to withdraw the extra-clauses then one cannot maintain the concept of ‘proof' as the basic concept of the definition; instead, I describe an alternative interpret ation based on the concept of a construction ‘performing' the operations indicated by a given sentence, and I show that it is not equivalent to the verificationist interpretation. I point out a redundancy in the internal -pseudo-inductive- structure of Kreisel's interpretation and I propose a way to resolve it. Finally, I develop the interpretation in terms of proofs from premises and show that a precise formulation of it must also make use of non-inductive clauses, not for the definition of the conditional but -surprisingly enough- for the definitions of disjunction and of the existential quantifier. 3 ACKNOWLEDGEMENTS During the production of this thesis I have received help from various people and institutions, to whom it is only fair that I express here my gratitude. Professor Luis Valdes firmly encouraged me to undertake postgraduate studies in Britain after graduating at the University of Murcia (Spain). His friendly unconditional support since then has been of inestimable value to me these last few years. Dr Colin Howson has been my supervisor since I came to the LSE Philosophy Department. He dealt with me patiently when my English was very far from fluent, and helped me in both academic and bureaucratic matters, always in a friendly and considerate way. Later I had the opportunity of being his teaching assistant for two years, and I learnt a great deal from the experience -both about logic and about teaching itself. He carefully read the final draft of the thesis, and made countless comments, concerning many clarification points as well as substantive issues. 4 Professor Moshe Machover taught me mathematical logic during the academic year 1992/93, and has closely followed my doctoral research since it started. To him I owe numerous suggestions and comments, some of which have played a crucial role in the development of the thesis. He has always being kind and patient to me, and I have learnt from him more than I can possibly express. Something for which I am grateful to these three scholars as well as to Dr Peter Urbach, is the enormous confidence that they have consistently shown in me and my prospects. To me, this has been of the greatest value. In May 1995 I gave a talk on the topic of this thesis at the Staff/Graduate Seminar of the Philosophy Departments of LSE and King’s College London, where I benefited from the questions and comments of several people in the audience. Finally, Dr Andrew Powell generously offered me full assistance while he was still a graduate student -albeit one senior to me, and infinitely much wiser. He read a good deal of my written work, and patiently discussed with me many of the topics of the thesis; when I finished the final draft, he reviewed it carefully and made comments on almost every 5 page. His observations, his criticisms and his illuminating expositions have no doubt made this thesis much better than it would have been otherwise. Moreover, from an early stage he also honoured me with a sincere friendship which I have enjoyed ever since. As for material help, the British Council and Caja de Ahorros del Mediterraneo (Spain) provided full financial support for both the college fees and my living expenses from October 1992 to September 1994 ( Becas CAM-British Council, Convocatoria de 1991). Later, the British Academy awarded me a postgraduate studentship which covered the college fees for the academic years 1994/95 and 1995/96 (Ref. BA94/3021/ LSE); and finally, the Spanish Ministry of Education and Science (presently ‘of Education and Culture’) awarded me a research grant which has being covering my living expenses since January 1995 (Ref. FP94 27484804). 6 TABLE OF CONTENTS A b s t r a c t ................................................................................................... 2 A cknowledgements ................... ............................................... 4 C h a p t e r 0. Introduction §0.1. The topic of this thesis and its significance 14 §0.2. Summary of contents 0.2.1. Chapter 1..................................................................... 17 0.2.2. Chapter 2..................................................................... 18 0.2.3. Chapter 3.................................................... 19 0.2.4. The ‘Conclusion' ......................................................... 20 §0.3. Original content of the th esis...................................... 21 §0.4. The intuitionistic ‘ideology' and the study of in- tuitionism today ........................................................................ 25 §0.5. The other main schools of constructive mathe matics 0.5.1. The logic of constructive mathematics ................. 27 0.5.2. The status of Markov's principle ................. 27 0.5.3. More on Markov's principle .................................... 29 0.5.4. Conclusion ................................................................... 30 §0.6. Notational conventions and preliminary notions 0.6.1. First-order languages ............................................... 31 0.6.2. Constructive sets ....................................................... 31 0.6.3. Constructive interpretations .................................. 33 0.6.4. More on constructive interpretations .................. 34 7 0.6.5. Constructive interpretations with a non-decid- able domain ............................................................................ 37 0.6.6. Conventions regarding quotations ........................ 38 C h a p t e r l . Th e S e a r c h fo r th e In t e n d e d In t e r p r e t a t io n : E x po sit io n §1.1. Brouwer's use of the logical constants 1.1.1. Introduction ................................................................ 40 1.1.2. Verificationist interpretations: meaning as provability ............................................................................... 41 1.1.3. Verificationist interpretations: meaning as problem-solving ..................................................................... 43 1.1.4. The operational interpretation .............................. 45 1.1.5. The verificationist versus the operational inter pretation ................................................................................. 47 1.1.6. The meaning of negation ......................................... 48 1.1.7. Negation and hypothetical constructions ............ 52 1.1.8. The meaning of the conditional ............................. 55 1.1.9. Disjunction, conjunction and ............................. 56 1.1.10. The quantifiers........................................................ 58 §1.2. Kolmogorov's interpretation 1.2.1. An interpretation in terms of mathematical pro blems........................................................................................ 61 1.2.2. Kolmogorov and Heyting ......................................... 62 1.2.3. The interpretation of the connectives.................. 63 1.2.4. The interpretation of the quantifiers ................... 65 8 §1.3. Heyting’s interpretation 1.3.1. Introduction ................................................................ 67 1.3.2. The verificationist point of view ............................ 67 1.3.3. Heyting and Kolmogorov ......................................... 70 1.3.4. The interpretation of the connectives .................. 71 1.3.5. The conditional and negation ................................. 73 1.3.6. The