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Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9* black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional cfiarge. Contact UMI directly to order. Bell & Howell Information and Learning 300 North Zeeb Road, Ann Arbor. Ml 48106-1346 USA 800-521-0600 UMI* LOGIC-AS-MODELING: A NEW PERSPECTIVE ON FORMALIZATION DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Roy T. Cook, B.A. ***** The Ohio State University 2000 Dissertation Committee: Professor Stewart Shapiro, Adviser Approved by Professor George Schumm Adviser Professor Neil Tennant Philosophy Graduate Program UMI Number 9982542 UMI UMI Microform9982542 Copyright 2000 by Sell & Howell Information and Leaming Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Leaming Company 300 North Zeeb Road P.O. 00x1346 Ann Arbor, Ml 48106-1346 ABSTRACT I propose a novel way of viewing the connection between mathematical discourse and the mathematical logician’s formalizations of it. We should abandon the idea that formalizations are accurate descriptions of mathematical activity. Instead, logicians are in the business of supplying models in much the same way that a mathematical physicist formulates models of physical phenomena or the hobbyist constructs models of ships. I first examine problems with the traditional view, and I survey some prior work in the spirit of the logic-as-model approach, including that of Gerhard Gentzen, Georg Kreisel, and Imre Lakatos. The utility of the present framework is then demonstrated in three case studies. First, this approach solves a purported problem with precise semantics for vagueness. Precise semantics supposedly err by replacing the defining characteristic u of vagueness (imprecision) with precise cut-offs, but on the logic-as-model approach this precision can be seen as an artifact serving to simplify the model but corresponding to nothing actually found in vague languages. Second, on the logic-as-model view the advantages claimed for branching quantifiers, i.e. providing the expressiveness of second-order languages while avoiding their semantic intractability, are illusory. On the one hand, second-order formulations of mathematical concepts are usually far more natural than branching quantifier formulations; on the other hand, branching quantisers are no more tractable than second-order languages. Finally, I examine the claim that Gottfried Leibniz’s (seemingly incoherent) infinitesimal calculus is nothing other than Abraham Robinson’s non-standard analysis. Although non-standard analysis is a useful model, there are other formalizations of Leibniz’s mathematics as fruitful as, yet incompatible with, the non­ standard approach, a fact hard to reconcile with the idea that logic provides accurate descriptions. After the case studies, I examine the objectivity of logic from the logic-as- model framework. This issue is difficult enough on traditional accounts of logic, and would seem to become even more difficult from the present point of view. I ll Nevertheless, a rich picture of the objectivity of logic is worked out with interesting new sorts of indeterminacy arising from the idea that logic provides models and not descriptions. IV Dedicated to my Mother ACKNOWLEDGMENTS I wish to thank my adviser, Stewart Shapiro, for intellectual support and guidance, infinite patience and understanding, and valuable friendship and help. Without his support, both scholarly and more generally, this dissertation would have been impossible. I also wish to thank Neil Tennant and George Schumm, who have been both great teachers and good friends, and have, along with Stewart, taught me practically everything I know about how philosophy ought be done. Thanks are also due to Randy Dougherty, Harvey Friedman, Tim Carlson, and the Ohio State University Department of Mathematics for giving me the mathematical background that is necessary for serious work in the philosophy of mathematics. I am also grateful for the intellectually stimulating environment that I have been a part of for the past six years. Jack Arnold, Robert Batterman, Emily Beck, VI Chad Belfor, Jon Cogburn, Julian Cole, Jon Curtis, Glen Haitz, Peter King, Robert Kraut, Bill Melanson, David Merli, George Pappas, Sarah Pessin, Lisa Shabel, Sheldon Smith, and Marshall Swain all deserve mention in this regard. Finally, I wish to thank Alice Leber for forgiving me when I spent too much time with research and too little time with her. VII VITA June 7, 1972 ..................................................... Bom - Fairfax, Virginia, USA. June, 1994 .........................................................B.A. Philosophy, Virginia Tech. June, 1994 ........................................................ B.A. Political Science, Virginia Tech. September 1990-Present ................................ Graduate Teaching Associate The Ohio State University PUBLICATIONS 1 “Monads and Mathematics: The Logic of Leibniz’s Mereology”, forthcoming in Stadia Leibnitiana. 2. “What Negation is Not: Intuitionism and 0 = 1’” , Analysis 60, pp. 5-11, [2000] (with Jon Cogbum). 3. review of Kreiseliana: About and Around Georg Kreisel, ed. Piergiorgio Odifreddi, forthcoming in Studia Logica (with Jon Cogbum). vui 4. “Hintikka’s Revolution; The Principles of Mathematics Revisited” , The British Journal o f the Philosophy of Science 49, pp. 309-316, [1998] (with Stewart Shapiro). 5. review of Structuralism and Structures: A Mathematical Perspective, by Charles Rickart, in Philosophia Mathematica 3, pp. 227-231, [1998] (with Stewart Shapiro). FIELDS OF STUDY Major Field: Philosophy IX TABLE OF CONTENTS Page Abstract...........................................................................................................................ii Dedication ...................................................................................................................... iv Acknowledgments ......................................................................................................... v Vita..................................................................................................................................vü Table of Contents ..........................................................................................................ix List of Figures ................................................................................................................ xii Chapters: [1] Introduction .........................................................................................................1 [1.1] The Logic-As-Model Approach .......................................................... 1 [1.2] Differences Between Logic-As-Model And Foundational Views .......................................................................16 [1.2.1] Statements Of Everyday Mathematics Are Logical Formulae ............................................................... 17 [1.2.2] Logical Work Can Lead To The Revision Of Mathematics ............................................... 22 [1.2.3] Philosophical Problems Can Be Reduced To Logical Problems ............................................... 27 [1.2.4] There Is A Single Correct Logic ........................................... 29 [1.2.5] Summing Up And Moving On .............................................. 33 [1.3] Gertiard Gentzen And Natural Deduction .......................................... 35 [1.3.1] Normal Form Proofs And The Hauptsatz ............................. 36 [1.3.2] Introduction And Elimination Rules ..................................... 43 [1.3.3] The Proper Continuation Of Gentzen's Project ....................49 [1.4] Georg Kreisel And Informal Rigour ................................................... 55 [ 1.5] Imre Lakatos, Rational Reconstructions, And Rigour .......................75 [1.5.1] Rational Reconstructions And Logic .....................................76 [1.5.2] Three Levels Of Formalization In Proof. ................................89 [ 1.6] The Next Four Chapters ......................................................................... 101 [2] Vagueness And Mathematical Precision .......................................................... 106 [2.1 ] Introduction ............................................................................................