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Marcus Rossberg Phd Thesis CORE Metadata, citation and similar papers at core.ac.uk Provided by St Andrews Research Repository SECOND-ORDER LOGIC: ONTOLOGICAL AND EPISTEMOLOGICAL PROBLEMS Marcus Rossberg A Thesis Submitted for the Degree of PhD at the University of St Andrews 2006 Full metadata for this item is available in Research@StAndrews:FullText at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/6407 This item is protected by original copyright Second-Order Logic: Ontological and Epistemological Problems Marcus Rossberg Submission for a PhD in Philosophy at the University of St Andrews Supervisor: Professor Crispin Wright Second supervisor: Professor Stewart Shapiro Date of submission: January 5th, 2006 Date of final examination: March 4th, 2006 Examiners: Dr Peter Clark Professor Ian Rumfitt i F¨urmeine Eltern, Irene und Wolfgang Rossberg, meine Großm¨utter, Hedwig Grosche und Ilse Rossberg, und gewidmet dem Andenken an meinen Großvater Paul Rossberg (1920 { 2004) For my parents, Irene and Wolfgang Rossberg, my grandmothers, Hedwig Grosche and Ilse Rossberg, and dedictated to the memory of my grandfather Paul Rossberg (1920 { 2004) ii Abstract In this thesis I provide a survey over different approaches to second-order logic and its interpretation, and introduce a novel approach. Of special interest are the questions whether (a particular form of) second-order logic can count as logic in some (further to be specified) proper sense of logic, and what epistemic status it occupies. More specifically, second-order logic is sometimes taken to be mathematical, a mere notational variant of some fragment of set theory. If this is the case, it might be argued that it does not have the \epistemic innocence" which would be needed for, e.g., foundational programmes in (the philosophy of) mathematics for which second-order logic is sometimes used. I suggest a Deductivist conception of logic, that characterises logical consequence by means of inference rules, and argue that on this conception second-order logic should count as logic in the proper sense. iii Acknowledgements My primary thanks go to my supervisors. Crispin Wright provided comments, crit- icism, discussion, support, and more criticism whenever I wanted or needed it. His dedication to my supervision, in particular in the final months, was beyond the call of duty. I also greatly benefitted from the comments and criticism I got from Stew- art Shapiro during his annual visits to St Andrews and when I went to visit him in Columbus in Spring 2004. Both my supervisors let me run with my own ideas when I needed the space to develop them, and demanded rigour and \tidying up" once I had run far enough. I am also indebted to my shadow supervisors Fraser MacBride and Roy Cook. The feedback I got from them, in particular in the the first couple of years, was invaluable. I was lucky enough to be able to write this thesis as a member of the Arch´e research centre. The active and collaborative atmosphere here creates what is prob- ably a unique environment for philosophical research. First and foremost amongst my fellows here I want to thank Philip \Kollege" Ebert and Nikolaj Jang Peder- sen who together with me were the first generation of Arch´e postgraduate students. We were soon joined by the next two \Arch´e Beavers", Ross Cameron and Robbie Williams. One cannot hope for better fellow Ph.D. students. Thanks guys! The thanks extends to the other members of Arch´e: Sama Agahi, Bob Hale, Paul Mc- Callion, Darren McDonald, Sean Morris, Agust´ınRayo, Andrea Sereni, and Chiara Tabet in the Neo-Fregean project, and my other colleagues in the centre, Elizabeth Barnes, Eline Busck, Richard Dietz, Patrick Greenough, Aviv Hoffmann, Carrie Jenkins, Dan L´opez de Sa, Sebastiano Moruzzi, Josh Parsons, Graham Priest, Anna Sherratt, S´oniaRoca, and Elia Zardini. The friendly, collaborative, but also philosophically intense atmosphere was not restricted to Arch´e, however; it is a also a characteristic mark of the whole postgrad- uate community in St Andrews. The Friday Seminar, which we all lived through together, provided an excellent training ground in all respects, philosophical and social (the latter in particular following the seminar, in the pub). Many of my fellow students here and friends outwith St Andrews stood by me while I was working on this thesis, and not only provided me with moral support, but also with philosophical input. Daniel Cohnitz has filled this role ever since we were undergraduate students together at the University of D¨usseldorf.The others, I hope, will forgive me if I just list their names here without further explanation: Ais- linn Batstone, Brandon Cooke, Joseph Diekemper, Iwao Hirose, Sune Holm, Amy Hughes, Kent Hurtig, Chris Kelp, Anne Manolakas, Brian McElwee, Aidan McG- lynn, Anneli Mottweiler, Cyrus Panjvani, Carlotta Pavese, Simon Robertson, Raf- faele Rodogno, Enzo Rossi, Markus Schlosser, Mog Stapleton, Dave Ward, Michael Weh, and, of course, the person I surely and inexcusably have forgotten to include in this list. A special thank-you goes to Linds Duffield for her fantastic support iv during the very stressful last few weeks before the submission of this thesis { weeks that did not seem to end. Much of the material in this thesis was first tested in the Friday seminar as well as the Arch´eresearch seminar. I am grateful for the many comments and criti- cisms I received. Earlier versions of some of the chapters were presented at various conferences, workshops and seminars all over Europe, including the Universities of D¨usseldorf,Helsinki, and Stockholm, the Ockham Society in Oxford, the SPPA con- ference in Stirling, the GAP.5 in Bielefeld, the LMPS03 in Oviedo, the FOL'75 in Berlin, and the ECAP5 in Lisbon. Many thanks to the audiences there. Chapter 5 profited greatly from a discussion with Kit Fine to whom I am indebted for his comments and suggestions. I am grateful for the financial support I received: for my tuition fees from the Arts and Humanties Research Board (AHRB { now AHRC), a full maintenance scholarship from the Studienstiftung des Deutschen Volkes [German National Aca- demic Foundation] from my second year on, and a travel award from the Russell Trust for a visiting scholarship at the Ohio State University at Columbus. I could not have written this thesis without the support of my family. There are no words for my gratitude towards my parents, Irene and Wolfgang Rossberg, and my grandparents, Hedwig Grosche and Ilse and Paul Rossberg. My grandfather Paul Rossberg sadly did not live to see the completion of my postgraduate studies. This thesis is for them. v Notational Conventions For the benefit of a consistent use of the logical symbols I have tacitly changed the symbols other authors use into the ones preferred here. Boolos' `!' and `$', for example, are replaced by `⊃' and `≡', respectively. Likewise, Quine's `(x)', for example, is replaced by `8x', and his notation using `.', `:', `.:', etc., as both symbols for conjunction and for scope distinctions is abandoned in favour of the nowadays more common use of `^' (for conjunction) and the use of parentheses (for scope distincitons). I use single quotations marks for mentioning an expression, and double quota- tions marks where the quoted expression is used; the latter case almost exclusively occurs for verbatim quotations from the literature and the use of a word in a non- literal sense or in a sense that I explicitly do not agree with. It should in all cases be clear from the context in which of these senses I uses the double quotation marks. I tacitly changed the quotations marks in verbatim quotations from the literature to accord to this rule where authors followed different conventions. (Quine and Boolos, for example, sometimes use double quotation marks for mentioning expressions.) Where longer passages are quoted verbatim from the literature, the quotation is separated from the main text by an extra wide margin, and quotation marks are omitted. Corner quotes, `p' and `q', are used as devices for quasi-quotation, as introduced in (Quine, 1955), x6. vi Contents 1 Introduction 1 1.1 The Project . 1 1.2 Outline of the Thesis . 5 2 The Formal System of Second-Order Logic 9 2.1 Preliminaries . 9 2.2 Language . 9 2.3 Deductive Systems . 11 2.3.1 Axiomatic System . 11 2.3.2 Natural Deduction . 14 2.4 Semantics . 16 2.4.1 Semantics for First-Order Logic . 16 2.4.2 Standard Semantics for Second-Order Logic . 17 2.4.3 Henkin Semantics . 18 2.5 Some Meta-Theoretic Results . 19 2.5.1 Standard and Henkin Semantics . 20 2.5.2 First-Order Logic . 20 2.5.3 Second-Order Logic with Standard Semantics . 22 2.5.4 Second-Order Logic with Henkin Semantics . 24 3 On Quine 26 3.1 Introduction . 26 3.2 Incompleteness and Branching Quantifiers . 29 3.3 Incoherence and Unintelligibility . 35 3.3.1 Russell's Paradox . 35 3.3.2 No Entity Without Identity . 39 3.4 Dishonesty: The Hidden Staggering Ontology . 42 4 Plurals 52 4.1 Some Critics . 52 4.2 The Plural Interpretation . 62 vii 4.3 Polyadic Predicates . 68 4.4 Still Wild . 75 4.5 Summary . 86 5 Ontological Commitment 89 5.1 Quine's Criterion . 89 5.2 Universals . 94 5.3 Second-Order Quantification . 95 5.4 Polyadic Predicates Again . 100 5.5 A New Criterion . 105 6 Semantic Incompleteness∗ 112 6.1 Introduction . 112 6.2 Logical Consequence . 114 6.3 Refining the Picture . 120 6.4 Some Arguments for Completeness . 127 6.5 Conclusion . 143 7 The Semantic Conception 145 7.1 Introduction .
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