WHAT DOES IT MEAN TO SAY THAT LOGIC IS FORMAL? by John Gordon MacFarlane A.B., Philosophy, Harvard College, 1991 M.A., Philosophy, University of Pittsburgh, 1994 M.A., Classics, University of Pittsburgh, 1997 Submitted to the Graduate Faculty of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2000 i Robert Brandom, Distinguished Service Professor of Philosophy (Director) Nuel Belnap, Alan Ross Anderson Distinguished Professor of Philosophy (Second Reader) Joseph Camp, Professor of Philosophy Danielle Macbeth, Associate Professor of Philosophy, Haverford College (Outside Reader) Kenneth Manders, Associate Professor of Philosophy Gerald Massey, Distinguished Service Professor of Philosophy ii WHAT DOES IT MEAN TO SAY THAT LOGIC IS FORMAL? John Gordon MacFarlane, PhD University of Pittsburgh, 2000 Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a pri- ori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topic-neutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this history can help us to see the sources of our disparate intuitions about logicality, and more importantly to sort these intuitions into central and adventitious ones. I argue that we have largely lost sight of the notion of formality (1) by which logic was demarcated in a central tradition from Leibniz through Frege—the intellectual home of most of the philosophical projects for which it matters how logic is demarcated. This historical perspective is especially useful in evaluating contemporary debates about the demarcation of logic, which often seem to turn on opposing but equally brute intuitions iii about logicality. As an illustration, I examine the popular permutation-invariance account of logicality, which is commonly motivated by appeal to sense (2) of formality. I present the account in a way that reveals a hidden lacuna, and I show how this lacuna might be filled by appealing to formality in sense (1). PREFACE The germ of this project was James Allen’s interesting seminar on the development of Aristotle’s logic. In what sense, I wondered, can the hodge-podge of argumentative advice and quasi-logical rules contained in the Topics be called a “logic”? This question led me to the contemporary literature on the demarcation of logic. One of the things that struck me immediately was the sheer variety of proposals. I wondered whether they were all aimed at the same target. I worked through some of this literature in an independent study with John McDowell. One of the books I read was John Etchemendy’s The Concept of Logical Consequence. Etchemendy argues that the hunt for a principled dividing line between logical and non- logical constants is a misguided by-product of Tarski’s account of logical consequence. This struck me as wrong. Surely the demarcational enterprise has deeper roots than that: even the medieval logicians distinguished between “formal” and “material” consequence. And so I began to investigate the history of conceptions of logicality, from Aristotle to the present. It soon became clear that there was an interesting story to tell about the persistent idea that logic is distinctively “formal.” There are many people to thank. Bob Brandom was an ideal dissertation director; his unflagging enthusiasm for the project and probing comments on my work in progress were invaluable. Danielle Macbeth went far beyond what is usually expected of an outside reader; I learned much from reading and reacting to drafts of her book on Frege’s logic and from our e-mail discussions of Kant and Frege. I thank Nuel Belnap for his detailed comments (especially on chapter 6) and for warning me away from many confusions, Joe Camp for iv v asking a crucial question that led me investigate the source of Kant’s logical hylomorphism, Ken Manders for referring me to a key passage in Frege’s “On the Foundations of Geometry: Second Series,” Jerry Massey for pointing me to Church’s review of Carnap’s Formalization of Logic and to his own work in the philosophy of logic, Nicholas Rescher for leading me to Trendelenburg and Couturat, Steve Engstrom for reading and discussing chapter 4, Rega Wood for comments on appendix A, James Conant for conversations about Kant, Russell, and Wittgenstein, and Lionel Shapiro and John Roberts for listening to my half-baked ideas and responding with encouragement and helpful comments. I also profited from the comments of many people who responded to talk versions of some of this material, including Joseph Almog, Lanier Anderson, Nick Asher, Paul Benacerraf, Ned Block, Tyler Burge, John Burgess, John Carriero, Charles Chihara, John Etchemendy, Hartry Field, Kit Fine, Peter Godfrey-Smith, Gil Harman, Mark Johnston, David Kaplan, Paolo Mancosu, Tony Martin, Calvin Normore, Gideon Rosen, Stephen Schiffer, Hans Sluga, David Sosa, Peter Unger, and Dan Warren. Finally, I thank Colleen Boyle for keeping my spirits up throughout. This dissertation is dedicated to the memory of my grandfathers, John C. MacFarlane and Gordon A. Wilson. I like to think that the very different lines along which they inspired me intersect in this document. Contents 1 INTRODUCTION 1 1.1 What is logic? . 1 1.1.1 Method, not subject matter? . 2 1.1.2 Enthymemes . 3 1.1.3 Analyticity, a prioricity, necessity . 4 1.1.4 Formal vs. material . 6 1.2 Why do we need a demarcation? . 7 1.2.1 Logicism . 7 1.2.2 Structuralism . 11 1.2.3 Explicitation . 14 1.3 Why formality? . 15 1.3.1 Pragmatic vs. principled demarcations . 16 1.3.2 Alternatives . 19 1.3.3 Historical importance of formality . 20 1.4 Why a historical approach? . 22 1.5 Prospectus . 27 2 DECOYS 31 2.1 Syntactic formality . 32 2.2 Schematic formality . 36 2.3 Grammatical formality . 41 vi CONTENTS vii 2.3.1 The immanence of grammar . 42 2.3.2 Identity . 44 2.3.3 Grammatical chauvinism . 46 2.4 Conclusion . 49 3 THREE NOTIONS OF LOGICAL FORMALITY 50 3.1 1-formality . 52 3.2 2-formality . 56 3.3 3-formality . 60 3.3.1 Schlick and Einstein . 62 3.3.2 Carnap . 63 3.3.3 Nagel . 65 3.4 Independence of the three notions . 65 3.5 Formality, generality, and topic-neutrality . 69 3.6 Philosophical significance of the three notions . 76 3.7 A puzzle . 77 4 KANT AND THE FORMALITY OF LOGIC 79 4.1 Kant’s characterization of logic . 81 4.1.1 Kant’s taxonomy of logics . 81 4.1.2 What does Kant mean by “general”? . 84 4.1.3 What does Kant mean by “formal”? . 87 4.1.4 Generality, formality, and Kant’s demarcation of logic . 91 4.2 The originality of Kant’s characterization . 95 4.2.1 The Wolffian school . 96 4.2.2 Descartes and Locke . 99 4.2.3 Leibniz . 102 4.2.4 Scholastics . 110 4.2.5 Crusius, Lambert, Tetens . 113 CONTENTS viii 4.3 The genesis of logical hylomorphism in Kant . 114 4.3.1 The definition of logic . 114 4.3.2 Material criterion of truth . 115 4.3.3 Remarks on the history of logic . 117 4.3.4 Evidence of the Reflexionen . 118 4.3.5 Precritical works . 119 4.4 Transcendental idealism and the formality of logic . 121 4.5 Kant as the source of modern logical hylomorphism . 127 4.5.1 In Germany . 127 4.5.2 In Britain . 128 4.6 Conclusion . 133 5 FREGE AND THE FORMALITY OF LOGIC 135 5.1 The status of logic in Frege’s early works . 137 5.2 Frege’s rejection of Kant’s Thesis . 143 5.2.1 Frege and Kant on logic and arithmetic . 143 5.2.2 Frege’s characterization of logic as 1-formal . 147 5.2.3 “The legend of the sterility of pure logic” . 149 5.2.4 “Logical form” in the FA . 151 5.2.5 Frege’s rejection of 3-formality . 152 5.2.6 “The most general laws of truth” . 157 5.2.7 How Frege resists the Kantian argument . 161 5.3 Is Frege’s logic 2-formal? . 165 5.4 Does Russell’s Paradox vindicate Kant’s Thesis? . 167 5.5 Frege’s influence . 170 6 PERMUTATION INVARIANCE AND LOGICALITY 173 6.1 Permutation invariance . 177 6.1.1 Categorial grammar . 178 CONTENTS ix 6.1.2 Typed presemantics . 180 6.1.3 Compositional semantics . 181 6.1.4 The permutation invariance criterion . 182 6.1.5 Advantages of this framework . 185 6.2 Logical notions and logical constants . 187 6.2.1 Presemantics and semantics . 187 6.2.2 Logical notions and logical constants .