The Cambridge Companion to FREGE
Each volume of this series of companions to major philosophers contains specially commissioned essays by an international team of scholars together with a substantial bibliography, and will serve as a reference work for students and non-specialists. One aim of the series is to dispel the intimidation such readers often feel when faced with the work of a difficult and challenging thinker. Gottlob Frege (1848–1925) was unquestionably one of the most important philosophers of all time. He trained as a mathemat- ician, and his work in philosophy started as an attempt to provide an explanation of the truths of arithmetic, but in the course of this attempt he not only founded modern logic but also had to address fundamental questions in the philosophy of language and philosophical logic. He is generally seen (along with Russell and Wittgenstein) as one of the fathers of the analytic method, which dominated philosophy in English-speaking countries for most of the twentieth century. His work is studied today not just for its historical importance, but also because many of his ideas are relevant to current debates in the philosophies of logic, language, mathematics and the mind. The Cambridge Companion to Frege provides a route into this lively area of research. New readers will fi nd this the most convenient detailed guide to Frege currently available. Advanced students and specialists will fi nd a conspectus of recent developments in the interpretation of Frege.
M ic h a e l P o t t e r is Reader in the Philosophy of Mathematics at the University of Cambridge and a Fellow of Fitzwilliam College. He is the author of Reason’s Nearest Kin (2000), Set Theory and its Philosophy (2004) and Wittgenstein’s Notes on Logic (2009 ).
Tom Ricketts is Professor of Philosophy at the University of Pittsburgh. He is the author of numerous articles on the develop- ment of analytic philosophy, especially Frege, Wittgenstein and Carnap.
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Continued at the back of the book
The Cambridge Companion to FREGE
Edited by
Michael Potter University of Cambridge
and
Tom Ricketts University of Pittsburgh
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Library of Congress Cataloguing in Publication data The Cambridge companion to Frege / [edited by] Michael Potter, Tom Ricketts. p. cm. – (Cambridge companions to philosophy) Includes bibliographical references and index. isbn 978-0-521-62428-2 – isbn 978-0-521-62479-4 (pbk.) 1. Frege, Gottlob, 1848–1925. I. Ricketts, Tom. II. Potter, Michael D. III. Title. IV. Series. b3245.f24c35 2010 193–dc22 2010011242
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Contents
List of contributors page ix Preface xiii Note on translations xv Chronology xvii 1 Introduction 1 Michael Potter 2 Understanding Frege’s project 32 Joan Weiner 3 Frege’s conception of logic 63 Warren Goldfarb 4 Dummett’s Frege 86 Peter Sullivan 5 What is a predicate? 118 Alex Oliver 6 Concepts, objects and the Context Principle 149 Thomas Ricketts 7 Sense and reference: the origins and development of the distinction 220 Michael Kremer 8 On sense and reference: a critical reception 293 William Taschek 9 Frege and semantics 342 Richard Heck
vii viii Contents
10 Frege’s mathematical setting 379 Mark Wilson 11 Frege and Hilbert 413 Michael Hallett 12 Frege’s folly: bearerless names and Basic Law V 465 Peter Milne 13 Frege and Russell 509 Peter Hylton 14 Inheriting from Frege: the work of reception, as Wittgenstein did it 550 Cora Diamond Bibliography 602 Index 628 contributors
Cora Diamond is University Professor and Kenan Professor of Philosophy Emerita at the University of Virginia. She is the author of The Realistic Spirit: Wittgenstein, Philosophy, and the Mind (1991) and the editor of Wittgenstein’s Lectures on the Foundations of Mathematics, Cambridge, 1939 (1976).
Warren Goldfarb is W. B. Pearson Professor of Modern Mathematics and Mathematical Logic at Harvard University and has been on the faculty there since 1975. His research interests cen- tre on mathematical logic, the development of analytic philosophy (particularly Frege, Russell, Carnap, Quine and Wittgenstein), the interrelationships between logic and philosophy and the issues in metaphysics and philosophical logic that are at the heart of the ana- lytic tradition. He edited Jacques Herbrand’s Logical Writings (1971), co-authored (with Burton Dreben) The Decision Problem: Solvable Classes of Quantifi cational Formulas (1979) and co-edited Kurt Gödel’s Collected Works , vol. III (1995), vols. IV–V (2003). His text- book Deductive Logic was published in 2003.
Michael Hallett has been a member of the Philosophy Department at McGill University since 1984. In that year he pub- lished Cantorian Set Theory and Limitation of Size , an historical analysis of Cantor’s work and of the emergence of modern set theo- ry, concentrating on the philosophical and conceptual difficul- ties with the power-set operation. His subsequent work centres on Frege, Hilbert and Gödel, with the geometrical and foundational work of Hilbert as its current focus. He is one of four general editors
ix x List of contributors of a project to publish (in six volumes) the notes of major lecture courses Hilbert gave on the foundations of mathematics and phys- ics between 1891 and 1933. The fi rst volume (on geometry) appeared in 2004.
Richard Heck is Professor of Philosophy at Brown University, where he has taught since 2005. He has worked on several aspects of Frege’s philosophy, as well as on general issues in philosophy of language, logic, mathematics and mind. He lives in Massachusetts with his wife, daughter and four cats.
Peter Hylton was educated at Kings College, Cambridge and at Harvard University. He is currently Professor of Philosophy and Distinguished Professor at the University of Illinois, Chicago. His publications include Russell, Idealism, and the Emergence of Analytic Philosophy (1990), Quine (2007) and numerous essays on the history of analytic philosophy: some of these essays are col- lected in Propositions, Functions, and Analysis (2005).
Michael Kremer is Professor of Philosophy at the University of Chicago. He received his PhD from the University of Pittsburgh in 1986 and taught at the University of Notre Dame prior to join- ing the University of Chicago faculty. He has published widely on early analytic philosophy, especially Frege, Russell and the early Wittgenstein, as well as philosophical logic and the philosophy of language. He is currently working on a book on Frege’s sense– reference distinction.
Peter Milne is Professor of Philosophy at the University of Stirling. He has published articles on the history and philosophy of logic, on formal epistemology and on the foundations of probability.
Alex Oliver is Reader in Philosophy at Cambridge University and a Fellow of Gonville and Caius College. He has written exten- sively on metaphysics, philosophical logic and the philosophy of mathematics.
Michael Potter is Reader in the Philosophy of Mathematics at Cambridge University and a Fellow of Fitzwilliam College. His List of contributors xi research interests span the philosophy of mathematics and the his- tory of early analytic philosophy. His books include Reason’s Nearest Kin (2000), Set Theory and Its Philosophy (2004) and Wittgenstein’s Notes on Logic ( 2009 ). He is also co-editor of Mathematical Knowledge (2007) and of a forthcoming collection on The Tractatus and Its History .
Thomas Ricketts has been a professor at the University of Pittsburgh since 2005, having previously held faculty appointments at Northwestern University, the University of Pennsylvania and Harvard University. His research interests focus on the develop- ment of analytic philosophy, especially Frege, Russell, Wittgenstein, Carnap and Quine.
Peter Sullivan is Professor of Philosophy at the University of Stirling. He has published articles on most of the leading fi gures in the early history of analytic philosophy and has a particular inter- est in Ramsey. He is co-editor of a forthcoming collection on The Tractatus and Its History .
William W. Taschek is Professor of Philosophy at the Ohio State University. His principal research is in the philosophy of language and the history of early analytic philosophy, with special emphasis on Frege, Russell and the early Wittgenstein.
Joan Weiner is the author of Frege in Perspective (1990) and Frege Explained (2004), along with numerous scholarly articles. She has been awarded fellowships by the Guggenheim Foundation, the American Philosophical Society and the Mellon Foundation, and a research grant by the National Science Foundation. She is currently Professor of Philosophy at Indiana University.
Mark Wilson works largely in the intersection between trad- itional philosophical concern and scientifi c practice; his recent book Wandering Signifi cance (2008) is devoted to such topics. He teaches philosophy at the University of Pittsburgh.
Preface
This volume has been many years in the making. It was begun by one of us (Thomas Ricketts), who commissioned most of its chap- ters. The other editor (Michael Potter) joined the project at a later stage, commissioning several more chapters and seeing the volume through to press. Although it differs in some respects from what either of us might have designed on his own, we are delighted to see it published now. We are grateful to the contributors for their patience and understanding during the volume’s long gestation: we hope they will judge the result worth the wait. M.D.P. T.R.
xiii
Note on translations
There are several words in German that Frege used with technical meanings. The various English translations of his work (and, as a result, the secondary literature in English) are not agreed about how to translate some of them. The following table shows the transla- tions that have generally been used in this volume. Quotations have sometimes, where the sense allows, been silently altered to conform to these conventions.
Frege This volume Other translations
Sinn Sense Meaning Bedeutung Reference Meaning, ( post-1891) nominatum, denotation Wertverlauf Value-range Course of values Vorstellung Idea or representation Begriffsschrift Conceptual notation Concept-script
The correct translation of Bedeutung is a matter of particular con- troversy for some commentators, and here complete uniformity has not been feasible.
xv
Chronology
8 November 1848 Frege born in Wismar 1866 Death of Frege’s father from typhus 1869–71 Frege attends University of Jena 1871–4 Frege attends University of Göttingen 1874 Frege begins teaching at University of Jena 1879 Publication of Begriffsschrift (Conceptual Notation ) Frege begins to receive a stipend from University of Jena 1884 Publication of Grundlagen der Arithmetik (Foundations of Arithmetic ) 14 Mar 1887 Frege marries Margarete Lieseberg 1893 Publication of fi rst volume ofGrundgesetze der Arithmetik (Basic Laws of Arithmetic ) 1896 Frege promoted to a professorship 1898 Death of Frege’s mother 16 June 1902 Russell writes to Frege about the contradiction 1903 Publication of second volume of Grundgesetze der Arithmetik 1904 Frau Frege dies 1918 Frege retires 26 July 1925 Frege dies in Bad Kleinen
For a list of Frege’s publications, please consult the bibliography at the end of this volume.
xvii
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Michael Potter 1 Introduction
Early life Frege was born in 1848 in Wismar, a small port on the Baltic coast in Mecklenberg. 1 His father, who ran a private school for girls there, died when he was eighteen, and his mother took over the running of the school in order to be able to provide for the university educa- tion of Frege and his younger brother. Frege was encouraged in this by a young teacher at his father’s school called Leo Sachse. Sachse had attended university in Jena, and Frege went there too in 1869, lodging in the same room that Sachse had rented there before him. Frege’s studies in Jena consisted mainly of courses in mathematics and chemistry. The only philosophy was a course on Kant’s critical philosophy given by Kuno Fischer. From Jena Frege went on to Göttingen, where he took further courses in mathematics and physics and wrote a dissertation, ‘On a Geometrical Representation of Imaginary Forms in the Plane’. His only philosophy course at Göttingen was one on the philoso- phy of religion given by Hermann Lotze. After fi ve semesters, Frege returned to Jena to submit a further dissertation for his venia docendi (i.e. licence to teach in the university). The title of this second dis- sertation was ‘Methods of Calculation based on an Extension of the Concept of Quantity’. Neither dissertation exhibits more than a passing interest in logic or the philosophy of mathematics. One of Frege’s mathematics lecturers at Jena, Ernst Abbe, acted as a sort of mentor, supporting him, for instance, in his efforts to
1 For information about Frege’s life I have relied throughout this Introduction on Lothar Kreiser, Frege: Leben, Werk, Zeit (Hamburg: Meiner, 2001 ).
1 2 Michael Potter gain promotion. But it is hard to fi nd anyone in Frege’s education who might count as a philosophical teacher of central importance. The nearest to a direct infl uence is perhaps Lotze, not because of his lectures on the philosophy of religion but because he published a book on logic in 1874. Dummett has convincingly argued 2 that an undated list of seventeen numbered observations about logic which has survived in Frege’s hand was written in response to reading Lotze’s book; internal evidence strongly suggests that these notes are probably among the earliest of Frege’s unpublished writings on logic to have survived (although perhaps not quite pre-dating Begriffsschrift , as Dummett suggested).3 In the notes, Frege makes a distinction, which was to be central to his thinking about logic throughout his career, between thoughts and ideas: a thought is something such that ‘it makes sense to ask whether it is true or untrue’, whereas ‘associations of ideas are nei- ther true nor untrue’. Truth is objective. As Frege puts it, ‘2 times 2 is 4’ is true, and will continue to be so even if, as a result of Darwinian evolution, human beings were to come to assert that 2 times 2 is 5. Every truth is eternal and independent of being thought by anyone and of the psychological make-up of anyone thinking it. 4 Frege does not yet quite say, as he would later, that the subject- matter of logic is truth, but he does say that logic ‘only becomes possible with the conviction that there is a difference between truth and untruth’. Following close on this, given that truth is objective, is that logic is not a branch of psychology. ‘No psychological inves- tigation can justify the laws of logic.’ But truth, which is on Frege’s presentation fundamental to logic, cannot be defi ned. ‘What true is,’ he says, ‘is indefi nable.’ Frege does not at this stage give an argu- ment to explain why truth is indefi nable, but he later held that any attempt to defi ne it would inevitably be circular, because one would have to understand the defi nition as being true . If what I have said about the dating of these notes is correct, then Frege formed some of his fundamental views about logic remarkably early. It is worth stressing, moreover, that the views just mentioned
2 M. Dummett, ‘Frege’s Kernsätze zur Logik’, in his Frege and Other Philosophers (Oxford: Clarendon Press, 1991). 3 See Frans Hovens, ‘Lotze and Frege: The dating of the “Kernsätze”’, History and Philosophy of Logic , 18 (1997 ), pp. 17–31. 4 PW , p. 174. Introduction 3 constitute a response to Lotze’s book, not a summary of it. It is true, for instance, that Lotze distinguished between logic and psych- ology, but his reason for doing so was that logic deals with the value of our thoughts whereas psychology deals with their genesis. This is obviously rather distant from Frege’s anti-psychologism, which was based on the objectivity of truth, not on its value. 5
Begriffsschrift Frege’s short book Begriffsschrift , which he published in 1879, marks the beginning of modern logic. The word ‘Begriffsschrift’ is not Frege’s own, but seems to have been coined by Humboldt in 1824: 6 it is usually translated ‘conceptual notation’ or ‘concept-script’. Here we shall call the book by its italicized German title and use the word unitalicized for the formal language it describes. The idea of a for- mal language is not itself new with Frege. But Frege’s Begriffsschrift has a number of features that were quite new in 1879. The ‘seventeen key sentences’ already show Frege treating logic as a subject whose central concern is truth, and regarding thoughts as of relevance to logic because they are what truth applies to. In the fi rst chapter of Begriffsschrift (‘Defi nition of the symbols’), Frege uses the term ‘judgeable content’ for what he previously called a thought. Moreover, he straightaway highlights an issue which was to remain of concern to him throughout his philosophical writings, namely that of identifying the structure of a judgeable content. Since what follows logically from The Greeks defeated the Persians at Plataea and what follows from The Persians were defeated by the Greeks at Plataea are identical, logic need not distinguish between these two propos- itions: they have the same judgeable content.
5 For the view that Frege should be seen as a neo-Kantian who was heavily infl u- enced by Lotze, see G. Gabriel, ‘Frege als Neukantianer’, Kant-Studien , 77 (1986 ), pp. 84–101. See also Hans Sluga, Gottlob Frege (London: Routledge and Kegan Paul, 1980 ). 6 See M. Beaney and Erich H. Reck (eds.), Gottlob Frege: Critical Assessments of Leading Philosophers (London: Routledge, 2005), vol II, p. 13. 4 Michael Potter
One of Frege’s innovations was to introduce a sign to mark the act of judging that something is the case. The sign he used was a ver- tical line which he called the judgement stroke. He also made use of a horizontal line which he called the content stroke, whose pur- pose was to turn what follows the stroke into a judgeable content. However, it is not entirely clear what this amounts to. A charit- able reader7 might see this as an implicit recognition that anything which expresses a judgeable content is of necessity complex, and hence in need of binding into a unity before it is capable of being judged. This is at any rate something which Frege was in his later writings keen to assert. A less charitable reader might think that if I have expressed a content then that is all there is to it: if the content I have expressed is judgeable, nothing more is needed to indicate that; if it is not, then preceding it with a stroke cannot make it so. Because in practice the vertical judgement stroke never occurs without being immediately followed by the horizontal content stroke, the combination of the two strokes inevitably came to be treated as a symbol in its own right. This is the origin of the turn- stile symbol that is ubiquitous in modern logic. However, it is worth stressing that this symbol, although it originated with Frege, is often now used in ways that he would not have recognized. In particular, Frege did not recognize a notion of conditional asser- tion, so would not have allowed the turnstile to be embedded, as in expressions such as A1 , A 2 , …, An B . The second major innovation which Frege’s conceptual notation encapsulates – and the one for which it is nowadays renowned – is a method for expressing multiple generality. However, Frege not only provides such a notation; he also displays a fi rm grasp of the prin- ciples that underlie it. He is clear, for instance, that in a quantifi ed expression such as ∀ x ∃ yRxy the letters ‘ x ’ and ‘ y ’ do not function like names. Frege conspicuously avoids the unfortunate usage inherited from mathematics which refers to them as variables: as he makes clear, they are not variable names but placeholders.
7 E.g. Peter Sullivan, ‘Frege’s logic’, in Dov M. Gabbay and John Woods (eds.), Handbook of the History of Logic , vol. III (Amsterdam: North-Holland, 2004), pp. 659–750. Introduction 5
If, in an expression (whose content need not be assertible), a simple or com- plex symbol occurs in one or more places and we imagine it as replace- able by another … then we call the part of the expression that shows itself invariant a function and the replaceable part its argument. 8 Notice, incidentally, that on this account predicates are a particu- lar kind of function, namely those derived from expressions whose content is assertible (i.e. from sentences). Frege’s choice of symbols shows awareness, too, of the desirabil- ity of notational economy. He has a sign for the universal quantifi er (nowadays always notated ∀), but he does not also have a sign for the existential quantifi er ∃, since ∃ x can easily be regarded as an abbre- viation for ~∀ x ~. The same economy is evident too in his choice of propositional connectives. He has signs for negation (nowadays ~) and for material implication (nowadays →) but not for the other connectives, which can be defi ned in terms of them. He also notes explicitly that he could just as well have used negation and conjunc- tion, although he stops just short of asserting that they are adequate to express all the others. Although he did not actually make use of the device of truth-tables in presenting his account, he might as well have done, as his presentation of the meanings of the logical connectives is explicitly truth-functional in character. The other thing for which the Begriffsschrift is especially not- able is the axiom system for predicate calculus contained in the second chapter (‘Representation and derivation of some judgements of pure thought’). He had already in the fi rst chapter formulated modus ponens From B → A and B derive A as well as the quantifi er rule From ∀ x (A → Φ(x )) derive A → ∀ x Φ(x ). Now he added the logical axioms, which he arranges in four groups: a →(b → a ) (c →(b → a ))→(( c → b )→(c → a )) (d →(b →a ))→(b →(d →a ))
8 Bs , §9. 6 Michael Potter