Translations from the Philosophical Writings of Gottlob Frege EDITED BY PETER GEACH Senior Lect~rrerin P/~ilosup/ry,C~ri~,ersity qf Bir~ni~!flt~rin AND MAX BLACK Profeaor of PhiIl~s,scrp/r~,Ci~rr~ell Cl~~ivcrsity BASIL BLACKWELL OXFORD . I 960 First Edition 1952 Second Edition 1960 PRINTED IN GREAT BRITAIN FOR BASIL BLACKWELL & MOTT LTD. BY THE COMPTON PRINTING WORKS (LONDON) LTD., LONDON, N.I AND BOUND BY THE KEMP HALL BINDERY, OXFORD PREFACE ONEaim of this volume is to make available to English readers FregeYsmere important logical essays, which have long been buried in various German periodicals (mostly now defunct). ~esidesthese we have given certain extracts from his Grund- gesetze der Arithmetik; these can be understood in the light of the essays, without the reader's needing to follow the chain of deduc- tion in the Grundgesetze. Special attention should be paid to Frege's discussion of Russell's paradox in the appendix to Vol. ii of the Grundgesetze. It is dis- creditable that logical works should repeat the legend of Frege's abandoning his researches in despair when faced with the paradox; in fact he indicates a line of solution, which others might well have followed out farther. The authorship of the various versions is stated in the table of contents. All versions have been revised with a view to uniform rendering of Frege's special terms; a glossary of these terms is supplied. Footnote flags such as A,B, relate to translators' footnotes; other footnotes are Frege's own. Acknowledgments are due to the editors of Mind and the Philosophical Review, for permission to use versions first published there. Acknowledgment is also due for use of the translations made from Vol. i of the Grundgesetze by P. E. B. Jourdain and J. Stachelroth (which were first published in the Monist, 1915-17), to the owners of the copyright, whom it has un- fortunately proved impossible to trace. Professor Ryle and Lord Russell have been most helpful by lending works of Frege that' were otherwise almost unobtainable. MAX BLACK. P. T. GEACH. NOTE TO SECOND EDITION WEhave made a number of corrections for this second impression. Mr. Michael Dummett was kind enough to check the whole translation and give us his advice. A new footnote on p. 243 refers to recent work on 'Frege's Way Out'. M. B. P. T. G. CONTENTS PAGE preface . v Note to Second Edition . vi Glossary . BEGRIFFSSCHRFT(Chapter I). Translated by P. T. GEACH. FUNCTIONAND CONCEPT.Translated by P. T. GEACH . ON CONCEPTAND OBJECT.Translated by P. T. GEACH ON SENSEAND REFERENCE.Translated by MAXBLACK . Illustrative Extracts &om Frege's Review of Husserl's Philosophie der Arithmetik. Translated by P. T. GEACH. A CRITICALELUCIDATION OF SOME POINTSIN E. SCHROE DER'S ALGEBRA DER LOGK. Translated by P. T. GEACH. WHATIS A FUNCTION?Translated by P. T. GEACH . NEGATION.Translated by P. T. GEACH . TRANSLATIONOF PARTS OF FREGE'S GRUNDG~ZE DER ARITHMETIK: Selections &om Volume I. Translated by P. E. B. JOURDAINand J. STACHELROTH. (Translated by P. T. GEACH)I1 (Vol. 11, §§13g-44,1467) FREGEAGAINST THE FORMALISTS(~01. II, §§86-137) . Translated by MAXBLACK FREGEON RUSSELL'SPARADOX (Appendix to Vol. 11) . Translated by P. T. GEACH GLOSSARY THEspecial terms used by Frege have been rendered as follows: Anschauung intuition, experiencc Art des Gegebenseins mode of prcsentation bedeutenl stand for Bedeutungl reference [occasionally: what . stands for] Begd concept [in the logical sense] Behauptungssatz declarative sentence beurtheilbarer Inhalt possible content of judgment eigentliche Zahl actual number [as opposed to a nunieral] Figur2 figure formal formal(ist) Gebild2 character, structure Gedankes thought Gegenstand object gewiihnliche (Bedeutung) customary (reference) gleich, GleichheitP equal, equality inhaltlich meaningful [an epithet of arith- metic interpreted in a non- formalist fashion] objektiv6 ob'ective Rechenspiel c1 culating game [~rithmeticon the formahst view] Rechnungsart arithmetical operation [addition, subtraction, etc.] Satz sentence, proposition, theorem, clause [according to context] The natural rendering of these words would be 'mean' and 'meaning'; this rendering is actually required for their occurrence in German works quoted by Frege, and for his own use of the words when alluding to such quotations. But 'meaning' in ordinary English often answers to Frege's Sinrt rather than Bedeutung. Russell's 'indicate' and 'indication' are barred because we need 'indicate' rather for attdeuten. The renderings given here seem to be the simplest means of expressing Frege's thought faithfully. Philosophical technicalities, like 'referent' or 'denotation' or 'nominatum,' would give a misleading impression of Frege's style. 'These two words are used to refer to supposedly meaningless marks, such as numerals are on the formalist theory. Both words must often also be rendered '(chess) piece.' Frege regards a 'thought' as sharable by many thinkers, and thus as objective. Frege remarks concerning these words that Be takes them to express strict identity, but other mathematicians disagree; 'equality' therefore seems preferable to 'identity.' It must be carefully noticed that the German words rendered 'object' and 'objective' are not connected in etymology or in Frege's mind. Concepts are fundamentally different from objects; but they are objective, i.e. not private to a particular thinker. ix x GLOSSARY Sinn sense Stufe level [of a concept or function] (unbestirnrnt) andeutenl indicate (indehtely) ungerade (Bedeutung) indirect (reference) [i.e. pertaining to words in oratio obliquo or virtual oratio obliaual1 a 'unsaturated' idea, image [regarded as essentially private] Wahrheitswerth truth-value Werthverlauf value-range, range of values [of a function] 1 Thic term is applied to pronouns (e.g. relative pronouns) and also to letters used as variables. (Frege Wedthe term 'variable'; cf. hi essay What is a Function?) This is Frege's term for such fragmentary expressions as '-conquered Gaul' or 'the capital of -'; and also for their Bedeutung, i.e. what they stand for. He may well have had in mind unsaturated molecules, which, without dissolution of their existing structure, can take up more atoms. To emphasize, as Frege does, the metaphorical nature of the term, we always write it in quotation marks. Contrasted--%th Begrif and Gedonke (q.v.). BEGRIFFSSCHRIFT a formalized Language of pure Thought modelled upon the Language of Arithmetic First published in 1879 I. EXPLANATION OF THE SYMBOLS THEsymbols used in the general theory of magnitude fall into two kinds. The first consists of the letters; each letter represents either an indeterminate number or an indeterminate function. This indeterminateness makes it possible to express by means of letters the general validity of propositions; e.g. : (a + b)c = ac + bc. The other kind contains such symbols as +, -, 4,o, I, 2; each of these has its own proper meaning.^ I adopt this fundamental idea of distinguishing two kinds of symbols (which unfortunately is not smctly carried out in the theory of magnitude*) in order to make it generally applicable in the wider domain of pure thought. Accordingly, I divide all the symbols I use into those that can be taken to mean various things and those that have a fully determinate sense. The first kind are letters, and their main task is to be the expression ofgenerality. For all their indeterminateness, it must be laid down that a letter retains in a given context the meaning once given to it. $ 2. Judgment A judgment is always to be +expressed by means of the sign This stands to the left of the sign or complex of signs in which the content of the judgment is given. ~fwe omit the little vertical p. 21 stroke at the left end of the horizontal stroke, then the * Consider the symbols 1, log, sin, Lim. A I render Bedeutung by 'meaning' or 'significance' throughout this work. since Frege had not yet begun to use it in his own special sense. Various other words, e.g. Begg ('concept') and Vorstellung ('idea') are also used in a less precise sense than he later gave to them. 1 2 TRANSLATIONS FROM THE WRITINGS OF GOTTLOB FREGE judgment is to be transformed into a men complex of ideas; the author is not expressing his recognition or non-recognition of the truth of this. Thus, let mean the judgment: 'unlike magnetic poles attract one another.' In that case A will not express this judgment; it d be intended just to produce in the reader the idea of the mutual attraction of unlike magnetic poles-so that, e.g., he may make inferences from this thought and test its correctness on the basis of these. In this case we qualify the expression with the words 'the circumstance that' or 'the proposition that.' Not every content can be turned into a judgment by prefixing 1- 1- to a symbol for the content; e.g. the idea 'house' cannot. Hence we distinguish contents that are, and contents that are not, possible contents ofjudgment.? As a constituent of the sign +the horizontal stroke combines the symbolsfollowing it into a whole; assertion, which is expressed by the vertical stroke at the left end ofthe horizontal one, relates to the whole thusforrned. The horizontal stroke I wish to call the content-stroke, and the vertical the judgment-stroke. The content-stroke is also to serve the purpose of relating any sign whatsoever to the whole formed by the symbols f~llowingthe stroke. The content of what follows the content-stroke must altuays be a possible content ofjudgment. A distinction of subject and predicate finds no place in my way of representing a judgment. In order to justify this, let me observe that there are two ways in which the content of two judgments may differ; it may, or it may not, be the case that all inferences that can be drawn from the first judgment when combined with * I use Greek uncials as abbreviations; if I give no special explanation of them I wish the reader to supply an appropriate sense.