History of Analytic Philosophy Series Editor: Michael Beaney Titles include: Stewart Candlish THE RUSSELL/BRADLEY DISPUTE AND ITS SIGNIFICANCE FOR TWENTIETH-CENTURY PHILOSOPHY 2 1 8- Annalisa Coliva 5-0 1 MOORE AND WITTGENSTEIN 20 Scepticism, Certainty and Common Sense ct - e n Gregory Landini on C FREGE’S NOTATIONS e v What They Are and How They Mean gra al P Sandra Lapointe y - BOLZANO’S THEORETICAL PHILOSOPHY ar br An Introduction Li y Omar W. Nasim ersit BERTRAND RUSSELL AND THE EDWARDIAN PHILOSOPHERS niv Constructing the World m U ol Graham Stevens kh c THE THEORY OF DESCRIPTIONS Sto o Nuno Venturinha (editor) d t e WITTGENSTEIN AFTER HIS NACHLASS s n e c Forthcoming: m - li o Andrew Arana and Carlos Alvarez (editors) c ANALYTIC PHILOSOPHY AND THE FOUNDATIONS OF MATHEMATICS ect. n n o Rosalind Carey c e v RUSSELL ON MEANING a gr The Emergence of Scientifi c Philosophy from the 1920s to the 1940s al p w. Giuseppina D’Oro w w REASONS AND CAUSES m o Causalism and Non-Causalism in the Philosophy of Action al fr George Duke ateri DUMMETT ON ABSTRACT OBJECTS ht m g Sébastien Gandon yri p RUSSELL’S UNKNOWN LOGICISM o C A Study in the History and Philosophy of Mathematics Anssi Korhonen LOGIC AS UNIVERSAL SCIENCE Russell’s Early Logicism and Its Philosophical Context 10.1057/9780230360150 - Frege's Notations, Gregory Landini 99778800223300__224477774411__0011__pprreexxiiii..iinndddd ii 1122//11//22001111 77::0055::0088 PPMM Douglas Patterson ALFRED TARSKI Philosophy of Language and Logic Consuelo Preti THE METAPHYSICAL BASIS OF ETHICS The Early Philosophical Development of G.E.Moore Sandra Lapointe (translator) 2 1 Franz Prihonsky 08- THE NEW ANTI-KANT 15- 0 2 Erich Reck (editor) ct - e THE HISTORIC TURN IN ANALYTIC PHILOSOPHY n n o C Maria van der Schaar e v a G.F. STOUT: ON THE PSYCHOLOGICAL ORIGIN OF ANALYTIC PHILOSOPHY gr al P Pierre Wagner (editor) y - CARNAP’S IDEAL OF EXPLICATION AND NATURALISM ar br Li y sit er v ni U m ol h k c o St o d t e s n e c m - li o c History of Analytic Philosophy ect. n Series Standing Order ISBN 978–0–230–55409–2 (hardcover) n o Series Standing Order ISBN 978–0–230–55410–8 (paperback) ec v (outside North America only) gra You can receive future titles in this series as they are published by placing a standing order. pal Please contact your bookseller or, in case of difficulty, write to us at the address below with ww. your name and address, the title of the series and one of the ISBNs quoted above. m w CHuamstopmsheirr eS RerGv2ic1e 6sX DSe, pEanrgtlmanednt, Macmillan Distribution Ltd, Houndmills, Basingstoke, al fro eri at m ht g yri p o C Also by Gregory Landini RUSSELL’S HIDDEN SUBSTITUTIONAL THEORY (OXFORD, 1998) WITTGENSTEIN’S APPRENTICE WITH RUSSELL (CAMBRIDGE 2007, 2009) RUSSELL (ROUTLEDGE, 2011) 10.1057/9780230360150 - Frege's Notations, Gregory Landini 99778800223300__224477774411__0011__pprreexxiiii..iinndddd iiii 1122//11//22001111 77::0055::0088 PPMM Frege’s Notations What They Are and How They Mean Gregory Landini 2 1 University of Iowa, USA 08- 5- 1 0 2 ct - e n n o C e v a gr al P y - ar br Li y sit er v ni U m ol h k c o St o d t e s n e c m - li o c ct. e n n o c e v a gr al p w. w w m o al fr eri at m ht g yri p o C 10.1057/9780230360150 - Frege's Notations, Gregory Landini 99778800223300__224477774411__0011__pprreexxiiii..iinndddd iiiiii 1122//11//22001111 77::0055::0088 PPMM © Gregory Landini 2012 All rights reserved. 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No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6–10 Kirby Street, London EC1N 8TS. 12 8- Any person who does any unauthorized act in relation to this publication 5-0 may be liable to criminal prosecution and civil claims for damages. 01 2 The author has asserted his right to be identified as the author of this work ct - e in accordance with the Copyright, Designs and Patents Act 1988. n n o First published 2012 by eC v PALGRAVE MACMILLAN gra Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, Pal registered in England, company number 785998, of Houndmills, Basingstoke, y - Hampshire RG21 6XS. brar Li Palgrave Macmillan in the US is a division of St Martin’s Press LLC, y 175 Fifth Avenue, New York, NY 10010. ersit v Palgrave Macmillan is the global academic imprint of the above companies ni U and has companies and representatives throughout the world. m ol Palgrave® and Macmillan® are registered trademarks in the United States, kh c the United Kingdom, Europe and other countries o St ISBN: 978–0–230–24774–1 d to e This book is printed on paper suitable for recycling and made from fully ns managed and sustained forest sources. 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Printed and bound in Great Britain by w w CPI Antony Rowe, Chippenham and Eastbourne m o al fr eri at m ht g yri p o C 10.1057/9780230360150 - Frege's Notations, Gregory Landini 99778800223300__224477774411__0011__pprreexxiiii..iinndddd iivv 1122//11//22001111 77::0055::0088 PPMM Dedicated to Nino Cocchiarella 2 1 8- 0 5- 1 0 2 ct - e n n o C e v a gr al P y - ar br Li y sit er v ni U m ol h k c o St o d t e s n e c m - li o c ct. e n n o c e v a gr al p w. w w m o al fr eri at m ht g yri p o C 10.1057/9780230360150 - Frege's Notations, Gregory Landini 99778800223300__224477774411__0011__pprreexxiiii..iinndddd vv 1122//11//22001111 77::0055::0099 PPMM 2 1 8- 0 5- 1 0 2 ct - e n n o C e v a gr al P y - ar br Li y sit er v ni U m ol h k c o St o d t e s n e c m - li o c ct. e n n o c e v a gr al p w. w w m o al fr eri at m ht g yri p o C 10.1057/9780230360150 - Frege's Notations, Gregory Landini 99778800223300__224477774411__0011__pprreexxiiii..iinndddd vvii 1122//11//22001111 77::0055::0099 PPMM Contents Preface ix 2 1 Author’s Note on the Use of Modern Logical Notations xi 08- 5- 1 0 2 1 Introduction 1 ct - e n n 2 Frege’s Basic Logics (without Wertverläufe) 15 Co e 2.1 Quantification theory versus CPLogic 15 av gr 2.2 Sentences are not names 29 al P 2.3 Judgeable contents 41 y - ar 2.4 Basic Law IV 45 br Li 2.5 Begriffsschrift and Grundlagen 49 sity 2.6 Grundgesetze 52 er v ni 2.7 Derivations of some theorems in the basic U m logic of the Grundgesetze 57 ol h k c o 3 The Ancestral 62 St o 3.1 The ancestral for objects 63 d t e 3.2 Proof of induction within CPLogic 65 ns e c 3.3 Cardinality as a second-level concept 72 m - li 3.4 The problem of infinity 81 o c ct. 4 Wertverläufe 84 ne n o 4.1 Numbers as objects 86 c e v 4.2 Grundlagen and Hume’s principle 98 gra 4.3 Missing IV in the Grundlagen 107 pal w. 4.4 I believe that for ‘extension of the concept’ we could w w write simply ‘concept’ 114 om al fr 5 Analysis and Recomposition 119 eri at 5.1 Free variables and the turnstile 120 m 5.2 Parts of senses and the informativity of logic 130 ght yri 5.3 Oratio Obliqua 144 p o C 5.4 Russell’s paradox of Sinn 150 vii 10.1057/9780230360150 - Frege's Notations, Gregory Landini 99778800223300__224477774411__0011__pprreexxiiii..iinndddd vviiii 1122//11//22001111 77::0055::0099 PPMM viii Contents 6 Engaging Problems 157 6.1 Urelements 157 6.2 The Ins and Outs of Frege’s Way Out 158 6.3 The argument for referentiality 168 6.4 Whence the contradiction? 174 6.5 Frege’s Academy 182 2 1 8- Notes 184 5-0 1 0 2 Bibliography 187 ct - e n Index 191 on C e v a gr al P y - ar br Li y sit er v ni U m ol h k c o St o d t e s n e c m - li o c ct. e n n o c e v a gr al p w. w w m o al fr eri at m ht g yri p o C 10.1057/9780230360150 - Frege's Notations, Gregory Landini 99778800223300__224477774411__0011__pprreexxiiii..iinndddd vviiiiii 1122//11//22001111 77::0055::0099 PPMM Preface I had the good fortune recently of attending an insightful lecture on 2 the ever-vexing origins of even our most basic arithmetic knowledge. 8-1 0 It afforded me an opportunity to remind everyone that some of us are 5- 1 0 satriel,l leoagcihci sutns.d Tohuebrtee darlye nhoats ma arnayth oefr uds ilfefeftr.e Bnut tc aomncoenpgt itohno soef thloagti cthisemre. ect - 2 n n In Principia Mathematica, Whitehead and Russell departed from Frege’s Co e original logicism, which maintained that numbers are objects. Principia’s av gr logicism is a no-classes and thus a no-numbers theory. Wonderful schol- al P arly controversies surround the logicist foundations of arithmetic. A y - ar great many of them originate with Frege, who, according to Benacerraf, br Li deserves the title of being both the first and also the last logicist. The sity many perspectives of Frege arise because his formal systems and nota- er v ni tions are beguiling. This book offers a new perspective on his notations U m and systems, comparing them to modern higher-order logic. ol h k Our new perspective is brought about by attention to three phases oc St of Frege’s formal logic: Begriffsschrift, Grundlagen and Grundgesetze. The o source of this new perspective is derived primarily from the work of ed t s n Nino Cocchiarella who for many years has argued that Frege’s theory ce of Wertverläufe is not a theory of sets but a theory of concept-correlation m - li o in which predication cannot be properly captured as set-membership. c ct. Cocchiarella developed this idea through the lenses of a formal recon- ne n o struction of Frege’s logic as a higher-order predicate calculus with con- c e v cept-correlates standing in for the referents of nominalized predicates. gra Using correlation as his guide, Cocchiarella formulated various type- pal w. free theories of attributes (properties and relations in intension) that w w skirt Russell’s paradoxes in a way that is well-motivated by the thesis om that predication is not membership. This book offers an investigation al fr and a vindication of the historical faithfulness of some of Cocchiarella’s ateri m reconstructions of Frege’s work. The results of the investigation, how- ht g ever, present Frege’s formal systems rather differently than that found in yri p Cocchiarella’s systems. In particular, much turns on our allowing spe- Co cial bindable structured variables and extensionality axioms for Frege’s hierarchy of levels of functions. This brings to the fore Frege’s unique formal language in which both type freedom and type regimentation are expressible. We find that Frege’s general theory of function-correlation rests on an identity ├ u ^ z´fz = fu that can only be properly represented ix 10.1057/9780230360150 - Frege's Notations, Gregory Landini 99778800223300__224477774411__0011__pprreexxiiii..iinndddd iixx 1122//11//22001111 77::0055::0099 PPMM