History of Analytic Philosophy Series Editor: Michael Beaney Titles include: Stewart Candlish THE RUSSELL/BRADLEY DISPUTE AND ITS SIGNIFICANCE FOR TWENTIETH-CENTURY PHILOSOPHY Annalisa Coliva MOORE AND WITTGENSTEIN Scepticism, Certainty and Common Sense Gregory Landini FREGE’S NOTATIONS What They Are and How They Mean Sandra Lapointe BOLZANO’S THEORETICAL PHILOSOPHY An Introduction Omar W. Nasim BERTRAND RUSSELL AND THE EDWARDIAN PHILOSOPHERS Constructing the World Graham Stevens THE THEORY OF DESCRIPTIONS Nuno Venturinha (editorr) WITTGENSTEIN AFTER HIS NACHLASS Forthcoming: Andrew Arana and Carlos Alvarez (editors) ANALYTIC PHILOSOPHY AND THE FOUNDATIONS OF MATHEMATICS Rosalind Carey RUSSELL ON MEANING The Emergence of Scientififi c Philosophy from the 1920s to the 1940s Giuseppina D’Oro REASONS AND CAUSES Causalism and Non-Causalism in the Philosophy of Action George Duke DUMMETT ON ABSTRACT OBJECTS Sébastien Gandon RUSSELL’S UNKNOWN LOGICISM A Study in the History and Philosophy of Mathematics Anssi Korhonen LOGIC AS UNIVERSAL SCIENCE Russell’s Early Logicism and Its Philosophical Context 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 (translatorr) Franz Prihonsky THE NEW ANTI-KANT Erich Reck (editorr) THE HISTORIC TURN IN ANALYTIC PHILOSOPHY Maria van der Schaar G.F. STOUT: ON THE PSYCHOLOGICAL ORIGIN OF ANALYTIC PHILOSOPHY Pierre Wagner (editorr) CARNAP’S IDEAL OF EXPLICATION AND NATURALISM History of Analytic Philosophy Series Standing Order ISBN 978–0–230–55409–2 (hardcover) Series Standing Order ISBN 978–0–230–55410–8 (paperback) (outside North America only) You can receive future titles in this series as they are published by placing a standing order. Please contact your bookseller or, in case of difficulty, write to us at the address below with your name and address, the title of the series and one of the ISBNs quoted above. Customer Services Department, Macmillan Distribution Ltd, Houndmills, Basingstoke, Hampshire RG21 6XS, England Also by Gregory Landini RUSSELL’S HIDDEN SUBSTITUTIONAL THEORY (OXFORD, 1998) WITTGENSTEIN’S APPRENTICE WITH RUSSELL (CAMBRIDGE 2007, 2009) RUSSELL (ROUTLEDGE, 2011) Frege’s Notations What They Are and How They Mean Gregory Landini University of Iowa, USA © Gregory Landini 2012 Softcover reprint of the hardcover 1st edition 2012 978-0-230-24774-1 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. 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. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2012 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010. Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries ISBN 978-1-349-32025-7 ISBN 978-0-230-36015-0 (eBook) DOI 10.1057/9780230360150 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress. 10 9 8 7 6 5 4 3 2 1 21 20 19 18 17 16 15 14 13 12 Dedicated to Nino Cocchiarella Contents Preface ix Author’s Note on the Use of Modern Logical Notations xi 1 Introduction 1 2 Frege’s Basic Logics (without Wertverläufe) 15 2.1 Quantification theoryversusCPLogic 15 2.2 Sentences arenott names 29 2.3 Judgeable contents 41 2.4 Basic Law IV 45 2.5 Begriffsschriftt andGrundlagen 49 2.6 Grundgesetze 52 2.7 Derivations of some theorems in the basic logic of theGrundgesetze 57 3 The Ancestral 62 3.1 The ancestral for objects 63 3.2 Proof of induction within CPLogic 65 3.3 Cardinality as a second-level concept 72 3.4 The problem of infinity 81 4 Wertverläufe 84 4.1 Numbers as objects 86 4.2 Grundlagen and Hume’s principle 98 4.3 Missing IV in the Grundlagen 107 4.4 I believe that for ‘extension of the concept’ we could write simply ‘concept’ 114 5 Analysis and Recomposition 119 5.1 Free variables and the turnstile 120 5.2 Parts of senses and the informativity of logic 130 5.3 Oratio Obliqua 144 5.4 Russell’s paradox of Sinn 150 vii viii Contents 6 Engaging Problems 157 6.1 Urelements 157 6.2 The Insand Outs of Frege’s Way Outt 158 6.3 The argument for referentiality 168 6.4 Whence the contradiction? 174 6.5 Frege’s Academy 182 Notes 184 Bibliography 187 Index 191 Preface I had the good fortune recently of attending an insightful lecture on the ever-vexing origins of even our most basic arithmetic knowledge. It afforded me an opportunity to remind everyone that some of us are still logicists. There are not many of us left. But among those that there are, each undoubtedly has a rather different conception of logicism. InPrincipia Mathematica, Whitehead and Russell departed from Frege’s original logicism, which maintained that numbers are objects. Principia’s logicism is a no-classes and thus a no-numbers theory. Wonderful schol- arly controversies surround the logicist foundations of arithmetic. A great many of them originate with Frege, who, according to Benacerraf, deserves the title of being both the first and also the last logicist. The many perspectives of Frege arise because his formal systems and nota- tions are beguiling. This book offers a new perspective on his notations and systems, comparing them to modern higher-order logic. Our new perspective is brought about by attention to three phases of Frege’s formal logic: Begriffsschriftt, Grundlagen and Grundgesetze. The source of this new perspective is derived primarily from the work of Nino Cocchiarella who for many years has argued that Frege’s theory of Wertverläufe is not a theory of sets but a theory of concept-correlation in which predication cannot be properly captured as set-membership. Cocchiarella developed this idea through the lenses of a formal recon- struction of Frege’s logic as a higher-order predicate calculus with con- cept-correlates standing in for the referents of nominalized predicates. Using correlation as his guide, Cocchiarella formulated various type- free theories of attributes (properties and relations in intension) that skirt Russell’s paradoxes in a way that is well-motivated by the thesis that predication is not membership. This book offers an investigation and a vindication of the historical faithfulness of some of Cocchiarella’s reconstructions of Frege’s work. The results of the investigation, how- ever, present Frege’s formal systems rather differently than that found in Cocchiarella’s systems. In particular, much turns on our allowing spe- 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