Table Of ContentHistory 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)
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You can receive future titles in this series as they are published by placing a standing order.
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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
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may be liable to criminal prosecution and civil claims for damages.
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in accordance with the Copyright, Designs and Patents Act 1988.
First published 2012 by
PALGRAVE MACMILLAN
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registered in England, company number 785998, of Houndmills, Basingstoke,
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ISBN 978-1-349-32025-7 ISBN 978-0-230-36015-0 (eBook)
DOI 10.1057/9780230360150
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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
