Frege's notations: what they are and how they mean

208 Pages·2012·5.114 MB·English

by  Gregory Landini

History of Analytic Philosophy
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THE RUSSELL/BRADLEY DISPUTE AND ITS SIGNIFICANCE FOR
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MOORE AND WITTGENSTEIN 20
Scepticism, Certainty and Common Sense ct -
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Gregory Landini on
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FREGE’S NOTATIONS e
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What They Are and How They Mean gra
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Sandra Lapointe y -
BOLZANO’S THEORETICAL PHILOSOPHY ar
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An Introduction Li
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BERTRAND RUSSELL AND THE EDWARDIAN PHILOSOPHERS niv
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ANALYTIC PHILOSOPHY AND THE FOUNDATIONS OF MATHEMATICS ect.
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RUSSELL ON MEANING a
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The Emergence of Scientifi c Philosophy from the 1920s to the 1940s al
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REASONS AND CAUSES m
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George Duke ateri
DUMMETT ON ABSTRACT OBJECTS ht m
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Sébastien Gandon yri
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RUSSELL’S UNKNOWN LOGICISM o
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A Study in the History and Philosophy of Mathematics
Anssi Korhonen
LOGIC AS UNIVERSAL SCIENCE
Russell’s Early Logicism and Its Philosophical Context
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ALFRED TARSKI
Philosophy of Language and Logic
Consuelo Preti
THE METAPHYSICAL BASIS OF ETHICS
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THE NEW ANTI-KANT 15-
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THE HISTORIC TURN IN ANALYTIC PHILOSOPHY n
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History of Analytic Philosophy ect.
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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)
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Frege’s Notations
What They Are and How They Mean
Gregory Landini 2
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© Gregory Landini 2012
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Dedicated to Nino Cocchiarella
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Contents
Preface ix
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Author’s Note on the Use of Modern Logical Notations xi 08-
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1 Introduction 1 ct -
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2 Frege’s Basic Logics (without Wertverläufe) 15 Co
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2.1 Quantification theory versus CPLogic 15 av
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2.2 Sentences are not names 29 al
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2.3 Judgeable contents 41 y -
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2.4 Basic Law IV 45 br
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2.5 Begriffsschrift and Grundlagen 49 sity
2.6 Grundgesetze 52 er
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2.7 Derivations of some theorems in the basic U
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logic of the Grundgesetze 57 ol
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3 The Ancestral 62 St
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3.1 The ancestral for objects 63 d t
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3.2 Proof of induction within CPLogic 65 ns
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3.3 Cardinality as a second-level concept 72 m - li
3.4 The problem of infinity 81 o
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4 Wertverläufe 84 ne
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4.1 Numbers as objects 86 c
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4.2 Grundlagen and Hume’s principle 98 gra
4.3 Missing IV in the Grundlagen 107 pal
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write simply ‘concept’ 114 om
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5 Analysis and Recomposition 119 eri
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5.1 Free variables and the turnstile 120 m
5.2 Parts of senses and the informativity of logic 130 ght
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5.3 Oratio Obliqua 144 p
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5.4 Russell’s paradox of Sinn 150
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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
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Bibliography 187 ct -
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Index 191 on
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Preface
I had the good fortune recently of attending an insightful lecture on
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the ever-vexing origins of even our most basic arithmetic knowledge. 8-1
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It afforded me an opportunity to remind everyone that some of us are 5-
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In Principia Mathematica, Whitehead and Russell departed from Frege’s Co
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original logicism, which maintained that numbers are objects. Principia’s av
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logicism is a no-classes and thus a no-numbers theory. Wonderful schol- al
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arly controversies surround the logicist foundations of arithmetic. A y -
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great many of them originate with Frege, who, according to Benacerraf, br
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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
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tions are beguiling. This book offers a new perspective on his notations U
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and systems, comparing them to modern higher-order logic. ol
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Our new perspective is brought about by attention to three phases oc
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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
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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
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in which predication cannot be properly captured as set-membership. c
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Cocchiarella developed this idea through the lenses of a formal recon- ne
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struction of Frege’s logic as a higher-order predicate calculus with con- c
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cept-correlates standing in for the referents of nominalized predicates. gra
Using correlation as his guide, Cocchiarella formulated various type- pal
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free theories of attributes (properties and relations in intension) that w
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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
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reconstructions of Frege’s work. The results of the investigation, how- ht
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ever, present Frege’s formal systems rather differently than that found in yri
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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
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