Russell's Hidden Substitutional Theory

Russell's Hidden Substitutional Theory

by Gregory Landini

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Overview

Russell's Hidden Substitutional Theory by Gregory Landini

This book explores an important central thread that unifies Russell's thoughts on logic in two works previously considered at odds with each other, the Principles of Mathematics and the later Principia Mathematica. This thread is Russell's doctrine that logic is an absolutely general science and that any calculus for it must embrace wholly unrestricted variables. The heart of Landini's book is a careful analysis of Russell's largely unpublished "substitutional" theory. On Landini's showing, the substitutional theory reveals the unity of Russell's philosophy of logic and offers new avenues for a genuine solution of the paradoxes plaguing Logicism.

Product Details

ISBN-13: 9780195116830
Publisher: Oxford University Press, USA
Publication date: 06/28/1998
Pages: 352
Product dimensions: 6.20(w) x 9.30(h) x 1.00(d)

Table of Contents

Introduction 3(1)
Quodlibet Ens Est Unum 3(4)
Overview 7(6)
PART I THE UNRESTRICTED VARIABLE 13(84)
1 Russell's Logicist Program
13(29)
Two Conceptions of Logicism: Frege and Russell
13(4)
Arithmetization
17(4)
Russell's Principle of Abstraction
21(9)
Logic as a Science
30(12)
2 The Logic of The Principles of Mathematics
42(27)
The Calculus for the Logic Propositions
42(6)
Russell's Definitions
48(4)
The Theory of Implication
52(2)
Quodlibet Ens Est Unum
54(3)
Denoting Concepts
57(6)
The Analysis of the Variable
63(6)
3 The New Theory of the Variable
69(28)
"On Fundamentals" Against Denoting Concepts
72(8)
An Argument Against Frege?
80(2)
The Variable as Primitive
82(7)
The Road to Substitution
89(8)
PART II TYPES AS LOGICAL GRAMMAR 97(102)
4 The Logic of Substitution
97(30)
Russell's Original Principles of Substitution
98(4)
The Basic Logic of Propositions
102(4)
Substitutional Principles
106(3)
Identity
109(3)
Proofs of Propositional Identities
112(15)
5 The "No Propositional Functions" Theory
127(19)
Substitution and Definite Descriptions
128(4)
Multiple Substitutions
132(3)
Comprehension and Identity
135(5)
Types as Logical Grammar
140(6)
6 The "No-Classes" Theory
146(30)
Classes as Extensional Propositional Functions
147(2)
Complex Prototypes and Extensionality
149(3)
The General Theory of Classes
152(13)
Comparison with Principia Mathematica
165(11)
7 The "No-Relations(e)" Theory
176(23)
Relations-in-Extension in Principia Mathematica
177(2)
Relations-in-Extension in the Substitutional Theory
179(4)
Cantor's Paradox of the Greatest Cardinal
183(7)
The Burali-Forti Paradox
190(9)
PART III RAMIFICATION 199(100)
8 Les Paradoxes de la Logique
199(35)
Three Paradoxes of Propositions
201(5)
Substitutional Manuscripts of April/May 1906
206(7)
Poincare's Vicious Circle Principle
213(3)
Logic without General Propositions
216(4)
The Statement Liar
220(4)
The Konig, Dixon, Berry, Richard, and Grelling Paradoxes
224(3)
Russell's "Mitigating Axiom"
227(4)
The Demise of "Les Paradoxes"
231(3)
9 Mathematical Logic as Based on the Theory of Types
234(21)
Orders of Propositions
235(5)
Substitutional Logic cum Orders of Propositions
240(6)
Predicativity and Reducibility
246(5)
Paradoxes of Propositions Avoided
251(4)
10 The Logic of Principia Mathematica
255(44)
The Formal System of Principia (cum *10)
255(3)
The Perils of Typical Ambiguity
258(9)
Orders within Types or Types within Orders?
267(5)
The Doctrine of the Unlimited Variable
272(3)
Poincare's Vicious Circle Principle
275(4)
The Philosophical Justification of the Type Part of an Order/Type Index
279(2)
The Philosophical Justification of the Order Part of an Order/Type Index
281(6)
The Multiple-Relation Theory of Judgment
287(4)
What Is Logic?
291(3)
What Logic Is Not
294(5)
Appendix A: Proof of the Peano Postulates 299(15)
Appendix B: Axioms, Theorems, and Definitions 314(11)
Bibliography 325(8)
Index 333

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