The Principles of Mathematics Revisited

Jaakko Hintikka

doi:10.1017/CBO9780511624919

This book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. It is therefore a book of critical importance to logical theory. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. The famous impossibility results by Gödel and Tarski that have dominated the field for the last sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with the existence of sets and other higher-order entities.

‘Hintikka’s new logic is highly original and will prove appealing to logicians, philosophers of mathematics, and mathematicians concerned with the foundations of the discipline.’

Source: Studia Logica

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Introduction
pp vii-xii
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1 - The functions of logic and the problem of truth definition
pp 1-21
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2 - The game of logic
pp 22-45
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3 - Frege's fallacy foiled: Independence-friendly logic
pp 46-71
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4 - The joys of independence: Some uses of IF logic
pp 72-87
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5 - The complexities of completeness
pp 88-104
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6 - Who's afraid of Alfred Tarski? Truth definitions for IF first-order languages
pp 105-130
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7 - The liar belied: Negation in IF logic
pp 131-162
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8 - Axiomatic set theory: Fraenkelstein's monster?
pp 163-182
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9 - IF logic as a framework for mathematical theorizing
pp 183-210
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10 - Constructivism reconstructed
pp 211-234
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11 - The epistemology of mathematical objects
pp 235-253
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Appendix (by Gabriel Sandu)
pp 254-270
- By Gabriel Sandu
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References
pp 271-280
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Index of names
pp 281-284
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Index of subjects and titles
pp 285-288
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The Principles of Mathematics Revisited

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Introduction
pp vii-xii

1 - The functions of logic and the problem of truth definition
pp 1-21

2 - The game of logic
pp 22-45

3 - Frege's fallacy foiled: Independence-friendly logic
pp 46-71

4 - The joys of independence: Some uses of IF logic
pp 72-87

5 - The complexities of completeness
pp 88-104

6 - Who's afraid of Alfred Tarski? Truth definitions for IF first-order languages
pp 105-130

7 - The liar belied: Negation in IF logic
pp 131-162

8 - Axiomatic set theory: Fraenkelstein's monster?
pp 163-182

9 - IF logic as a framework for mathematical theorizing
pp 183-210

10 - Constructivism reconstructed
pp 211-234

11 - The epistemology of mathematical objects
pp 235-253

Appendix (by Gabriel Sandu)
pp 254-270

References
pp 271-280

Index of names
pp 281-284

Index of subjects and titles
pp 285-288

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