Reductionism is one of those philosophical myths that are either enthusiastically embraced or wholeheartedly rejected. And, like all other philosophical myths, it rarely gets serious consideration. "Reasoning About Theoretical Entities" strives to give reductionism its day in court, as it were, by explicitly developing several versions of the reductionist project and assessing their merits within the framework of modern symbolic logic. Not since the days of Carnap's "Aufbau" has reductionism received such close attention (albeit in a necessarily restricted and regimented setting such as that of modern mathematical logic). As such this book fills a void in the philosophical literature and presents a challenge to every would-be (anti-)reductionist.
Author(s): Thomas Forster
Series: Advances in Logic 3
Publisher: World Scientific
Year: 2003
Language: English
Pages: 98
Title ......Page 3
Copyright ......Page 4
Preface ......Page 5
Contents ......Page 7
1. Introduction ......Page 9
1.1 Interpretations ......Page 16
2. Definite Descriptions ......Page 21
2.1 Formal definition of the interpretation ......Page 23
2.2 Functions from singular descriptions ......Page 26
2.3 Definite descriptions and modal realism ......Page 27
2.3.1 Carnap's treatment ......Page 29
3.1 Congruence relations ......Page 31
3.2 Extending the language ......Page 33
3.2.1 Implementations of languages with a canonical simulation ......Page 36
3.2.2 An illustration: utilities ......Page 38
3.2.3 Some standard mathematical examples ......Page 40
3.3 Second-order and higher-order theories ......Page 41
3.3.1 Third- and higher-order virtual entities ......Page 43
4. Cardinal Arithmetic ......Page 45
4.1 The languages of set theory and arithmetic ......Page 46
4.2 The canonical simulation ......Page 47
4.2.2 Multisets of cardinals and the multiplicative axiom ......Page 49
4.2.3 Ramsey Theory ......Page 52
4.3 Virtual illfounded sets ......Page 53
4.3.1 Irredundant trees ......Page 54
5.1 Doubly virtual cardinals ......Page 57
5.2.1 The parts that virtualism cannot reach ......Page 60
5.2.1.2 Exponentiation ......Page 61
5.3 Untyped invariant arithmetic ......Page 62
5.4 Implementation-insensitivity ......Page 64
5.5 Iterated virtuality and reflection ......Page 69
6. Ordinals ......Page 71
6.1 The elementary theory of wellorderings ......Page 72
6.2 The language of ordinal arithmetic ......Page 77
6.3 Ordinals of wellorderings of sets of ordinals ......Page 79
6.3.1 What does the T function do? ......Page 82
6.3.2 Hartogs' theorem ......Page 86
6.4 Implementations of ordinal arithmetic ......Page 87
6.4.1 The Russell-Whitehead implementation and Scott's trick ......Page 88
6.4.2 The Von Neumann implementation ......Page 89
6.4.2.1 The Burali-Forti paradox for Von Neumann ordinals ......Page 91
Bibliography ......Page 93
Index of Definitions ......Page 96
Index ......Page 97
