Plural logic / Alex Oliver and Timothy Smiley.

Format:

Book

Author/Creator:

Oliver, Alex.

Contributor:

Smiley, T. J. (Timothy John)

Language:

English

Subjects (All):

Logic.

Physical Description:

xiv, 336 pages ; 24 cm

Place of Publication:

Oxford : Oxford University Press, [2013]

Summary:

Alex Oliver and Timothy Smiley provide an account of plural logic. They argue that there is such a thing as genuinely plural denotation in logic, and expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists.

Contents:

1 The project 1

1.1 Plural phenomena 1

1.2 Plurals in mathematics and logic 4

1.3 Strategies for a logic of plurals 7

1.4 Manoeuvres of a conservative logician: a case study 9

1.5 Plan of the book 12

2 History 15

2.1 Distributive and collective predication 16

2.2 Mill 19

2.3 Frege 20

2.4 Lesniewski 22

2.5 Russell 25

2.6 Russell to Boolos 30

3 Changing the subject 33

3.1 Changing the subject to sets 34

3.2 Uniformity 35

3.3 Against the naive version of changing the subject 37

3.4 Changing the subject and the predicate 37

3.5 The pain of paradox 40

3.6 Changing the subject is simply not on 42

3.7 Changing the subject in practice 42

Appendix. Events to the rescue? 44

4 Predicative analyses 51

4.1 Russell's theory of plural descriptions 52

4.2 Other predicative analyses 56

4.3 The equivocity objection 58

4.4 Boolos's 'reciprocal illumination' 60

4.5 Boolos's second-order representation of plurals 61

4.6 Boolos and equivocity 63

4.7 Rumfitt's purified Boolosian scheme 64

Appendix. Dummett and Frege on plurals 65

5 Terms-singular and plural 73

5.1 Terms 74

5.2 Varieties of singular term 76

5.3 Varieties of plural term 78

5.4 The Russellian idea of singular term 80

5.5 Nested terms 83

5.6 Empty terms 86

5.7 Predication 89

6 The indeterminacy of plural denotation 93

6.1 Two accounts of denotation 94

6.2 Plural descriptions: some elementary facts 95

6.3 Which account is correct? 96

6.4 Dissenting voices I 97

6.5 Free relatives and wh-questions 99

6.6 Dissenting voices II 101

6.7 Indeterminacy 102

7 Some basic ideas of plural logic 105

7.1 Variables and quantification 106

7.2 Inclusion 108

7.3 Distributive predicates 112

7.4 Collective predicates 113

8 Plural descriptions 119

8.1 A theory of descriptions 120

8.2 Formalizing the theory, definability, and ineliminability 123

8.3 Exercises for the reader 125

8.4 Superplurals 127

Appendix. Sharvy's theory of descriptions 129

9 Multivalued functions 139

9.1 Varieties of function 140

9.2 Mathematicians and logicians 143

9.3 Functions and relations 145

9.4 The ambiguity objection 147

9.5 Proposals for eliminating them 150

10 Lists 153

10.1 Lists as terms 153

10.2 Term-forming 'and' 154

10.3 Lists as strings 158

10.4 Places and positions 162

10.5 Terms vs strings in tire literature 165

10.6 Analyses assessed 167

Appendix. In defence of multigrade predicates 172

11 Singular logic 181

11.1 Topic neutrality 182

11.2 Syntax 189

11.3 Axioms 191

11.4 Metatheorems 192

11.5 Semantics 193

Appendix. Soundness and completeness proofs 197

12 Mid-plural logic 207

12.1 Ideas 208

12.2 Syntax 211

12.3 Axioms 212

12.4 Metatheorems 213

12.5 Semantics 214

12.6 Relation of mid-plural logic to singular logic 218

12.7 The algebra of plurals 219

Appendix. Soundness and completeness proofs 222

13 Full plural logic 233

13.1 Syntax 233

13.2 Semantics 235

13.3 Expressive power 236

13.4 Partial axiomatization 240

13.5 Comprehension 242

13.6 Choice 243

14 Cantorian set theory 245

14.1 Plurals and sets 246

14.2 Cantor's collections 248

14.3 The empty set 250

14.4 Singletons 254

14.5 Ur-elements 258

14.6 A superstructure, not a foundation 260

14.7 Iterative Cantorian set theory 264

14.8 Using the plural in developing conventional set theory 269.

Notes:

Includes bibliographical references (pages [301]-312) and index.

ISBN:

0199570426

9780199570423

OCLC:

813004867