Foundations of Intensional Semantics

This book provides a systematic study of three foundational issues in the semantics of natural language that have been relatively neglected in the past few decades. It focuses on the formal characterization of intensions, the nature of an adequate type system for natural language semantics, and the formal power of the semantic representation language. The theory proposed offers a promising framework for developing a computational semantic system that is sufficiently expressive to capture the properties of natural language meaning while remaining computationally tractable. Written by two leading researchers in the field, Foundations of Intensional Semantics will be of interest to students and researchers in formal semantics, computational linguistics, logic, artificial intelligence, and the philosophy of language.

Chris Fox is a Reader in the Department of Computer Science at the University of Essex. In addition to numerous papers, his previous publications in the area of computational semantics include The Ontology of Language: Properties, Individuals, and Discourse (2000).

Shalom Lappin is Professor of Computer Science at King’s College, London. He has published extensively on issues in computational linguistics and formal grammar, and his books include Local Constraints vs. Economy (with David Johnson, 1999), Fragments: Studies in Ellipsis and Gapping (edited with Elabbas Benmamoun, 1999), and The Handbook of Contemporary Semantic Theory (edited, Blackwell, 1996).

PREFACE

XI

LIST OF ABBREVIATIONS

XV

1 INTRODUCTION

1

(12)

1.1 Montague's Intensional Logic

4

(3)

1.2 Architectural Features of IL

7

(2)

1.3 Structure of the Book

9

(4)

2 ALTERNATIVE APPROACHES TO FINE-GRAINED INTENSIONALITY

13

(25)

2.1 An Algebraic Representation of Possible Worlds Semantics

13

(6)

2.2 Two Strategies for Hyperintensionalism

19

(4)

2.3 Thomason's Intentional Logic

23

(2)

2.4 Bealer's Intensional Logic

25

(3)

2.5 Structured Meanings and Interpreted Logical Forms

28

(2)

2.6 Landman's Data Semantics

30

(1)

2.7 Situation Semantics and Infon Algebras

31

(2)

2.8 Situations as Partial Models

33

(1)

2.9 Topos Semantics

34

(3)

2.10 Conclusion

37

(1)

3 INTENSIONS AS PRIMITIVES

38

(18)

3.1 A Simple Intensional Theory

39

(5)

3.2 Types and Sorts

44

(1)

3.3 Abstraction and Application

45

(1)

3.4 PT: An Untyped Theory

46

(7)

3.5 Intensionality in FIL and PTCT

53

(1)

3.6 Conclusion

54

(2)

4 A HIGHER-ORDER, FINE-GRAINED INTENSIONAL LOGIC

56

(14)

4.1 Introduction

57

(1)

4.2 Fine-Grained Intensional Logic

57

(7)

4.3 A Semantics for FIL

64

(5)

4.4 Conclusion

69

(1)

5 PROPERTY THEORY WITH CURRY TYPING

70

(40)

5.1 PTCT: A Curry-Typed Theory

72

(2)

5.2 PTCT: Syntax of the Basic Theory

74

(1)

5.3 A Proof Theory for PTCT

75

(9)

5.4 Example Proof

84

(1)

5.5 Extending the Type System

85

(9)

5.6 Intensional Identity vs. Extensional Equivalence

94

(2)

5.7 A Model Theory for PTCT

96

(7)

5.8 Types and Properties

103

(1)

5.9 Separation Types and Internal Type Judgements

104

(4)

5.10 Truth as a Type

108

(1)

5.11 Conclusion

108

(2)

6 NUMBER THEORY AND CARDINALITY

110

(13)

6.1 Proportional Cardinality Quantifiers

111

(2)

6.2 Peano Arithmetic

113

(2)

6.3 Number Theory in FIL

115

(1)

6.4 Proportional Generalized Quantifiers in FIL

115

(1)

6.5 Number Theory in PTCT

116

(2)

6.6 Proportional Generalized Quantifiers in PTCT

118

(1)

6.7 Presburger Arithmetic

119

(1)

6.8 Presburger Arithmetic in PTCT

120

(1)

6.9 Conclusion

121

(2)

7 ANAPHORA AND ELLIPSIS

123

(11)

7.1 A Type-Theoretical Approach to Anaphora

124

(3)

7.2 Ellipsis in PTCT

127

(3)

7.3 Comparison with Other Type-Theoretical Approaches

130

(3)

7.4 Conclusion

133

(1)

8 UNDERSPECIFIED INTERPRETATIONS

134

(16)

8.1 Underspecified Representations

135

(9)

8.2 Comparison with Other Theories

144

(4)

8.3 Conclusion

148

(2)

9 EXPRESSIVE POWER AND FORMAL STRENGTH

150

(13)

9.1 Decidability and Completeness

150

(2)

9.2 Arguments for Higher-Order Theories

152

(1)

9.3 Arguments against Higher-Order Theories

153

(2)

9.4 Self-application, Stratification and Impredicativity

155

(1)

9.5 First-Order Status and Finite Cardinality

156

(5)

9.6 Relevance of PTCT to Computational Semantics

161

(1)

9.7 Conclusion

161

(2)

10 CONCLUSION

163

(9)

10.1 Montague Semantics and the Architecture of Semantic Theory

163

(1)

10.2 Algebraic Semantics and Fine-Grained Alternatives to MS

164

(1)

10.3 A Conservative Revision of MS

165

(1)

10.4 Enriching Property Theory with Curry Typing

166

(1)

10.5 An Intensional Number Theory

167

(1)

10.6 A Dynamic Type-Theoretic Account of Anaphora and Ellipsis

168

(1)

10.7 Underspecified Interpretations as A-Terms of the Representation Language

169

(1)

10.8 PTCT and Computational Semantics: Directions for Future Work

170

(2)

BIBLIOGRAPHY

172

(9)

AUTHOR INDEX

181

(2)

SUBJECT INDEX

183

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Foundations of Intensional Semantics

Summary

Author Biography

Table of Contents

Supplemental Materials