Foundations of Intensional Semantics Chris Fox and Shalom Lappin Foundations of Intensional Semantics Foundations of Intensional Semantics Chris Fox and Shalom Lappin c 2005 by Chris Fox and Shalom Lappin blackwell publishing 350 Main Street, Malden, MA 02148-5020, USA 9600 Garsington Road, Oxford OX4 2DQ, UK 550 Swanston Street, Carlton, Victoria 3053, Australia The right of Chris Fox and Shalom Lappin to be identified as the Authors of this Work has been asserted in accordance with the UK Copyright, Designs, and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs, and Patents Act 1988, without the prior permission of the publisher. First published 2005 by Blackwell Publishing Ltd 1 2005 Library of Congress Cataloging-in-Publication Data Fox, Chris, 1965- Foundations of intensional semantics / Chris Fox and Shalom Lappin. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-631-23375-6 (hardcover : alk. paper) ISBN-10: 0-631-23375-X (hardcover : alk. paper) ISBN-13: 978-0-631-23376-3 (pbk. : alk. paper) ISBN-10: 0-631-23376-8 (pbk. : alk. paper) 1. Semantics. 2. Semantics (Philosophy) 3. Intension (Logic) 4. Semantics–Data processing. I. Lappin, Shalom. II. Title P325.F657 2005 401’.43–dc22 2005010277 A catalogue record for this title is available from the British Library. 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For further information on Blackwell Publishing, visit our website: www.blackwellpublishing.com For my parents Contents Preface xi List of Abbreviations xv 1Introduction 1 1.1 Montague’s Intensional Logic 4 1.2 Architectural Features of IL 7 1.3 Structure of the Book 9 2Alternative Approaches to Fine-Grained Intensionality 13 2.1 An Algebraic Representation of Possible Worlds Semantics 13 2.2 Two Strategies for Hyperintensionalism 19 2.3 Thomason’s Intentional Logic 23 2.4 Bealer’s Intensional Logic 25 2.5 Structured Meanings and Interpreted Logical Forms 28 2.6 Landman’s Data Semantics 30 2.7 Situation Semantics and Infon Algebras 31 2.8 Situations as Partial Models 33 2.9 Topos Semantics 34 2.10 Conclusion 37 viii Contents 3Intensions as Primitives 38 3.1 A Simple Intensional Theory 39 3.2 Types and Sorts 44 3.3 Abstraction and Application 45 3.4 PT: An Untyped Theory 46 3.5 Intensionality in FIL and PTCT 53 3.6 Conclusion 54 4AHigher-Order,Fine-Grained Intensional Logic 56 4.1 Introduction 57 4.2 Fine-Grained Intensional Logic 57 4.3 A Semantics for FIL 64 4.4 Conclusion 69 5Property Theory with Curry Typing 70 5.1 PTCT: A Curry-Typed Theory 72 5.2 PTCT: Syntax of the Basic Theory 74 5.3 A Proof Theory for PTCT 75 5.4 Example Proof 84 5.5 Extending the Type System 85 5.6 Intensional Identity vs. Extensional Equivalence 94 5.7 A Model Theory for PTCT 96 5.8 Types and Properties 103 5.9 Separation Types and Internal Type Judgements 104 5.10 Truth as a Type 108 5.11 Conclusion 108 6Number Theory and Cardinality 110 6.1 Proportional Cardinality Quantifiers 111 6.2 Peano Arithmetic 113 6.3 Number Theory in FIL 115 6.4 Proportional Generalized Quantifiers in FIL 115 6.5 Number Theory in PTCT 116 6.6 Proportional Generalized Quantifiers in PTCT 118 6.7 Presburger Arithmetic 119 Contents ix 6.8 Presburger Arithmetic in PTCT 120 6.9 Conclusion 121 7Anaphora and Ellipsis 123 7.1 A Type-Theoretical Approach to Anaphora 124 7.2 Ellipsis in PTCT 127 7.3 Comparison with Other Type-Theoretical Approaches 130 7.4 Conclusion 133 8Underspecified Interpretations 134 8.1 Underspecified Representations 135 8.2 Comparison with Other Theories 144 8.3 Conclusion 148 9Expressive Power and Formal Strength 150 9.1 Decidability and Completeness 150 9.2 Arguments for Higher-Order Theories 152 9.3 Arguments against Higher-Order Theories 153 9.4 Self-application, Stratification and Impredicativity 155 9.5 First-Order Status and Finite Cardinality 156 9.6 Relevance of PTCT to Computational Semantics 161 9.7 Conclusion 161 10 Conclusion 163 10.1 Montague Semantics and the Architecture of Semantic Theory 163 10.2 Algebraic Semantics and Fine-Grained Alternatives to MS 164 10.3 A Conservative Revision of MS 165 10.4 Enriching Property Theory with Curry Typing 166 10.5 An Intensional Number Theory 167 10.6 A Dynamic Type-Theoretic Account of Anaphora and Ellipsis 168 10.7 Underspecified Interpretations as λ-Terms of the Representation Language 169 x Contents 10.8 PTCT and Computational Semantics: Directions for Future Work 170 Bibliography 172 Author Index 181 Subject Index 183 Chapter 5 Property Theory with Curry Typing In Chapter 4 we presented a conservative revision of Montague’s IL in which we introduced an alternative characterization of inten- sions while retaining the Church typing and higher-order nature of the system. One of the main reasons for employing higher- order type systems is that they contain functional types, which are required to provide adequate semantic representations for sev- eral syntactic categories. A unified treatment of NPs is possible if they are assigned the type of generalized quantifier (functions from properties to propositions, or truth-values). Adjectival and adverbial modifiers correspond to functions from properties to properties (or sets). Sentential modifiers are interpreted as func- tions from propositions to propositions (truth-values). As we observed in Chapter 1, IL is developed within an inten- sional version of Church’s (1940) Simple Theory of Types (STT) employing the typed λ-calculus with intension and extension forming operators, and modal operators. Gallin (1975) simplifies this system by replacing the intensional and extensional opera- tors with the basic type s. Barwise and Cooper (1981) invoke a set- theoretic counterpart of STT to develop an account of generalized quantifiers in natural language, and this framework is applied by, PropertyTheorywithCurryTyping 71 among others, Keenan and Stavi (1986), Keenan and Westerståhl (1997), and Lappin (2000a). Cooper (1996) uses a version of the typed λ-calculus to construct a situation theoretic treatment of generalized quantifiers. Groenendijk and Stokhof (1990, 1991) in- tegrate their dynamic logic into a variant of Montague’s IL in order to represent certain kinds of discourse anaphora. In this chapter we propose a more far-reaching departure from MS in which we sustain the radical intensionalist perspective of FIL within a first-order property theory enriched with a flexible Curry type system. We refer to this theory as Property Theory with Curry Typing (PTCT). Like FIL, PTCT permits fine-grained specifications of meaning. Unlike FIL, it is expressed in a first- order framework, and it supports polymorphic, separation, and product types.24 In developing a radical intensionalist semantics within a first-order framework we are pursuing an approach that bears some connection with ideas suggested by Bealer (1982); Turner (1987, 1992), and Zalta (1988). However, in contrast to this earlier work, PTCT has a robust Curry type system with fully articulated proof and model theories. Through its use of Curry typing, PTCT contains the full range of functional types. Moreover, it allows for limited (non-iterated) polymorphism, so that it captures the fact that certain natural language expressions, like coordination, correspond to functional types that apply to a variety of distinct argument types. However, quantification in the language of well-formed formulas in PTCT is first-order. Quantification over functional entities and types is expressed through quantification over terms, which are elements of the domain. The model theory is an extension of a standard ex- tensional model for the untyped λ-calculus, with the additional expressive power of Curry types added through a distinct com- ponent of the language of terms. The logic remains first-order in character. We prove that the basic PTCT logic (without number theory) is sound and complete. 24 Separation types yield a form of subtype. 72 Property Theory with Curry Typing 5.1 PTCT: A Curry-Typed Theory There are various ways in which we can view PTCT.Itcanbe regarded as a development of PT (Turner 1992) (see Chapter 3). The main difference being that PTCT has a fully-fledged language of types, whereas PT typically mimics types using properties. The addition of types requires changes to the syntax of the language, and the proof theory, in order to give the appropriate behaviour to expressions in which types appear. These additions ensure that we can represent quantified propo- sitions that explicitly restrict the domain of quantification. For example, if we wish to represent the sentence (109) John believes everything that Mary believes. then the quantifier representing everything can be restricted to range only over propositions. In PT, such restrictions can only be expressed in the language of wffs. There are other significant differences. PT has a universal type. It has been argued that this is appropriate for dealing with poly- morphic phenomena such as conjunctions, gerunds and infini- tives (Chierchia 1984; Chierchia and Turner 1988). However, we find this approach unduly permissive in that it imposes no con- straints on the relevant types of the arguments and conjuncts.