Lexical Meaning in Context: A Web of Words

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Lexical Meaning in Context This is a book about the meanings of words and how they can combine to form larger meaningful units, as well as how they can fail to combine when the amalgamation of a predicate and argument would produce what the philosopher Gilbert Ryle called a “category mistake”. It argues for a theory in which words get assigned both an intension and a type. The book develops a rich system of types and investigates its philosophical and formal implications, for example the abandonment of the classic Church analysis of types that has been used by linguists since Montague. The author integrates fascinating and puzzling observations about lexical meaning into a compositional semantic framework. Adjustments in types are a feature of the compositional process and account for various phenomena including coercion and copredication. This book will be of interest to semanticists, philosophers, logicians, and computer scientists alike. n i c h o l a s a s h e r is Directeur de Recherche CNRS, Institut de Recherche en Informatique de Toulouse, Universit´e Paul Sabatier, and Professor of Philosophy and of Linguistics at the University of Texas at Austin. He is author of Reference to Abstract Objects in Discourse (1993) and co-author of Logics of Conversation (2003) with Alex Lascarides.

Lexical Meaning in Context A Web of Words NICHOLAS ASHER CNRS, Institut de Recherche en Informatique de Toulouse and University of Texas at Austin

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107005396
C

Nicholas Asher 2011

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978-1-107-00539-6 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

page viii

Preface

PART ONE

FOUNDATIONS

1

1

Lexical Meaning and Predication 1.1 Types and presuppositions 1.2 Different sorts of predication 1.3 The context sensitivity of types 1.4 The main points of this book

3 4 10 18 21

2

Types and Lexical Meaning 2.1 Questions about types 2.2 Distinguishing between types 2.3 Strongly intensional types 2.4 Two levels of lexical content 2.5 Types in the linguistic system

25 25 27 34 44 49

3

Previous Theories of Predication 3.1 The sense enumeration model 3.2 Nunberg and sense transfer 3.3 Kleiber and metonymic reference 3.4 The Generative Lexicon 3.5 Recent pragmatic theories of lexical meaning

61 62 64 69 71 87

PART TWO

95

4

THEORY

Type Composition Logic 4.1 Words again 4.2 The basic system of types

97 101 103

vi

Contents 4.3 4.4 4.5

Lexical entries and type presuppositions The formal system of predication A categorial model for types

106 114 121

5

The Complex Type • 5.1 A type constructor for dual aspect nouns 5.2 Some not-so-good models of • types 5.3 The relational interpretation of • types 5.4 Subtyping with •

130 130 137 149 160

6

• Type Presuppositions in TCL 6.1 How to justify complex type presuppositions 6.2 Applications 6.3 • Types and accidentally polysemous terms

163 163 169 185

PART THREE

189

DEVELOPMENT

7

Restricted Predication 7.1 Landman’s puzzle 7.2 More puzzles 7.3 Extensional semantics for as phrases 7.4 A new puzzle 7.5 As constructions in TCL 7.6 Proper names in as phrases revisited 7.7 An aside on depictives

191 193 194 195 197 201 211 213

8

Rethinking Coercion 8.1 Re-examining the data 8.2 Coercion and polymorphic types 8.3 Discourse and typing 8.4 Discourse-based coercions in TCL

214 214 219 236 240

9

Other Coercions 9.1 Noise verbs 9.2 Coercions from objects to their representations 9.3 Freezing 9.4 Cars and drivers, books and authors 9.5 Verbs of consumption 9.6 I want a beer 9.7 Evaluative adjectives 9.8 Coercions with pluralities 9.9 Aspectual coercion and verbal modification

246 246 247 248 249 251 252 256 261 262

Contents

vii

10

Syntax and Type Transformations 10.1 The Genitive 10.2 Grinding 10.3 Resultative constructions 10.4 Nominalization 10.5 Evaluating TCL formally

272 272 280 281 284 296

11

Modification, Coercion, and Loose Talk 11.1 Metonymic predications 11.2 Material modifiers 11.3 Loose talk 11.4 Fiction and fictional objects 11.5 Metaphorical predication

300 300 301 305 309 312

12

Generalizations and Conclusions 12.1 Integrating ordinary presuppositions 12.2 Conclusions: a sea of arrows

315 315 318

PART FOUR References Index

321 323 331

CODA

Preface

Just over fifty years ago with the publication of “Two Dogmas of Empiricism”, W. V. O. Quine launched a persuasive and devastating attack on the common sense notion of word meaning and synonymy, according to which two terms were synonymous just in case they had the same meaning. Quine’s legacy continues to hold sway among much of the philosophical community today. The theory of word meaning is often thought either not to have a subject matter or to be trivial—dog means dog. What else is there to say? Well, it turns out, quite a lot. Linguists like Charles Fillmore, Igor Mel’cuk, Maurice Gross, Beth Levin, Ray Jackendoff, James Pustejovsky, and Len Talmy— to mention just a few, as well as researchers in AI who have built various on-line lexical resources like WORDNET and FRAMENET, have provided rich and suggestive descriptions of semantic relations between words that affect their behavior. And this has led to several proposals for a theory of word meaning. Against this rich descriptive background, however, problems have emerged that make it not obvious how to proceed with the formalization of lexical meaning. In particular, something that is commonly acknowledged but rarely understood is that when word meanings are combined, the meaning of the result can differ from what standard compositional semantics has led us to expect: in applying, for instance, a property term ordinarily denoting a property P to an object term ordinarily denoting a, the content of the result sometimes involves a different but related property P applied to an object b that is related to but distinct from the original denotation of a. While the choice of words obviously affects the content of a predication, the discourse context in which the predication occurs also affects it. The trick is to untangle from this flux a theory of the interactions of discourse, predication, and lexical content. That is what this book is about.1 1

I owe many people thanks for help with this book: Alexandra Aramis, Alexis, Elizabeth, and Sheila Asher, Tijana Asic, Christian Bassac, David Beaver, Stephano Borgo, George

Preface

ix

In this book, I argue that the proper way to understand the meaning of words is in terms of their denotations and the restrictions that other words impose on them. And it is the latter that govern how words interact semantically. I begin with the widely accepted observation according to which a predication will succeed only if the selectional restrictions the predicate imposes on its arguments are met. I provide an analysis of selectional restrictions by assigning words types. Meeting a selectional restriction is a matter of justifying a lexical presupposition, the presupposition that a term has a certain type. This analysis yields a theory of lexical meaning: to specify the type and the denotation of a word is to give its lexical meaning. The mechanisms of presupposition justification developed in dynamic semantics in recent years lead to an account of how predication adds content to the “ordinary” contents of the terms involved, which will provide my account of meaning shifting in context. The theory I will develop in this book has implications for compositional semantics, for example for the architecture of verbal and nominal modification. It also unifies analyzes in compositional semantics of presuppositions with my analysis of type presuppositions; for instance, the presuppositions of factive verbs or definite noun phrases are just special cases of type presuppositions. The idea that there are non-trivial semantic interactions between words that affect the content of a predication is intuitive and perhaps obvious. But working out a precise theory, or even an imprecise one, of this phenomenon is difficult. I begin with some basic questions, distinctions, and observations. What is a word? In some sense the answer is obvious: words are the things dictionaries try to define. On the other hand, the answer is not so simple. Words in many languages come with inflection for case, for number, for gender, among other things. Furthermore, there are morphological affixes that can transform one word into another like the nominalization affixes in English: an Bronnikov, Robin Cooper, Denis Delfitto, Pascal Denis, Tim Fernando, Pierdaniele Giaretta, John Hawthorne, Mark Johnson, Hans Kamp, Chris Kennedy, Ofra Magidor, Alda Mari, Claudio Masolo, Bruno Mery, Friedericke Moltmann, Philippe Muller, David Nicolas, Barbara Partee, Sylvain Pogodalla, Franc¸ois Recanati, Christian Retoret, Antje Rossdeutscher, Sylvain Salvati, Magdalena Schwager, Stuart Schieber, Torgrim Solstad, Tony Veale, Laure Vieu, Kiki Wang, Laura Whitten, and participants of the seminars on lexical semantics at the University of Verona, the University of Texas at Austin, the University of Stuttgart, and the Summer Institute of the Linguistic Society of America at Stanford, where some of this material was presented. I want especially to thank Hans Kamp and members of the SFB 732 at the University of Stuttgart for their generous invitation to spend three months there to work on this project during the summer of 2008. Special thanks are also due to Julie Hunter and Renaud Marlet who reread much of the manuscript and offered many helpful comments and to James Pustejovsky, who got me to work on the subject of coercion and dot objects in the first place. Finally, I’d like to thank Andrew Winnard, Sarah Green, Gillian Dadd, Alison Mander, and Elizabeth Davey from Cambridge University Press for their help with the manuscript. The book is dedicated to Tasha, my darling little cat who didn’t manage to live to see the end of this project.

x

Preface

affix like -ion turns a verb like afflict into the noun affliction. Morphological affixes and prefixes can often affect the meaning of a word; they can also determine how their host words combine with other words, as we shall see later on in this book. Even inflections like the plural are not always semantically innocent. Thus, the notion of a word quickly becomes a theoretical term; the meaningful parts of the lexicon may include things that we ordinarily would think of as bits of words, and basic word stems (the elements to which affixes and prefixes attach) may not end up looking like ordinary words at all. Despite these complications, I will continue to speak (loosely) of words. What is it to give the meaning of a word? There are a number of answers in the literature on lexical semantics or theories of word meaning. Cognitive semanticists like Len Talmy and Tom Givon, among others, think that meanings are to be given via a set of cognitively primitive features—which might be pictorial rather than symbolic. According to these semanticists, a lexical theory should provide appropriate cognitive features and lexical entries defined in terms of them. Others in a more logical and formal framework like Dowty (1979) (but also Ray Jackendoff, Roger Shank, and other researchers in AI) take a specification of lexical meaning to be given in terms of a set of primitives whose meanings can be axiomatized or computationally implemented. Still others take a “direct” or denotational view; the function of a lexical semantics is to specify the denotation of the various terms, typically to be modelled within some model theoretic framework. All of these approaches agree that a specification of lexical meaning consists in the specification of some element, whether representational or not, formal or not, that, when combined with elements associated with other words in a well formed sentence, yields a meaning for a sentence in a particular discourse context. Whatever theoretical reconstruction of meaning one chooses, however, it should be capable of modelling inferences in a precise manner so that the theory of lexical meaning proposed can be judged on its predictions. In addition, the theoretical reconstruction should provide predictions about when sentences that are capable of having a truth value are true and when they are not. This drastically reduces the options for specifying lexical meaning: such a specification must conform with one of the several ways of elaborating meaning within the domain of formal semantics; it must specify truth conditions, dynamic update conditions of the sort familiar from dynamic semantics (Kamp and Reyle (1993), Groenendijk and Stokhof (1991), Asher (1993), Veltman (1996)), or perhaps provability conditions of the sort advocated by Martin-L¨of (1980) and Ranta (2004), among others. For proponents of a direct interpretation of English, a denotational approach to lexical meaning suffices. Most semanticists, however, use a logical language

Preface

xi

to state the meanings of natural language expressions. The logical representations of sentential meanings are typically called logical forms. Within such a framework a lexical entry for a word should specify a logical representation that when combined together with the contributions of other words in a well-formed sentence will yield a logical form with well-defined contents. I shall follow formal semantic tradition and use a logical language with a welldefined model theoretic interpretation to provide as well as to construct logical forms.2 Thus, at a minimum, lexical semantics should be concerned with the lexical resources used to construct logical forms in a language with a precise model theoretic interpretation. But what are those resources? Clearly the syntactic structure of a clause is necessary for constructing a logical form for the clause, but that is not the province of lexical semantics. One that is, however, is argument structure. Most words—verbs, adjectives, nouns, determiners, and adverbs—have arguments, other words or groups of words, that they combine with; and the meaning of such words must specify what other kinds of words or groups of words they can combine with to provide larger units of meaning. But an account of lexical meaning must do more than this; it must also specify what the process of combination is when the representation of one word meaning combines with other word meaning representations licensed by their argument structures. It must couple its representation of a word’s meaning with a mechanism for combining this representation with the representations of the meanings of its arguments or of the words to which it is an argument. The construction of logical form and the lexical resources used to construct it thus inevitably involve the notion of predication; when one bit of logical form functions as an argument to another, a predication relation holds between a property denoting term and its argument. A satisfactory theory of lexical meaning must yield an account of predication, and the choice of a model of predication affects the choice of how to represent lexical meanings. I turn now to a basic formal model of predication and the representation of lexical meaning.

2

Cognitive semantics lexical theories will not figure in this book, because they do not really have the resources to provide logical forms for sentences capable of defining truth conditions or update conditions. G¨ardenfors (1988) has provided a formal model of the cognitive semantics view of lexical meaning by taking the cognitive features to form the basis of a vector space. Lexical meanings are then represented as vectors or sets of vectors in this space. Such a theory can give us a potentially interesting measure of similarity in meaning by appealing to distances between points in this feature space. Certain lexical inferences can also be accounted for as G¨ardenfors (1988) shows. But the compositional problem, that is, the problem of showing how these meanings compose together to get meanings of larger units, is unsolved, and it is not at all obvious how one could solve it within the vector or feature space framework for anything more than the simplest of fragments of natural language.

PART ONE FOUNDATIONS

1 Lexical Meaning and Predication

To build a formal model of predication and to express lexical meaning, I will use the lambda calculus. The lambda calculus is the oldest, most expressive, and best understood framework for meaning representation; and its links to various syntactic formalisms have been thoroughly examined from the earliest days of Montague Grammar to recent work like that of de Groote (2001), Frank and van Genabith (2001). Its expressive power will more than suffice for our needs.1 The pure lambda calculus, or λ calculus, has a particularly simple syntax. Its language consists of variables together with an abstraction operator λ. The set of terms is closed under the following rules: (1) if v is a variable, then v is a term; (2) if t is a term and v a variable, then λvt is also a term; (3) if t and t are terms, then the application of t to t , t[t ], is also a term. We can use this language to analyze the predication involved when we apply a predicate like an intransitive verb to its arguments. The meaning of an intransitive verb like sleeps is represented by a lambda term, λx sleep (x); it is a function of one argument, another term like the constant j for John that will replace the λ bound variable x and yield a logical form for a larger unit of meaning under the operation of β reduction. β reduction, also known as β conversion, is a rule for inferring one term from another. β reduction is the formal counterpart in the λ calculus of the informal operation of predication. One can also think of reduction as the rule governing application, and so I shall call it the rule of Application.2 I’ll write such a rule in the usual natural deductive format. 1

2

There are other formalisms that can be used—for instance, the formalism of attribute value matrices or typed feature structures with unification. This formalism, however, lacks the operation of abstraction, which is crucial for my proposals here. Besides Application, there are other rules standardly assumed for the λ calculus—for example, α conversion, which ensures the equivalence of bound variables, rules for equality, and the following rules which validate a rule of Substitution that I shall introduce subsequently:

4

Lexical Meaning and Predication

• Application: λxφ[α] φ( αx ) The λ calculus as our representational language tells us in principle what our lexical entries should look like. For example, if we decide that a word like cat is a one place predicate, then our lexical entry for this word should have the form λx cat (x), where cat is an expression in our language for logical forms that will, when interpreted, assign the right sort of denotation to the word and contribute to the right sort of truth conditions for sentences containing the word. Of course, there are lots of decisions to be made as to what cat should be exactly, but we will come back to this after we have taken a closer look at predication.

1.1 Types and presuppositions Sometimes predications go wrong. This is something that lexical semantics has to explain. (1.1) a. ?That person contains an interesting idea about Freud. b. That person has an interesting idea about Freud. c. That book contains an interesting idea about Freud. d. That person is eating breakfast. e. That book is red. f. #That rumor is red. g. # The number two is red. h. # The number two is soft. i. # The number two hit Bill. j. The number two is prime. k. John knows which number to call. l. *John believes which number to call. The predications in (1.1f,g,h) or (1.1i) are malformed—each contains what Gilbert Ryle would have called a category mistake. Numbers as abstract objects can’t have colors or textures or hit people; it’s nonsensical in a normal • t = t → t[t ] = t [t ] • t = t → f [t] = f [t ] • t = t → λx t = λx t Church (1936) shows how to encode Boolean functions within the λ calculus, once we have decided on a way of coding up truth functions.

1.1 Types and presuppositions

5

conversation to say something like the number two is red, soft, or that it hit Bill.3 The mismatch between predicate and argument is even more blatant in (1.1l). One has to exercise some care in understanding why a predication like (1.1a) sounds so much odder than (1.1b–d). In some sense people can contain information: spies have information that they give to their governments and that counter-spies want to elicit; teachers have information that they impart to their students. But one can’t use the form of words in (1.1a) to straightforwardly convey these ideas. The predication is odd; it involves a misuse of the word contain. If it succeeds at all in making sense to the listener, it must be subject to reinterpretation. It’s important to distinguish between necessary falsity and the sort of semantic anomaly present in (1.1a) and (1.1f–i). In the history of mathematics, many people, including famous mathematicians, have believed necessarily false things. But competent speakers of a language do not believe propositions expressed by a sentence with a semantically anomalous predication. (1.1a) or (1.2c,d) are semantically anomalous in a way that (1.1b–d) or (1.2a,b) below are not.4 (1.2) a. Tigers are animals. b. Tigers are robots. c. #Tigers are financial institutions. d. #Tigers are Zermelo-Frankel sets. Many philosophers take (1.2a) to be necessarily true and (1.2b) to be necessarily false.5 Nevertheless, according to most people’s intuitions, a competent speaker could entertain or even believe that tigers are robots; he or she could go about trying to figure this out (e.g., by dissecting a tiger). It is much harder to accept the possibility, or even to make sense of, a competent speaker’s believing or even entertaining that tigers are literally financial institutions, let alone ZF style sets. Thinking about whether a competent speaker could entertain or believe the proposition expressed by a sentence gives us another means to distinguish between those sentences containing semantically anomalous expressions and those that do not. 3

4 5

As the attentive reader may have already guessed, besides “normal” conversations, there are also “abnormal” discourse contexts—contexts that would enables us to understand these odd sentences in some metaphorical or indirect way, or that even enable us to reset the types of words. More on this later. Thanks to Dan Korman for the first two examples. The reason for this has to do with a widely accepted semantics of natural kinds due to Hilary Putnam and Saul Kripke, according to which tigers picks out a non-artifactual species in every possible world.

6

Lexical Meaning and Predication

The reason why (1.1a), (1.1f–i) or (1.2c,d) are semantically anomalous, while the other examples above are not, is that there is a conflict between the demands of the predicate for a certain type of argument and the type of its actual argument. People aren’t the right type of things to be containers of information, whereas tapes, books, CDs, and so on are. Rumors aren’t the right type of things to have colors, and tigers aren’t the right type of things to be sets or financial institutions. We can encode these humdrum observations by moving from the pure lambda calculus to a typed lambda calculus. The reason why some predications involve misuses of words, don’t work, or require reinterpretation, is that the types of the arguments don’t match the types required by the predicates for their argument places. (1.1a) involves a misuse of the language. Contain, given the type of its direct object, requires for its subject argument a certain type of object—a container of information; and persons are of not of this type—they don’t contain information the way books, journal articles, pamphlets, CDs, and the like do. On the other hand, there is no such problem with (1.1b); books are the sort of object that are containers of information. (1.1c) is also fine, but that is because the verb have doesn’t make the same type requirements on its arguments that contain does. The typed lambda calculus, developed by Church (1940), assumes that every term in the language has a particular type. This places an important constraint on the operation of Application. Assume that every term and variable in the lambda calculus is assigned a type by a function type. • Type Restrictied Application: λxφ[α] φ( αx ) provided type(x) = type(α). λxφ[α] is undefined, otherwise. In what follows, I’ll encode type with the usual colon notation; α: a means that term α has type a. The typed lambda calculus has many pleasant semantic and computational properties. This has made it a favorite tool of compositional semanticists since Montague first applied it in developing his model theoretic notion of meaning in the sixties. I will model predication as type restricted application and lexical entries as typed lambda terms. This will require that each term gets a type in a given predicational context. Moreover, each term will place restrictions on the type of its eventual arguments. The data just discussed indicates that the set of

1.1 Types and presuppositions

7

relevant types for a theory of predication and lexical meaning are quite finegrained; in the next chapter we will see how this data and other data lead to the hypothesis of a great many more types than envisaged in Montague Grammar or standard compositional semantics. Before addressing questions about types, I want to investigate some implications of Type Restricted Application for a theory of predication. There is a compelling analogy between the way types and type requirements work in the typed lambda calculus and the linguistic phenomenon known as presupposition. Linguists take certain words, phrases, and even constructions to generate presupposed contents as well as “proffered” contents; the latter enter into the proposition a sentence containing such items expresses, whereas the presupposed contents are conceived of as constraints on the context of interpretation of the sentence. For instance, in (1.3) The dog is hungry the definite determiner phrase (DP) the dog is said to generate a presupposition that there is a salient dog in the discourse context. If such a presupposition is satisfied in a discourse context, the presupposition is said to be bound; if it cannot be bound, the presupposition is accommodated by making the supposition that the discourse context contains such a salient dog. However, there is a certain cost to such suppositions; if there really is no salient dog in the context, (1.3) is difficult to interpret. Frege and Strawson proposed that in cases where no salient dog can be found, a sentence like (1.3) cannot be literally interpreted and fails to result in a well-formed proposition capable of having a truth value.6 This view of presupposition, though it has its detractors, is well established in linguistics and has received a good deal of empirical support and formal analysis (Heim (1983), van der Sandt (1992), Beaver (2001)). Type Restricted Application says something very similar to the doctrine of presupposition: a type concordance between predicate and argument is required for coherent interpretation. If an argument in a predication cannot satisfy the type requirements of the predicate, then the predication cannot be interpreted and fails to result in a well formed logical form capable of having a truth value. There are other similarities between presupposition and type requirements. A common test for presupposition is the so-called projection test: presuppositions “project” out of various operators denoting negation, modality, or mood. So if the type requirements of a predicate are a matter of presupposition, then semantically anomalous sentences like (1.1g,h) should remain anomalous 6

This is known as the “Frege-Strawson” doctrine.

8

Lexical Meaning and Predication

when embedded under negation, interrogative mood or modal operators. This is indeed the case: (1.4) a. # The number two could have been red. b. # Is the number two soft? c. # The number two didn’t hit Bill.7 The sentences in (1.4) all convey presuppositions that are absurd and that cannot be met—namely, that the number two is a physical object. Other tests for presuppositions concern the non-redundancy of presupposed content and the inability to make certain discourse continuations on presupposed content.8 These tests apply to type requirements of predicates as well. It is not redundant to say the abstract object two is prime instead of two is prime, and it seems impossible to make discourse continuations on the type requirements, since the latter are not even propositional contents. Thus, it seems that the type requirements of predicates provide a kind of presupposed content. I shall call these type presuppositions. Two features of presuppositions will be very important for the study of predication in this book. The first is the variability among terms that generate presuppositions to license accommodation. It is standardly assumed that the adverb too generates a presupposition that must be satisfied in the given discourse context by some linguistically expressed or otherwise saliently marked content. Thus, in an “out of the blue” context, it makes no sense to say (1.6) Kate lives in New York too. even though as a matter of world knowledge it is clear that the presupposition of too in this sentence is satisfied—namely, that there are other people besides Kate who live in New York. Even if the proposition that there are other people besides Kate who live in New York is manifestly true to the audience of (1.6), (1.6) is still awkward, unless the presupposed content has been made salient somehow in the context. The presupposed, typing requirements of the predicates in (1.1) and (1.4) resemble the behavior of the presupposition of too; they 7

8

A presuppositional view should allow that this sentence has a perfectly fine reading where the negation holds over the type requirements as well. But typically such readings are induced by marked intonation. If this sentence is read with standard assertion prosody, then it is as anomalous as the rest. The continuation test says that one cannot elaborate or explain or continue a narrative sequence on presupposed content. Thus, one cannot understand the example below as conveying that John regretted that he yelled at his girlfriend and that then after fighting with her he went to have a drink.

(1.5) John regrets that he fought with his girlfriend. Then he went to have a drink.

1.1 Types and presuppositions

9

have to be satisfied in their “predicative” context in order for the sentences containing them to receive a truth value. Accommodation of these type presuppositions is impossible. The sentences that fail to express a coherent proposition capable of having a truth value do so, because the relevant type presuppositions cannot be satisfied, given that the arguments and predicates therein mean what they standardly mean and have the types that they standardly do. On the other hand, some presupposition introducing phrases like possessive DPs readily submit to accommodation. For instance, Sylvain’s son presupposes that Sylvain has a son, but this information is readily accommodated into the discourse context when the context does not satisfy the presupposition. (1.7) Sylvain’s son is almost three years old. Other definite descriptions can be satisfied via complex inferences. The example below, which features a phenomenon known as “bridging,” features such an inference; the definite the engine is “satisfied” by the presence of a car in the context—the engine is taken to be the engine of the car: (1.8) I went to start my car. The engine made a funny noise. In the following chapters we will see cases of type presuppositions that can either be satisfied in complex ways like the bridging cases or can be accommodated via a “rearrangement” or modification of the predicative context, if the latter fails to satisfy the type presuppositions in a straightforward way. Figuring out when presupposed typing requirements can be accommodated and when they cannot will be a central task of this book. Another important property of presuppositions is their sensitivity to discourse context. For instance, if we embed (1.7) in the consequent of a conditional, the presupposition that projects out from the consequent can be bound in the antecedent and fails to project out further as a presupposition of the whole sentence (1.9): (1.9) If Sylvain has a son, then Sylvain’s son is almost three years old. A similar phenomenon holds for type presuppositions. Consider (1.4a) embedded as a consequent of the following (admittedly rather strange) counterfactual. (1.10) If numbers were physical objects, then the number two could have been red. The presupposition projected out from (1.4a) is here satisfied by the antecedent of the counterfactual and rendered harmless. Thus, category mistakes for the most part must be understood relative to a background, contextually supplied set of types, a background that may itself shift in discourse.

10

Lexical Meaning and Predication

1.2 Different sorts of predication Having introduced types as part of the apparatus of predication, let me come back to predication itself. I have spoken so far of predication as a single operation of applying a predicate to its arguments. But in fact predication takes many forms in natural languages, some particular to particular languages, others more general. Even among ordinary predications, linguists distinguish between: • predication of a verb phrase to a subject or a transitive verb to an object • adjectival modification with different types of adjectives—e.g., evaluative adjectives like good rock, bad violinist, material adjectives like bronze statue, paper airplane, and manner adjectives like fast car, slow cigar • adverbial modification and modification of a verb phrase with different prepositional phrases or PPs—e.g., the distinction between load the wagon with hay and load the hay on the wagon. Beyond these are more exotic forms of predication: • metaphorical usage (extended predication) (1.11) John is a rock. • restricted predication (1.12) John as a banker makes $50K a year but as a plumber he makes only $20K a year. • copredication (1.13) The lunch was delicious but took forever. (1.14) The book has a purple cover and is the most intelligible introduction to category theory. (1.15) #The bank is rising and specializes in IPOs. • loose predication (1.16) That’s a square (pointing to an unpracticed drawing in the sand). • resultative constructions (1.17) a. Kim hammered the metal flat. b. * Kim hammered the metal gleaming. (1.18) depictives a. Pat swims naked. b. *Pat cooks hot. • the genitive construction

1.2 Different sorts of predication

11

(1.19) a. Kim’s mother b. Kim’s fish • noun noun compounds (1.20) a. lunch counter b. party favor Each one of these forms of predication presents its own challenges for lexical and compositional semantics; the lexical theory must assign to the words in these constructions the right sort of meaning and postulate the right sort of composition rules for predication so as to get the right result. In addition, a lexical theory must specify what morphological processes and elements affect meaning and how; it must give those processes and elements a meaning. A lexical theory using the typed lambda calculus can provide the right sort of picture to tackle these issues. Let’s consider these forms of predication in a bit more detail. Loose predication is a difficult and well-known problem in philosophy.9 But other forms of predication mentioned above, which linguists think also provide challenges for lexical theory, have not received so much philosophical scrutiny or formal analysis. Copredication, for instance, which is a grammatical construction in which two predicates jointly apply to the same argument, has proved a major challenge. Languages, as we shall see in the next chapter, distinguish between events and objects; the predicates that apply the one type do not apply in general to the other type literally. It turns out that some objects, however, are considered both events and physical objects in some sense. Consider, for instance, lunches. Lunches can be events but they are also meals and as such physical objects. As a result, lunch supports felicitous copredications in which one predicate selects for the event sense of lunch while the other selects for the physical object or meal sense. (1.21) Lunch was delicious but took forever. It turns out that many words behave like lunch in (1.21) and denote objects with multiple senses or aspects. I will call predications like those in (1.21) aspect selections, and I will analyze these predications as predications that apply to selected aspects of the object denoted by the surface argument. In trying to account for instances of copredication that involve aspect selection like (1.21), standard, typed theories of predication and lexical semantics confront some difficult if not unanswerable questions. How can a term have two incompatible types, as is apparently the case here? How can one term 9

Loose predication is related to vagueness, and vague predication might be considered another form to be studied. But I shall not do that here.

12

Lexical Meaning and Predication

denote an object or set of objects to which apply two properties demanding different, even incompatible types of their bearers? It would have to be the case then that such an object must have, or belong to, two incompatible types. But how is that possible? Proponents of standard type theory have only one clear recourse, and that is to claim that terms associated with two incompatible types are ambiguous. But that deepens the mystery about copredications involving aspect selections: if lunch in (1.21) is ambiguous between a physical object reading and an event reading, then we must disambiguate the term in one way to make sense of the first predication but disambiguate it in a second way to make sense of the second predication; and the problem is that, on the surface at least, we have only one term to disambiguate—we have to choose a disambiguation, but such a choice will inevitably cause one of the predications in (1.21) to fail. At this point we might try a strategy of desperation and postulate a hidden “copy” of the problematic term, rewriting (1.21) in effect as (1.21 ) Lunch was delicious but lunch took forever. This copying strategy now allows the proponent of standard type theory to proceed to disambiguate the two occurrences of lunch in different ways allowing the two predications to succeed. But the promise of the copying strategy is shortlived. Copying expressions will get us incorrect truth conditions in many cases. Consider (1.22), where last applies to events, while tasted applies only to objects (you can’t taste events except metaphorically): (1.22) A lunch was gingerly tasted by Mary and then lasted three hours. The copying strategy forces us to interpret (1.22) as (1.23) A lunch was gingerly tasted by Mary and then a lunch lasted three hours. It’s easy to see that (1.22) and (1.23) have different truth conditions; (1.22) is true only in those situations where Mary gingerly tasted the same lunch that lasted three hours, while (1.23) can be true in situations where Mary gingerly tastes one lunch but another lunch lasts three hours. Thus, it is not obvious how to deal with examples of copredication even from the standpoint of compositionality, if one’s lexical theory produces a rich system of types. Montague himself noted that there were copredications that were puzzling even within his much more impoverished system of types. In (1.24) temperature seems to have two aspects, one of which is a number on a scale, while the other is a function from times to numerical values. (1.24) The temperature is 90 and rising.

1.2 Different sorts of predication

13

We’ll see many more examples of copredications and how to analyze them in chapters 5 and 6. Restricted predications like the main predication in John as a banker makes $50K a year seem to predicate properties of certain parts or aspects of the “restricted argument” (in the example at hand, John). But it is not clear in what sense we should understand the word part. In the example, it’s not as though we’re predicating the making of a salary to some physical part of John, as in John’s right arm has a bruise. Restricted predication thus introduces some puzzles of its own. Resultative constructions are another form of complex predication. (1.17a), for example, involves two predications on the term the metal—one by the adjective flat and one by the verb.10 But this construction also introduces a third predication, which features the causal relation between the hammering and the flatness of the metal. Genitive constructions also introduce a predication in which some relation is predicated of the objects described by the two noun phrases or DPs that make up the genitive construction. Sometimes this relation is given by the head noun as in (1.19a) but sometimes it is not as in (1.19b). The predication in the genitive also seems to add to or to change the content of the words in the construction. Finally, the semi-productive predication construction in English, known as noun-noun compounding, also seems to add to or transform the meanings of its constituents—sometimes in radical ways so as to produce idioms whose meaning is not derivable from the meanings of its constituent terms like party favor. Besides these forms of predication, there are other factors that can influence the content of a predication. One is number. Some predicates with plural arguments require a collective or group predication, which requires the argument to be described in a certain way. Consider (1.25) a. The students surrounded the building. b. The students mowed the whole meadow. (1.25a,b) exemplify collective predications where a property is predicated of the whole set of students but not of each student individually. Contrasting with collective predication is distributive predication, which occurs when a property or relation is predicated of each element of a plural group, as in (1.26) The boys each worked hard. There is a grammatically marked distinction between plurality and singularity, about which most lexical theories have nothing to say. Nevertheless, they 10

Syntacticians take the structure of this sentence to be quite complex, involving what is called a “small clause,” the metal flat, and the verb hammer.

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Lexical Meaning and Predication

should, because an account of the lexical meaning of the word disperse or surround must mark it as requiring an argument that, if plural, must be interpreted collectively. In many predications, interactions between the type presuppositions of the predicate and the types of its arguments also affect the content of the predication. (1.27) provides examples of this phenomenon, which is known as coercion in the literature. Coercion is so called because it appears that one word coerces another word (usually the second word is an argument of the first) to have a different meaning from its usual one. (1.27) a. good lunch, good children b. Mouse isn’t very tasty unless you’re a cat. c. John started a cigarette, started a car, began the sonata, finished the novel. d. John liked the dress with the flowers/ liked the garden with the flowers. The phenomenon observed with the adjectival predications in (1.27a) is a very general and diverse one. The predications therein show how adjectival modification can affect the type and meaning of the resulting noun phrase. A good lunch is one that tasted good. Pustejovsky (1995) and others have proposed that an adjective like good selects a component of the meaning of its argument— roughly, its purpose or telic role. Nevertheless, as many have noted, such adjectives also apply to arguments that don’t have purposes. For instance, children in and of themselves don’t have fixed purposes; yet when good modifies children, we understand different things: when someone says those are good children, we understand that the children are behaving well or that they have certain laudatory dispositions to behavior. There is a subtle, though undeniable, shift in meaning in these predications. A theory that simply says that good denotes the property of being good, that children denotes children and lunch denotes a meal and that says nothing about how these meanings combine in predication other than that the objects denoted by the one term have the property denoted by the other cannot make any headway explaining these nuances in meaning. Unfortunately, many philosophers and some linguists still hold such a theory to be true (for instance, see Fodor and Lepore (1998)). (1.27b) shows how the bare singular use of a count noun can in many circumstances change the type of the noun phrase from count to mass. This transformation is known as grinding in linguistics. The examples in (1.27c) show how aspectual verbs coerce their arguments into denoting some sort of event. Aspectual verbs require or presuppose that their direct object is some event involving their subject; when their direct objects are not event-like, a felicitous coercion sometimes occurs, and we infer defeasibly that some sort of

1.2 Different sorts of predication

15

activity involving the subject of the aspectual verb serves as its internal argument. Thus, we understand John started the car as John’s started the running of the car’s engine. To start a cigarette is typically to start to smoke a cigarette. (1.27d) shows that coercions can happen with prepositional phrases—the dress with the flowers has at least one interpretation where a representation of flowers is stitched, printed, or drawn on the fabric of the dress, while the garden with the flowers does not have that interpretation, at least not nearly so saliently.11 As we shall see, there are subtle differences with respect to the presuppositions in the typing requirements of various aspectual verbs and other coercing predicates. Are such coercions really part of lexical semantics? That is, is it a defeasible but a priori inference that if John started the car, John started the engine of the car or that if Julie enjoyed the book, then (defeasibly) she enjoyed reading it? Do such inferences follow solely from one’s linguistic mastery of the language? Fodor and Lepore think that none of these inferences belong to lexical semantics but are rather part of encyclopaedic or world knowledge. However, most people can distinguish between the largely automatic interpretations that these predications seem to entail and those that require more conscious effort. One might take that to be a mark of the information as being present even during predication rather than inferred afterwards using background, nonlinguistic beliefs. It is notoriously difficult to distinguish between what is properly a part of lexical meaning and what is world knowledge. Quine’s attack on lexical meaning can be seen as starting from the point that one cannot make this distinction in a principled way. Part of the difficulty is that, to some extent, the division between word meaning and world knowledge is a theory-internal distinction. For instance, if you’re an externalist for whom the meanings of two singular or natural kind terms t and t are determined by their reference, it may be a fact of meaning that t = t or not. Thus, water is H2 0 would be a fact of meaning, and hence analytic, on such a view! Despite these difficulties, there is are tests one can use to see whether it is certain information conventionally associated with particular word meanings rather than just general world knowledge that gives rise to these inferences. For one thing, it seems pretty clear that the infierences given in (1.27c) are tied to particular predicates, particular verbs. Let’s suppose that cigarette, like lunch, always has associated with it a possible event reading. It should then be possible to access that appropriate event reading with other predicates that take events.

11

This example is due to Marliese Kluck.

16

Lexical Meaning and Predication

(1.28) a. Nicholas’s smoking of that cigarette will begin in 2 minutes. b. Nicholas’s cigarette break will begin in 2 minutes. c. ??Nicholas’s cigarette will begin in 2 minutes. It’s quite clear that (1.28c) is semantically strange. The event associated with cigarette in enjoy the cigarette, begin the cigarette, finish the cigarette, just isn’t available with other event predicates. This strongly suggests that there is some particular conventional meaning that issues from the predication of the properties these verbs denote to the objects denoted by their arguments that isn’t available in other predicational contexts. That is, the eventuality of smoking isn’t just accessible with any predication involving cigarette; it is the result of combining cigarette as an object or internal argument of an aspectual verb or a verb like enjoy. It is not only that such inferences are tied to particular verbs; they are tied to them independently of what the verb’s object is. Consider the use of a nonsense word like zibzab. 12 To say (1.29) John enjoyed the zibzab is to say that John enjoyed doing something to the zibzab. At this point, it’s really hard to understand how the inference to an event reading is part of world knowledge. It becomes clear that it is an a priori truth that to enjoy something is to be involved in some interaction with it—some eventuality. When the direct object argument of a verb like enjoy does not denote an eventuality as part of its standard meaning, coercion introduces somehow an appropriate eventuality. This militates strongly for placing coercion within the realm of linguistic knowledge, not contingent factual information about what the world is actually like. A further question concerns how this eventuality involved in predications like those in (1.27c) is specified. While we’ll see that this is not true in all cases, some nouns like cigarette help to specify the eventuality induced by coercion when they are in the direct object of a coercing verb. To see this, consider replacing the word cigarette with the relevant part of its entry in Webster’s New World College Dictionary, which should at least roughly have the same content as the word cigarette. (1.30) a. Nicholas enjoyed a cigarette. b. Nicholas enjoyed a small roll of finely cut tobacco wrapped in thin paper. 12

Thanks to Chris Kennedy for these sorts of examples.

1.2 Different sorts of predication

17

Speakers immediately get the defeasible interpretation of (1.30a) where Nicholas smokes a cigarette but not in the second case. These alternations appear pretty systematic with a small class of words like book, novel, sonata, and so on, indicating that indeed such defeasible interpretations are a part of lexical semantics.13 Coercions, as Aristotle said of all familiar things, are easy to see but hard to understand. Some linguists have argued that in fact coercions are what they appear to be. They indicate that the meaning of terms fluctuates from context to context, and some have taken the moral of these observations to be some sort of radical contextualism about meaning (for instance, Recanati (2004, 2002, 2005)). Pustejovsky (1995) also seems to endorse such a view in emphasizing the “generativity” of the lexicon. But these conclusions do not follow from the evidence. They are also vastly counterintuitive: when I say that I enjoyed the cigarette, does the word cigarette now all of a sudden change its meaning to mean smoking a cigarette? It does not seem so. Fodor and Lepore (1998) and more recently Cappelen and Lepore (2004) correctly, in my view, crticize such an approach to coercion in their criticisms of Pustejovsky (1995). Furthermore, despite many claims that the lexicon is in fact generative or context sensitive in some radical way, I do not know of any formally worked out proposal of this view.14 As we shall see in chapter 3, the generative lexicon of Pustejovsky (1995) has static lexical entries that do not change during coercion. When it comes to technical developments, I shall show that in fact basic word meanings cannot change if we are to be able to derive any predictions at all about lexical meaning. If that approach to coercion is wrong, however, what is the right approach? Coercion is a ubiquitous, attested phenomenon in natural language. One has to be able to give an analysis of it in any remotely viable theory of predication and lexical meaning. In order to explain the data one has to do one of two things: either one has to develop a theory of lexical meaning where the lexical entries themselves change in context, or one has to complicate one’s notion of predica13

14

Laura Whitten and Magda Schwager independently observed to me that using such dictionary definitions is problematic, because Gricean maxims would predict that there are some special reason that the more complex formulation is used. This special reason would block the standard associations with the content. But if the inference here concerns nonlinguistic knowledge, we wouldn’t expect the flouting of the Gricean maxim to block such an inference. A Gricean explanation of why the inference fails for (1.30b) occurs precisely because that inference is based on lexical content, not world knowledge. Although one possibility for formalisation would be in a connectionist approach, where word meanings are thought of as vectors of strengths of associations with other words that get recalculated every time the word occurs. This is very far from either Pustejovsky’s generative lexicon or Recanati’s relevance theory approach. It’s also philosophically and conceptually extremely unsatisfying, as such an approach doesn’t begin to tell us anything about lexical meaning or lexical inference or about how meanings compose together.

18

Lexical Meaning and Predication

tion and logical form. Since I do not see how the first option can be developed in any detail that does justice to the composition of sentential or discourse meaning from lexical content, I will investigate the alternative that coercion phenomena call for a reanalysis of predication. When I say that I enjoyed a cigarette, the word cigarette does not change its meaning but what I enjoyed is doing something with the cigarette. That is, coercions involve a more complex act of predication than one might have thought. The reason that this is so, I will argue, is that in coercions type presuppositions have to be accommodated and, as with accommodation at the level of contents, type accommodation typically introduces new material into logical form.15 Viewed from this perspective, coercion is not really a problem about meaning change in the lexicon; it’s a problem about compositionality—about how lexically given meanings combine together in the right sort of way. I argue for a similar conclusion for dual or multiple aspect nouns like lunch, book, temperature, and so on; the process of justifying type presuppositions with the types assigned to these nouns will complicate the process of logical form construction and may add new content to logical form. To account for many features of predication, the logic of meaning composition has to be rethought and revamped considerably from the standard approach to predication that underlies Montague Grammar. This is the task to which I devote myself for most of this book.

1.3 The context sensitivity of types We want a theory of lexical information that offers a framework within which empirical research will yield a correct account of lexical content. I have argued that a lexical theory has to do two things to reach this goal: give an account of the meanings of lexical items and an account of the operation of predication needed to derive truth conditions for clauses. But in order to capture the observations and intuitions of most of those who have worked in the field of lexical semantics, we need to do this in a particular way: we need to construct a theory of lexical meaning and predication that can exploit features of the discourse context. In particular this means a context sensitive theory of typing. Observations that confirm this last claim have been around for years. Nevertheless, there have been few attempts in the literature to account for these observations. Lascarides and Copestake (1995) noticed that the event readings of the object of a verb like enjoy can depend on discourse factors. Normally enjoy coerces its object or theme argument that has the type book into an expression 15

GL says little about predication. Because it fails to carry through on either way of analyzing the data, we shall see that GL fails to account for coercion.

1.3 The context sensitivity of types

19

that involves an event of reading the book as in (1.31a). But this reading depends on the assumption that Julie is a person, someone capable of reading the book, because this reading is not available with (1.31b). It becomes available if (1.31b) occurs in the context, say, of a fairy tale in which goats have been established as capable of reading. This is a matter of linguistic knowledge, of how the discourse context affects the analysis of (1.31b), not a matter of world knowledge—it is completely unintuitive to assume as a general matter of world knowledge that goats can read in fairy tales (it has to depend on the particular fairy tale!). (1.31) a. Julie enjoyed the book. b. The goat enjoyed the book. Most stories about coercion (Pustejovsky (1995), Nunberg (1979), Egg (2003), Asher and Pustejovsky (2006)) assume that the object argument of enjoy is some sort of eventuality, which is the result of a typing adjustment due to a clash between the type of argument the verb demands and the type of argument that is in fact its direct object; the verb enjoy requires an event as object argument and so coerces the direct object into giving an argument that is of some event type. Regardless of the details of how this coercion process actually works, (1.31a,b) shows two things. First, the inference from enjoy the book to enjoy reading the book must be defeasible. Second, the fact that we can get the reading that the goat enjoyed the book in (1.31b), given a discourse context in which goats talk, shows that the typing and typing adjustment rules must be sensitive to information in discourse. Danlos (2007) has shown that aspectual verbs are also sensitive to discourse context. Aspectual verbs take some sort of eventuality as an object or theme argument. The GL framework claims that these eventualities are given by the lexical entries of nouns. But Danlos’s examples show that this is not the case: (1.32) a. ??Yesterday, Sabrina began with the kitchen. She then proceeded to the living room and bedroom and finished up with the bathroom. b. Yesterday Sabrina cleaned her house. She began with the kitchen. She then proceeded to the living room and bedroom and finished up with the bathroom. c. Last week Sabrina painted her house. She started with the kitchen. She then proceeded to the living room and bedroom and finished up with the bathroom. The examples in (1.32) show that the eventuality is not, at least in all cases, given by the lexical entry of a noun in the theme argument of the verb or some

20

Lexical Meaning and Predication

adjoined PP. When a discourse is not clearly about any particular eventuality as in (1.32a), the coercions induced by the presence of aspectual verbs are rather bad; there is no way to recover an associated eventuality to serve as the theme argument of the aspectual verbs. On the other hand, when the discourse is clearly about some eventuality, then these aspectual verbs naturally get associated with that eventuality. Skeptics at this point might say that this is not a matter of linguistics at all but cognitive psychology and world knowledge.16 The judgements in (1.32) might be a matter of the interpreter’s being primed with a recent mentioning of some event that can be used to fill in the needed event arguments to the aspectual verbs. However, if these judgements were merely a matter of priming, then the structure of the prior discourse shouldn’t affect the judgements. Yet it does. Contrast (1.32c) with (1.32d): (1.32d) ?Last week Julie painted her cousin’s house. Then this week she started with the kitchen. She then proceeded to the living room and bedroom and finished up with the bathroom. This discourse is once again much more problematic and resembles (1.32a) in that it’s no longer clear what is the event argument of the aspectual verbs start, proceed, and finish up. (1.32c) and (1.32d) form a minimal pair and provide strong evidence that furnishing an event argument to these aspectual verbs is a linguistic matter that involves an interaction between the discourse context and how it is structured, lexical meaning, and the construction of the meaning of an individual clause. It is not a matter of language independent world knowledge or of psychological priming. It’s not just the aspectual verbs whose coercive force is sensitive to discourse context. Discourse context can push us to reinterpret many expressions:17 (1.33) a. I went to the gallery after the robbery. The elephant had been stolen. b. I went to the zoo after the robbery. The elephant had been stolen. Discourse context also affects the interpretation of other forms of predication. The genitive is one such predicational construction where there is a relation inferred to hold between variables or constants introduced by the DPs in that construction. Sometimes this relation is specified by a relational noun in the genitive construction—e.g., John’s mother. However, Asher and Denis (2004) note that discourse-based information can override the relation provided by the lexical meaning of relational nouns: 16 17

Jerry Fodor put this point to me on a related matter after a lecture on discourse structure at Rutgers in 2005. These examples are inspired by examples from Marliese Kluck.

1.4 The main points of this book

21

(1.34) [Context: Picasso and Raphael both painted a mother with a child.] Picasso’s mother is bright and splashy —a typical cubist rendition. It makes Raphael’s mother look so somber. And where no relational noun is involved, discourse context can also affect how to interpret the underspecified relation introduced by the genitive: (1.35) a. All the children were drawing fish. b. Suzie’s salmon was blue. As Asher and Denis (2004) and a number of other researchers (Vikner and Jensen 2001, Partee and Borshev 1999) have argued, the genitive construction seems to be one that is partially determined by the lexical entries of the nouns involved—and Asher and Denis argue that in particular it is the types of the nouns involved that help determine the relation. But the examples above show once again that the meaning of this construction is discourse sensitive in a way that previous analyzes of the interaction between discourse and the lexicon (cf. Lascarides and Copestake 1996) were unable to capture. These examples of discourse sensitivity show something of much more general interest as well. The eventualities that are inferred to be the arguments of the aspectual verbs in, for example, (1.32b,c) are inferences that are not based on general conceptual or world knowledge but on information that is conveyed linguistically in prior discourse. This must be so, since if these inferences were driven by non-linguistic knowledge, then (1.32a) should be fine when it isn’t. We haven’t shown, of course, that these mechansims cannot be guided by nonlinguistic information; but we have shown that they are not themselves simply a matter of world knowledge. The discourse sensitivity of the mechanism shows us something that the simple predicational examples cannot do straightforwardly (though see the last paragraph of the last section of this chapter); the coercion mechanisms are part of the linguistic system.

1.4 The main points of this book Examples of coercions with simple predication like those in (1.27a,c) have received an extensive discussion in the lexical semantics literature (Pustejovsky (1995), Kleiber (1999), Egg (2003), Recanati (2005)). But a satisfactory formal analysis of coercion, copredication, or restricted predication and how these phenomena affect the composition of content is lacking. Further, no one to my knowledge has really investigated the interactions between discourse and lexical meaning of the sort introduced in the previous section. I will develop a

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Lexical Meaning and Predication

theory of predication and lexical meaning and an account of how these interact with discourse structure in this book. I will use the theory to investigate phenomena like copredication, restricted predication, and coercion. The guiding idea, already implicit to some extent in Pustejovsky (1995), but made much more explicit in Asher and Pustejovsky (2006), is that almost all words will have single and simple lexical entries. Words like the nouns cat, lunch, book or the verbs kill, read, and master denote simple properties or relations, and so accordingly the logical forms that specify their denotations are very simple. For instance, cat has the lexical entry λx cat(x), while master is a simple transitive verb that denotes a relation between its subject (or external) and object (or internal) arguments. If words have simple entries and make simple contributions to truth conditional content, they come with a rich amount of information about the types assigned to the lambda terms and the variables within them. These types will guide predication and be responsible for fine-grained differences in lexical meaning. When words are combined together to form clauses, sentences, and discourses, the types associated with various terms will interact in complex ways. I will introduce operations of type adjustment in response to type mismatches between predicate and argument that correspond to the accommodation of a type presupposition, or more generally speaking the justification of such a presupposition. I will tell a similar story about morphological processes with semantic import; in effect the application of semantically rich morphology to a word stem is also a matter of type driven predication, which may bring with it various type adjustments. The effects of these type adjustments at logical form is that the logical form for the clause will contain elements that are not present in the lambda terms for the constituent words themselves. Predication involves not only applying a function to an argument but also operations of adjustment corresponding to type presupposition justification. Coercion and the sort of problematic copredications that I introduced earlier will invoke particular sorts of presupposition justification. As we will see, however, predication is not simply a matter of putting well formed lexical meanings together and adjusting them when they do not fit; type information is also to some extent dependent on the discourse context. The theory I will present will provide a framework for investigating the context sensitivity of type assignments. While I will use the framework of the typed lambda calculus, I will extend the typed lambda calculus beyond the usual set of simple types and functional types to include two complex types. These complex types furnish the basis for my analysis of problematic cases of copredication and coercion. One complex type, the • type, will be used to analyze terms of which we can predicate

1.4 The main points of this book

23

properties of two different aspects of the same thing (what I call aspect selection). Most of these terms are nouns, and I will call them dual aspect nouns. I will argue that this kind of predication requires a special metaphysical conception of the objects whose aspects are the bearers of the properties predicated. The type adjustment with dual aspect nouns is, in some sense, just a shifting of emphasis or a reconceptualization of the very same object. Coercions like those induced by verbs like enjoy, shift the predication entirely away from the original object to some other object of a different type—for instance, an eventuality associated with the original object. Such coercions, unlike aspect selection, do not affect the way the objects denoted by the term that is subject to the coercion are counted or individuated. I will model coercions with another sort of complex type, something that I shall call a polymorphic or dependent type. I will provide rules for dealing with the types I introduce in the analysis of predication when they occur either as type presuppositions of predicates or as types on arguments of predicates that require some other type. Rules for type presupposition justification will allow us to select the appropriate type of the argument for the requirements of the predication at hand. In addition I will show how the type system is sensitive to the discourse context. By integrating a theory of discourse structure and discourse contexts from earlier work (Asher (1993), Asher and Lascarides (2003)) within the theory of predication, I will show how discourse can transform and constrain type assignments and type transformations. The type system is dynamic and evolves as discourse proceeds in a way similar to the way that linguists and philosophers have argued that the semantics of discourse evolves dynamically. My approach crucially distinguishes the logical forms constructed during predication from the types that guide and constrain predication. When we accommodate, say, a type presupposition of a predicate that demands an event as an argument but is given something of type physical object, justifying the presupposition will require not only an adjustment in types but, typically, an adjustment at the level of logical form as well. We need types to construct logical form, but we also need the logical forms as distinct semantic citizens, for it is they, not the types, that are the vehicles of model theoretic content. I will argue for a “two stage” or two level semantics for lexical meaning: a level with the usual intensions for the expressions of logical form, and a level with a proof theoretic semantics for the types. The nature of types and the argument for my two level theory will occupy much of the next chapter. The development of the system of complex types and the two-stage theory of semantics and its applications will occupy the rest of the book. Besides the analyzes of copredication and coercion, I will show how the system yields an analysis of restricted predication and the genitive

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Lexical Meaning and Predication

construction. Though topics like metaphor and poetic license outrun the scope of this book, I will tentatively offer an application of my system to these topics at the end of the book. We’ll see that types in the theory of predication are closely linked to metaphysical principles of individuation and counting. Thus, the types used to guide predication will be of a quite general nature. The system of types, however, involves more types than those just needed for checking predication. Finegrained differences in types can affect the content of a predication and can account for at least some analytical entailments—entailments that are a priori and follow from the meanings of the “non logical” words of natural language. More speculatively, types will provide a linguistic foundation for a theory of concepts and of internally available contents and inferences. The approach is thus anti-Quinean (and also contra Fodor and Lepore (1998), Cappelen and Lepore (2004)). Nevertheless, I have taken to heart the warnings of Fodor and Lepore about lexical semantics. For instance, my type-driven theory of predication is agnostic about lexical decomposition beyond what is demanded by morphology and syntax.

2 Types and Lexical Meaning

The typed lambda calculus and its operation of type restricted application are familiar to anyone who has worked in formal semantics. But there are largely unexamined questions about the nature of types, their relations to formulas of logical form, and the effect of rules of type shifting on logical form. We need to look at these questions in detail.

2.1 Questions about types Let us first turn to questions about types. For one thing, are our types all atoms or are there types that have a structure and that are constructed from “type constructors” together with other types? Another question is, what is the interpretation of our types? Montague Grammar has an answer to our questions. Montague Grammar starts with two basic types, the type of entities e and the type of truth values t and then closes the collection of types under the recursive rule that if a and b are types, then so is a ⇒ b.1 The type a ⇒ b is one that, given an argument of type a, produces an object of type b. Montague Grammar converts these extensional types into intensional types as follows: if a is an extensional type, then s ⇒ a is its intensional correlate, where s is the type of worlds or more generally indices of evaluation. Montague Grammar has an extensional set theoretic model of types: the primitive types are identified by their inhabitants, the set of objects of that type relative to some domain of interpretation, while the set 1

The notation → for the functional type constructor is standard. But as I will distinguish this functional type constructor from its related cousin, the implication constructor for logical forms, → and from the usual way of defining a function ( f : a → b), I will use the slightly nonstandard ⇒.

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of functional types over types a and b is modelled as the function space or set of all functions from a into b, { f : a → b}. When Montague developed his theory, his use of the typed lambda calculus served a logical purpose. Turing (1937) had shown that the untyped lambda calculus had a model in the set of computable functions, but the application of such a theory in formal semantics was problematic. When terms of the untyped lambda calculus include the standard truth functional operators essential to semantics, it is easy to form terms like λx¬x[x], which is the property of not applying to oneself—the Russell Property. Applying the Russell property to itself (note that we’re using untyped Application here) produces the following result, which is uninterpretable when negation is understood as in classical logic: (2.1) λx¬x(x)[λx¬x(x)] = ¬(λx¬(x)[λx¬x(x)]) The typed lambda calculus avoids this problem, since the Russell Property does not have a consistent type in the typed lambda calculus. It is, in other words, not a well-formed term. The typing of expressions allows one to combine the lambda calculus with the operators of classical logic; it also ensures that the theory is consistent if set theory is. The typed lambda calculus avoids paradox in a simple and pretty much cost free way. Since the work of Dana Scott and Gordon Plotkin in the early seventies (Scott (1972)), we have abstract models of the type-free lambda calculus. But they have certain drawbacks for the purpose of studying natural language semantics. The models used by Scott require that the values of lambda terms be continuous functions in the sense that one can compute their value in the limit given some long enough run of values. But it is precisely the operators of classical logic like ¬, ∀, or ∃ that fail to be continuous in the requisite sense. While the untyped lambda calculus has various uses in mathematics and computer science,2 there are compelling linguistic reasons to adopt a typed lambda calculus in constructing logical forms. The models of a Montagovian typed lambda calculus together with the standard quantifiers and connectives are unproblematic set theoretic constructions. Furthermore, the typed lambda calculus provides a tight connection between syntactic categories and semantic 2

For example, in representing fixed points or recursion. In recursion, a function or term takes itself as an argument. Turner (1989) has argued that in natural language terms, in particular property terms, can take themselves as arguments as well:

(2.2a) Being nice is nice. (2.2b) It’s bad to be bad. I will argue in chapter 10, however, that examples like (??) don’t argue against a typed notion of predication.

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types, as has been shown in the field of categorial grammar. There is a natural homomorphism from categories to types that paints a compelling picture of the interaction between syntax and semantics and yields a simple computational process for the derivation of a logical form for a clause. A third reason is that the typed lambda calculus has just the right expressive power. There is a natural distinction between properties and objects crucial to the intuitive conception of predication (where one predicates a property of an object)— properties are functions whereas objects are the arguments of functions. It’s less obvious how to model traditional predication in an untyped system, since the basic distinction between objects and properties isn’t reflected in the semantics of expressions. A typed system yields a natural explanation of semantic well-formedness and semantic anomalousness too, while untyped systems don’t. Nevertheless, typed systems are expressive enough to define the operations relevant to the analysis of predication as well as semantic correlates of the morpho-syntax of word formation. Examples include: nominalization— the operation from higher types into e (e.g., is square −→ the property of being square), grinding (the operation that takes a count noun like rabbit and transforms it into a mass), the operation from type e into a quantifier type to handle coordinations like John and three students went to the party, the operation from types of transitive verbs as types of first-order relations into higher types (should we wish it), the operation from individual denoting terms into collection denoting terms, and the transformation from mass/count arguments into activity/telic eventualities (e.g., drink wine vs. drink three glasses of wine). In a minute, we are going to complicate Montague’s vision of types considerably. I turn now to some reasons for this move.

2.2 Distinguishing between types An analysis of predication that uses a rich system of semantic types is only promising if there is good evidence that languages encode such a system. Luckily, there is evidence that information about types is conventionalized; it even affects the case system in some languages. There are many ontological categories that languages reflect in their grammar: the distinction between questions and propositions, for instance, or the distinction between abstract objects and concrete or physical objects. These distinctions permit us to distinguish between the malformed predications in (1.1f–j) and well-formed predications in which the types of the predicate and its arguments match.

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2.2.1 Events vs. objects Many other distinctions between types of objects exist. What follows is a partial list. The distinction between eventualities (events, processes and states) and non-eventualities (other sorts of objects) allows us to predict general patterns of predication. Events occur at times but objects don’t; adverbs of manner of motion go well with events, but their adjectival counterparts fail to predicate felicitously of many noneventualities, as the minimal pairs below in (2.2) and (2.3) demonstrate: (2.2) a. John’s birth occurred at 10 am this morning. b. #John occurred at 10 am this morning. (2.3) a. The tree grew slowly. b. ?The tree was slow. Languages like Japanese reflect the distinction between eventualities and objects in the grammar within the system of particles. Chinese reflects the distinction between eventualities and objects within the system of classifiers. Linguists have also shown that languages are sensitive to the differences among eventualities. Dowty (1979) and Vendler (1957), among others, have observed that punctual events like achievements don’t go well with adverbials that express the fact that the event they modify took place over an extended interval of time. On the other hand, activities fit well with such adverbials.3 (2.4) a. #John died for an hour. b. John ran for an hour. Thus, further distinctions between different sorts of eventualities help explain patterns of predication involving temporal adverbials. The distinction between states and other eventualities shows up in other parts of the grammatical system. For instance, verbs that denote states in general do not accept the progressive form in English, while verbs that denote events do. (2.5) a. #Samantha is knowing French. b. Samantha is running. c. Arnold is dying. Japanese marks a distinction between animate and inanimate in its predicative verb structure, which is related to the difference between event types and non-event types (but does not exactly correspond to it). • iru(x), x: animate • aru(x), x: inanimate (event) 3

See also Krifka (1987), Rothstein (2003) for more discussion of this data.

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2.2.2 Locations vs. objects Another example of a grammatically grounded distinction between types in natural language concerns the distinction between locations and physical objects. Places are fixed elements in the terrestial reference frame, while objects typically have a complex internal structure and can move with respect to the terrestial reference frame. Some evidence for this distinction comes from Basque, where the grammar encodes differences between location and objects via two genitive cases -ko and -ren; locations in general easily take the genitive -ko but not -ren, while objects in general do the reverse. Aurnague (1998) distinguishes the following types: places (e.g., valley, field, river, mountain, hill), objects (e.g., apple, glass, chair, car), and mixed objects (e.g., house, church, town hall). Of particular interest are the “mixed objects” and the behavior of their expressions in Basque. The terms for mixed objects readily accept both forms of the Basque genitive. So if we accept the encoding hypothesis for Basque, mixed objects like houses would appear to belong to two types, or two ontological categories, at the same time—location and physical-obj—neither of which is a subtype of the other (it is neither the case that the properties associated with physical objects are inherited as properties of places nor that the properties associated with places are inherited as properties of physical objects). (2.6) Maite dut etxeko atea haren paretak harriz eginak direlariak. (Michel Aurnague p.c.) I like the door of (locational genitive) the house the walls of (physical object genitive) which are made of stone. Prepositions in English serve to distinguish between places or locations and physical objects, though the distinctions are less clear cut than in Basque (Asher 2006).

2.2.3 Mass vs. count The mass/count distinction is another type distinction marked in many languages. Chinese marks this distinction in its system of classifiers, whereas other languages like English mark the distinction with determiners and to some extent within nouns (although many nouns can receive both a count and noun interpretation). Certain determiners in English are designated as mass determiners—e.g., much, as in much water, much meat. They do not go with count nouns in general—e.g., much person, much people are malformed. Other determiners like many, every, the apply both to count nouns and mass nouns but

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require special interpretations when put with mass nouns.4 Thus every water, many waters must range over contextually given portions of water or perhaps kinds of water. Languages also distinguish “determiner phrases” (DPs) denoting quantized portions of matter like many waters or the water from DPs with ordinary count nouns. (2.7) You can take two piles and put them together to make a bigger pile. (2.8) Two little waters make one large water. (in a restaurant) Piles pick out portions of matter as does the expression two waters. But you can’t put two dogs together to make a bigger dog. This last observation holds for all count nouns.

2.2.4 Kinds vs. individuals Another universal distinction in language is the distinction between kinds and individuals. Kinds are often expressed in English with a bare plural noun phrase (e.g., cats, numbers, people) but can also be expressed with other constructions: (2.9) The Mexican fruit bat is common in this area. Linguists take the definite noun phrase in sentences like (2.9) to refer to a kind rather than to range over individual members of the kind. They argue, quite sensibly, that the predicate is common in this area (but also others like is extinct, is widespread) cannot hold of an individual but only of kinds or species. Thus, most languages encode a three way distinction between the types of masses, countable individuals, and kinds. Chinese, once again, encodes the distinction between kinds and individuals in the classifier system. (2.10) Moby Dick shi yi tiao / *zhong jing. Moby Dick be one Cltail /*Clkind whale. Moby Dick is a whale (individual).

2.2.5 Containers vs. containables A more subtle type distinction encoded in the system of prepositions within English involves containers and containables. In general anything that desig4

Borer (2005a) argues that the mass/count distinction is to be located in a classifier-like syntactic projection and that nouns are by default mass. She cites as evidence the fact that nouns without classifiers in Chinese are interpreted as mass.

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nates a specific volume or enclosure can be a container. Many physical objects can serve as containers, although some cannot. (2.11) b. c. d.

a. The water is inside the pitcher. The keys are inside the car. John put the keys inside his pocket/inside the drawer. # John threw the keys inside the air. (Versus: John threw the keys in the air.) e. # John put the wine inside the water. (Versus: John put the wine in the water.)

2.2.6 Plurality Plurality also introduces type distinctions. Research on plurals, as I briefly outlined in chapter one, distinguishes at least two types of plural predication: distributive and collective. Distributive predication occurs when a property or relation is predicated of each element of a set as in (2.12) The boys each worked hard. On the other hand, (2.13a,b) exemplify collective predications where a property is predicated of the whole set of students but not of each student individually. (2.13) a. The students surrounded the building. b. The students mowed the whole meadow. Sometimes collective predication occurs with singular nouns (and so this semantic phenomenon must be distinguished from the syntactic phenomenon of number). (2.14) The committee is meeting in the lounge. (collective predication) Some predicates, finally, put no requirements on how their plural arguments are to be understood. 5 (2.15) Three girls danced with four boys. (2.15) makes no claims about whether the boys distributively or collectively danced with the girls, only that there were three girls and four boys and that dancing went on between them. 5

These are called cumulative predications.

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The distinctions between types of predication mark distinctions in type presuppositions. For instance, an account of the lexical meaning of the word disperse or surround must mark it as requiring an external argument that is of group or collective type, which means that the argument must denote a group of individuals or range over groups of individuals. Verbs like disperse or surround do not go well with inherently distributive quantifiers like most students, whereas they apply perfectly to plural noun phrases that can be interpreted as denoting groups: (2.16) a. ?Most students surrounded the building.6 b. The students in the square surrounded the building. Other predicates like work must be interpreted distributively and impose type presuppositions on their arguments to that effect.

2.2.7 Types and specific lexical items Some verbs have quite specific type requirements encoded in their selectional restrictions. (2.17) a. John weeded (mulched, hoed . . . ) the garden (lawn, area, tomatoes, peas, plants . . . ). b. John hoed (mulched) the weeds. c. # John mulched, hoed, weeded the water. d. #John shoveled the closet. You can’t weed, hoe, or mulch certain types of locations—bodies of water, for instance. Locations that can be weeded have to have dirt or soil in them. And you can only shovel a location open to the elements. This is not a matter of world knowledge but a matter of grammar, broadly construed: one can perfectly well imagine someone cleaning a body of water like a lake of floating algae or water plants, but we don’t call that weeding. Given the fine-grained type distinctions made by the language, it should not be surprising that when predicates combine with arguments of different types, the meaning of the predicate shifts in the resulting predication. Consider the following. (2.18) a. John swept (shoveled, wiped . . . ) the closet (room, walkway, kitchen, fireplace, floor, counter . . . ). b. John swept (shoveled, wiped . . . ) the dirt (debris, manure, sand, slush, litter, shavings, cinders, dust . . . ). 6

Note that the partitive DP most of the students does admit of a collective interpretation, unlike the straight quantified DP most students.

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Predications involving sweep have a different content depending on whether the direct object is a location, a place, or a surface (we can assimilate all of these here to the type location), or whether the direct object is a portion of matter. The resulting meanings for sweep, for instance, are so different that they don’t license copredication or ellipsis: (2.19) b. c. d.

a. John swept the kitchen and Mary the entryway. John swept the dust and Mary the leaves. #John swept the kitchen and Mary the leaves. # John swept the kitchen and the dust.

Another example of an apparent meaning shift in a predicate because of a shift in type of its argument comes from communication verbs. The meaning of verbs like whisper, whistle, whine, etc. varies with respect to whether it has an object argument; furthermore, the type of this argument can also affect the verbal meaning. (2.20) a. John shouted (whispered, whistled, whined . . . ). (activity) b. John shouted (whispered, whistled, whined . . . ) a warning. (accomplishment) c. John shouted (whispered, whistled, whined . . . ) at the animal. (accomplishment or activity) d. The bullets whistled past John. (accomplishment) Such so called verbal alternations have occupied linguists for many years.7 They offer us another means for seeing how languages encode a sophisticated system of types. But they also clearly pose a challenge for lexical semantics that resorts to types: how can we account for the shifts in the meaning of a verb given its different arguments? Other well-known verbal alternations involving prepositions show subtler shifts in meaning. (2.21) b. c. (2.22) b. c. d. 7

a. John treated Mary to dinner. John treated Sam for cancer. John treated the cancer. a. John loaded the hay on the wagon. John loaded the wagon with the hay. John sprayed the paint on the wall. John sprayed the wall with the paint.

For a comprehensive bibliography and discussion, see Levin (1993).

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To state the generalizations as to how these predications behave, we must recognize distinct types accorded to arguments of these transitive verbs. Predicating the verb treat of a living entity such as a person, animal, or even a plant means something quite different from when the verb is predicated of a direct object that denotes a disease. The type of the object of the prepositional phrase modifying the verb also will make a difference to what the predication ends up expressing; for instance, we infer that all of the hay is on the wagon in (2.22a) but not in (2.22b), whereas (2.22b) conveys that the wagon is fully loaded, or at least prepared, whereas this is not the case in (2.22a). Sensitivity to argument type is not just a property of verbs and prepositions. We’ve seen how determiners interacting with different nouns can deliver different types of DPs. Adjectives have a similar behavior. Consider the difference in the meaning of the adjectives flat (due to Partee (1984)) and red in the following: (2.23) b. c. d. (2.24) b. c. d.

a. flat tire flat beer flat water flat country a. red meat red shirt red pen red wine

I could go on and on with the list of type distinctions made in language and that affect the contents of predications. There is a lot of evidence that natural languages encode a rich system of types that are used to guide predication. For each type distinction, I will try to establish syntactic or lexical alternations to distinguish between types. That is, positing a type distinction in a lexical theory will require linguistic evidence: there must be a linguistic construction that accepts expressions of one type but not the other. This is at least a minimal condition.

2.3 Strongly intensional types Supposing that we enrich the type system with a suitable collection of subtypes of e, can we still retain Church’s and Montague’s simple set theoretic model of types? The answer is no. The set theoretic model isn’t right for several reasons. First, we need to distinguish between a type and the type’s extension or inhabitants in order to have a coherent notion of subtyping. Consider, for example,

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the relation between the type of physical properties (physical-object ⇒ t, or p ⇒ t) and the type of all first-order properties e ⇒ t. It is intuitively obvious that the type of physical properties is a subtype of the type of all first-order properties. But if we think of types as sets and identify the relation of subtyping with the relation of subset, then this intuitive connection is lost: the set of functions from physical objects to truth values is disjoint from the set of functions from entities generally to truth values, as long as there are some entities that are not physical objects. That is, these two sets of functions have distinct domains and so their intersection is 0, even though the physical properties are subfunctions of the first-order properties. No element of the set of inhabitants of p ⇒ t is an element of e ⇒ t. But we can’t simply do without the notion of subtyping. We need the usual subtyping relation between physical properties and first-order properties to state a suitable form of application, according to which the application of a determiner, which requires its first argument to be some first-order property, to a noun phrase denoting a physical property or physical kind yields a semantically well-formed content. The set theoretic interpretation of functional types is a disaster for the compositional construction of logical forms once we posit subtypes of e. We might try to reinstate the set theoretic interpretation by taking p ⇒ t and e ⇒ t to be sets of partial functions. This gets things wrong for a different reason. Consider the obvious truth that physical properties like height, weight, mass, length, and so on are not properties of abstract objects. Yet on the partial functions view, they are and necesssarily so! They are simply everywhere undefined on the domain of abstract objects. Can we limit our interpetation of types then to be just certain sets of partial functions? Perhaps, but which ones? Do we want to consider as a physical property some property that is only defined for very few types of physical objects? More importantly, if we exclude the empty partial function as an inhabitant of any functional type, then our sets of “nice” partial functions may not be closed under operations that we would like for types, like joins and meets (or unions and intersections if you think set theoretically). Furthermore the empty partial function is just ∅. So if we ban empty partial functions from the set of partial functions, the set of types loses its lattice and even semi-lattice structure. This threatens the whole structure of the lexical type hierarchy and with it the analysis of predication and lexical meaning. The whole attempt to save the set theoretic model begins to look very gerrymandered. Another reason we need to move away from the extensional conception of types is that sets are not a faithful representation of some types that we need. Consider, for example, the set of fictional entities. Let us take seriously the idea that fictional objects are indeed fictional and so they don’t exist; they are not

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part of the domain of interpretation. Further, fictional objects not only don’t exist in the actual world; they don’t exist in possible worlds either—i.e., other ways in which the world could be. A fictional creature like a hobbit is not at all like the sister that I might have had; the latter exists in a possible world, the former does not. That’s what it is to be fictional. So on a view that identifies types with their inhabitants, types corresponding to fictional objects and the absurd type would be the same type, since they have the same extension or the same set of inhabitants, namely the empty set. But types of fictional objects are intuitively distinct from ⊥. Whether a term describes a fictional character or not certainly appears to make a difference as to how predications are understood. Within fiction, there is no question of checking or wondering whether the predication actually results in a literal truth. It is even quite controversial among philosophers who have written on fiction whether terms that appear to refer to fictional entities refer in fact to anything at all. On the other hand, fictional talk differs from metaphorical or loose talk; fictional talk is literal—the trees in The Lord of the Rings literally speak (see 2.25) whereas in metaphorical talk (2.26) the predications aren’t to be taken at face value. (2.25) Look! the trees are speaking. (Lord of the Rings) (2.26) These trees are really speaking to me. I’m going to paint the living room green. To make sense of this difference in predicational behavior, we should distinguish a type of fictional objects. And it should be distinct from the absurd type no matter what the circumstances of the actual world are. This leads me to adopt the thesis that types are neither to be identified with their actual inhabitants (extensions) nor even their possible inhabitants (standard semantic intensions). They are “hyper-intensional.”8 If types are intensional entities, they are not intensions as semantics standardly conceives of them—i.e., as functions from indices (possible worlds or sequences consisting of a world, time, context, and other appropriate elements) to extensions.9 Modulo a certain understanding of fictional objects, I have established that types aren’t to be identified with extensions, intensions, or sets thereof. So where do types fit into an ontology of abstract entitites? We need to think about the relations between the following sorts of abstract entities: • types 8 9

Reinhard Muskens in Muskens (2007) also argues for a hyper-intensional construal of types. The extension (at an index) of an individual constant is an individual; the extension of a 1-place predicate is a set; and an extension of a closed formula is a truth value.

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• concepts • properties Given the relatively well-understood analysis of properties in formal semantics and pragmatics as semantic intensions (or as functions from indices to extensions), types cannot be properties. In addition, properties are typically understood to be mind-independent entities, entities whose existence is not dependent on the existence of minds. Types, however, given their role in guiding predication, are part of the conceptual apparatus necessary for linguistic understanding. They are mind-dependent representations of mind-independent properties and individuals. This leads us to the hypothesis that types are concepts, which I take to be mind-dependent entities as well. This hypothesis has some promising support. Concepts come at different levels and granularities. They form a hierarchy, just as types do. There are concepts of what it is to be a property, what it is to be an individual, what it is to be a physical object, and so on. There are also much more specific concepts: the concept of red, the concept of S´egol`ene Royale or of Hillary Clinton. Like types, concepts are the internal, mind dependent reflection of mind independent properties and individuals they are concepts of. Concepts (and types) have their own “internal” semantics which has to “track,” in the appropriate way, the properties and individuals they are concepts of. It is in virtue of such tracking that a concept is a concept of some object or property. For example, the concept of red tracks the property of being red. A concept red of the color red is triggered by something at a particular location l in the conceiver’s visual field, typically when the conceiver is perceptually aware of something at l that is in fact red in color. Though this tracking is generally reliable, it can occasionally fail. For instance, if there were something wrong with a conceiver’s visual apparatus or the circumstances of the perceptual event were very non-standard, the concept red could be introduced as holding of some object when the object of which the conceiver is perceptually aware is in fact green in color. Another way the tracking could go wrong is that the object is in fact red but the concept fails to be triggered. Although making this notion of “tracking” precise is a book-length project on its own, I can say a few words here. I understand this notion in terms of how the rules for the application of the concept in the conceptual system function. In fact it is these rules that define the concepts and give them their content. These rules look something like natural deduction rules. A concept has certain “introduction rules” and certain inference rules that it licenses. The introduction rules stipulate that an object must satisfy certain conditions for its falling under a certain concept; these conditions may be determined by the sensory system

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of the organism or by other associated concepts. Talk of satisfying the application conditions of a concept can be replaced by the notion of something’s being provably of that type, giving an intuitionist flavor to the interpretation of types.10 There are more complex combination rules as well that determine how one concept interacts with others. It is these same rules that determine whether an object is of a basic type like cat or not. Type presuppositions and the rules for type presupposition justification are instances of rules of type and concept combination. This system of rules supplies what computer scientists call an internal proof theoretic semantics of a term and these rules define or give the content of concepts and a fortiori of types. This internal semantics contrasts with the mind-external, denotational semantics of the terms, which involve real world objects, properties, and so on. The two are connected by the tracking mechanism. Linking concepts and types together helps us understand both: concepts get a rigorous framework from type theory, while types are now linked to the agent’s sensory interactions with his environment and as well as the interactions between other concepts/types in the linguistic system. It is plausible that humans in a given speech community share concepts and a type system. They must do so in order to communicate, to exchange information.11 The internal semantics conceived as a system of proof or computation rules allows us to make sense of a shared conceptual system. If your concept of red and my concept of red have the same internal semantics, then we can be said to have the same concept of red, and similarly for other concepts. We can prove of two such proof or computational systems whether they are the same or not, using several different criteria. The crudest one is input/output equivalence; roughly two systems are input/output equivalent, just in case they give the same results in the same cases. With respect to the conceptual/type system, this would mean that the same linguistic actions and judgments are observed in the same contexts. Demonstrably, members of the same speech community have systems that are largely input/output equivalent. There is also the criterion of trace equivalence, where for each computation, the same sequence of actions is observed in the two systems. Finally, there is the criterion of bisimulation, according to which for each point in the computation there are the same possible continuations. Because the rules that make up the internal content of concepts are in general defeasible, concepts cannot determine reference to mind independent entities or properties independently of the context of their application. My notion of a concept thus differs from Frege’s notion of a sense.12 On a standard Fregean 10 11 12

More on this below. A proof of this fact is to be had in Lewis (1969). Peacocke (1992) uses concepts as constituents of thoughts to account for informativeness and

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view, senses compose together to yield thoughts, which determine on their own the truth conditions of sentences and discourses. I need a different view. A proposition is the result of the compositional interpretation of logical forms for the words that make up the sentence or discourse. But the compositional interpretation of a logical form results for me, as for most semanticists, in an intension—a function from indices to truth values. Since concepts don’t determine extensions, let alone intensions, concepts cannot be the constituents of propositions.13 Philosophers might take intensions to be simply formal standins for what propositions “really are.” But even then, if sentences are typically about mind-independent objects and the properties and relations these objects stand in, then the “real propositions” such sentences express will not contain concepts either— at least not of the sort I have in mind, mind-dependent entities with an internal semantics. More concretely, consider basic referential expressions: indexicals, proper names, demonstratives. The content of these terms, intuitively, has to do with the individuals they denote, not some proof object or set of rules for defeasibly determining whether a given object is in their denotation. The content of the type associated with you consists of rules for determining who the audience is in a particular context. But that’s not the contribution of you to the content of a clause in which it occurs. Its semantic content in this sense is the audience itself. Forceful externalist arguments given by Kripke (1980), Putnam (1975), and others show that our concepts associated with names of individuals and natural kinds do not suffice to determine the extensions or intensions of these expressions. If one looks to the behavior of such terms in modal contexts, there is compelling evidence that their meanings are not in general determined by “what is in the head” of a competent speaker of the language. In this respect too, types resemble concepts; they are tied via the expressions they type to properties and real world entities, but they are not identical to properties or real world entities, nor to sets thereof. They are part of our conceptual apparatus used to guide predication. To show how the externalist arguments affect types, let’s consider a typical Twin Earth scenario, familiar from the externalist literature cited above. Oscar on Earth and his twin “Twin Oscar” on Twin Earth speak syntatically identical languages and are type identical down to their molecular constitution. In

13

Frege style puzzles about the substitution of coreferential terms; he takes concepts to be something like Fregean senses. In fact, on the standard semantic conception of propositions, propositions don’t have “constituents” except in a set theoretic sense—and these would be sets of n-tuples of worlds, other indices, and truth values.

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keeping with general physicalist principles then, they have the same internal make up, the same thoughts, the same conceptual system. In particular their linguistic judgements about semantic well-formedness will be the same. Thus, when Oscar and Twin Oscar each interpret the strings water is wet and water is a tree, they assign the same syntactic form and the same semantic types to each expression; for the first string they will each construct a coherent logical form using the tools of the theory of predication, while for the second they will not. But whereas they marshal the same type system and conceptual resources when dealing with their languages, the languages of Oscar and Twin Oscar are different—they have a different semantics. On Earth water picks out the kind H2 0, or real water, whereas on Twin Earth the string water has a different semantics—it denotes a chemical compound distinct from H2 0, which, so the story goes, is XYZ or “twin water.” Water in English picks out a different substance from water in Twin English. Thus water makes dramatically different contributions to truth conditions in sentences of English and Twin English like: (2.27) Water is H2 0. (2.27) is true (and necessarily true) in English but false (and indeed necessarily false) in Twin English. Such Twin Earth scenarios are well established in the philosophical literature, and intuitions about them are relatively robust. They constitute powerful evidence that as internal reflections of properties and individuals, concepts, and types are not identical with mind-independent properties or individuals nor do they determine them, although they are associated with them through the tracking mechanism. If concepts, and a fortiori types, are not constituents of propositions, they can nevertheless compose together. Making the linguistically relevant types a subset of the set of concepts allows us to use the logical framework of types to explore concepts. Types associated with properties are functions from one type into another; when given an appropriate type as argument, types associated with properties return a new type. We can even compose types or concepts together to give us types associated with propositions or semantic intensions.14 I shall call the type corresponding to a proposition a thought. If there are as many types as there are distinct word stems in the language, then the hypothesis that types are concepts and compose together to yield thoughts gains in plausibility. Let us look a bit more closely at the composition of thoughts. Once we 14

Composition allows us to talk of concepts as constituents of thoughts, though this talk should be interpreted with care. It is true that concepts compose together to form thoughts, but that does not necessarily mean that the finished product will actually contain those concepts. Nevertheless, because I shall identify thought contents at least in part with the derivation of the thought from its constituents through composition, that’s pretty close to constituency.

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distinguish between subtypes of the type e of entities non-extensionally, we must think of the complex types in a more fine-grained way as well. Just as we have different subtypes of the type e, we also have different subtypes of e ⇒ t, or even t itself, the type of propositions. For instance, the concept or type red associated with the nominal modifier red combines with other concepts to form more complex concepts like red pen, red apple, and so forth. This is their “proffered” internal content that they contribute to the thoughts of which they are constituents. The nominal modifier red has a type that is a subtype of (e ⇒ t) ⇒ (e ⇒ t). Apple itself has a type that is a subtype of e ⇒ t. Later we will see how these “proffered,” as opposed to presupposed types affect the calculation of logical form. It is important to separate these proffered types from the type presuppositions of these terms. Both red and apple have type presuppositions on their arguments in logical form. For red its argument must be a physical object; for apple its λ bound variable must be an offspring of a plant. This allows such terms to enter into semantically well formed predications that are necessarily false (on an externalist semantics) like (2.28) Apples are vegetables (2.29) Those are apples (said by someone pointing to pears) but prevents the following from being well-formed predications when the words are understood literally: (2.30) #Apples are rocks. (2.31) #Those are apples. (said by someone pointing to cars) Here’s an approximation of the logical forms I shall assign to red and apple, with their type presuppositions expressed as restrictions on the λ bound arguments: • λP: e ⇒ t λx: p (red(x) ∧ P(x)) • λy: plant-offspring apple(y). This means that apple is defined when it applies to fruits other than apples to yield a truth value (it yields the truth value “false” in such cases).15 The fine-grained type apple is a function that when applied to an individual returns a “proof” that the object is an apple, if that object meets the conditions of application, and it returns a “refutation” that the object is an apple, 15

This is illustrative only. Nothing hangs on the particular choice of the type presupposition for apple, only that it must have one.

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if it fails to meet the conditions of application.16 It is this fine-grained type that is the internal proffered content of the term apple and that enters into the composition of thoughts. The same is true of all primitive types. The basic, primitive types like cat, dog, animal are analogous to the defined types like integer in mathematical applications of the λ calculus. The difference is that concepts and a fortiori types don’t seem to have easy definitions in the ordinary sense, although there are computations that humans have mastered that enables them to use concepts/types correctly. When I write where v is a variable, v: apple, I mean that the object denoted by v relative to an assignment meets the application conditions that are constitutive of the type apple; thinking of these application conditions as a proof or computation, this means that when the computation takes the value of v as input, it terminates and yields the value true. The notion of “object” within the internal semantics also requires explanation, as these are individual concepts.17 What about the fine-grained type red? red is a functor that when applied to a subtype of p returns another subtype of p—exactly which subtype of p depends on the fine-grained type of its argument. The type red thus has an argument place in it for a physical type. When the functor is applied to apple, we get the physical subtype: • red(apple) We now define the fine-grained type and internal semantics of the word red vis-`a-vis the polymorphic type red. red is a polymorphic type because the particular type of red depends on the type of the modifier’s argument. We may go further and require red to be a dependent type, as in (2.32) John hates everything red—red meat, red apples, red shirts, and so on. The exact type given by red will in this case depend on the value of the variable. While most of the time adjectives can be understood as having polymorphic types in this book, we will need dependent types to handle quantificational examples like (2.32). red presupposes that its argument be a subtype of p. If the modifier’s argument meets this type presupposition, then red(α), the application of red to α, is the value. Then the word red has the following type in this context: 16

17

This differs from what advocates of Type Theory like Martin-L¨of (1980) and Ranta (2004) take the denotation of a λ term to be; in type theory the denotation of a term is the collection of “positive” proofs. This corresponds closely to the notion of a classical extension. My view allows it to be the case that we can neither determine that something is an apple or that it is not an apple, but I will not investigate these refinements here. Cardelli (1988) takes an object is essentially a collection of functions or attributes with values, but there are other construals of individual concepts that are compatible with this approach.

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• (α ⇒ t) ⇒ (red(α) ⇒ t) The way to understand the internal semantics of the term is again to use the proof theoretic idea, in the large sense of proof. Red takes any type that gives proof application conditions for entities that are of type α p and returns a new type that gives application conditions for what it is to be a red α. The fine-grained types of verbs are similar to red in that they are functors taking a sequence of types (the sequence represents their arguments) and returning another type. It is natural to think of verbs as the “glue” that binds the other types in a thought together. This means we must introduce more complex types for verbs than is usual in standard formal semantics. As we’ll see, it is the DP rather than its head noun that furnishes the type of the argument to the verb, since determiners are often responsible in a language like English for saying whether the argument of a verb is count or mass, or a plural that is collectively interpretable or only has a distributive interpretation. The type of a verb takes DP types (or the types associated with the DP’s values) as arguments to give us a proposition. Thus, intransitive verbs will be functors from one DP type to propositions, while transitive verbs will be functors from two DP types. We will represent differences in verb alternations by the differences in the output that a particular instance of the polymorphic or dependent type yields. To figure out what a DP type is, we must turn our attention to determiners. Their conceptual role is to state relations between types in terms of the individual concepts that satisfy the application rules for these types. Each determiner with a distinct logical function will give rise to a distinct determiner concept. For instance, the determiner every has a fine-grained type every that when applied to a subtype of count e returns a functional type that, given a subtype of count, returns a thought, or a subtype of the type of propositions. The thought expressed by applying every first to α and then to β is the thought that every individual concept that satisfies the application rules of α also satisfies the application rules of β. The internal semantics of every is a type theoretic version of a Barwise and Cooper (1981) generalized quantifier. An intransitive verb or VP type takes such a generalized quantifier type as an argument and provides it with a subtype of e ⇒ t to yield a thought. For example, the sentence in (2.33a) has the corresponding thought described by the fine-grained type in (2.33b): (2.33) a. Every dog barks. b. (every(dog))(bark) c. ∀x(dog(x) → bark(x)) From (2.33b) we determine the type of bark. It is the function from DP types α

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to the fine-grained type α(bark), which is a subtype of the type of propositions. This generalizes straightforwardly to other grammatical categories. This develops the idea proposed by Howard (1980), the so-called “CurryHoward isomorphism,” according to which each lambda term corresponds to a type, which encodes a proof. These proofs determine the internal meaning of the term and its natural language correlate. To go back to our example, the λ term corresponding to the English expression every dog is: • λP∀x(Dog(x) → P(x)) The type of this term is (e ⇒ prop) ⇒ prop, where prop is the type of propositions. The interpretation of the term itself is one which is a function from proofs to proofs; given a property P or a proof from individuals to truth values, the lambda term denotes a proof that every dog has that property P. To apply this term to the lambda term λx bark(x) successfully is to produce a proof procedure for proving that every dog barks. The internal semantics is given in terms of the proof theoretic apparatus of the logical system, not in terms of some externally given values provided by an independent model. The notion of proof can be suitably generalized to an abstract mathematical sort of construction of the sort that intuitionists have worked on for many years, known as Topos theory.

2.4 Two levels of lexical content Let me summarize the discussion so far. I’ve argued that types are concepts, mind-dependent entities with a fine-grained content. Types have an internal semantics that is given at least in part in terms of the rules by which they combine with other types; they are proof theoretic objects.18 Types can be associated with other types—what we might think of as traits; each trait we associate with a type is a constraint on the introduction rule for the type. If there are concepts other than the linguistic types that I focus on here, I speculate that they have the same sort of internal content, specified by rules of combination, introduction, and elimination. Types as proof objects provide an internal semantics for natural language sentences and discourses that complements the external semantics given by ordinary intensions. We can even distinguish between different conceptualizations of the same physical object or of the same property, so in some respects the structure of types has the capacity to make finer distinctions in meaning 18

For similar ideas, see Fernando (2004)’s automata theoretic analysis of verb meanings, which amounts to a constructive theory for their types.

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than intensional semantics can.19 But the purpose of the internal semantics is not the same as that of the external semantics. Intensions are the soul of a theory of meaning—they are needed to determine truth, reference, and other external, semantic properties that link language to the world we talk about. Types and their adjustments are the heart of a theory of predication and responsible for other properties of meaning. The construction of logical form depends on type checking, and it is the clash of general, high-level types that lies behind failures of predication. This is semantics but it is an internal matter, something that speakers do “in their heads.” Types, and type presuppositions in particular, are responsible for our intuitions about semantic well-formedness and, I will argue below, for analytical entailments and for generalizations about word meaning. Types also serve a computational and theoretical need in lexical semantics and in the theory of predication. They simplify considerably the task of checking whether a syntactically acceptable sentence is semantically well formed or not; the property of semantic well-formedness for a sentence corresponds to the property of producing a normalized logical form for it, a logical form in which no more applications can be performed; and in a typed system with types like the one I will propose, this property is not only decidable but polynomial in complexity.20 Much of this book will be devoted to examining the interactions between internal semantics given by types and the external semantics for logical forms. For, as I’ll argue below when I look in detail at models of complex types, it is precisely in the rules that determine how type shifts from complex types to constituent types affect the logical form that various complex types will distinguish themselves. In other words, a division between the internal semantics and the type system and the external semantics allows us to get at the right semantics for complex types, since the semantics for these types is in part determined by what they do to logical form. It’s not usual in semantics to postulate two systems of meanings for words and the complex constituents made out of them. Martin-L¨of’s intuitionistic Type Theory (Martin-L¨of (1980), Ranta (2004)) identifies types and proofs and the semantic contents of terms; the internal semantics thus emerges as the only meaning of the logical forms, and the external semantics drops out. The meaning of a term in Type Theory is just the collection of its proofs. 19

20

In classical semantic systems, the interpretation of types needs to be distinguished from that for logical forms, since types do not convey enough information to provide truth conditions. Montague Grammar’s simple set theoretic model of types assigns types a very impoverished semantics compared to that given to logical forms. The types of the extensional typed lambda calculus do not, for instance, distinguish between any entities or between the truth values. Even in an intensional setting, Montagovian types do not distinguish between propositions or between different one place properties, for instance—all one place properties are of the same type. One complication to this claim concerns nominalization. I discuss this in chapter 10.

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Given the arguments in the previous section concerning the independence of the semantics of concepts and thoughts on the one hand and the semantics of propositions on the other, we cannot simply replace our intensional semantics of terms in the lambda calculus with the internal semantics of types. This move would run afoul of very basic intuitions about semantic content, as I argued above. To collapse the level of types and its conceptual, internal semantics with the level of logical forms and its intensional semantics is to miss the intentional nature of language in the Husserlian sense, according to which the semantic denotations of most expressions consist in mind-independent entities. The system of types and mind-dependent concepts is per force no substitute for external, mind-independent contents.21 There are also technical problems with the conception of meaning at the core of Type Theory. Consider a statement that has no proof, say because it is false. Its semantic value is the same as a sentence whose proof cannot be constructed say because of a type clash of the sort we saw in the last chapter. There seems to be no way in Type Theory to distinguish between statements that have no proof because they are false and statements that are semantically anomalous. Since this distinction is fundamental to the approach taken in this book, I cannot follow the philosophical and conceptual view laid out in Type Theory. But I can use their technical developments to fashion my internal semantics, while rejecting Type Theory’s conceptual and philosophical claims about meaning. Crucial to my enterprise is the distinction between objects in the internal semantics and objects in the external semantics. With this we can use a type theoretic view of the internal semantics to explain semantic anomalousness. If there is no proof that results from applying a type corresponding to a property to an internal object or individual concept, this is because there is a type inconsistency between the type assigned to the object and the type presupposed by the function representing the property applied to it. Statements without proofs in an internal semantics are those with a type inconsistency, and it is the lack of any positive proof in the internal semantics that defines semantic anomalousness. This “internal falsity” is of course distinguished from ordinary falsity in the external semantics; if a λ term is false in this sense in the internal semantics, it cannot even be evaluated in the external semantics. But a statement’s having a non-empty set of proofs from its internal semantics does not entail external truth outside of mathematical contexts; it may not 21

It seems to me that the simplest statements about the external world like you’re hungry pose problems for purely proof theoretic approaches to meaning. Externalist arguments of the sort examined in the previous section constitute powerful evidence that concepts, and a fortiori types, our internal mental reflections of real world properties and things, don’t suffice to determine denotations or intensions. A theory that identified the meanings of terms with the internal semantics of types has no means for accounting for these arguments.

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even be consistent with external truth, as I’ve already argued.22 Furthermore, my internal semantics distinguishes between the types needed for constructing a proof of well-formedness (a normalization)—these are the types involved in type presuppositions, and the fine-grained types assigned to lexical expressions as part of their proferred content. Hence, there is in prinicple room in the internal semantics for a distinction between semantic anomalousness and (internal) falsity. Type Theory isn’t the only monostratal proposal for lexical meaning. There’s also what I call the Sortal Theory. The Sortal Theory claims that type information is really just a part of the language of logical forms that is singled out by selectional restrictions. Type information is really just information about sorts. Semantic ill-formedness is the result of using the semantic and proof theoretic apparatus of the language of logical forms to derive a contradiction based on meaning postulates about sorts. At first glance, the distinction between the Sortal Theory and a typed theory of logical forms looks to be one of notational variance. The typed lambda term of the bistratal theory I propose in (2.34a) has the very similar looking Sortal Theory analogue in (2.34b) (φα is the object level formula associated with the type α). (2.34) a. λx: α ψ(x) b. λx (ψ(x) ∧ φα (x)) But here appearances deceive. Sorts, as predicates in logical form, have a set theoretic semantics; and they provide no constraints on the construction of logical form. Without any types at all, we have all the inconveniences of doing compositional semantics in a type-free environment. Furthermore, without types, we have no account of semantic ill-formedness, which my theory accounts for in terms of the failure of the type presuppositions of terms to be justified and hence for a failure of normalization. According to the Sortal Theory, there aren’t any syntactically well-formed sentences that are semantically ill-formed; those sentences that are predicted to be ill-formed because of type clashes on the typed view are just false on the Sortal Theory. The Sortal Theory can no longer explain the projection of selectional restrictions either. These problems count, to my mind, as strong evidence against the Sortal Theory. Matters are no better with coercion. Selectional restrictions are just like any other predicate on the monostratal Sortal Theory; so the literal meaning of 22

There is here a delicate but well-studied matter about how the internal semantics deals with the acquisition of new information about internal objects. Computer scientists have labelled this the problem of side-effects.

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most of the cases of coercion that we have seen is inconsistent. Thus, on the Sortal Theory, coercion must be a matter of content revision. But I have no idea how such a revision would go, and, I suspect, neither does anyone else. The Sortal Theory has nothing promising to say about coercions. Those who don’t believe in a linguistic level of lexical meaning separate from general world knowledge will be attracted to the Sortal Theory. Quineans or philosophers with a minimalist view of lexical conent will think that the Sortal Theory makes entirely the right identification between coercion and belief revision. But it seems to me to be an entirely inadequate alternative to the two levels of lexical content I have set out here. The two-level theory of lexical content has tantalizing implications for a general theory of content. Types offer an interesting approach to problems that motivate the Fregean picture of language by developing a two-factor theory of content in a rigorous way. An internal semantics for the types will allow us to construct an “internal semantics” for language, which could be the internal reflection of externalist, model theoretic conceptions of content. That is, a proof theoretic, internal semantics gives content to concepts and thoughts which are linked through the logical forms to the externalist semantics of intensions.23 We have seen that we might want the meaning of the type hesperus to come apart from the meaning of the name Hesperus. In this way we can in principle make sense of the informativeness of certain true identity statements as well as acknowledge their necessary truth. Indeed, having two semantics associated with formulas (one albeit indirectly) gives us the tools to make sense of puzzles about belief as well as paradoxes of informativeness.24 We want to have something like a Fregean picture where two distinct types or concepts may correspond to the same referential meaning or intension. Unlike Frege’s senses, however, types and concepts do not determine the referential or standard, intensional meaning of terms. At best there is a homomorphism from concepts to sets of intensions that preserve the structures relevant to predication—i.e., the relations between types.

23

24

Many people have followed the view in the philosophy of mind (see for instance Burge (1979), Tye (1999), Dretske (1981)) that the contents of thoughts like the contents of many expressions in natural language are often determined by external factors, but it is hard to deny that there is an internal component to thoughts as well. For the beginnings of such an exploration, see Asher (1986, 1987, 1989).

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2.5 Types in the linguistic system 2.5.1 Type checking in predication Types are designed to affect semantic well-formedness and felicitous predication. The general idea is that predicates place type presuppositions on their arguments, which their arguments must satisfy or at least be compatible with. In the lambda calculus representations of word meanings that I shall use, almost every word is a predicate with one or more lambda abstracted variables. So most words have type presuppositions. But not all types play this role. There is good evidence that finely individuated types do not play a role in the type checking relevant to predication. Consider again these examples from the first chapter: (2.35) b. c. d.

a. Tigers are animals. Tigers are robots. #Tigers are financial institutions. #Tigers are ZF sets.

One might hypothesize that types at the upper end of the type hierarchy are relevant to type checking. animal is pretty clearly at the upper level of the type hierarchy (D¨olling 2000). robot is a subtype of artifact, which is also at the upper level. And most type hierarchies would take animal and artifact to be incompatible types. This assumption would predict that (2.35b), (2.35c), and (2.35d) are all equally semantically anomalous. But they are not, even though they are all false, indeed necessarily false, given the widely accepted externalist semantics for natural kinds like tigers. The clash between the type demands of the predicate and the type of its argument in these examples has to do with rather deep metaphysical principles like individuation and counting conditions. Sets, especially of the ZF kind, have simple individuation criteria based on the axiom of extensionality and are all built out of a single object, the null set, on the standard, cumulative conception. Tigers share nothing with ZF sets with respect to individuation conditions. Financial institutions are more abstract than standard physical objects— they are relational structures that depend on various physical individuals but not any particular ones (the employees of the institution can change, so can its physical locations while the financial insititution continues). While the individuation conditions of financial institutions are murkier than those for ZF sets, once again tigers are not counted or individuated in anything like the way that financial institutions are. Robots and animals, on the other hand, are close enough in terms of identity and counting conditions that we can make sense of (2.35b); we can imagine a competent speaker believing such a proposition.

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(2.35c,d) are much harder to make sense of without a big background story. And, as one would expect if one adopts the presuppositional perspective, a big enough background story may lead to a type coercion making the predication once again acceptable. The type checking associated with well-formedness of predication is sensitive to those types that have different individuation and counting conditions. A semantic anomaly will result only when the type requirements of the predicate and argument give rise to incompatible individuation conditions. The discussion of the type presuppositions of (2.35b) suggests that the type presupposition of the predicate are robots must be simply that the argument is of type p or physical object. In other words, in order for the predication to succeed, the argument can be any sort of physical object; it is not restricted to things of type artifact, let alone robot. Thus, the type presupposition that robot contributes as a check on predication is quite general. The types with distinctive individuation conditions relevant to type checking are somewhat diverse from the perspective of standard type hierarchies. They include very general types—like physical object, informational object or abstract object, mass, count, plurality, singularity, and eventuality. But there are also relatively fine-grained distinctions in types that predication is sensitive to: the distinction between state, activity, accomplishment, and achievement, as well as the distinction between different types of locations, and the distinction between portions of matter and normal countable individuals. Assuming that seemingly closely related types like activity and accomplishment are interchangeable, for instance, leads to a type clash; adverbials like in an hour and for an hour are sensitive to the defining and differentiating features of these types of eventualities and accept one type but not the other for their arguments. I shall assume that such distinctions also point to distinctive differences in individuation conditions. On the other hand, it’s hard to find a predicate similarly sensitive to the defining and differentiating features of tigers and robots. If only types with distinctive individuation conditions perform an important role in the type checking of predication and are part of the type presuppositions of a term, what role do the more fine-grained types play in the grammatical system? Fine-grained types and the internal semantics can affect logical form when combined with functor types. These functor types may provide different fine-grained value types when they take arguments of different fine-grained types. For instance, in section 2, I distinguished between:

2.5 Types in the linguistic system (2.36) b. c. d. e.

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a. red(apple) red(shirt) red(pen) red(meat) red(wine)

Each of the modifying predications in (2.36) are distinct. When we predicate a color of an apple, we predicate it typically of the skin. We can call an apple truly red even if most of it, its flesh, is white. On the other hand, this won’t do for red shirt. A shirt is red only if the entire shirt, or very nearly so, is red. We may call a pen red if it writes in red ink; the color of the visible surface of the pen may not be relevant at all for that predication to succeed. Finally, red meat refers to a kind of meat, e.g., beef, lamb, and the like, which may not be at all red on the surface, say, when it shows up on the dinner plate. While the adjective red like other color words is typically analyzed as an intersective adjective (according to which the red X is both an X and red), it appears to become subsective (according to which the red X is an X but not necessarily red) when combined with meat or pen. That is, the following sentences are not necessarily false: (2.37) The red pen is not red. (2.38) That piece of red meat is not red. (2.39) This red wine isn’t really red; it’s more of a very deep purple. Even though there is no type incompatibility here between the type of the color adjective and the type of the common noun it modifies, there is a subtle shift in meaning and semantic behavior of the adjective noun combination. This observation extends to many adjectives, in particular gradable adjectives: thus small robot has a different meaning from small mountain, small hill, and so on. This shift arises from the interaction of the fine-grained types or internal semantics of words, I contend. These subtle shifts in the resulting predication make it plausible that in principle each word stem, even each word, could give rise to a distinct concept, a distinct fine-grained type that can play a role in the grammatical system. Without such fine-grained differences, we cannot explain the differences in meaning noted above. The fine-grained types associated with verbs also find a use in the grammatical system. Recall, for instance, the difference between the two transitive uses of a verb like load in John loaded the wagon with hay (the wagon is completely full) versus John loaded hay on the wagon (the wagon is not necessarily completely full). By feeding different types into the functor type, we can output different types of propositions that describe different relationships between the

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wagon and the hay. Of course, this is only a framework for a worked out analysis, but it is general enough to accommodate, for instance, a sophisticated approach like that of Krifka (1987). We must thus distinguish two uses of types in the grammatical system. To check the felicity of predications, we must resort to type presuppositions, while to understand the shifts in meaning, we resort to the fine-grained types associated with words. The fine-grained types are part of the type hierarchy, and to check whether one meets the type presuppositions of a predicate it suffices to use the relation on types. For instance, suppose that Hesperus and Phosphorus, two names for the planet Venus, give rise to two different concepts, two distinct fine-grained types. For the purposes of type checking in predication, even for the predicate of identity, the distinctness of these concepts or finegrained types doesn’t matter. What matters is the following subtyping chain. (2.40) hesperus, phosphorus heavenly-body physical-object e. If we have a predicate P that requires the relevant argument to be occupied by something of type physical-object, then the predication Pt should succeed if t has the type hesperus; the type presuppositions of P should be met by any subtype of physical-object, in particular hesperus (or phosphorus).

2.5.2 Lexical inference Fine-grained types also provide a basis for lexical inference or analytical entailment. The issues of lexically based inference and analytical entailment are murky ones, and one can study the problems of constructing logical forms and the phenomena concerning predication that motivate this book without taking a position on them. The data concerning lexically driven inferences are also controversial. Fodor and Lepore (1998), for instance, question the robustness or legitimacy of the data as really part of semantics. Among most working linguists and lexicographers, however, the intuitions are strong if not entirely systematic that there are many analytical entailments. The question I want to pursue briefly here is whether, and if so how, the type mechanisms I have devised for the construction of logical form give us purchase on the often debated but little resolved issues of lexical inference and analytical entailment. In so doing, I shall also show how the internal semantics distinguishes analytic truth and analytic falsity from semantic well-formedness and semantic ill-formedness. Roughly, while it is the type presuppositions and the mechanisms for justifying them that determine well- or ill-formedness, it is the connections between the fine-grained types of proferred content that determine analytic truth or falsehood.

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Simply studying predication and arriving at a logical form tells us little about analytical entailments. Analytic entailments are entailments of the logic underlying logical forms; they are valid and so true in all models for the language. But unless we add certain information based on the links between non-logical words, we reduce the class of analytic entailments to the set of logical consequences in the language of logical form. If the lexicon is to add anything to the class of analytic entailments, it must do so in virtue of one of the following possibilities. (1) Certain lexical entries for words are logically complex and thus give rise to entailments using the logic validated by the models for the logical forms. (2) The lexicon contains axioms relating the meanings of words and thus adds to the logical theory of compositional semantics. (3) The rules for the analysis of predication, in particular the rules for handling complex types and the relations between types, add information to logical form that the underlying logic can exploit to provide analytic entailments beyond the entailments provided by the underlying logic alone. Linguists and philosophers have for the most part pursued the first track under the rubric of providing a “decompositional” semantics (Jackendoff (1990) comes prominently to mind). Even our quite minimal hypothesis that every word stem or root gives rise to a distinct type leads to at least some decomposition: distinct words with the same stem but formed using semantically rich morphological affixes and suffixes should have common bits of logical form; that is, the logical form of some words of the same root must be constructed out of the logical forms of at least one of the other words of the same root. For instance, many linguists have argued that sink’s syntactic behavior as a transitive but also intransitive verb indicates that its lexical meaning consists of two facts, one describing an underspecified action on the part of an agent, and the second describing the resulting state of that action, namely that the theme of the sinking is sunk. This yields a host of analytic entailments, some of which are not completely trivial: (2.41) a. The enemy sank the ship with a torpedo −→ the enemy used a torpedo to sink the ship −→ the ship sank −→ the ship was sunk. b. The enemy sank the ship with a torpedo −→ the enemy’s torpedoing the ship was the cause of its sinking. Asher and Lascarides (1995, 2003) explore such decompositional analyzes for a variety of verbs, including causatives and psych verbs. Here is the decomposition of Asher and Lascarides (2003) for sink. It involves a causal relation between two facts—one determined by an underspecified action marked with a question mark, and the other by a property of being sunk instantiated in a state that is posterior to the action.

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(2.42) sink: λxλeλy(?(y, x, e) CAUSE ∃s(sunk(x, s) ∧ e ≺ s)) To handle the alternation between causative and intransitive sink, Asher and Lascarides assume that unsaturated arguments of the causing fact can be existentially closed off at the final stages of constructing the logical form for a clause containing such a verb. This decompositional analysis, together with analyzes for use and gerunds, validates all the entailments in (2.41). Levin (1993) contains a comprehensive list of verbs which invite similar decompositional analyzes. They include verbs like:25 bake, build, braid, brew, burn, carve, cast, chirp, cook, crochet, dig, draw, drill, fold, hum, knit, mumble, murmur, paint, sing, shoot, sketch, weave, shout, whisper, whistle, write, sink, break, open, close...

The following examples indicate how these very different verbs obey the same syntactic alternations as sink, suggesting a similar analysis. (2.43) a. The bottle broke. (intransitive/causative transitive) b. John broke the bottle. (2.44) a. The window opened suddenly. (intransitive/causative transitive) b. Mary opened the window suddenly. Asher and Lascarides (2003) generalize their decomposition to the class of psych verbs. They also all have an underlying causal structure, where some unspecified action by the subject causes the patient (the direct object of the psych verb) to have the state denoted by the past participle form of the verb. Thus, all actions denoted by psych verbs decompose into an event with a particular result state involving an adjectival form of the root lexeme. Such a decomposition obviously does not give a complete analysis of a verb like bother (it couldn’t since it relies on the meaning of bothered to specify the meaning of the verb), but it is a useful way of portraying the similarity in meaning of psych verbs. Decomposition in this limited sense can show how the same basic lexical meaning can lead to several syntactic realizations and thus serves a useful purpose. These limited decompositions also provide for limited analytical entailments. Nevertheless, the decompositional approach has received substantial criticism (Fodor 1971). Fodor assumed that if a decompositional approach to lexical semantics, for example one in which kill is analyzed as cause to die, is psychologically real, then it should take longer to process words with more com25

For a treatment of some of these, see van Hout (1996) or Bittner (1999).

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plex meaning representations than those with simpler representations. However, many experiments have shown that this is not the case for the pair kill and die. Note, however, that kill does not support syntactic alternations like the ones above.26 Strongly decompositional approaches to word meaning have in general failed dramatically to provide plausible results for lexical meanings.27 One major problem is that it is not clear what the primitives out of which all lexical meanings are built should be. In a few areas of lexical research such as the work on spatial and temporal expressions, decomposition has led to more promising results. Researchers have largely agreed on the need for certain topological primitives (contact and weak contact, or equivalently parthood and weak contact) and geometrical ones. For time, the primitives required to construct linear orders are well known and axiomatized. More recently, the topological and even geometrical primitives have been axiomatized within a project to analyze the meanings of spatial, more precisely topological, prepositions (Aurnage and Vieu (1997), Vieu (1997), Muller (2002)). Such work is extremely important for lexical semantics, because it gives a concrete meaning to the primitives, determining what sorts of models for the logical form language are the admissible ones (Asher and Vieu (1995), Donnelly (2001)). Such an axiomatization yields a wealth of analytic entailments that go well beyond those given by the axiomatization of the first-order connectives and quantifiers. There have also been, following Jackendoff’s seminal work, thorough investigations of motion verbs (Asher and Sablayrolles (1995), Muller and Sarda (1999)), though correspondence theorems continue to be lacking for the motion verb primitives of goal, source, and path. In most other areas of lexical meaning, the sort of thorough investigation that lexical research on temporal and spatial expressions has delivered is lacking. Axiomatizations are at best partial, and more often than not characterizations of lexical meaning rely on notions that are part of the background language in which one characterizes the reference of these terms and hence the models; and this frequently amounts to little more than a restatement of intuitions. 26

27

It is true that kill takes different adverbials, in particular adverbials involving process like sadistically, with a knife, and so on from die. These, so the argument goes, modify the process of killing (Pustejovsky (1995)). Pustejovsky (1995) takes these adverbials to be evidence that kill involves a process and a result state. This is not really convincing, however. What these adverbial modifiers show is that there must be a type distinction between events of killing and events like dying, not that there must be a decomposition involved, though indeed it does seem to be an analytical entailment that if Sam killed Pat then Pat died. However, recent experiments by Gail McKoon on activity, stative, change of state, and indirect causation verbs does provide evidence for the hypothesis that there is increasing representational complexity for these types as we go from one category on the left to the next on the right, as hypothesized by Levin and Rappaport (1994). McKoon discussed these experiments during her seminar at the LSA summer school in Stanford, California, 2007.

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There is of course a reason why axiomatizations in these areas don’t exist. In the case of space and time or space-time, the mathematical structures underlying the meanings of various words in natural language were clear thanks to thousands of years of reflection on these topics by philosophers, mathematicians, and physicists. It was relatively easy, though still not straightforward, to build a qualitative theory that characterizes that structure as it’s reflected in language. But there’s no guarantee that other areas of lexical meaning are amenable to such treatment. There have been hasty general treatments of lexical primitives like Shank’s (1974) conceptual dependency paradigm. Such a description might lead to an abstract underlying structure and then perhaps an axiomatization. But one would need a much more detailed descriptive analysis, and there are no guarantees that an abstract structure or that an axiomatization would emerge therefrom. I don’t know what the abstract structure of buying and selling is; looking at formal treatments in ontology of various concepts, one gets the feeling that either there are no interesting structures worth axiomatizing or that research has somehow gotten off on the wrong track. It may simply be utopian to think that every part of the lexicon will end up looking like the parts concerning spatial and temporal expressions. Perhaps the best we can hope for is a partially decompositional analysis with some primitives properly understood. A type system encodes analytic entailments without full axiomatization or decomposition. Some entailments are just type presuppositions. For instance, (2.45) Kim is a bachelor −→ Kim is male, Kim is adult, Kim is human (2.46) Kim is not a bachelor −→ Kim is male, Kim is adult, Kim is human are natural inferences most people would make. I take these inferences to show that bachelor puts detailed type presuppositions on its argument when used predicatively. To say exactly what these are, we need to appeal to an operation on types , which is a meet operation and is defined by appealing to the given subtyping relation : Definition 1 σ τ = the greatest lower bound of σ and τ in the partial order

on types. I will assume that for any two types σ and τ, there is at least one type that is a subtype of both and that’s the type ⊥. So the term bachelor looks like this: λx: adult male Bachelor(x), where the typing on x provides the type presuppositions. These presuppositions translate back out into logical form via the following, natural principle: • Type Presupposition Inferences

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Suppose ψ dynamically implies28 δ(t) where t has the type presupposition α. Then ψ dynamically implies φα (t), where φα is the translation in logical form of the type α. Since each word stem gives rise to a type, there is no problem about assuming the existence of a φα . Type Presupposition Inferences enables us to infer that if Kim is a bachelor, then Kim is male and Kim is adult, as desired. Moreover, these are plausibly analytic entailments. On the other hand, not all analytical entailments based on lexical meaning are presuppositional. Some, in fact many, have to do with the fine-grained types that are part of proferred, internal content. The type tiger is a subtype of animal, and so that should be an analytic entailment. But being an animal is not presupposed of the bound variable in the logical form for tiger. If it were, then it would be as crazy to think that tigers are robots, as it is crazy to think that they are financial institutions or ZF sets. This entails that there are semantically well-formed sentences that are analytically false. Similar remarks apply to bachelor. While Kim is a bachelor entails that Kim is not married, Kim is not a bachelor fails to entail that Kim is unmarried. Thus, unmarried, assuming such a type exists, is not part of the type presuppositions of bachelor. But it is an analytic entailment based on lexical meaning that bachelors are unmarried. To express this entailment using the type hierarchy, we exploit the subtyping relation on types. If α β then anything of type α is also of type β, and this is itself a conceptual truth. To capture such entailments we need to be able to translate from the type hierarchy into logical forms or into the language of thoughts introduced in the last section. It is commonly assumed that is a partial ordering on the type hierarchy. And so a type hierarchy T together with the usual axioms for partial ordering will allow us to deduce subtyping relations. I’ll call this deduction relation, T , for a given type hierarchy T . We can now provide an obvious method for producing analytic entailments: • From Types to Entailments: Suppose for a type hierarchy T , T α β. Then: 1. it is a conceptual truth that every(α, β) 2. it is an analytic truth that ∀x(φα (x) → φβ (x)) From Types to Entailments provides a simple way of inferring analytic entail28

Dynamic notions of implication carry bindings from premises to conclusions. In particular in dynamic semantics ∃xφ dynamically implies φ(x). For details on a dynamic system, see Groenendijk and Stokhof (1991).

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ments. For instance, we can easily encode the entailment that a bachelor is unmarried via the following type constraints:29 • bachelor unmarried Type hierarchies and From Types to Entailments yield a host of analytic truths. (2.47) b. c. d. e. f.

a. Tigers are animals. Chairs are physical objects. Stories are informational objects. Everything that has a color is a physical object. A physical property is a property. If it’s Tuesday, it can’t be Wednesday.

(2.47a–d) are straightforward consequences of the type hierarchy used to check predication. (2.47e) is perhaps a less straightforward consequence of the type system. Expressions that pick out days of the week pick out different days. This is a matter of what these expressions mean. The type hierarchy encodes these facts by using the meet operation : • tuesday monday = ⊥ There are some plausibly analytic entailments that the type hierarchy doesn’t encode directly. For example, consider the entailment from (2.48) Kim killed Sandy to (2.49) Sandy died. To capture this entailment, we would have to suppose that the complex type or thought kill(kim, sandy) is a subtype of die(sandy). But this would predict that adverbs that go with die can then go with kill, and this is not the case; one can say that Sandy died peacefully, but one can’t use these verbal modifiers in the same way with kill: (2.50) Kim killed Sandy peacefully. 29

If types are concepts, then linking types via the relation really amounts to conceptual analysis, a time honoured approach to philosophy. In a type hierarchy entailments can be captured without requiring any definitional reduction of one type to some Boolean combination of other types. Nor do we have to take on board the dubious entailments of a profligately decompositional approach.

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To capture such analytic entailments without resorting to decomposition, we need to look at the rules of application for Kim’s killing Sandy in our internal semantics of types. Since the rules that sanction the application of the type of Kim’s killing Sandy to a situation will also sanction the application of the type Sandy’s dying to that situation, the analytic entailment seems secure. Similarly, one can argue that Kim bought a computer from Sandy analytically entails Sandy sold a computer to Kim and vice versa, even though neither fine-grained type is a subtype of the other.30 Note that these observations do not lead to the generalization that any term of propositional type that has a non empty set of proofs as its internal content is an analytic statement. Once again, the presence of such proofs in general demonstrates only that the typing constraints involved in the term are coherent; it entails only that the term of propositional type is capable of having an (external) truth value. However, if we can show using with the fine-grained internal meanings for the primitive types that any proof object in the internal meaning of α is also a proof object for β, we have an analytic entailment. The type hierarchies linguists use are a simplified expression an approximation of those fine-grained internal meanings. Thus, we get to the following generalization of From Types to Entailment, where I represents the proof relation over the type meanings. • If x: α I x: β, then φα analytically entails φβ . Many analytic entailments follow simply from the type system we need to analyze predication. Others follow from the types associated with the proferred internal contents of lexical expressions. Types and their internal semantics thus offer a simple way of modeling lexical entailments without having to resort to an extraneous set of meaning postulates or dubious lexical decompositions. I will argue below that in cases of coercion or copredication, new entailments based on lexical meaning and predication can arise from the adjustments in types needed to satisfy type presuppositions and to make the predication work. By using type presupposition justification rules to add information 30

If these two fine-grained types were subtypes of each other, we would expect them to have the same inhabitants. So any event of Sandy’s selling a car to Kim is identical to an event of Kim’s buying a car from Sandy and vice versa. This would predict that any adverbial modification of one should hold of the other. But that isn’t true. For instance, (2.51a) does not entail (2.51b), nor does (2.52a) entail (2.52b):

(2.51) a. Kim bought a car from Sandy for Bo. b. Sandy sold a car to Kim for Bo. (2.52) a. Kim bought a car from Sandy with a credit card. b. Sandy sold a car to Kim with a credit card.

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to logical form, we end up modeling all sorts of entailments that don’t follow simply from the meanings of the syncategorematic or logical words of the language. Some of them could even be defeasible entailments such as start a cigarette ∼ | start to smoke a cigarette. For these it is the mechanisms of predication that introduce the entailment into logical form. Assuming that every lexical root has its own unique type as I do thus doesn’t preclude having a non-trivial theory of lexical inference encoded in the type hierarchy. Subtyping relations between types enable us to encode entailments based on lexical meaning. And having a type associated with each word root doesn’t preclude a decompositional analysis of various words in terms of the root type—consider the case of psych verbs, whose meaning is analyzed in terms of their adjectival correlates. It’s thus a logically unsupported step to conclude, as Fodor and Lepore (1998) do, that an anti-decompositionalist approach to lexical semantics precludes any interesting theory of lexical or analytic entailment. This is clearly not enough to lay the issue of analytical entailment to rest, but it shows some of the resources of a typed approach.

3 Previous Theories of Predication

With this discussion of some of the philosophical background for a typed theory of lexical meaning, I now turn to examining some precursors of the theory to be proposed here. All of these precursors take some sort of stand concerning the representation of lexical meaning and the sort of phenomena that are the focus of this book—various forms of predication both at the clausal and morphological levels, copredication, and coercion. A basic issue concerning lexical representation is lexical ambiguity. Some people distinguish between monomorphic and polymorphic languages, which correspond to two different ways of thinking about ambiguity. A monomorphic language (or lexicon for a language) is one in which each word has a unique type and syntactic category. Ambiguous words on this view are analyzed as words with the same orthography but each with a different sense. This is the view found in Montague Grammar or even in HPSG, and it corresponds to an approach to ambiguity according to which ambiguous expressions are represented by the set of their disambiguations. A polymorphic language (or lexicon for a language) is one in which each word may have multiple types or underspecified types that may be further specified during the composition process. Underspecification is a tool or method devised by linguists working on ambiguity in sentential and discourse semantics. Underspecification produces underspecified logical forms for ambiguous discourses For instance, suppose we have to give the truth conditions of a sentence with multiple quantifiers like (3.1): (3.1) Every nurse examined a patient. (3.1) has two readings: one in which the universal quantifier introduced by every nurse takes wide scope over the existential quantifier introduced by a patient, and one in which the existential quantifier has wide scope over the universal quantifier. An underspecified logical form for (3.1) would introduce

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“holes” in the formula (Reyle 1993, Bos 1999), keeping the λ terms for the two quantifiers separate from the verb but also introducing relations between the terms to encode constraints on how λ conversion should go—for instance, the underspecified logical form for (3.1) will encode the fact that both quantifiers determine arguments for the main verb. But the scope of operators is only one source of underspecification. There are many anaphoric or context dependent phenomena in natural language—for instance, pronouns, tenses, presuppositions—for which a determinate semantics cannot be supplied independently of the discourse context. Underspecified logical forms are used to represent the semantic contribution of these words and morphemes. Take a pronoun like he. Its meaning is supplied by the antecedent it is linked with in a particular discourse context. One cannot specify a lambda term conveying the meaning of such an expression without resorting to underspecification. However, with underspecification, specifying a lambda term is relatively straightforward; Asher (1993), for instance, has the following lambda term for he— λP∃x(P(x) ∧ x =?), where ? marks the “hole” to be filled in by a term provided by the pronoun’s antecedent.1 Thus, underspecification provides a natural and powerful tool for representing certain kinds of ambiguity in lexical semantics. The λ calculus extends naturally to a treatment of such underspecification, as Pogodalla (2004) argues.

3.1 The sense enumeration model The most orthodox model of lexical meaning is the monomorphic, sense enumeration model, according to which all the different possible meanings of a single lexical item are listed in the lexicon as part of the lexical entry for the item. Each sense in the lexical entry for a word is fully specified. On such a view, most words are ambiguous. This account is the simplest conceptually, and it is the standard way dictionaries are put together. From the perspective of a typed theory, this view posits many types for each word, one for each sense. Predication may select for a subset of the available senses for a predicate or an argument because of type restrictions. While conceptually simple, this approach fails to explain how some senses are intuitively related to each other and some are not. This is not only a powerful intuition shared many linguists and lexicographers; there are also linguistic tests to distinguish between word senses that are closely related to each other and those that are not. Words or, perhaps more accurately, word occurrences 1

See Pogodalla (2004) for a completely typed version of this expression in which holes are introduced as a special type.

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that have closely related senses are called logically polysemous, while those that do not receive the label accidentally polysemous or simply homonymous. Cruse (1986) suggests copredication as a test to distinguish logical polysemy from accidental polysemy: if two different predicates, each requiring a different sense, predicate properties of different senses of a given word felicitously, then the word is logically polysemous with respect at least to those two senses. Another test is pronominalization or ellipsis: if you can pronominalize an occurrence of a possibly ambiguous word felicitously in a context where the pronoun is an argument of a predicate requiring one sense while its antecedent is an argument of a predicate requiring a different sense, then the word is logically polysemous with respect to those senses.2 Contrast (3.4a–b) and (3.4c–e): (3.4) a. #The banki specializes in IPOs. Iti is steep and muddy and thus slippery. b. #The bank specializes in IPOs and is steep and muddy and thus slippery. c. Lunch was delicious but took forever. d. He paid the bill and threw it away. e. The city has 500,000 inhabitants and outlawed smoking in bars last year. Bank is a classic example of an accidentally polysemous word. As (3.4a–b), show, both the pronominalization and copredication tests produce anomalous sentences, which confirm its status as accidentally polysemous. On the other hand, lunch, bill, and city are classified as logically polysemous, as (3.4c–e) witness that they pass the tests of copredication and pronominalization. The distinction between accidental and logical polysemy isn’t absolute. There are degrees of relatedness that the felicity of copredications and pronominal tests reflect. Contrast (3.4d–e) with (3.5a,b), for instance: (3.5) a. ?My janitor uses a brush and so did Velazquez. b. ?The city outlawed smoking in bars last year and has 500,000 inhabitants. 2

Of course, there are other tests for ambiguity, such as the so-called contradiction test, according to which a particular sentence with an ambiguous word ω may be consistently judged both true and not true in the same context since an interpretation may access different senses of ω. Read with contrastive stress, sentences like

(3.2) The bank specializes in IPOs (3.3) The book is beautiful can be understood as true and false in the same context. Nevertheless, the contradiction test is not a sure fire indicator of real ambiguity. Books are both informational and physical in some sense even though we can select each one of these aspects within a predication. We can also access both meanings simultaneously.

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(3.5a,b) aren’t as good as (3.4c–e); (3.5a,b) are rather zeugmatic. Their zeugmaticity results from different factors; (3.5b) as opposed to (3.4e) involves the same two lexical senses but in a different order. This suggests that copredications are subject to discourse effects. On the other hand, (3.5a) involves two relatively unrelated senses of brush—paint brushes and cleaning brushes have for the most part very different properties—different functions, sizes, and so on. In some languages like French, these two senses have different translations, pinceau and brosse. These senses don’t cohere together in the way that the distinct senses for lunch, bill, book, and city do. Though a full explanation promises to be complex, sense enumeration models have no way of explaining the differing degrees of success that copredications appear to have. Sense enumeration models fare even worse with the phenomenon of coercion. We have seen that coercions are context sensitive; the exact meaning of a phrase like begin with the kitchen is often determined by discourse context. According to the sense enumeration model, this would require begin or kitchen to have a distinct sense for each one of the contextually specified meanings of begin with the kitchen. This seems highly unintuitive; further, it threatens to make a specification of lexical meaning practically impossible, as discourse contexts can coerce such phrases in unpredictable ways.

3.2 Nunberg and sense transfer In contrast to the sense enumeration model of word meaning is the view that lexical semantics consists of a set of lexical entries, while pragmatics furnishes a set of maps from one lexical meaning to another. These maps specify, inter alia, coercions. Frege’s theory of meaning includes coercions. He thought that intensional operators like verbs of saying or of propositional attitudes induced a meaning shift of the terms within their scope. He stipulated that the meaning of terms when they occur within the scope of an intensional operator shifted from their reference to their sense. The intensional operator coerces a meaning shift, a shift from customary meaning to non-customary meaning. Predication for Frege had to involve type checking in order to discard the application of, say, a belief relation to the simple extension of its clausal argument. Depending on whether one subscribes to the many levels of sense interpretation of Frege or not, one can say that such coercions are carried through the sense hierarchy as the depth of embeddings under intensional operators increases. Generalizing Frege’s strategy, Nunberg (1979, 1995) proposes that lexical meanings are subject to shifts in predication. A lexical entry specifies a denotation for a referring expression or for a predicate according to Nunberg.

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While Nunberg isn’t completely explicit about what the specifications of lexical meanings look like, his ideas fit within the standard framework in which the denotations of terms are expressed by lambda terms.3 These terms will then have their standard interpretation in a model. For instance, the word cat will have as its lexical entry an expression λx cat(x) and its interpretation in the intended model will be that function that for any given world w and time t returns 1 if x is assigned as a value something that is a cat at w and t, and returns 0 otherwise. Sometimes such lexical entries can be transformed via a general rule like grinding or by a more specialized rule transferring the normal interpretation of a term to some salient associated entity when predication demands this. Nunberg applies such a notion of transfer to definites and indexicals (thus it would appear that Nunberg adopts a version of a type driven theory of predication). Some examples Nunberg and others have found to motivate this view are quite vivid:4 (3.7) I’m parked out back. (3.8) The ham sandwich is getting impatient. The basic idea is intuitive. In these examples, applying a particular predicate whose argument is required to have type α to an argument whose type is β, where α β = ⊥ forces either the argument term or the predicate term or both to change their meanings so that the predication can succeed. For instance, ham sandwiches can’t literally get impatient; and if I’m standing in front of you, I can’t literally be out back. So what happens is that we shift the meaning of the terms so that the predications succeed: it’s my car that is parked out back, and it’s the guy who is eating the ham sandwich who is getting impatient. The problems lie in the details. When exactly is the sense transfer function introduced? The transfer function could always be optionally available but that would lead to vast overgeneration problems: (3.9) The ham sandwich that hasn’t been eaten is on the counter. (3.9) would be predicted to have a reading on which the eater of the ham sandwich that hasn’t been eaten is on the counter. Let’s suppose with Sag (1981) that Nunberg’s sense transfer functions work 3 4

For instance, see Sag (1981) for a formal development of his view. Note that the “parking” examples are felicitous with all sorts of subjects:

(3.6) a. John is parked out back. b. The students are parked out back. c. Most students park out back.

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on lexical entries for common noun phrases or N . The result we want for (3.8) is clear. The ham sandwich should have the following logical form: (3.8’) λP the(x) ( f (ham sandwich)(x) ∧ P(x)) where f is the transfer function mapping ham sandwiches to people who are eating them or who have just eaten them. The problem is that we only become aware of the need for a type adjustment mechanism in building the logical form of the sentence when we try to put the subject noun phrase together with the verb phrase, and at that point it is no longer straightforward to add the transfer function to work over the contribution of the common noun.5 Indeed, if we think of the process of predication here as building up a denotation for the sentence in compositional fashion, then it is hopeless to try to apply the transfer function where Sag and Nunberg would like; the common noun denotation is simply no longer available at this stage of composition. If we think of the process of predication as building a logical form rather than a denotation, then the contribution of the common noun is in principle available, but specifying the scope of the transfer function is not straightforward. It cannot be always a common noun that should be shifted in coercion cases. We have to generalize Sag’s assumption to get an analysis of (3.7): either the full DP or the verb will have to undergo sense transfer. But if it is the entire noun phrase or, in the language of syntax, the DP (determiner phrase) that is shifted, we get incorrect truth conditions. Consider (3.10):6 (3.10) George enjoyed many books last weekend. A straightforward application of the Nunberg strategy yields the following logical form: (3.10’) f (many books(x)) ∃e(enjoy(g, x, e) ∧ last weekend(e)) The transfer function shifts the meaning of the entire quantifier, and so presumably the quantifier ranges over the elements in the image of the transfer function, namely some sort of eventualities. Of course, we don’t know what those are. Perhaps, for the sake of concreteness, we could assume that they are events of reading books. But if (3.10’) says that there were many book reading events that George enjoyed over the weekend, this is compatible with there being just one book that George read over and over again that weekend. Clearly, this is not a possible reading of (3.10), and so we cannot implement Sag’s proposal for Nunberg’s idea straightforwardly for all coercions without a lot more 5 6

See Egg (2003) for even more extreme examples, where the coercion isn’t clearly needed until subsequent sentences of discourse are processed. Thanks to George Bronnikov for this example.

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work. In fact, this observation about (3.10) will constitute a problem for many extant accounts of coercion, Pustejovsky’s GL account included, as we’ll see below. Nunberg himself did not investigate coercions involving enjoy. But the DPshifting approach has general empirical problems. Consider the difference between the two minimal sentences in (3.11): (3.11) b. (3.12) b.

a. Everyone is parked out back and is driving an old volvo. # Everyone is parked out back and is an old volvo. a. ??John was hit in the fender by a truck and will cost a lot to repair. John’s car was hit in the fender by a truck and will cost a lot to repair.

(3.11a) is fine but relies on the fact that the subject DP does not shift. Were there shifting going on, then (3.11b) should be good, but it is plainly not. Similarly if the DP had actually shifted meaning, then (3.12a) should have a reading that is equivalent to (3.12b), but it does not. (3.12a) is simply semantically anomalous. Kleiber (1999) argues that the sense transfer model as I’ve specified it doesn’t pass the anaphoricity and copredication tests. For example, consider the examples below where anaphor and its antecedent are forced by the predication contexts to have two different senses linked by a transfer function of the sort Nunberg countenances. (3.13) a. George Sand est lue par beaucoup de monde, bien qu’elle soit disparue depuis longtemps (George Sand is still read by many people even though she died long ago). b. ?George Sand est lue par beaucoup de monde, bien qu’ils ne soient plus e´ dit´es (George Sand is still read by many people even though they (the books she wrote) are no longer in print). Taking their cue from such examples, Kleiber and others reject Nunberg’s analysis of logical polysemy using reference shifters. They conclude that examples like (3.7) and (3.8) point to a different sort of phenomenon from the logical polysemy exemplified by the classic coercion cases involving aspectual verbs.7 To avoid some of these problems, the sense transfer model can resort to 7

Kleiber’s observations have some force. Nevertheless, once again judgments in acceptability reveal a not completely black and white distinction. The following examples from Abusch (1989) show that Kleiber’s examples don’t give us the full story. In some cases we can get anaphoric links between two expressions whose meanings are related by a sense transfer function of the sort that Nunberg envisions.

(3.14) a. The mushroom omelet left without paying, because he found it inedible. b. ? The mushroom omelet is eating it with gusto.

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an alternative transfer strategy: shift the meaning of the verbs rather than the meaning of common nouns or DPs.8 This can in principle avoid the problem noted with (3.10) and with the problem noted by Kleiber. But this shift leads to another set of difficulties. Consider the following ellipsis facts.9 (3.16) b. c. d.

a. I’m parked out back, and Mary’s car is too. ? I own a car that is parked out back and Mary’s car does too. Mary enjoys her garden and John his liquor. ?Mary enjoys digging in her garden, and John his liquor.

If we transfer the sense of parked out back to the property of owning a car that is parked out back to account for the parking examples, then the ellipsis in (3.16a) should be odd or it should give rise to the following reading: Mary’s car does not own a car that is parked out back as well. But the ellipsis is fine, and it lacks this reading. Similarly if we transfer the sense of enjoy to something like enjoy digging to make sense of the predication that Mary enjoys her garden, then we should predict that the ellipsis in (3.16c) should be bad much the way (3.16d) is. But the ellipsis in (3.16c) is fine. If we do not apply meaning transformations at the level of denotations, we must spell them out at the level of logical form. And we can’t do this in the obvious way without predicting that (3.16a,b) are equivalent—similarly for (3.16c,d). That is, we cannot simply replace the contribution of parked out back, which I’ll assume is a 1-place predicate here, to one that looks like this: λx∃z (parked-out-back(z)∧owns(x, z)). This means that simply applying transfer functions to verbal meanings is unworkable. Even if we can get this proposal to work using techniques like underspecification,10 many questions about this proposal still remain. For instance, is the And it appears that sometimes copredication works as well in these cases. (3.15) is somewhat zeugmatic but passable, I think: (3.15) Plato is one of my favorite authors and is on the top shelf to the right. 8 9 10

Nunberg (1993) suggests this as a possibility. Thanks to Alexandra Aramis for suggesting these sorts of examples. If one makes use of underspecification and of ways of resolving underspecifications in ellipsis which are not part of Nunberg’s proposal and which were not really understood at that time, one gets a much more plausible proposal, which is in fact that of Egg (2003).

• λx parked-out-back(x) −→ λx∃z (parked-out-back(z)∧?(x, z)) • λyλx enjoy(x, y) −→ λxλy∃z (en joy(x, z)∧?(z, y)) The ? stands for a relation that must be filled in from the discourse context. Crucially, as Asher (1993) argues, such underspecifications in ellipsis contexts can be resolved one way in the source of the ellipsis and another way in the target. This explains why (3.16a,c) are good and distinct from the degraded ellipses in (3.16b,d). This gets us something like the right logical form for sentences such as (3.10) and other examples of coercion.

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shift at the level of logical form really to be described as a change of meaning of the verb in these predicational contexts? Is it really conceptually plausible that these verbs shift their meaning but that other words do not? Exactly which sentential constituents can be arguments of the sense transfer function? Exactly what triggers the application of the function? A more serious problem with this account is that it doesn’t answer the question, “Why this meaning transformation and not another?” There are no constraints on when a sense transfer function can be introduced at all. It’s clear that one can’t invoke an arbitrary function whenever predications break down, for then we could not predict that any sentences are semantically anomalous. The whole type system has become otiose in that case. But neither can we appeal to some function that’s salient in the context. Here’s an example from Kleiber (1999), where it’s perfectly clear what the transfer function should be in the context that would make this predication acceptable. The function takes us from pianos to the noise they make or from the verb parvenir to a verbal meaning which means something like make a sound that comes. Nevertheless, we cannot get (3.17) to mean that we heard the sound of the piano which came to us floating over the waters of the lake. The sense transfer model has no explanation why. (3.17) Nous entendˆımes le piano, qui nous parvenait flottant par-dessus du lac (We heard the piano which came to us floating over the waters of the lake). More generally, there’s no account of what would validate the application of transfer functions. Why should we be permitted to make such transfers in some cases but not in others? The sense transfer model, moreover, doesn’t address the distinction between logical and accidental polysemy. We need a more semantically based account to handle this distinction. Nevertheless, postulating a set of maps amongst lexical meanings is a good idea. The sense transfer model just doesn’t develop it at the right level. In the theory I propose, something like these maps will operate over types and will be part of a lexical entry.

3.3 Kleiber and metonymic reference Kleiber (1999) proposes another model under the rubric of “metonymic reference” to account for coercion.11 This model assumes part–whole relationships to be the basis of coercion and many examples of predication. Properties of some parts of objects can be “coerced” into predications of the whole. In 11

Kleiber’s model is similar to Langacker’s notion of “active zones.” See for, instance, Langacker (1990).

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Kleiber’s model, we can include as parts of an object things commonly associated with it. But this makes the part–whole relation no longer the usual one, and so the model becomes rather mysterious. We clearly don’t want to consider John’s car to be part of John in any standard mereological sense! (3.18) a. Paul est bronz´e (Paul is tanned). b. Les am´ericains ont d´ebarqu´e sur la lune en 1969 (The Americans landed on the moon in 1969). c. Le pantalon est sale (The trousers are dirty). d. Le stylo est rouge (The pen is red). e. John was hit in the fender by a truck. Modifications of some of these examples show that part–whole relations sometimes figure in copredications: (3.19) a. Paul est bronz´e et tr`es athl´etique (Paul is tanned and very athletic). b. Les am´ericains ont d´ebarqu´e sur la lune en 1969 et ont men´e une sale guerre en indochine (The Americans landed on the moon in 1969 and waged a dirty war in Indochina). c. Le pantalon est sale et trou´e (The trousers are dirty and torn). d. Le stylo est rouge et tr`es cher (The pen is red and very expensive). However, the part whole relation can’t cover all the cases of logical polysemy— recall (3.18e). To consider another example, it’s not clear that we want to say that part of the lunch was delicious and part of the lunch took forever in (3.4c), repeated below: (3.4c) Lunch was delicious but took forever. While there is no doubt that predication may take advantage of a part–whole relation, one needs more details as to how the compositional construction of logical forms proceeds in Kleiber’s case as well. When does the metonymic shift occur and why does it occur? Kleiber’s view isn’t sufficiently developed in order for us to be able to answer these questions. The examples of metonymy are not examples of semantic, type coercion. It is in principle possible from the perspective of conceptual coherence that, for instance, the entire population of America landed on the moon in 1969. Of course, that’s false and unbelievable, but that’s not a semantic matter. In the Nunberg examples or the examples of coercion involving aspectual verbs, something like sense transfer or type coercion has to take place for semantic reasons: the predicate and the argument can’t as they stand combine without producing a semantic anomaly. Kleiber nevertheless extends his model to deal with coercion cases involving

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aspectual verbs. Kleiber sensibly enough postulates a temporal interval argument for begin that has a well-defined initial part and then a continuation (typically expressed by an accomplishment verb with telicity and durativity). For any direct object or internal argument of begin not expressed by a phrase including an accomplishment verb, there must be a homomorphism from the accomplishment structure to the internal argument. This implies that begin must have something like an accomplishment as its object. But now how does this relate to the metonymic model? It is not obvious that the relationship between an eventuality (Laurence’s smoking a cigarette) and one of its participants (the cigarette) is one of part and whole, since the cigarette as a spatio-temporal entity is not included completely in the event, nor is the event included in the cigarette. Still less is it clear that such constraints are sufficient to capture many relevant cases. Consider for example: (3.20) a. Paul a commenc´e de d´econstruire son livre. Marie a commenc´e le sien aussi (Paul began to deconstruct his book. Mary began hers too). b. Paul a commenc´e de d´econstruire son livre. Marie a commenc´e a` d´econstruire le sien aussi (Paul began to deconstruct his book. Mary began to deconstruct hers too). One can construct a homomorphism from the deconstruction of the book to a temporal structure that captures accomplishments. Then Kleiber’s model would predict that (3.20a) has a reading that is a paraphrase of (3.20b). But it does not.

3.4 The Generative Lexicon The last approach to polysemy that I shall treat here is by far the most developed of the ones I have surveyed. That is the model given by the Generative Lexicon or “GL.” The present approach draws its inspiration from many of the ideas in GL as developed by Pustejovsky (1995) and others. GL is motivated by the phenomena of coercion and logical polysemy. It aims to preserve compositionality as much as possible while giving an account of these phenomena beginning with a single lexical entry for logically polysemous words. Pustejovsky (1995) thinks that specific lexical meanings are generated in the meaning composition process from interactions of type constraints and something like a sens g´en´eral, a general, perhaps underspecified meaning for all the uses of the word that can be specialized via predication. The hard work is to figure out an appropriate format for a sens g´en´eral and the procedures that in appropriate contexts will specify this underspecified meaning.

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1. Argument Structure: Specification of number and type of logical arguments. 2. Event Structure: Definition of the event type of a lexical item. Sorts include state, process, and transition. 3. Qualia Structure: A structural differentiation of the predicative force for a lexical item. 4. Dot Objects: they are invoked to explain copredications. In Pustejovsky (1995) they are part of the qualia structure. 5. Lexical Inheritance Structure: this part of the lexicon specifies the type hierarchy. GL avails itself of the formalism of attribute value matrices, or AVMs, combined with types known as typed feature structures (TFS), a very good formal account of which can be found in Carpenter (1992). In effect, a TFS can be thought of as a type with a bit of information attached via “attributes” whose values are more types. Formally, in the language of type theory, this amounts to associating with the “head” type of the TFS a record or collection of types. Since there is a natural translation from feature structures to first-order formulas, the use of TFSs for lexical entries allows one to combine type information with information that naturally goes into the logical form of a lexical entry. The two most innovative aspects of GL are qualia structures and dot objects. Following Moravscik (1975), GL postulates that many nouns contain information about other objects and eventualities that are associated with the denotation of the noun in virtue of something like the Aristotelian explanatory causes, or αιτια. The qualia structure of a word specifies four aspects of its meaning: • • • •

constitutive: the relation between an object and its constituent parts; formal: that which distinguishes it within a larger domain; telic: its purpose and function; agentive: factors involved in its origin or in “bringing it about.” ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎣

⎤

⎥⎥⎥ door(x,y) ⎥⎥⎥ ⎥⎥⎥ const : aperture(y) ⎥⎥⎥ ⎥⎥⎥ formal : physobj(x) ⎥ telic : walk through(e’,w,y) ⎥⎥⎥⎥⎥⎥ ⎥⎦ agentive : make(e,z,x)

Figure 3.1 Qualia structure of door

3.4 The Generative Lexicon ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎣

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⎤

⎥⎥⎥ cigarette(x) ⎥⎥ const : tobacco(x, y) ⎥⎥⎥⎥⎥⎥ formal : physobj(x) ⎥⎥⎥⎥⎥ telic : smoke(e, x, y) ⎥⎥⎥⎥⎥⎥ agentive : roll(e’, x, y) ⎥⎦

Figure 3.2 Qualia structure of cigarette

Pustejovsky and his students hypothesize that in predications like (3.21), the verbs begin and enjoy, which require an eventuality for their internal arguments, select as their arguments the eventualities encoded in the qualia structures of nouns like cigarette. (3.21) a. begin a cigarette (i.e., smoking) b. enjoy the book (i.e., reading) According to GL, the observations about coercion with verbs like enjoy, begin, and finish indicate a rich type structure in which objects of a given type α conventionally associate with objects of other types relevant to an object of type α’s production or function; this type structure is accessible to and guides the composition process. Pustejovsky (1995) also uses observations about coercions involving certain prenominal adjectival modification constructions to motivate qualia-based typing information: (3.22) a. fast car b. fast motorway c. fast water To explain these data, Pustejovsky and Boguraev (1993) argued that fast selects for the telic-role of the NP head, sometimes acting coercively. For most of the above Adj-N combinations, we arrive at interpretations such as “cars that are fast moving” and “motorways that permit fast traffic.” Bouillon (1997) discusses a wide range of phenomena which are handled by reference to qualia in such constructions. Another application of qualia-based typing has been to account for certain denominal verb formation cases, as discussed in Hale and Keyser (1993). Hale and Keyser’s canonical examples are illustrated in (3.23) below. (3.23) a. John put the books on the shelf. / J. shelved the books. b. John sent the letter by fax. / J. faxed the letter. c. John put the wine in bottles. / J. bottled the wine.

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GL proposes the following analysis of these observations: the noun-to-verb transformation is licensed in just those cases where the noun’s telic role involves an object that matches the verb’s direct object. For example, from the noun shelf, whose telic makes reference to the relation of holding books (and related objects), the verb shelve is licensed, because its direct object shares the argument referenced in the noun’s telic value. Climent (2001) discusses this idea in more detail. I turn now to a critical examination of the framework. First of all, we can quickly dispense with the idea of meaning generation during the composition process. There is no generation of meaning in the process of meaning composition in GL; meaning composition simply selects components of lexically specified meanings in the analysis of the coercion or copredication phenomena. The exploitation of qualia in the purported explanation of the various phenomena just listed simply focuses on a particular component of the lexical meaning of a noun and unifies this with the typing requirement of a predicate of the noun phrase or higher projection thereof (e.g., DP). The generative lexicon should rather have been called the “selectional” or “specificational” lexicon. Nevertheless, a selectional process in meaning composition is an interesting idea. How far does it take us? It turns out that the notion of a selectional meaning composition process is not well developed in GL and breaks down rather rapidly in the face of recalcitrant data and formal scrutiny. First, it is often difficult to understand exactly what the values of the qualia are supposed to be. Is the telic role of a shelf, its purpose, really just to hold books? Presumably not. Shelves can hold all kinds of things—wine glasses, ski boots, outboard motors, clothes. So we should predict that sentences like (3.24) a. John shelved his sweater b. Mary shelved her glasses have a similar meaning to John shelved the books, but if (3.24a–b) make sense at all, they involve a shift to a metaphorical sense of shelve—something similar to shelve an idea. Furthermore, this analysis predicts denominal verbs that don’t exist: (3.25) a. John poured the wine in the glasses. / # John glassed the wine. b. John put the roast in the oven. / #John ovened the roast. The analysis for denominal verb formation vastly over generalizes. The criterion proposed by GL isn’t sufficient for explaining the observations. And GL’s claim that having the right qualia structure is a necessary condition for denominal verb formation isn’t right either. For example, the classic examples of

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denominal verbs like lunch and snack have intransitive uses in which there are no direct objects that could be referenced in the telic roles associated with the nouns lunch, snack, and so on. An additional problem is that the qualia values seem to shift depending on the examples analyzed. For instance, to get the right meaning for fast car, which is a modal meaning (has the ability to go fast), we have to say that the car’s telic role is to be able to drive or to be able to transport people. But if we modalize the value of the telic quale, then we should expect John enjoyed the car to mean something like John enjoyed the ability of the car to drive or to transport people, and we wouldn’t expect the much more natural reading—John enjoyed driving / looking at the car. Similarly for the telic role of motorway. In spite of these worries, the telic roles of many nouns are nevertheless quite clear. It is much less clear what the constitutive or formal qualia are for many objects. Consider, for instance, the constitutive quale for door vs. tobacco. For tobacco the constitutive is something like an Aristotelian material cause, the matter out of which the object is constructed, but that is hardly the interpretation of the constitutive quale for door, which GL takes to be an aperture. Apertures are not matter—they are precisely the lack of matter. These remarks might seem quibbling, but there is a general point behind them. If the qualia are not defined in a precise way, then the theory can postulate anything for the values of qualia and thus loses explanatory power. The general unclarity concerning what the qualia structure of a given noun is extends from the particular values of qualia to their types. What exactly is the type of the qualia object? This is a legitimate question for GL since it makes heavy use of types. An aspectual verb like begin seems to take an event type as its object argument or at least some sort of intensional event description, as it is perfectly coherent to say: (3.26) a. John began to read that book, but he never finished (reading) it. b. John began that book, but he never finished it. If the logical form of the first clause of either (3.26a) or (3.26b) were something like (3.26’) ∃e∃e (begin( j, e, e ) ∧ read that book(e)) then we would predict that (3.26a,b) imply that there was a reading of the book and that that event was incomplete or unfinished, which seems quite problematic, and perhaps even contradictory. Assuming that the infinitival contributes a property of events or event types as the argument of begin avoids this problem

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and invites a straightforward intensional analysis of (3.26a,b) along the lines of well-known modal analyzes of the progressive (Dowty 1979, Asher 1992).12 On the other hand, we don’t have imperfective paradox-like phenomena with the verbs enjoy or finish. If it’s true that I enjoyed smoking my cigarette, then I did smoke my cigarette. Notice also that enjoy doesn’t, unlike the aspectual verbs, subcategorize for infinitival clauses, at least in English. (3.27) b. c. d. e.

a. Nicholas enjoyed smoking a cigarette. Nicholas enjoyed a cigarette. #Nicholas enjoyed to smoke a cigarette. Nicholas finished a cigarette. #Nicholas finished to smoke a cigarette.

However, in GL the distinction between event types and events isn’t clear at all. Some interpreters of GL (Egg 2003) have taken the object of the coercion verb to be an event type; others (Asher and Pustejovsky (2006), or Copestake and Lascarides 1996)) have taken the arguments to be events. The truth seems to lie somewhere in the middle, and to depend on the verb. Our discussion of event types versus events as qualia values brings us to problems in the compositional semantics. In fact, the interaction between compositional semantics and lexical semantics is hardly ever discussed in GL, and this leads to problems—not surprisingly if you believe, as I do, that a theory of lexical meaning has to also specify a method of meaning composition or an account of predication. For instance, some authors like Kleiber have claimed that GL does not account for the data concerning anaphoric availability. That is, the sort of coercions induced by aspectual verbs are alleged to fail the pronominalization test. Suppose we introduce an element into discourse during coercion that has the type of an eventuality, as the telic roles for the AVMs for book, cigarette, and other artifacts do in GL. Then we should predict these eventualities to be accessible for future anaphoric reference, if none of the well-known semantic or syntactic principles barring anaphoric links obtain. However, the marginal acceptability of (3.28a) translated from Kleiber (1999) and (3.28b) appear to put this hypothesis in jeopardy. (3.28) a. Paul has started a new book. ??It (that) will last three days. b. ?Paul enjoyed his new book, though it was quick. It’s clear what is intended in these examples: an anaphoric reference to the reading event. Of course, if we distinguish between the types of objects for 12

Assuming that an infinitival provides an event type or a property of events would also be in keeping with the compositional semantics of infinitivals. See Asher (1993) for a discussion.

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aspectual verbs like start and verbs like enjoy as I did above, then accounting for the difference in acceptability of (3.28a,b) is straightforward. Other examples, however, show that an event is anaphorically available in coercion examples, when the discourse context is right. (3.29) a. Paul has started a new book, but that won’t last.13 b. Paul has started a new book. His reading will take him three days. (3.28a,b) and (3.29a,b) all require an event denoting expression to be accessible to the pronoun. I take it that (3.29a,b) are acceptable. The contrastive construction in (3.29a) makes the event more salient. The use of a definite description in (3.29b) also makes it easier to pick up the event of Paul’s doing something with the book. However, other, similar uses of aspectual verbs as in (3.30) also support anaphoric reference to events without difficulty when the discourse context is of a special kind. (3.30) Last week Julie painted her house. She began with the kitchen. That didn’t take very long. She then proceeded to the bedroom and the living room. That took forever, because she painted friezes on the walls in those rooms. In this use of aspectual verbs anaphoric reference to the events of painting the rooms is easy. Such examples also pass the copredication test when we use relative clauses: (3.31) Yesterday Julie painted her house. She began with the kitchen which didn’t take very long. She then proceeded to the bedroom and the living room, which took forever, because she painted friezes on the walls in those rooms. These examples show that the pronominal test with coerced eventuality readings does succeed, though the reasons for this success have to do not with qualia but with complex interactions between discourse structure and clausal logical form. It’s not clear what, if anything, GL predicts about the possibility of referring anaphorically to an event within the qualia structure. It had better not be the case that all the eventualities mentioned in the qualia are all available as anaphoric antecedents. That would give plainly false predictions. A more sensible hypothesis, empirically speaking, is that the terms denoting eventualities in the qualia structure only become available as anaphoric antecedents when they become the arguments of the coercion predicate, giving them as it were some sort of linguistic realization. But when that happens, does 13

Thanks to Elizabeth Asher for this example.

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Previous Theories of Predication ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎣

⎤

⎥⎥⎥ cigarette ⎥⎥⎥ ⎥⎥⎥ sem arg1 : [1] ⎥⎥⎥ ⎥⎥⎥ Pred : cigarette ⎡ ⎤ ⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ smoke ⎥⎥⎥ ⎥⎥⎥ telic : ⎢⎢⎢⎢⎢ Pred : smoke ⎥⎥⎥⎥⎥ ⎥⎥⎥⎥⎥ ⎢⎢⎣ sem arg1 : [2] ⎥⎥⎦ ⎥⎥⎥⎥⎥⎦

Figure 3.3 Partially rewritten qualia structure of cigarette

that mean that the original object of the coercion predicate is no longer linguistically realized? Is it no longer available for anaphoric reference? That seems plainly wrong. Figuring out a principled view of what is going on, however, will require much more attention to the details of composition and interpretation than is available in GL. Let’s look at some of the formal details. Qualia structures appeal to the formalism of typed feature structures, or TFSs. A typed feature structure is a function from a type to a set of pairs of attributes and values. These values may either be constants (these are in fact types), variables with indices that are to be identified with values of some other attributes, or typed feature structures themselves. Values are thus recursively defined. The standard language of feature structures has a modal semantics (Blackburn (1997), Lascarides et al. (1996)). The models for TFSs correspond to directed acyclic graphs (DAGs). Following Blackburn (1997), one can define the semantics of TFSs by thinking of features as modal operators that label arcs between the nodes of a DAG types as propositions holding at nodes, and constraints on types as conditionals. Notice that the types themselves don’t have any particular structured semantics in this analysis. All types, that is, are considered as atoms. This is already an important limitation: we cannot state type presuppositions for higher types. Despite the appeal of the ideas in GL, there is more than a bit of “grunt work” to get everything to work out as advertised, as many have noted. The exposition in Pustejovsky (1995) is rather lax and informal and uses first-order formulas, as we saw above in the examples of qualia structures, instead of the proper formalism for TFSs. To represent the feature structure for cigarette properly, we would need to rewrite the qualia structure for cigarette as in figure 3.1. In addition, we need the value of the telic quale in the qualia structure for cigarette to be defeasible, since it may be that when we say that Max enjoyed the cigarette, he enjoyed eating it (it might be a chocolate cigarette). To reflect this fact we would need to add the default type assignments of Lascarides et al. (1996).

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We have now expressed qualia properly in TFSs, but we are still not quite done. The operation of unification over TFSs is a binary operation that returns a new TFS where the old values in the common arguments of two TFSs are replaced with their meets (greatest lower bounds in the type hierarchy) and then adds all the argument value pairs that are not shared. Typed unification is unification checked by typing; for instance, we can unify the feature structure for cigarette with the feature structure for enjoy, provided that the object argument’s type in the TFS for enjoy (the object argument in the TFS for enjoy is just a variable that will be replaced under unification with the feature structure for cigarette) has a meet with cigarette (the type) that is not the absurd type ⊥. By hypothesis, however, the object argument for enjoy is an event, while cigarette is a subtype of physical object and has no non-absurd meet with the event type. So the composition crashes here, unless we can coerce either the type of the appropriate argument slot of the verb or the type of cigarette. GL’s idea is that in this case, one of the qualia types of cigarette should be substituted for cigarette. But how is this done? How do we in fact shift the predication to hold of the telic value when we have a phrase like enjoy the cigarette? How exactly is the predication in coercion cases involving aspectual verbs or verbs like enjoy supposed to work in terms of unification? Does one unify the variable in the argument place of the verb with something of the appropriate type in the AVM or feature structure associated with the element that is supposed to fill the argument position? Something like this must be going on, but such a rule is not part of unification standardly construed. Neither a defeasible version of unification like that in Lascarides et al. (1996) nor GL itself provides rules suitable to the analysis of coercion. The unification formalism underlying TFSs is not by itself sufficient to get the appropriate bit in the logical form as the argument for a coercing predicate like enjoy. We need some way of manipulating the feature structure so that we get the event in the telic quale as the argument of enjoy. In feature structure terms, this means stipulating a special lexical rule that transforms the original feature structure into one of the appropriate sort. This now looks very much like the sense transfer function of Nunberg. Alternatively, we can see the feature structure with qualia to be a disjunction of a primary sense (the basic word meaning) and associated senses (the meanings associated with the qualia); predication selects among the associated senses when necessary. Now that we have brought the structures of GL into familiar territory, we see that they have the familiar problems. The qualia constrain but also pack Nunberg’s sense transfer function into the lexical entries for nouns. That strategy brings with it the problems that we’ve seen already with the sense transfer function. For example, this strategy threatens to get the wrong truth conditions

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for sentences with coercions and quantification like (3.10) repeated below; we want the coercion to affect the argument of enjoy but we don’t want the quantifier many books to be coerced into the quantifier many book readings, even though the coercion based on a type clash can only be identified once we are at the stage of combining the whole DP with the verb. (3.10) George enjoyed many books last weekend. Putting the sense transfer function into the noun seems to be precisely the wrong strategy to deal with coercions, since doing so shifts the domain of quantification in (3.10) on a standard compositional analysis of the DP many books. This wrongly predicts that (3.10) is true if there are many events of reading one book over the weekend. We should rather think of the transfer as doing something to the verb as I argued earlier. Perhaps one could implement something like this idea with special lexical rules for TFSs (although all GL approaches are, as far as I know, mute on this subject), but it would look extremely ad hoc and be highly non-compositional. Hopefully, there is a better alternative! Pustejovsky and Moravscik exploit the Aristotelian explanatory causes of traditional metaphysics to handle coercion. Aristotle of course takes the αιτια to be universal features of being. However, it doesn’t take long to realize that qualia aren’t a universal feature of all types of substances in todays “common sense” metaphysics. Types associated with terms that denote natural kinds— e.g., water, h2 o, wood, gold, birch, elm, penguin, etc. do not plausibly have any associated agentive or telic qualia.14 Pustejovsky (2001) tries to turn this problem into advantage for GL by defining the type artifact as any type whose associated feature structure has agentive and telic qualia. For such substances their origin or the event of their construction as well as their purpose are pertinent to understanding what they are. This certainly seems plausible. So it should follow that only artifacts give rise to the sort of coercions with the aspectual verbs and verbs like enjoy that motivate GL. Nevertheless, even this restricted qualia hypothesis runs into trouble, and on two counts. First, it would seem that one can enjoy some natural substances even without turning them into artifacts; climbers can enjoy a cliff; many people can enjoy the mountains or the sea, the beach, the forest, the wilderness, the wide open spaces, and so on. Nevertheless, these would all appear to be objects that are not artifacts. On the other hand, for some artifacts like door, the associated agentive and telic qualia don’t work with verbs like enjoy in the way that the associated qualia of 14

At least not unless the terms “acquire” a use or purpose in a particular context. Pustejovsky (p.c.) noted occurrences of spoiled water, which would indicate that the water was spoiled for some purpose like a lab experiment.

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cigarette do. What would the telic role of door be? One might think that it is to be walked through, or closed. But if I say (3.32) John enjoyed the door this sounds very strange; and if it means anything, it certainly doesn’t mean that he enjoyed closing the door or walking through it. In a somewhat similar vein, consider (3.33) John enjoyed the vegetable garden. Vegetable gardens are artifacts; they are created by human for the purpose of growing food, as their name implies. Nevertheless, it’s hard to get a reading for this according to which John enjoyed growing vegetables in the vegetable garden or that he enjoyed creating the vegetable garden. At least equally available is the much more generic “enjoyed looking at” reading. Such counterexamples abound with artifacts. Kitchens, bedrooms, lamps, forks, knives are all artifacts that can be enjoyed, but the GL predicted reading of, say, John enjoyed the lamp—the reading that John enjoyed turning on the lamp or using it to illuminate something—is not there. The list of artifacts that don’t behave in the way GL predicts is much longer than the list of those that do.15 It does not help, as Pustejovsky and Bouillon (1995) or Godard and Jayez (1993) do, to appeal to aspectual clashes between the demands of the aspectual verbs or enjoy and the eventualities contained in the qualia to explain away counterexamples to GL’s account of coercion. They argue that the qualia for door, lamp, or kitchen involve eventualities that do not meet the demands of the aspectual verbs for some sort of accomplishment as an object argument; therefore, it’s no surprise that the qualia don’t show up as the values of the coercion. But notice that the aspectual verbs or enjoy are relatively catholic in their requirements on the type of eventuality they can take. (3.35) a. John enjoyed/began/finished/started swimming/sleeping/drinking. (activities) b. John enjoyed/began/finished/started swimming three laps/crossing the street/drinking three beers. (accomplishments) 15

Here is another counterexample from Kleiber (1999) with respect to the coercion data. GL should predict that (3.34a,b) have at least one common reading, since one thing one typically does with bulletin boards is read them. Nevertheless, (3.34a) can’t be readily interpreted as (3.34b).

(3.34) a. Paul a commenc´e le tableau d’affichage (Paul began the bulletin board). b. Paul a commenc´e a` lire le tableau d’affichage (Paul began to read the bulletin board).

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These verbs don’t take states or achievements as arguments. One could try to argue that kitchen’s, door’s, and lamp’s telic or agentive qualia are all states or achievements. But this seems a dubious stipulation. A plausible telic quale for lamp is to illuminate an area; and this is, by the usual aspectual tests, an activity. It would appear that the agentive quale of all of these objects is some sort of a creation event, which is an accomplishment. Thus, either the eventualities associated with the telic and agentive qualia of these terms are not the intuitive ones (in which case what use are they?) or the theory fails to make the right predictions for a vast number of ordinary cases. In any case, the inadequacy of qualia to make the right predictions does not just affect the account of coercions involving aspectual verbs. It also threatens the account of adjectives. Consider (3.22c). Water that is moving fast is the desired interpretation of this adjectival modification, but it is one which does not seem to involve reference to any of the inherent qualia of water. As Fodor and Lepore (1998) point out, and as we’ll see in detail later, there are many types besides those given in a classic qualia structure that are relevant to coercion. More importantly, GL is far too inflexible to handle certain examples that show that what eventualities are selected is sensitive to information already present in the context derived from preceding discourse. (3.36) Paul a commenc´e de tapisser sa chambre. Marie a commenc´e la sienne aussi (Paul has started to wall paper his room. Mary has started hers too). These examples pattern with examples like (3.30), in which the particular eventuality selected as the argument of the aspectual verbs is inferred from the discourse context. There are other ways types get affected as well. In GL only the verb is involved in the coercion of its arguments; for instance, the aspectual verbs and enjoy coerce their object argument to be of event type, and the event type is specified by one of the qualia. But this is empirically incorrect. For instance, the subject of an aspectual verb may determine what the type of its theme or event argument is. (3.37) a. The janitor has begun (with) the kitchen. b. The cleaners have started the suits. c. The exterminator has begun (with) the bedroom. In each of these there is a coercion to a particular type of event for the object argument of the aspectual verb. Yet it is obvious that the noun phrases in object

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position do not by themselves supply the requisite events, as the minimal pairs in (3.38a–c) lack the salient readings of (3.37a–c).16 (3.38) a. Jan has begun (with) the kitchen. b. Kim and Pat have started the suits. c. Sam has begun (with) the bedroom. We need information about the subject or agent of the aspectual verb to get the preferred readings for (3.37a–c). In GL, it’s totally unclear how the agent of the verb affects the selection of the event argument. GL cannot explain the effects we have noted. And for those who would argue that these are readings based on “pragmatic” inferences, they are at least as robust as the readings of the coercion cases that motivate GL. Secondly, they don’t depend on anything in the “context” other than the arguments of the verb! To be sure, these inferences are defeasible as well as the ones for the basic qualia. As far as I can tell, there’s no argument to say that qualia are part of lexical semantics if these aspects of meaning are not.17 There are many more instances of coercion that don’t fit in very well in the GL framework of qualia. The coercions in (3.39) involve the (defeasible) inference to a particular type of object of drink, namely alcohol. This phenomenon is known as “lexical narrowing.” (3.39) a. Chris likes to drink. b. Chris drinks all the time. c. Chris is a heavy drinker. d. Let’s have a drink. Qualia cannot model this sort of coercion, as they are tied to nouns denoting substances—mainly certain kinds of artifacts. We need to generalize the type system considerably in order to capture this data. Other coercion phenomena, like the metonymic examples of Kleiber, don’t find a proper home in GL. In Chinese, there are clear examples of part–whole shifts in the classifier system, where we shift from kinds to members of the kind. Here is an example where there is an anaphoric link between two DPs, and a metonymic shift due to the classifiers.18 16 17 18

Of course, (3.38a–c) have readings on which the agents have begun constructing or fabricating the objects. For a similar criticism of GL, see Vespoor (1996). This example is due to Laurent Pr´evot; see Pr´evot (2006).

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(3.40) a. ni kan zhe zhu hua. you look that CL.plant flower (Look at that flower.) b. jian liang duo gei wo ba. cut two CLbloom give me EXCL (Cut two blooms for me.) One might attempt to assimilate these metonymic coercions to the qualia via the constitutive role, but then the constitutive ceases to have a clear definition— is it truly the material cause or not? In addition it’s far fetched to suppose that a qualia structure will list all the parts of an artifact as complex as a passenger jet, for example. Vikner and Jensen (2002) have argued that perhaps qualia might be used to interpret the English Saxon genitive construction. They hypothesize that the qualia tell us what relation there is between the noun phrase or DP with the genitive marking and the head noun. However, here too an appeal to qualia doesn’t help very much. A quick survey shows that the number of relations involved between these two expressions is vast indeed. (3.41) a. Bill’s mother b. Mary’s ear c. Mary’s team d. The girl’s car e. The car’s design f. Mary’s cigarette (i.e., the cigarette smoked by Mary) g. Bill’s cake (the cake baked by Bill or the cake eaten by Bill) h. The wine’s bottle i. A mother’s boy j. The rapist’s victims k. Japan’s economy l. The economy’s sluggishness m. The economy’s performance n. Sunday’s meeting These examples illustrate that the relation between the DP in the genitive and the head noun is often determined by the meaning of the nouns in the construction—sometimes by the argument structure of a relational noun or a deverbal noun. With respect to GL, qualia appear to affect the genitive if the head noun is not intrinsically relational—e.g., (1.19e) (telic), (1.19d) (formal), (1.19f,g) (telic or agentive)—cf. Vikner and Jensen (2002). On the other hand, the head noun and the noun in the DP with genitive case can affect the relation (not just the head N), and furthermore, many relations present in the genitive construction don’t fall under any qualia — e.g. (1.19k,n) (spatial or temporal

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location), (1.19l) (“predicational”), 1.19c (set membership), (1.19b) (ownership or control). Qualia are at best a partial predictor of what’s going on with the genitive construction. The determination by the syntax/semantics interface of the relation involved in the interpretation of the genitive is defeasible. Information from the discourse context can overrule, for example, the determination of the relation by a relational noun like mother: (3.42) [Context: Picasso and Raphael both painted a mother with a child.] Picasso’s mother is bright and splashy —a typical cubist rendition. It makes Raphael’s mother look so somber. Some might object that this example involves extrananeous elements. The terms mother instead of Madonna suggest a special meaning. But Recanati (2004) offers the following example to show that contextual effects override the semantic import of relational nouns for the treatment of the genitive: (3.43) a. Ok. I just heard from John and Bill who are working at the old folk’s home. b. We need some supplies right away. c. John’s grandmother needs a wheelchair, and Bill’s grandmother a bedpan. Here we can perfectly well access the interpretation where John and Bill are in charge of certain aged persons. And it overrides the relational interpretation of grandmother and grandfather. GL’s introduction of qualia structures to analyze coercion looks very different from Nunberg’s sense transfer model. But in the end it packs the sense transfer function into the lexical entry for nouns, which just gives the wrong results when we implement it precisely. Moreover, the data show that the qualia model is insufficiently general and too inflexible to succeed. What about dot objects, another idea within GL? The underlying idea is again intuitive; copredications indicate that a word may have two independent senses in some sense at the same time. But this intuition is in need of development as well as defense. And once again, we don’t have the right formal framework to investigate this intuition. Pustejovsky’s introduction of dot objects to model copredication is an additional overload on the overtaxed AVMs with the typed feature structure and unification-based framework. Dot objects actually introduce considerable complexities into the type formalism and involve a rather subtle metaphysical analysis, as I’ll show in detail in the next chapter.

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In Pustejovsky (1995), one can glean a proposal for exploiting terms of complex • type. The proposal is to take an object of complex type and then “select” one of the constituent types at hand. It is in fact a simple coercion story but uses a different apparatus from qualia. Suppose, as Pustejovsky does, that book is a complex object of type informational object • physical object. By using such projection operations, we can coerce book to the appropriate type for examples like: (3.44) a. The book is interesting. b. The book weighs five pounds. On the other hand, it’s not at all clear, since it appears that the projection operation is destructive, how one can either account for copredication as in (3.45a) or the anaphoric coreference in (3.45b), where the pronoun it refers back to an informational type object while the predication in the main clause forces book to be of type physical object or p for short. (3.45) a. The book is interesting but very heavy to lug around. b. John’s Mom burned the book on magic before he could master it. In (3.45a), on the other hand, we need to use the informational type to handle the first predication and the physical type to make the second predication go through. But once we retype the book from physical object • informational object or p • i to, say, i, then how can we recover the physical type? This problem receives no solution in the GL model of dot objects. However, matters are worse. Suppose p p • i = ⊥; that is, suppose that the complex type physical object • information and the type physical object are inconsistent, and so their meet is ⊥. Then, GL’s theory of dot objects goes inconsistent. After selection, what happens is that instead of (3.46) book(x), x : p • i (where x : p • i is an assignment of the type p • i to x) you get, after one of the selection rules, (3.47) book(x), x : p But this is inconsistent with the basic typing rules for book. If one decides that the meet of the two types is consistent, it would still produce a funny typing of book. It would imply, for instance, that the i part of this retyped p•i object is not there or empty, and that would lead to very strange interpretations. Moreover, we would not be able to handle simple sentences like (3.45). Something has to be done to fix this.

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We have now surveyed both empirical and formal problems with GL. Despite its intuitive appeal and its undoubted success in making coercion and copredication central problems of lexical semantics, GL is in big trouble. The core claim that meanings are generated during the composition process is at best misleading. The posited complex lexical structures consisting of qualia and dot objects either don’t make sense as they stand or fail to make the right predictions on a relatively massive scale. It’s also not clear how to work out the ideas in the TFS formalism. Much of what Pustejovsky (1995) wants to do with dot objects, for instance, doesn’t make sense in the typed feature structure formalism, where types are atoms—whatever dot types are, they are not atoms. The TFS formalism supplies neither rules nor a semantics for such type constructors as •. And so no analysis of copredication is forthcoming in such a formalism. TFSs also fail to do justice to coercion, because it provides no general rules for doing what has to be done to get coercion to work. The main problem with GL is that it postulates an elaborate theory of lexical meaning without providing a theory of predication and meaning composition over that implicit in the theory of unification. To get an adequate explanation of the data, we need to do both. We also need to think much harder about the conceptual underpinnings of dual aspect nouns and the types that help us understand them. GL gives us a rich set of problems to think about, but we will have to build a replacement theory pretty much from the ground up. And we will have to use a different framework from that of TFSs.

3.5 Recent pragmatic theories of lexical meaning In contrast to GL’s attempt to locate the mechanisms for coercion and aspect selection within the lexical semantic entry for particular words, there is a pragmatic approach to such phenomena, exemplified on the one hand by Stanley and Szabo’s hidden variables approach (Stanley and Szabo (2000)) and on the other by relevance theorists Sperber and Wilson (1986), Recanati (2002), or Carston (2002). Broadly speaking, such approaches attempt to analyze phenomena such as coercion or aspect selection as involving pragmatic reasoning. For Stanley and Szabo the pragmatic reasoning involves finding contextually salient values for hidden variables; for the relevance theorists, it involves a process of enrichment of logical form. (3.48) It’s raining. (3.49) I’ve had breakfast.

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By postulating a hidden location or temporal variable, Stanley and Szabo can account for the conveyed content that these sentences typically have in a particular discourse context. For instance, uttering it’s raining conveys the content that it’s raining at some salient place of utterance. I’ve had breakfast conveys that I’ve had breakfast during some salient interval of time. Discourse context can of course affect salience: in (3.50), for instance, the salient location is that of the mountains in (3.50a), while it is a location on the way to the mountains in (3.50b). (3.50) a. We can’t go into the high mountains today. It’s snowing too hard, and we won’t be able to see anything. Let’s go for a hike around here instead. b. We can’t go into the high mountains today. It’s snowing too hard and the roads are a mess. Stanley and Szabo provide, however, no mechanism for determining the salient locations with which to bind the variable they introduce in logical form.19 The Stanley–Szabo proposal can be adapted to deal with coercions.20 One would need to postulate for each coercing predicate λxφ two hidden variables, one higher-order relational variable and another that will end up being related to x. Enjoy, which is understood to take an event as a direct object, would now look something like this: (3.51) λyλeλz enjoy(z, e) ∧ R(e, y) When no coercion is present R could be resolved to identity, but when the direct object of the verb involves denotations to non-event-like objects, then R would be resolved to some contextually salient property. So, for instance, enjoying playing a soccer game intuitively involves no coercion, and the variable introduced by the DP that ranges over playing soccer game events would be identified with the event variable that is the direct object argument of enjoy. However, Laura enjoyed a cigarette would involve a coercion; in this case the variable introduced by the DP ranging over cigarettes would be related via some contextually salient relation R like smoking to get the logical form: (3.52) ∃y∃e (enjoy(l, e) ∧ smoke(e, y) ∧ cigarette(y)) This isn’t quite right, as the lexical entry doesn’t specify that enjoy is a control verb. Additional stipulations, however, will fix the problem. 19 20

For examples like (3.50), one could appeal to the formulation of relevance in van Rooij and Schulz (2004) to define the salient binder for the variable. In effect Egg (2003) develops such a proposal. This is just the proposal used in the underspecified approach I suggested would be a possible option for the sense transfer strategy.

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This approach has an advantage over GL or the sense transfer view as developed by Sag and Nunberg, in that it gets the quantification cases right: enjoying many books does not, on this view, mean that one enjoys many events of reading books. On the other hand, this generalization of the Stanley–Szabo approach has several fundamental flaws. The first is that it has nothing to say about when coercion takes place. The typed approach I will provide does: when there is a type mismatch, there will be an attempted type adjustment, which, if successful, will in turn lead to a justification of the type presupposition. Without this sort of restriction on coercion due to type mismatch, the Stanley–Szabo approach predicts interpretations for enjoyed playing the soccer game that do not fit the facts. It predicts the possibility of all sorts of strange relations between this event and the event that is actually enjoyed—like, for instance, that the event enjoyed was different from the one that involves the playing of the soccer game. Such interpretations are not attested at all, regardless of context. In addition, consider the following example, due to Vespoor (1996): (3.53) a. Last night, my goat went crazy and ate everything in the house. b. At 10 p.m., he started in on your book. b . At 10 p.m., he began to eat your book. b . #At 10 p.m. he began your book. (3.53b ) is plainly bad even though we are primed in the context to understand that the eventuality to be coerced to is an eating event. The salient value for the hidden variable is clearly salient. Thus, the account is too weak and predicts coercions where there are none, unless we add to it a type-driven theory of predication. This approach also operates at the wrong level. Not all coercions are alike. While begin, start, and finish all coerce some arguments —e.g., start a cigarette, begin a cigarette, and finish a cigarette, they are not all equally happy in their coercive capacity with other direct objects. And one and the same ergative verb may license coercion in its internal or direct object argument, whereas it does not in its intransitive form. The aspectual verbs all work this way. (3.54) b. c. d. e. f. g. h.

a. I started your book this morning. I began your book this morning. I finished your book this morning. #The book started/finished this morning. #The book finished this morning. The reading of the book started this morning. The reading of the book finished this morning. The meeting started/began/finished this morning

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Begin, start, and finish all accept event denoting DPs in their theme argument in their intransitive use (3.54f–h), but they do not coerce non-event denoting DPs in that position into DPs that denote events (3.54d,e). In contrast, they support coercion of their theme argument position when they are used transitively. It’s very unclear how simply postulating a variable at the level of logical form will be able to explain these observations. Coercions are sensitive to actual lexical items and not to general pragmatic principles. Aspectual verbs like stop and finish have a clearly related meaning, but their behavior with respect to eventuality coercion is quite different. (3.55) b. c. d. e. f. g. h.

a. Mary finished eating the apple. Mary finished the apple. Mary stopped eating the apple. Mary stopped the apple. Jules finished smoking the cigarette. Jules finished the cigarette. Jules stopped smoking the cigarette. Jules stopped the cigarette.

Whereas (3.55a,b) have interpretations in common (in (3.55b) Mary could have finished doing other things to the apple), (3.55c,d) do not. (3.55d) only has a reading that (3.55c) lacks, in which Mary stops some physical motion of the apple, like its rolling off the table. (3.55a,c) have similar meanings, although (3.55c) does not entail that she ate the whole thing whereas (3.55a) does. (3.55b,d) have quite dissimilar meanings: finish the apple cannot mean stopping some physical motion of the apple. Aspectual differences between these verbs can account for the differences between the similar meanings of (3.55a) and (3.55c), but the differences between (3.55b) and (3.55d) cannot be so explained.21 Further, stop and finish don’t license the same coercions. (3.56) a. John has finished the garden (the kitchen). b. #John has stopped the garden (the kitchen). (3.56) is perfectly natural and means that John has finished doing something to the garden, like preparing it or digging it up. It’s very unclear what (3.56) means at all in an “out of the blue” context. There are similar though less striking differences between start and begin. Further, the coercion effects of these verbs are less easily interpretable with many consummable direct objects than the coercion effects of finish. 21

As far as I know no one has looked at these minimal pairs.

3.5 Recent pragmatic theories of lexical meaning (3.57) b. c. d. e. f.

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a. Mary started eating her apple. ?Mary started her apple. Mary started eating her sandwich. Mary started her sandwich. Mary began eating the apple. ??Mary began the apple.

These examples sound more or less good depending on their arguments. I have a harder time getting the eating event associated with fruits with these verbs than with constructed foods like sandwiches or candy.22 When start or begin take natural foodstuffs as their direct objects, I do not find that the coercions pick up an eating event, even when the example is primed in the context. Finally, the aspectual verb end hardly seems to induce event coercion at all. (3.58) b. c. d. e.

a. ?Mary ended the cigarette. ??Julie ended the apple.23 Alexis ended the sonata with a flourish. Lizzy ended the sonnet with a wonderful image. Mary ended the meeting (the discussion, the negotiations).

When we use the verb finish with a consummable, a coercion is possible; but this is not the case when the coercing predicate involved is end. The coercive capacities of end are quite restricted, in fact to things that are informational artifacts or entities that are or have eventualities of some sort as aspects (e.g., things like films). Event coercion is very much a matter of the verb’s fine-grained meaning, in particular its presuppositions, not a general pragmatic mechanism. The Stanley–Szabo approach attempts to explain coercion at the wrong linguistic level. Further evidence that this last charge is warranted comes from the observation that coercion seems to be linguistically constrained. Consider these examples due to James Pustejovsky: (3.59) a. Isabel has a temperature. b. Check Isabel’s temperature. c. Every physical object has a temperature. 22

23

I didn’t find any citations on Google for starting apples, starting peaches, or the like. And begin the fruit sounds even weirder to me. I did find one citation involving starting a peach candy. With vegetables like carrot, there are many citations involving products derived from carrots like carrot juice, but no citations for started a carrot by itself. There seem to be no sensible uses of this on Google, though someone might attempt to convey the proposition that Mary killed the apple by using this sentence. There seems to be an alternative to this aspectual use of end for all sorts of objects that can be destroyed, which is put an end to. This might be responsible for blocking the coerced reading of (3.58a,b).

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(3.59a) conveys that Isabel has a fever, an abnormally high temperature, whereas of course (3.59b,c) do not. Why should pragmatic processes of enrichment be blocked in these cases? It is not clear what sort of answer there could be, unless it has to do with the conventional meanings of words and, more importantly, certain linguistic, more specifically type, constraints on coercion. If the Stanley–Szabo account does not seem promising for coercion, it has even less to say about dual aspect nouns. It has no account of the complex cases of copredication that motivated GL’s introduction of “dot objects” or Cruse’s idea of perspectives. Finally, it has no account of predication failure. If hidden higher-order variables are present in all predications, we should be able to coerce verbs like hit. But that doesn’t seem to be possible. (3.60) Bill hit the number two. The Stanley–Szabo view is not the only pragmatic approach to lexical meaning. Relevance theorists (Sperber and Wilson (1986), Recanati (2004, 1993)) attempt to capture these inferences by appealing to a pragmatic process of free enrichment. Interpreters will add to the content of (3.48) and (3.49) in order to make the information conveyed optimally relevant, according to the principle of relevance. Free enrichment is a local, optional process in which literally encoded meanings are added to by a pragmatic inferential process to form enriched meanings that then combine to form the proposition that expresses the conveyed truth conditions of a sentence. Relevance theorists point to phenomena like lexical narrowing as support for their theory. Similar examples resembling lexical narrowing involve the specialization of a broad general meaning to a particular argument. (3.61) a. The ironing board is flat. b. My neighborhood is flat. c. My country is flat. It is indeed true that each of these uses of flat specializes a general meaning, as we can see from our ellipsis test. (3.62) ?This ironing board is flat and my country is too. However, these cases can fit into the Stanley–Szabo approach, if we assume that flat involves a hidden degree variable (Kennedy (1997)); the subject of the predication could in principle dictate a particular degree of flatness relevant to the subject’s type. In fact, given Kennedy’s work, it appears relatively straightforward to give a semantic account of the differences in (3.61). Nevertheless, relevance theorists can also attack examples like (3.61) using the principle of relevance.

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There are a number of problems with the proposal of free enrichment (and processes that relevance theorists take to be similar, like Nunberg’s sense transfer). First of all, free enrichment and sense transfer, as they’re usually formulated, predict the wrong truth conditions for many coercion cases involving coercion (recall the discussion of (3.10), just as Sag’s (1981) precisification of Nunberg’s sense transfer model does. Second, if these processes are optional, they will have the same problems as the Stanley–Szabo account in overgeneralizing cases of coercion. Finally, general pragmatic mechanisms do not account for the sensitivity of coercion and aspect selection to particular words and particular predicational contexts. The data I have provided here strongly suggests that coercion and aspect selection have to do with the semantics of particular words; they are driven by type mismatches, not by optional, general pragmatic mechanisms, though the latter may be eventually needed to fill out underspecified constraints provided by the justification of type presuppositions. A final difficulty with the relevance-based accounts is the lack of precision in the notion of relevance.24 The formulation of the principle of relevance requires an appeal to unknown mechanisms of inference and processing cost. It’s unclear how these mechanisms, or an appeal to speaker intentions, can constrain predication in a precise fashion. For instance, we should expect on the enrichment view that we could always make sense of predications in which there are, from my perspective, type clashes. There should be little difference between the classic coercion cases and cases where no coercion is possible such as some of the examples in (1.1), repeated below: (1.1g) The number two is red. One could imagine a speaker intending to convey with (1.1g) that the number two guy in some procession is wearing red. But such a pragmatic interpretation of (1.1g) just doesn’t seem possible because of the constraints put on predication by the meanings of the individual words. Pragmatic approaches also fail to say anything relevant about the cases of coercion like John enjoyed his glass of wine or Mary enjoyed her cigarette in out of the blue contexts. On the free enrichment approach, sense transfer or the hidden variable approach, we should either expect no definite answer in out of the blue contexts or variation. However, speakers are remarkably constant in their interpretations of such sentences. Such theories have difficulty in saying anything about the interaction between semantics, predication, and the pragmatic principles they use to analyze coercion. There simply isn’t enough detail 24

See Asher and Lascarides (2003) or Merin (1999) for a more detailed look at Relevance Theory.

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in extant pragmatic theories to see how they could make substantive predictions or how they could exploit the semantic content of lexical items. In short, there is no pragmatic substitute for a lexical semantic theory. Let us take stock. We have seen that previous accounts in general do not provide formally or materially adequate accounts of phenomena that a lexicon theory must address. They all attempt to shift the meaning of various terms to account for coercion. Nunberg and the pragmatic approaches leave the shifting mechanism completely unconstrained, providing little explanatory force, while GL makes the shifting mechanism too rigid to account for the data and appears to get the wrong truth conditions for quantified sentences involving coercion. Turning to copredications, none of these theories has an account that begins to get the facts right. More generally none of these theories can account for how previous discourse context can affect predication. The approach I will flesh out in the next chapters attempts to remedy these weaknesses while retaining some of the flexibility of pragmatic approaches.

PART TWO THEORY

4 Type Composition Logic

I have surveyed the main, extant proposals for lexical semantics and have argued that these do not meet the demands for a satisfactory theory of lexical meaning. I propose a new framework here, a context sensitive model of typedriven lexical semantics, that builds on the insights of previous accounts but attempts to remedy their inadequacies. This framework, the type composition logic, or TCL, assigns to each word stem a type. Some word stems like stone, which are ambiguous between a verbal and a nominal meaning, will have a complex type to represent that ambiguity. Such word stems may not have a distinct logical form until the syntactic environment or a morphological suffix1 selects for the verbal or nominal sense. The lexical entries for word stems will take seriously the idea that predicates place type presuppositions on their arguments, presuppositions that must either be satisfied by the types of the arguments or accommodated if the predication is to be semantically well-formed. I will thus distinguish between type presuppositions that predicates place on their arguments and the types that the term arguments introduce as part of the proffered content. When putting terms together in a predication—be it in a standard predication, a nominal or verbal modification, the application of a determiner to a noun phrase, or even semantically rich morphological processes—extra contributions to logical form can arise when there is a type clash between a predicate and one of its arguments and a type presupposition must be accommodated. This means the account of predication must involve more than the simple application of a property to an object or a relation to its terms. It must involve accommodation mechanisms that will introduce material into logical form to make the predication succeed in certain cases. In addition, I will make these mechanisms sensitive not only to the presuppositions introduced within local predication—the application of 1

For example, in German Bank is ambiguous just as its English homynym is; however, plural morphology allows us to distinguish B¨ancke (benches) from Banken (financial institutions).

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the predicate to its arguments—but also to the discourse context in which this predication takes place—hence the “context sensitivity” in the context sensitive model of lexical content. My proposal brings the theory of predication and lexical content into line with a general tendency towards dynamics in semantics and pragmatics that has developed over the last quarter century. Dynamic semantics has proved extremely useful in analyzing the semantics of anaphoric elements in language like pronouns and tenses and of expressions that generate presuppositions that must be integrated into the given discourse context (e.g., Kamp (1981), Heim (1982, 1983), Asher (1986, 1987), van der Sandt (1992), Asher (1993), Kamp and Reyle (1993), Groenendijk and Stokhof (1991), Veltman (1996), Beaver (2001)), and extensions of this idea to include richer ideas of discourse structure and pragmatics have also proven useful (Asher and Lascarides (2003)). Dynamic semantics treats the meaning of a formula as a relation between information states, an input information state, and the output information state. The input information state represents the content of the discourse context to date while the output information state represents the content of the previous discourse context integrated with the content of the formula. The interpretation of a discourse involves the relational composition of constituent sentences’ relational meanings. In dynamic semantics for natural languages, as well as in the dynamic semantics for programming languages, the interpretation of a formula can either function as a test on the input context or can transform the context. So, for example, John is sleeping in dynamic semantics would yield a formula that functions as a test on the input context, which we can think of as a set of elements of evaluation. If an element of evaluation verifies the proposition that John is sleeping, then it is passed on to the output context; if it does not verify the proposition, it does not become part of the output context. Operators like conditionals form complex tests on an input context C: an element of evaluation e will pass the test defined by If A then B just in case any output o from A given e will yield an output from B (given o as an input). Some sentences, for instance those containing indefinite noun phrases, output a context that is distinct from the input one. They transform elements of the input context; in particular they reset or extend the assignment functions that are parts of elements of the context to reflect the information they convey. On a view that treats assignments as total functions over the set of variables, an indefinite has the action of resetting an assignment that is part of the point of evaluation for formulas, as in Tarskian semantics. On a view where assignments are treated as partial functions, the interpretation of an indefinite extends the assignment with a value to the variable introduced by the indefinite

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in logical form. This reset or extended assignment becomes part of the output context. In dynamic semantics presuppositions constitute a particular sort of test on the input context. Consider a sentence like (4.1) Jack’s son is bald. The presupposition generated by a definite noun phrase like Jack’s son (namely, that Jack has a son) must be satisfied by the input context, if the interpretation of the rest of the sentence containing the definite is to proceed. One way of satisfying the presupposition in (4.1) occurs is for it to be already established in the context of utterance of (4.1) that Jack has a son. This can occur, for instance, when (4.1) is preceded by an assertion of Jack has a son. Presuppositions can also be satisfied by contexts within the scope of certain operators, as in (4.2), even though it has not been established in the discourse context that Jack has a son:2 (4.2) If Jack had a son, then Jack’s son would be bald. In dynamic semantics the content of the antecedent of the conditional is encoded into the input context for the interpretation of the consequent and the presupposition it generates. Thus, the satisfaction of the presupposition by the antecedent of the conditional in (4.2) means that the presupposition places no requirement on the input context to the whole conditional, the context of utterance. The satisfaction of the presupposition by elements of the discourse context entails that the presupposition does not “project out” as a requirement on the context of utterance. Thus (4.2) is consistent with the assertion that in fact Jack has no son. On the other hand, dynamic semantics predicts that if we change (4.2) just slightly so that the antecedent does not provide a content that satisfies the presupposition, the presupposition will project out as a requirement on the input context to the whole conditional: (4.3) If Jack were bald, then Jack’s son would be bald too. What happens when a presupposition cannot be satisfied by the discourse context? It depends on what sort of presupposition is at issue. Some presuppositions, such as those introduced by definite noun phrases, are easily “accommodated.” In dynamic semantic terms this means that the input context 2

One of the great successes of Dynamic Semantics has been to show that the behavior of presuppositions introduced by material within the consequent of a conditional follows straightforwardly from the conception of the conditional as a complex test on the input context and thus offers a solution to the so-called projection problem.

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is altered in such a way so that the presupposition is satisfied. Other presuppositions, such as that generated by the adverbial too, are much less easily accommodated, as I mentioned at the beginning of this book. Given that operators like conditionals can add “intermediate” contexts between the context of utterance and the site where the presupposition is generated, we need a theory of where and how to accommodate in case the input context does not satisfy the presupposition. In some theories of presupposition that operate on semantic representations like that of van der Sandt (1992), accommodation simply involves adding a formula for the presupposition to an appropriate part of the representation for the discourse.3 Van der Sandt stipulates that one tries to accommodate at the outermost context first and if that fails then one tries to accommodate at the next outermost context. The constraints on accommodation are that the addition of the presuppositional material be consistent with the discourse context. So, for instance, one cannot add the presupposition that Jack has a son to a context where it is established that Jack does not have a son. I will develop a theory of presupposition satisfaction and accommodation for type presuppositions that is related to the clausal theory of presupositions as developed in van der Sandt (1992). To implement a theory of presupposition at the level of types, we need three things: a theory of types, a theory of the relations between types, and an account of type presupposition satisfaction and type presupposition accomodation. The theory of type presupposition satisfaction is simple. Normal predication involves a test on the types of the expressions involved in the predication. If the type of the functor and its argument are related in such a way that the argument’s type is a subtype of the type presupposition placed by the functor on its argument, then the predication is licensed and the construction of logical form can proceed via application. That is, the presupposition induced by the functor is satisfied. Matters become more complex when complex types like • are involved; a type presupposition might be satisfied by one of the constituent types of a • type. I will study this issue in detail in chapter 6. On the other hand, when such presuppositions are not satisfied, there are several ways to accomodate the presupposition. One way is reminiscent of classic accommodation at the clausal level: we simply add the type requirements of the predicates to the types of the arguments via the least upper bound operation . But there are other sorts of accommodation that depend, for instance, on the 3

In other theories like that of Heim (1983), the accommodation procedure is not really well developed; but see Beaver (2001) for a detailed account of accommodation in a Heimian approach to presupposition.

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types of the arguments and the types of the predicate. In chapter 8 we will look at a particular type of accommodation that occurs when coercion takes place. We must integrate type presuppositions within our representation of the lexical meaning of words. We will do this by adding another parameter to our terms, one for type contexts that will carry information about the type presuppositions of various terms. As we shall see, applying a predicate to its arguments may change the extant presuppositions or indeed the logical forms of the predicate or the argument if accommodation is involved. I will set out in this chapter and in chapter 6 a composition logic that determines how the types of lambda terms can combine, and how type presuppositions can be accommodated. First, however, I set out a preliminary set of types and the relation of subtyping on them.

4.1 Words again With the adoption of a dynamic framework at the lexical level, we must revisit the issue of lexical ambiguity and the units of lexical semantics. Traditionally, ambiguity is a matter of individuating expressions. In the structural case, individuation is often provided on independent syntactic grounds. For example, morphosyntax provides at least two derivations for sing, dance, and drink— one on which these lexical items are analyzed as intransitive verbs, one on which they are treated as transitive verbs. These different expressions are represented via bracketing, labelled trees, syntactic terms, or some other means, with different representations standing for different expressions. For lack of independent natural candidates, the “structures” underlying lexically ambiguous expressions like bank are then represented in some arbitrary fashion. According to this traditional picture, it is expressions, not surface forms, that serve as inputs to semantic interpretation; there is a sharp distinction between “logical” polysemy and “accidental” polysemy, in that the latter is invisible to semantics. The traditional treatment of accidentally polysemous words is analogous to the treatment of pronouns in traditional semantics: the choice of the pronoun’s antecedent is “invisible” to the interpretative component; traditional semantics solves this problem by indexing the pronoun with its antecedent as an input to logical form construction. The use of a dynamic approach with underspecification however, allows one to investigate the resolution of anaphoric constructions as part of the interaction between underspecified, compositional content and discourse context.4 A similar observation should be made at the 4

The literature concerned with DRT is replete with such investigations. See, for instance, Kamp and Reyle (1993).

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lexical level. Many word stems in English like inquire, hear, own, woman, and sister come with a determinate type structure, which also determines their argument structure. But other word stems do not. Word stems like stone, sink, boat, and dog can be determined to be of nominal or verbal type depending on their local “predicational” environment.5 Because there is a close connection between types and syntactic categories, the choice of type will determine the syntactic category and will allow such word stems to combine with functional or closed class lexical elements like determiners, or verbal morphology.6 Accidentally polysemous words like bank in English, which can be disambiguated by the predicational context and the selectional restrictions of the predicates they combine with, are a special case of word stems with an underspecified type. Below I will introduce a restricted set of disjunctive types in the system; roughly, if α and β are subtypes of some determinate type, then the disjunctive type α ∨ β is also a type. Such disjunctive types can serve as least upper bounds of two types, when upper bound for the two constituent types exist in the hierarchy. But even when this doesn’t occur, the semantics postulates disjunctive “objects” that serve to model underspecifications or accidental ambiguities. For instance, stone will denote in the internal semantics a disjunctive object involving a basic type stone and a transitive verbal type as one of its other disjuncts.7 Intuitively, this object does not correspond to a determinate type in the hierarchy, however. The word stem bank will have be assigned the disjunctive object financial institution ∨ river border ∨ transitive verb. This use of disjunctive objects is analogous at the type level to the use of underspecification at the level of compositional content; we can exploit the predicational context and discourse context to disambiguate lexical entries with disjunctive types, just as those who use underspecification at the compositional level use discourse context to disambiguate anaphors or relative operator scopings. For example, bank in the following sentences must be a verb; only if it combines with the infinitival complementizer to do we get a grammatical sentence and a 5

6 7

Borer (2005a,b) has an extreme version of this view, on which all open class lexical elements have in effect an indeterminate type. She conceives of the content of such words as being outside the linguistic system. TCL can accommodate the motivating cases for her view, without denying that the lexicon also contains entries for lexical stems with a determinate type structure. In a review of Borer’s work, Potts uses data from tagging of large corpora to argue that in fact most English words have some determinate type structure. He notes that of the 40,000 word types in the Brown corpus, only 11.5% are even 2-ways ambiguous. Marantz (1997) and Angelika Kratzer in an unpublished manuscript contain competing approaches to Borer’s view but also maintain that it is morphosyntax and the functional lexemes and their projections that determine argument structure, not the lexicon. Plural morphology can also help in picking out the appropriate type of the item, as in the case of German Bank and its two, different plural forms that provide a disambiguation. This follows the suggestion of Asher and Denis (2005).

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well-formed predication: John wanted to bank his check that very day or Mary wanted to bank the airplane.

4.2 The basic system of types Let us begin by examining simple, functional, and disjunctive types, as well as the types that carry presuppositions. Thus, we have • Simple or Primitive Types: e, t, physical object, etc. • Presuppositional Type: Π, another base type that carries the type presuppositions of terms. • Disjunctive Types: If σ, τ, and ρ are types, σ ρ and τ ρ, then (σ ∨ τ) is a type. • Functional Types: If σ and τ are types, then so is (σ ⇒ τ). • Quantificational Types: If σ is a simple type, and τ is any expression denoting a type and x is a variable ranging over types, then ∃x σ τ is a type.8 To illustrate, a term t is of this quantificational type if there is a subtype x of σ such that t is of type τ[x]. The set of simple types, ST, forms the core of the system. It contains the basic types countenanced by Montague Grammar, e, the general type of entities, and t, the type of propositions, along with a finite set of subtypes of e and a countable set of subtypes of t. Another distinguished subtype is ⊥, the absurd type. When there is no type in the type hierarchy γ such that α γ and β γ, α ∨ β represents a disjunctive object that is the internal semantics of an accidentally polysemous term that must be resolved to assign the term a determinate type. Functional types represent properties. ST comes with the subtyping relation , which forms a semi-lattice over ST with ⊥ at the base.9 Using on the simple types, we define a greatest lower bound operation for elements of ST. Definition 2 Greatest Lower Bound: α α = β iff β α and β α and there is no γ  β such that β γ and γ α and γ α . has the usual properties—e.g., idempotence, commutativity, and α β iff α β = α. TCL captures incompatibility between types in terms of their 8 9

x is, I realize, close to an individual level variable x. I strived for typographic consistency making all type formulas in small caps. Hopefully this will not cause too much confusion. As we will see, the subtyping relation as defined in the next section will entail that  is not the supremum of the lattice. In fact  is not a type. Note that the fact that  is not a type in the hierarchy does not stop us from using the tautology  in logical forms. These are quite different objects.

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common join, ⊥. We can also define a dual to greatest lower bound, least upper bound, or : α  β = γ iff γ is the least general type in the hierarchy such that α  γ and β  γ = γ. Note that  may not be always defined, since there may be no type that is the least upper bound of arbitrary types α and β

4.2.1 Subtyping In the previous chapter, we saw that the standard set theoretic model of types fails to provide a coherent notion of subtyping for functional types, once we admit a rich set of subtypes of the type of entities e. To summarize the difficulty, recall that according to set theory, the set of physical properties or functions of type p ⇒ t, that is, the set of all functions from objects of physical object type to propositions, and the set of first-order properties or functions of type e ⇒ t (the set of all functions from entities to propositions) are disjoint, even though p e in the lattice of simple types and even though every function in p ⇒ t is a subfunction of some function in e ⇒ t. There is no coherent notion of subtyping for higher-order types, where subtype is understood as subset, once we admit multiple subtypes of e. Type theory and the categorial models that I develop below provide a coherent notion of subtyping for all types, in the sense that, together with the rules of the simple, typed λ calculus, they generate a consistent logic or system of proof. We need such a notion of subtyping to specify an appropriate rule of application for β reduction: roughly one can apply a λ term λxφ to a term t if the type of t is a subtype of the type of x. I will specify subtying using a restricted, intuitionistic notion of deduction or proof for types, Δ . • From subtyping to logic: α ST β α Δ β In particular, the model will verify: Fact 1

Subtyping for functional types: α α β β (α ⇒ β) (α ⇒ β )

Subtyping for functional types implies that e ⇒ t p ⇒ t. This makes sense from a proof theoretic or computational point of view: if you have a proof that given a proof of an entity, you have the proof of some proposition, then you have a proof that given a proof of an entity of a particular type (say

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a physical object), you have a proof of a proposition. But we cannot derive p ⇒ t e ⇒ t. This seems not to get us what we want for our type hierarchy, since this implies, on the usual conception of first-order properties, that the type of first-order properties is a subtype of the type of physical properties.10 In light of this, we must re-examine what we mean by a first-order property. In a system with many subtypes of e, something is a first-order property just in case it is a function from some subtype of e into the type of propositions. To spell this out, our types must be defined in a second-order language for types. The type of first-order property would thus not be what we naively take it to be, namely e ⇒ t, but rather something that is implied by all function types taking as inputs subtypes of e and returning a proposition. That is, the type of a first-order property is: (4.4) ∃x e (x ⇒ t) Anything from whose type declaration we can “prove” (4.4) is a first-order property. To get anywhere, we must provide subtyping rules for existentially quantified types. To get a sensible notion of subtyping as deduction, my subtyping rules follow the standard introduction and elimination rules for ∃. In particular, where A is any type expression with an occurrence of β and B a type expression where β does not occur, then • Type theoretic ∃ introduction: β α A (∃x α A( Xβ )) • Type theoretic ∃ “exploitation”: β α, A B (∃x α A( xβ )) B This enables us to get the right facts about first-order properties. In particular, take the λ expression for black dog, whose course grained, denotational meaning is a function from physical objects to propositions. The NP has the type p ⇒ t, from which we can easily prove (4.4) using the ∃ introduction rule. We can now combine black dog with a determiner whose type presupposition 10

This has disastrous consequences for the construction of logical form. Consider the rule of application in the λ calculus which is like Modus Ponens—given a type α and a type α ⇒ γ, we get γ. Now take the case of a determiner which is something of type (e ⇒ t) ⇒ ((e ⇒ t) ⇒ t) and it must combine with something of p ⇒ t. We have by assumption that e ⇒ t p ⇒ t. But we cannot now apply the determiner meaning to its restrictor; application is not sound in this case, just as β α does not allow us to conclude: β → γ, α γ.

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on its first argument is that given by (4.4). We also have the general type of physical properties, ∃x p (x ⇒ t), the general type of informational properties, ∃x i (x ⇒ t), and so on. The subtype hierarachy for these will be the intuitive one. (4.4) is the type presupposition of anything that intuitively takes a first-order property as an argument—e.g., a determiner or DP. Any expression that expresses a particular first-order property will satisfy this presupposition in the sense of entailing it. Thus: Fact 2 Any ordinary physical property (e.g., mass, shape, weight, color, etc.) is a first-order property and any property of informational objects (e.g., the property of being interesting, intelligible, etc.) is a first-order property. In addition, applying a physical property to an object of non-physical type is not defined (yields a type clash), and similarly applying a property defined only on entities of abstract object type, i.e., of type i, to something of type p is not defined.

4.3 Lexical entries and type presuppositions In the simply typed lambda calculus, type checking is done automatically during the moment of application. In the system developed here, however, a clash between the type requirements of a predicate and the types of its arguments may require adjustments to the predication relation and to logical form. Doing this directly within the typed λ calculus led Asher and Pustejovsky (2006) to unwanted complexity, and so I have chosen a different route, separating out those operations involving type presupposition justification from the core of the simply typed λ calculus. To pass presuppositions through properly from predicates to arguments, I add a presuppositional parameter to each type as de Groote (2006) and Pogodalla (2008) do to handle dynamic contexts.11 Each term has an extra argument for a presupposition element that can be modified by the lexical item. For instance, the standard lexical semantic entry for tree looks like this: (4.5) λx: p tree(x) 11

Since I’m not trying to embed dynamic semantics in the λ calculus, I do not resort to their continuation style semantics. They add two parameters of interpretation, but I shall add only one. I use standard dynamic semantics for passing type values across discourse spans. Nevertheless, everything I do here should be fully compatible with other approaches to dynamic semantics.

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The predicate’s type presupposition is that its argument must have type p. My revision to the entry for tree adds another λ bound variable π of type Π, which ranges over presuppositional elements. These presuppositional elements carry typing requirements from a predicate to its arguments. The addition of a type presupposition encoding parameter to terms complexifies the system of types in a predictable way. The type of propositions in TCL moves from Montague’s t to Π ⇒ t, and DPs and determiners shift predictably, given the more complex types for first-order properties. Although we cannot yet capture completely the presuppositions of determiners, let’s suppose for now that the type presuppositions of determiners are quite weak at least with respect to the higher-order variables: these range over all types that satisfy the definition of a first-order property in (4.4). Thus all DPs will have the general type: (∃x e (x ⇒ (Π ⇒ t)) ⇒ (Π ⇒ t)). I’ll abbreviate this type to dp. I’ll abbreviate the polymorphic property argument for a DP, ∃x e (x ⇒ (Π ⇒ t)), to be simply 1 (for a first-order property). Accordingly, determiners will have the type 1 ⇒ (1 ⇒ (Π ⇒ t)). Because the type presuppositions of the higher-order variables don’t vary with the lexical entry, I’ll assume from now on that all capital variables of the form P, Q, R, P1 , etc. are of type 1. Scripted variables P, Q, etc. will range over modifiers or objects of type 1 ⇒ 1, or mod. Capital Greek letters Φ, Ψ, etc. will range over objects of DP type. Let’s go back to the entry for nouns. In a standard typed system a noun is an argument of modifier forming an NP. An NP denotation furnishes an argument to a determiner to form a DP. DPs in turn take verbs or VPs as arguments. But the behavior of type presuppositions shows that the view of semantic composition that is encoded in the standard functional types for these categories is not quite right. Modifiers only combine with nouns if they satisfy the type presuppositions of the latter; modification by any subsective adjective preserves the type of the noun it syntactically modifies.12 For example, heavy must justify the typing requirements of dog; that is, a heavy dog is still a dog. Or to take a more extreme example, a red number, to the extent that it can be interpreted at all even given contextual help, still seems to be a number, whatever else it might be. Thus, the type presuppositions of the noun must percolate to the modifier. In order to pass the presupposition from the noun to the modifier, I will make 12

That is, anything that satisfies an NP in which a subsective adjective X applies to an NP Y always satisfies Y as well. This is problematic for non-subsective modifiers: wooden nutmegs aren’t real nutmegs (though they look very much like them), and former soldiers are normally no longer soldiers. Nevertheless, even former soldiers might carry the same type relevant to the predication as current soldiers. We will revisit this issue in chapter 11, where I shall argue that these modifiers are more subsective than they might seem.

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use of the presupposition parameter.13 Nevertheless, I will also follow tradition and assume that modifiers take first-order properties as arguments of the sort traditionally associated with nouns. Thus, letting P be a variable of modifier type, the functional type from first-order properties to first-order properties, the lexical entry for tree looks like this:   (4.6) λPλxλπ P(π ∗ argtree 1 : p)(x)(λvλπ tree(v, π ))

argtree stands for the “first” argument position of the predicate; this argument 1 position is always associated with the referential or denoting argument; the first argument position of a transitive verb will be its subject, while the second argument position will be filled by its direct object or internal argument. π ∗ argtree 1 : p says that, in addition to the type constraints π already encodes, whatever fills the first argument position of tree must have a type that justifies the type p. This includes the variable v in the lexical entry (4.6), but it may also include a variable contributed by the modifier. So summing up, (4.6) says that a noun like tree takes a modifier as an argument in order to pass along its type presupposition requirements to the modifier. In order for the modifier to combine with the noun, it must justify the presupposition that is appended via * to its presupposition parameter argument. The modifier also takes a simple first-order property, in this case λvλπ tree(v, π ), as an argument. This way of doing things may look complicated in comparison to the usual type declarations in the λ calculus, but we will need the flexibility provided by this formalism when we have to justify presuppositions. Presupposition parameters range over lists of type declarations on argument positions for predicates. To save on notational clutter, I’ll often just write argi when the predicate is obvious, or just list the type when the argument position and predicate is obvious. By integrating this λ term with a determiner and a verb, more type presuppositions will flow onto the presupposition parameter π. I haven’t said what the type of x is explicitly in the entry for tree. Because we want to allow for flexible justification of presuppositions, the type presupposition of x should be very weak. I shall take it to be of type e. When these presuppositions are to remain local to the term but not already specified, I’ll use the standard notation of the λ calculus for readability, but these presuppositions should be understood as generated by the predicate and stored in the local presuppositional parameter. Thus, the final entry for a tree will be:   (4.7) λP: mod λx: e λπ P(π ∗ argtree 1 : p)(x)(λvλπ tree(v, π ))

In practice, I will omit the typings of terms when they are obvious. More gen13

This acts just like the left context parameter in de Groote (2006).

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erally, for any noun Nn, we have the following entry, where PreNn encodes the type presuppositions of Nn. (4.8) λP: mod λz: e λπ P(π ∗ PreNn )(z)(λuλπ Nn(π, u)) Given these abbreviations, the general type of nouns shifts from the standard e ⇒ t to mod ⇒ 1. For NPs I will have the relatively simple type schema, α ⇒ (Π ⇒ t), where α is a subtype of e. I suppose that when no adjective is present, P in (4.8) applies to the trival subsective adjective, λPλx: eλπP(π)(x), and so gives a predictable type for a simple NP. I deal with multiple adjectival modifiers through type raising to form a complex modifier; the exact logical form for such raised NPs will be a matter of meaning postulates.14 In general predicates must pass their typing presuppositions onto their arguments. More specifically, a VP should pass typing presuppositions to its subject, and a transitive verb should pass its typing requirements to its object. For example, in The book is heavy, the type requirements made by the verb phrase on its subject argument should be satisfied or accommodated in order for the predication to succeed. In keeping with standard assumptions about the syntax–semantics interface, this means we must rethink the entries for our lexical entries to track these presuppositions and to put them in the right place. Consider an intransitive verb like fall. We want its type presuppositions to percolate to the subject. We can do this by assuming that an intransitive verb, or more generally a VP, is a function from DP denotations to propositions.15 This function will feed the appropriate type presuppositions to the subject DP in the same way that nouns feed their presuppositions to their modifiers. In order to do this properly, we must make our verbal predicates relatively catholic as to the type of DP they accept. The specific type requirements that verbs will place on their arguments will not be placed on the DP itself, but be placed rather in the presupposition slot so that it may be propagated to a local justification site and justified there—i.e., on the individual argument term furnished by the DP, if that is possible. If the higher-order types 1, dp etc. shift in type, it is usually because of a shift of one of their constituent types, from one subtype of e to another. It is quite difficult to shift higher-order terms directly, as we can see from the lack of coercions, say, from propositions to questions or from propositions to properties. 14

15

The issue of multiple adjectival modification gets complicated when we have a subsective combined with a non-subsective adjective as in fake, yellow gun or yellow, fake gun. It seems that both of these NPs have the reading of denoting an object which is a fake gun and yellow; nevertheless, standard compositional accounts will not be able to provide this reading for fake, yellow gun without resorting to meaning postulates as well. This applies the same trick used to handle transitive verbs taking generalized quantifier arguments—which makes the latter functions from DP denotations to VP denotations.

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We need presupposition checking on the type of the individual variable bound by the determiner in the DP in order to make the system function well. For instance, we need this to explain how the ambiguity in the meaning of an accidentally polysemous word like bank gets resolved when it combines with a VP like offered subprime loans in: (4.9) Many banks offered subprime loans in those days. The quantifier in (4.9) quantifies over banks that are financial institutions, not river banks. This happens because the type presupposition of the VP is passed directly onto the variable that many binds and because this variable inherits the disjunctive type, river-bank ∨ financial-institution. I assign the type to the noun bank, the type mod ⇒ ((river-bank ∨ financial-institution) ⇒ (Π ⇒ t), which means that the variable bound by the determiner in the DP will get the type river-bank ∨ financial-institution and it is the type of the variable that will be disambiguated by the type presupposition of the VP on its subject argument, which is financial-institution. The alternative, that the ambiguity is resolved at the level of the higherorder properties of the NP or DP directly, gives the wrong results. To see this, consider what would be, on this view, the type of many banks. It should be a function from properties of banks, considered either as river banks or financial institutions, to propositions. A property of banks is presumably a disjunctive type formed from two functional types, one of the form river-bank ⇒ (Π ⇒ t) and one of the form financial-institution ⇒ (Π ⇒ t). Given our reconstrual of subtyping and interpreting type disjunction as akin to logical disjunction, the type of the NP is then equivalent to ((river-bank financial-institution) ⇒ (Π ⇒ t)), which means that the variable bound by the determiner gets the type river-bank financial-institution. This last type is, however, presumably equivalent to the type ⊥ and so the quantification in (4.9) is vacuous, which goes completely contrary to intuitions and is a disastrous prediction. We might try to salvage this approach by making the type presuppositions of the VP hold only at the level of the DP type itself. This type presupposition would be (financial-institution ⇒ (Π ⇒ t)) ⇒ (Π ⇒ t), while the DP’s type is ((river-bank financial-institution) ⇒ (Π ⇒ t)) ⇒ (Π ⇒ t). Given Subtyping for Functional Types, the type presupposition of the VP on the subject is a subtype of the DP’s type, and so should be satisfied. But now we don’t resolve the ambiguity! The DP continues to quantify over riverbanks and over financial institutions, and the word bank remains ambiguous! This again goes contrary to intuitions. Thus, for an intransitive verb like fall, we have the following lexical entry where π ∗ argfall 1 : p specifies the type assignment for fall on its first argument

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place to be of type p or of physical type. This first argument place is ultimately filled by the variable bound by the quantifier that is the translation of the DP in subject position. The variables that will end up as subject place arguments for the verb can be determined from the syntax of the λ term for the predication. In particular, the variable introduced by the DP Φ will be one; and in the representation below, y is one as well.   (4.10) λΦλπ Φ(π ∗ argfall 1 : p)(λy: pλπ fall(y, π ))

The type of an intransitive verb or a VP, which in Montague Grammar has the type e ⇒ t, now has the more complex type, dp ⇒ (Π ⇒ t), or the functional type from general DP types to propositional types. Transitive verbs look very similar. The presuppositions of the predicate must percolate to the internal (direct object) arguments as well as the external (subject) arguments, which brings us to the following translation for a verb like hit. (4.11) λΦλΨλπ Ψ(π ∗ arg1 : p){λx: pλπ Φ(π ∗ arg2 : p)(λy: pλπ hit(x, y, π ))} I’ll usually abbreviate this to: (4.12) λΦλΨλπ Ψ(π ∗ arg1 : p){λxλπ Φ(π ∗ arg2 : p)(λyλπ hit(x, y, π ))} The typing on x and y, which I’ve omitted in (4.12), are easily inferrable from the typing on the first and second arguments of the verb; they are both of type p. Such lexical entries for verbs permit a lexical treatment of a wide scope reading for the direct object or internal argument quantifier. The following, slightly different entry for the verb suffices: (4.13) λΦλΨλπ Φ(π ∗ arg2 : p){λyλπ Ψ(π arg1 : p)(λxλπ hit(x, y, π ))} Let’s now go back to determiners. All singular determiners require their head noun phrases to be first-order properties. Most determiners, however, also carry type presuppositions on the sort of individuals they quantify over. For instance, the mass determiner expressions in English much and a lot of require that the elements in the domain of quantification be masses or portions of matter, another subtype of e. For instance, much requires the variable it binds to range over masses. The determiner a on the other hand requires the variable it binds to range over quantized or countable objects. These presuppositions interact with the presuppositions conveyed by the NPs with which determiners combine. Consider (4.14) I’ll have a Chardonnay (a water, a coffee).

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(4.14) means that I’d like a particular quantity of Chardonnay, water, or coffee— a glass, a cup, perhaps even a bottle. Conversely, mass determiners impose portion of matter readings: (4.15) There was a lot of rabbit all over the road. (4.15) means that there was a considerable quantity of rabbit portions of matter all over the road. This leads me to suppose that the determiner imposes a type presupposition on the individual argument that interacts with the type presupposition of the NP. In the next section, we’ll see how. Linguistically, this is interesting, because it appears that many NPs are in fact underspecified with respect to the mass/count distinction; the mass/count distinction is imposed by the determiner in English (and in other languages, like Chinese, by the classifier).16 Just as with complex clausal presuppositions, there is an order in which these presuppositions must be integrated into logical form. In particular, the presupposition introduced by the determiner must be satisfied or accommodated by the NP prior to dealing with the presupposition introduced by the predicate that makes up the nuclear scope of the determiner. This leads to an explanation of the judgements in (4.16). (4.16) a. A water has 12 ounces in it. b. *Much water has 12 ounces in it. Once we have integrated the mass presupposition on the variable bound by the determiner, it can no longer go with the obligatorily count property in the VP. Further evidence for the ordering of presuppositions and their justification comes from looking at complex cases of coercion. Consider (4.17) a. The Chardonnay lasted an hour. b. *Much Chardonnay lasted an hour. The preferred reading of (4.17a) is that a particular quantity of chardonnay participated in some event (presumably drinking) that lasted an hour. We provide the whole noun phrase some sort of event reading only after the determiner presupposition has been integrated into logical form and been justified by the predicate in the restrictor. The fact that lasted an hour requires for successful coercion a quantized event leads to uninterpretability when the bound variable has received the type mass from the determiner. These observations lead to the following entry for a : (4.18) λP: 1λQ: 1λπ ∃x(P(π ∗ arg1P : count)(x) ∧ Q(π)(x)) 16

In this TCL agrees with the “exoskeletal” approach of Borer (2005a,b).

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When building up a λ term for a DP, we will typically get a sequence of type presuppositions on the variable bound by a determiner: those given by the verb will have to be justified jointly with those given by the head NP and by the determiner. It is instructive to see what would happen if, instead of the proposal given here, we restricted the variables P and Q in the case of the determiner a to the type count ⇒ (Π ⇒ t). How could we apply such a determiner to a simple physical property like λxλπ dog(x, π ∗ p)? There is no subtyping relation between count first-order properties and physical properties, and so the application of a count determiner denotation to a physical property is not sound. As we’ll see below, we can use accommodation to adjust type presuppositions so that application may proceed. Roughly, accommodation allows us to take both type presuppositions and take their meet and assume this meet is the type of the argument and the λ bound variable it is supposed to replace in the operation of application. So far so good. But given our general facts about subtyping for functional types, the meet of a count property presupposition and a physical property will not be (count p) ⇒ (Π ⇒ t) but rather: (count  p) ⇒ (Π ⇒ t). This meet makes a wrong prediction about what the variable of a DP like a dog ranges over: the prediction is that it ranges over anything that is an inhabitant of count  p and this includes abstract objects, masses, etc. This does not at all accord with intuitions. But I see no other way of making the alternative way of construing the type presuppositions of determiners work out.17 Thus, a determiner places only weak type presuppositions of the sort I have given here on its property arguments and places the more particular type presuppositions on the bound variable the determiner introduces. The lexical entries chosen determine how presuppositions percolate through the derivation tree, and they predict that presuppositions will be justified typically locally to the argument’s typing context (the π that determines the typing of the argument). For instance, when we combine a determiner and an NP to form a DP that is an argument to another predicate φ, then φ conveys presuppositions to the NP or the DP’s restrictor. If such a presupposition is justified in the restrictor of a DP, it will also be justified in the nuclear scope. The converse, however, does not hold. This leads to a distinction between global justification (justification in the restrictor) versus local justification (justification in the nuclear scope). As we shall see, there is a preference for binding justifications of presuppositions at this “global” level, but this preference can be overridden 17

The same reasoning that I have given here applies to the view of types as partial functions, though I have argued this is not a viable alternative to TCL’s strongly intensional view of types.

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when it is impossible to justify a presupposition at that site.18 Furthermore, notice that just as with clausal presuppositions (see (4.19) where his father is bound to the presupposed content of the proper name John), one presupposition can justify another: the type presuppositions on the nouns car and tree justify, indeed satisfy, the presupposition of the verb hit, which wants a subject and object argument of type p in (4.20). (4.19) John’s son loves his grandfather. (4.20) A car hit a tree. Our understanding of subtyping via deduction or entailment enables us to extend and accordingly and  through the higher types. We can use our general notion of subtyping to derive easily some welcome facts. For one, we can prove that any specific noun is a subtype of noun,19 any specific modifier is a subtype of mod,20 any specific NP is a subtype of np,21 any specific DP is a subtype of dp,22 and so on, which is all to the good.

4.4 The formal system of predication In this section, I introduce the formal system of predication in TCL that exploits the notion of presupposition projection via λ bound parameters and the lexical entries elaborated in the previous section. This includes a composition logic with complex types, a logic that tells us how to compose bits of logical form together, taking account of type presuppositions, to form the logical form for a clause or a sentence. In standard formal semantics, a composition logic is used to build logical forms for clauses or sentences of a natural language. It assumes a syntactic structure for the clause to be composed and basic lexical entries for each word, with perhaps some morphological decomposition. A composition logic 18

19

20 21 22

The way I have set up the lexical entries, together with the preference for global binding, predicts as a default a principle that was stipulated in Asher and Pustejovsky (2006). The principle is the Head Typing Principle: Given a compositional environment X with constituents A and B, such that the logical forms of A and B, A and B have the types: A: α, B: β with α β = ⊥, then if A is the syntactic, lexical head in the environment, then the typing of A must be preserved in any composition rule for A and B producing a type for X  . The Head Typing Principle tells us that the head type in the construction should remain. For example, tree has the type mod ⇒ (p ⇒ (Π ⇒ t)), while noun is of the type mod ⇒ ∃x(x ⇒ (Π ⇒ t)). Assume that the type corresponding to tree is given. Then assuming mod we need to show that ∃x(x ⇒ (Π ⇒ t)), which follows from p ⇒ (Π ⇒ t) by ∃ introduction. This is trivial, since all modifiers are of type mod by definition. Once again this is proved by ∃ introduction. This is trivial, since specific DPs are all of the type 1 ⇒ (Π ⇒ t).

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exploits the rules of the lambda calculus to produce a normalized logical form giving the semantic content of the clause in a logically perspicuous language, usually that of higher-order intensional logic. The semantics of terms in the usual composition logic is just that of higher-order intensional logic, and so the composition process is not isolated as a special system. Since even first-order intensional logic is of course undecidable, we cannot prove that the composition process has any interesting computational properties, unless we separate out the composition logic from the general semantic framework. If we separate the compositional logic from its standard intensional, external semantics, and endow it with its internal interpretation, then we can show that the process of constructing logical form does have interesting computational properties. Formulas in our TCL have an external semantics given by the intended interpretation of the formulas of higher-order intensional logic, but these formulas also have an internal semantics via their types. This semantics is used to guide and to verify the composition process. A predication is well formed iff it has a normalization, and two formulas are equivalent just in case they reduce to the same normal form. The categorial interpretation of the types provides the internal semantics for our formulas. This internal semantics does not of course determine the external semantics, but it does capture certain general features of the external semantics—for instance, the aspectual nature of nouns that have a • type. We will check that the rules of TCL are sound with respect to the categorial interpretation. Because TCL concerns only type assignments, shifts in logical form due to type assignments and β reductions that are guided by a syntactic tree, the logic will turn out to have some interesting properties, which I’ll examine in chapter 10. Our lexical entries furnish the terms of TCL. These terms are formulas of λ2 , the higher-order typed lambda calculus. The rules of TCL are designed to deliver normalizations for logical forms for clauses or sentences, given a syntactic parse of the clause or sentence. Thus, TCL derivations are a quite restricted subset of all possible derivations of λ2 . The syntactic parse is a tree structure and dictates how the normalization procedure should go and also dictates what should be the types associated with the terms of the leaves: we begin with the leaves and then compute the result of combining lower nodes. The close correspondence between syntactic categories and semantic types, which has been part of semantics since the beginning of Montague Grammar and is part of compositional interpretation, means that normalization will succeed unless an unresolvable type clash occurs during application. TCL rules divide into those that deal with types generally, the “simple rules of TCL,” and those rules that are specific to handling presuppositions. The first simple rule for TCL is Application and its dual Abstraction. I have adopted

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a slightly more permissive version of application than was introduced in the first chapter where the argument and λ bound variable were required to be type identical. Instead I have allowed, given our complex system of types, a version in which one can do Application if the argument is a subtype of the λ bound variable or vice versa. This makes sense in terms of presupposition theory: if one can deduce the presupposition from what is given in the context, then this is a classic case of binding, and there should be no adjustment necessary to the presupposition or to the context to which it is applied. φ( xt ) is the result of replacing every occurence of x in φ with an occurence of t. Below I state Application by applying λ term to a term with a type assignment— t: τ means that t has type τ. (4.21) Application λx: αφ[t: γ], λxφ: α ⇒ β, γ α φ( xt ): β (4.22) Abstraction x: α, t: β λxt: α ⇒ β Application is a slight generalization of Modus Ponens and incorporates as well the principle of strengthening of the antecedent; it corresponds to the inference valid in classical, intuitionist ,and linear logic:23 φ ψ ψ → χ, φ χ In addition to these two rules, it is convienient to add the derived rule of Substitution. Let −→β denote the reduction relation between formulas using the rules of TCL. Then: (4.23) Substitution t −→β t 

φ −→β φ( tt ) I turn now to the rules for presuppositions. The rule Binding Presuppositions provides an account of binding with respect to lexical type presuppositions encoded in π. Suppose our context π contains the type α that is supposed to be placed on some argument position i of a predicate but that argument position already has the type γ assigned “locally” and γ α.24 It is clear that 23 24

In effect, I have incorporated here a version of subtype coercion. See Luo (1999). In practice, the local type assignment will always be the last on the string of type assignments appended to the local presuppositional parameter, or at least outside those typing requirements imposed by other terms.

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the local typing on the argument position satisfies the type requirement α. At this point the type presupposition α, like a bound, clausal presupposition, just “disappears”. In the rule below, I assume that the type assignments on argiP can either occur within a concatenated sequence or in separate parts of logical form. (4.24) Binding Presuppositions γ α, argiP : α, argiP : γ argiP : γ Sometimes presuppositions can’t be bound. With clausal presuppositions, when a presupposition cannot be satisfied, its content is accommodated or added to a place in the discourse structure if it is consistent to do so. Something similar holds in TCL for type presuppositions. Suppose that an argument position i for some predicate P has type α but there is a presupposition imposed on it from another term that it have type β. In this case, the local presupposition parameter π will look like this: π ∗ argiP : α ∗ argiP : β. If α β  ⊥, then the type presupposition will simply be added to the typing of t. This is what the rule of Simple Type Accommodation states: (4.25) Simple Type Accommodation

α β  ⊥, argiP : α ∗ argiP : β argiP : α β Binding is a special case of Simple Type Accommodation (since α β = α, if α β). There is a general consensus that clausal presuppositions should be bound locally and accommodated typically globally (i.e., at a level that outscopes all proffered content). Simple Type Accommodation, however, is very close to binding, and so we will treat both here locally. In general, type presuppositions are to be justified locally; only when this fails must the justification take place at some other place in the derivation tree. Simple Type Accommodation makes sense of the interactions of determiner type presuppositions and the type presuppositions of the NPs that go in for the restrictor arguments. Consider (4.14), for instance, repeated below. (4.14) I’ll have a Chardonnay. Let’s suppose that Chardonnay has the type presupposition wine.25 That is, I 25

Actually Chardonnay will most likely have a complex type that allows for the following sorts of copredications:

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will suppose that if someone tries to apply the word Chardonnay to something like a rock or even a liquid like milk, he simply has failed to make a predication (and hasn’t understood the meaning of the word). Thus Chardonnay has the lexical entry λPλπ P(π ∗ wine)(λxλπ chardonnay(x, π )). In integrating this with the trivial modifier, and then the resulting first-order property expression with the determiner a and using Simple Type Accommodation to take care of the type presuppositions, I predict the variable introduced by the DP a Chardonnay to have the type of wine count portion of wine, which is, intuitively, what is desired—the noun phrase is a property of properties of quantities or portions of wine. The same applies mutatis mutandis to other interactions between the type presuppositions of determiners and the presuppositions of their arguments. Let’s now turn to a derivation for a simple sentence like (4.27) to illustrate our simple rules. (4.27) A car hit a tree. I begin with the determiner entry which is a count determiner. From here on out I will abbreviate the type count to ct. (4.28) a. λP: 1λQ: 1λπ1 ∃v: e (P(π1 ∗ arg1P : ct)(v) ∧ Q(π1 )(v)) b. Here again is the entry for tree: : p)(w)(λvλπ tree(v, π )) λP: mod λw: e λπ P(π ∗ argtree 1 We must combine tree and the trivial modifier, λP: 1λx1 : eλπ1 P(π1 )(x1 ), using Application and Substitution several times (I will put the argument to which Application is about to apply in square brackets for readability): (4.29) b. c. d. e. f. g.

  a. λPλwλπ P(π ∗ argtree 1 : p)(w)(λvλπ tree(v, π ))[λPλx1 λπ1 P(π1 )(x1 )] λwλπλPλx1 λπ1 P(π1 )(x1 )(π ∗ argtree : p)(w)[λvλπ tree(v, π )] 1   λwλπλx1 λπ1 λvλπ tree(v, π )(π1 )[x1 ](π ∗ argtree 1 : p)(w) λwλπλx1 λπ1 λπ tree(x1 , π )[π1 ](π ∗ argtree : p)(w) 1 λwλπλx1 λπ1 tree(x1 , π1 )(π ∗ argtree 1 : p)[w] λwλπλπ1 tree(w, π1 )[π ∗ argtree 1 : p] λwλπ tree(w, π ∗ argtree : p) 1

Combining the result of that reduction with the determiner, another use of Application yields a formula for the DP (since type(λw: pλπ2 tree(w, π2 )) 1): (4.26) Chardonnay has enjoyed international acceptance due to its versatility both in growing and in wine production, and is, for many casual drinkers, the quintessential white wine. (slightly adapted from http://www.wisegeek.com/what-is-chardonnay.htm). But I will abstract away from the issue of complex types here.

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tree : ct)[v] ∧ (4.30) λQλπ1 ∃v (λwλπ2 tree(w, π2 ∗ argtree 1 : p)(π1 ∗ arg1 Q(π1 )(v))

Two more uses of Application and Substitution get us: : ct ∗ argtree (4.31) λQλπ1 ∃v (tree(v, π1 ∗ argtree 1 1 : p) ∧ Q(π1 )(v)) At this point we can use Simple Type Accommodation and resolve our presuppositions on the variable v as well as on the variable w (noting that p ct  ⊥). : p ct) ∧ Q(π1 )(v)) (4.32) λQλπ1 ∃v (tree(v, π1 ∗ argtree 1 Let’s now build the VP. In adding the verb’s contribution to the VP, I do not specify the types of x and y below as they can be inferred from the typing on the first and second arguments of hit.   hit   (4.33) λΦλΨλπ Ψ(π∗arghit 1 : p)(λxλπ Φ(π ∗arg2 : p)(λy1 λπ hit(x, y1 , π ))) tree [λQλπ1 ∃v (tree(v, π1 ∗ arg1 : p ct) ∧ Q(π1 )(v))]

That is, the verb hit requires that its arguments be of physical object type. But first let us combine the verb’s contribution with the DP logical form in square brackets. Since Φ is of type dp, we can use Application:  tree (4.34) λΨλπ Ψ(π ∗ arghit 1 : p)(λxλπ λQλπ1 ∃v (tree(v, π1 ∗ arg1 : p ct)  hit   ∧Q(π1 )(v))(π ∗ arg2 : p)[λy1 λπ hit(x, y1 , π )])

We use Application and Substitution to eliminate the higher-order variable Q (which is of type 1):  tree (4.35) λΨλπ Ψ(π ∗ arghit 1 : p)(λxλπ λπ1 ∃v (tree(v, π1 ∗ arg1 : p ct)    hit ∧λy1 λπ hit(x, y1 , π )[π1 ][v])[π ∗ arg2 : p])

(4.35) now reduces via 3 instances of Application and Substitution (using the bracketed arguments) to:   hit tree : p ct) (4.36) λΨλπΨ(π ∗ arghit 1 : p)(λxλπ ∃v (tree(v, π ∗ arg2 : p ∗ arg1  hit ∧ hit(x, v, π ∗ arg2 : p)))

We notice now that the second argument of hit and the first argument of tree are the same (the variable v). So we can now reduce this formula via an instance of Binding and Substitution to:   tree (4.37) λΨλπΨ(π ∗ arghit 1 : p)(λxλπ ∃v (tree(v, π ∗ arg1 : p ct)  ∧ hit(x, v, π )))

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Let’s now continue the derivation by constructing a logical form for the subject DP. (4.38) a. λP: 1λQ: 1λπ∃y(P(π ∗ arg1P : ct)(y) ∧ Q(π)(y)) [λuλπ car(u, π ∗ argcar 1 : p)] b. which reduces to: λQλπ∃y (car(y, π ∗ argcar 1 : p ct) ∧ Q(π)(y)) We now integrate the subject DP’s logical form with that of the VP in (4.36c) to get:   tree (4.39) a. λΨλπ Ψ(π ∗ arghit 1 : p)(λxλπ ∃v (tree(v, π ∗ arg1 : p ct)    car ∧ hit(x, v, π )))[λQλπ ∃y (car(y, π ∗ arg1 : p ct) ∧ Q(π )(y))]

b. Application yields:  hit λπλQλπ ∃y (car(y, π ∗ argcar 1 : p ct) ∧ Q(π )(y))(π ∗ arg1 : p)   tree  [λxλπ ∃v (tree(v, π ∗ arg1 : p ct) ∧ hit(x, v, π ))] c. Application and Substitution yield:   λπλπ ∃y (car(y, π ∗ argcar 1 : p ct) ∧ λxλπ ∃v (tree(v, π ∗   hit argtree 1 : p ct) ∧ hit(x, v, π ))(π )(y))[π ∗ arg1 : p] d. Another use of Application and Substitution yields: car   λπ∃y (car(y, π ∗ arghit 1 : p ∗ arg1 : p ct) ∧ λxλπ ∃v (tree(v, π ∗  hit argtree 1 : p ct) ∧ hit(x, v, π ))[π ∗ arg1 : p][y]) e. Another two uses of Application and Substitution yield: car hit λπ∃y (car(y, π ∗ arghit 1 : p ∗ arg1 : p ct) ∧ ∧∃v (tree(v, π ∗ arg1 : p∗ hit argtree 1 : p ct) ∧ hit(y, v, π ∗ arg1 : p))) We can now check the presuppositions of hit on its subject. Hit requires its first argument to be of type p. This presupposition has to be satisfied by y, which it clearly is. Thus (4.39e) reduces via Binding and Substitution to: car tree : ct (4.40) λπ∃y (car(y, π ∗ argcar 1 : p ct) ∧ ∃v (tree(v, π ∗ arg1 : p ∗ arg1 p∗) ∧ hit(y, v, π)))

Let us now look at an example of adjectival modification involving the intersective adjective heavy. Heavy has the lexical entry in (4.41a). Note that it does not impose special typing requirements on its property argument P or on x:

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(4.41) a. λP: 1 λx: e λπ (P(π )(x) ∧ heavy(x, π ∗ argheavy : p)) b. Applying the lexical entry for tree to (4.41a), we obtain: heavy λzλπλPλxλπ (P(π )(x) ∧ heavy(x, π ∗ arg1 : p)) tree   (π ∗ arg1 : p)(z)(λuλπ tree(u, π )) c. After several uses of Substitution and Application, we get: tree : p∗ λzλπ (tree(z, π ∗ argtree 1 : p) ∧ heavy(z, π ∗ arg1 heavy

arg1 : p)) d. Binding and Substitution reduce the presuppositional contexts: tree : p)) λzλπ (tree(z, π ∗ argtree 1 : p) ∧ heavy(z, π ∗ arg1 In (4.41b), we had two compatible typings on the same variable. When we attempt the same derivation for the modification heavy number, we get an irresolvable type clash in the presuppositions. In particular, we will have the following constraints on the parameter π: heavy

: abstract ∗ arg1 (4.42) π ∗ argnumber 1

: p.

We cannot justify the type presuppositions of the noun and the adjective with any of our rules. TCL thus predicts the attempted modification in heavy number to crash; no well-formed lambda term corresponds to this noun phrase and it has no proof-theoretic interpretation.

4.5 A categorial model for types I turn now to the semantic setting for this calculus. Church’s simple set theoretic model of types is not available to us because of problems with the subtyping relation and the hyperintensional nature of types as proof objects. There are two ways we can proceed with alternative models. One is to follow out the proof theoretic intuition that I laid out in identifying types with concepts. The proof theoretic paradigm has developed into Type Theory thanks largely to the work of Martin-L¨of (1980), and Ranta (2004), as well as Girard (1971). Work by Luo and his collaborators (Luo (1994), Luo and Callaghan (1998)) shows how to analyze the existentially quantified types within that formalism, originally introduced by Strachey in the 1980’s and elaborated in considerable detail by Reynolds (1983). The other way is to examine a more abstract model of proof or computation using category theory. I take a categorial construction to be a model for types and as an abstract way of specifying a certain conception of the proof theoretic, internal semantics for λ terms. But we can also use it to reflect the structure of objects in the type. In this section, I briefly review some of the basics of category theory and show how a proof theoretic

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interpretation is reflected in the categorial model, and I show the soundness of the basic rules for subtyping in the model. For a detailed presentation of the relevant material, see Asperti and Longo (1991)’s very beautiful introduction to category theory or Goldblatt (1979). Those who don’t want to look at these details can skip this section. The category theoretic notion of a closed cartesian category, or CCC, has the resources within which to model the quantifier free types introduced so far and furnishes a connection with the underlying logic of Type Theory (Lambek and Scott (1986)). It does not give a model of the full system of quantified types (Chen and Longo (1995)), nor does it give us by itself a satisfactory analysis of subtyping. Nevertheless, we can begin with CCC’s. To understand what a CCC is we need some definitions. Categories are a collection of objects and maps between objects that are related both to graphs and deductive systems as explained below. Definition 3 A graph consists of a set of objects (points) and a set of arrows between points (edges). In addition, there are two maps from Arrows to Objects, one labeled Domain and the other Co-Domain, which pick out the collections of first arguments and second arguments of arrows respectively. Definition 4 A deductive system D is a graph in which for each object A in D, there is an arrow 1A , which is the identity map on A, and for each pair of arrows f : A → B and g : B → C, g ◦ f : A → C, the composition of g with f , is defined. Objects can be identified with formulas and arrows with proofs. The closure under composition gives us a rule of inference. Definition 5 A category is a deductive system where ∀ f : A → B, g : B → C and h : C → D: • f ◦ 1A = f = 1 B ◦ f • (h ◦ g) ◦ f = h ◦ (g ◦ f ) In category theory one can define certain constructions over objects in a category. Products, for instance, are one such construction. Definition 6 Let A be a category and a, b be objects in A. Then a × b is the categorial product of a and b, which comes with two arrows π1 : a ×b → a and π2 : a ×b → b such that for any arrows f : c → a and g: c → b in A there is a unique h: c → a × b such that h ◦ π2 = g and h ◦ π1 = f . Category theorists typically display a categorial product using the following

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diagram where all the arrows commute: c D f zzzz DDDDg z h DDD zz D"  |o z z / a a × b π2 b π1 In the category of Set, product constructions correspond to Cartesian products. Other constructions of interest in category theory are those of an initial and terminal object. Definition 7 An object 0 is an initial object in a category C iff for all objects c in C, there is a unique arrow f : 0 → c in C. An object 1 is a terminal object in C iff for all objects c in C, there is a unique arrow f : c → 1 in C. Terminal objects are the duals of initials. Duality in category theory reverses the directions of the arrows in a construction. Categorial coproducts are the dual of products and correspond to disjoint union in set theory. They give rise to the following commuting diagram: c z< O bDDD f zzz DDg z h DDD zz D z z / a+b o a π2 b π1 A cartesian category is one that contains a terminal object and that is closed under the product construction. Duality ensures that a cartesian category also contains coproducts. Another notion from category theory that we need in order to give the basic model of types is the notion of an exponential, which corresponds to the function space construction in set theory. Definition 8 Let C be a cartesian category with objects a, b in C. The exponent of a and b is object ba in C together with a map eval: ba × a → b such that for all maps f : c × a → b there exists a unique h: c → ba such that (h × id) ◦ eval = f . Note that h × id: c × a → ba × a. Exponent objects of the form ba have a domain Do(ba ) = a and a co-domain Co(ba ) = b. A cartesian closed category, or CCC, is a cartesian category that is closed under the exponent operation. If the basic objects of a CCC are the basic types, then the exponent objects correspond to the functional types; the initial object corresponds to ⊥ and its dual is the terminal object, though it does not correspond to a type in the type hierarchy. Furthermore, coproduct

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provides us with a natural model of disjunctive types and disjunctive objects. There are many mathematical examples of CCCs. In terms of logic, we have conjunction and arrow corresponding to the type operations of product and exponent. More specifically, we can model both the introduction and elimination rules for ∧ and →. Since a CCC is closed under products, given a and b, we have a × b and b × a, which corresponds to the conjunction introduction rules on formulas that correspond to types: A, B A∧ B. The projection π1 given with a product object corresponds to a conjunction elimination inference rule: A ∧ B A. Similarly the projection π2 yields A ∧ B B . Using these, we can easily show a × b  b × a (where  denotes isomorphism), so conjunction is commutative. Turning to the → rules, suppose ac and bc are exponent objects, then ac × bc is isomorphic to (a × b)c ; this corresponds to the rule on formulas C → A, C → B C → (A ∧ B). The map eval given with exponent objects provides a categorial model of modus ponens, while the unique function h defined for exponent objects provides a model for conditional proof. This suffices to show: Fact 3 The positive implicational fragment of intuitionistic logic has a model in a CCC (e.g., Lambek and Scott (1986)). We can include an interpretation of the type quantifiers within a categorial model once we move to a construction known as a topos, thanks to the work of William Lawvere.26 A topos is a category with a terminal object, a subobject classifier, and closed under pull backs (Crole (1993), Asperti and Longo (1991)). While subobjects are a category theoretic rendition of the familiar set theoretic notion of subset (see Asperti and Longo (1991)), pull backs, or fibre products, are special category theoretic constructions that will prove extremely useful when we come to complex types. Here is their definition. Definition 9 Let C be a category and let a, b, c be objects in C, with morphisms r : a → c, t : b → c. The pull back with respect to r and t, denoted a×c b, is an object in C with two morphisms π1 : a×c b → a and π2 : a×c b → b satisfying t ◦ π2 = r ◦ π1 , such that for any d ∈ C and morphisms f : d → a and g : d → b satisfying t ◦ g = r ◦ f , there exists a unique morphism h: d → a ×c b such that f = π1 ◦ h and g = π2 ◦ h. If a category C admits of pull backs for all arrows r and t of C, then it admits 26

For a very nice explanation of how quantifiers are interpreted in a topos, see Goldblatt (1979).

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of products for all objects a, b of C. Here is a picture of a pull back: d

f h

" a ×c b

π1

'/

a

g

.

π2

  b

r  /c t

It is the structure of the whole pull back with its different maps that tells us about the objects in the type. In the category of set, a pull back forms sets of those pairs in a Cartesian product a × b that project down to the same element in c via the maps given from a to c and from b to c. It thus yields a set of equivalence classes of pairs. Fact 4 Within a topos the standard laws of intuitionistic quantification are sound.27 I will specify a particular kind of topos to model the type system and to provide the internal semantics of λ terms of TCL. This topos, call it T , includes all the basic types in ST, the simple semi-lattice of types provided for the langauge.28 The basic subtype relation of ST is modelled by stipulating that our topos include an arrow from α to β whenever ST contains α β. Given that T is a category, this ensures that the subtyping relation over simple types is transitively closed. A topos is also closed under product, coproduct and pull back as well as their duals, as well as the structures needed to interpret quantification. Thus, the topos T contains all functional types and all restricted quantified types. To extend the basic relation of subtype given by ST to more complex types, I will build a partial order within T using the arrows that model ST. I will use a restriction of the deduction relation given by Fact 4 to capture the subtyping relation.29 Essentially, this restricts the arrows that are used to define the subtyping relation. I associate the logical formula T r(α) with the type α. Principally, I need to eliminate theorems that would cause trouble for the subtyping 27 28 29

For details, see Goldblatt (1979). In subsequent chapters, I shall add a limited number of complex types as elements of the topos. There are in fact many notions of subtype that one can explore in a categorial setting. One might be to use retracts, as defined in Asperti and Longo (1991). But this seems to not give us a sensible notion of deduction, which I take to be important in defining the subtyping relation.

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relation; e.g., since A → A, then using deduction to model subtype without restrictions would give us β α ⇒ α, for any α and β.30 We don’t want that! Let stand for the consequence relation defined in Fact 4, and let Δ and Δ be defined as follows: Definition 10

Restricted deduction: α β ∈ ST T r(α) → T r(β) ∈ Δ Δ, T r(α) T r(β) and Δ  T r(β) T r(α) Δ T r(β)

Since the underlying logic is sound with respect to a topos, then so is Δ , as long as Δ itself is consistent. Definition 11

Subtyping: α β iff α Δ β.

As to the interpretation of and , let a b be the product type of a and b, which models conjunction. Given this interpretation, a b a (and b) and always exists. I will model a  b, when it exists in the type hierarchy, as the disjunctive object a ∨ b or coproduct a + b, and ⊥ as the initial object.  corresponds to the terminal object, but as is evident from the definition of subtyping, we cannot prove for any type α that α , because in the underlying logic

. Hence, I will not take  to be an element of the type hierarchy, though the categorial models all contain terminal objects. Many pleasing facts follow from the definition of subtyping and Δ . First, it is simple to prove a version of cut for Δ : X, T r(α) Δ T r(β), Y, T r(β) Δ T r(γ) X, Y, T r(α) Δ T r(γ) This means that the subtyping relation is transitive. It is also reflexive.31 In view of the internal semantics, proof theoretic equivalence implies identity and so is also anti-symmetric, as desired. Furthermore, because the subtyping relation is now based on a notion of inference within T , we can validate all of our basic rules. Fact 5 Application, Abstraction, and Substitution are all sound in a CCC and hence in the topos T .32 30 31 32

Thanks to Zhaouhui Luo for pointing this out. This follows in Δ because  is not a type. Interpret Application via Eval, Abstraction via the unique function h associated with a exponent object. Substitution is the identity rule.

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The rule from From Subtyping to Logic yields a reasonable logical relation for . In particular, Fact 6 The rule Subtyping for Functional Types is sound in a CCC and hence in T .33 The definition of subtyping also yields a converse of the Subtyping for Functional Types Rule. Fact 7

(a) α ⇒ β α ⇒ β iff α α and β β . (b) α ∨ β α ∨ β iff (α α or α β ) and (β β or β α ) (c) α β α β iff (α α and β β ) or (α β and β α )34

This implies that only functional types of the same -arity are subtypes of other functional types; more generally, is a structure preserving morphism over exponents, products and coproducts in a CCC or T . Fact 8

Simple Accommodation is sound in a CCC and hence in T .35

These facts show the quantifier free fragment of TCL to be sound and make clear the natural correspondence between category theoretic models and the proof theoretic interpretations of Type Theory or the looser application rule systems that I have appealed to in discussing the nature of concepts. We shall see extensions of this correspondence as we move on to consider more complex categorial constructions. Finally, using the constructions in a topos, we can prove: Fact 9 Type theoretic ∃ Introduction and Type theoretic ∃ Exploitation are sound in T .36 If the quantifiers are bounded in such a way as to be restricted to finite collections of types, bounded existential quantifiers reduce to finite disjunctions 33

34

35

36

The rule assumes that a a and b b . Given the intuitionistically valid deduction A ⇒ A, B ⇒ B , A ⇒ B A ⇒ B , since T r(a ) Δ T r(a) and T r(b ) Δ T r(b) by assumption, and assuming A ⇒ A, B ⇒ B  A ⇒ B , we have A ⇒ B Δ A ⇒ B . Consider the case where α and β are simple types and assume that α ⇒ β α ⇒ β . We must have proven α ⇒ β from α ⇒ β and this requires a Δ proof from α to α and from β to β , which can only be given by the subtyping relation in Δ. We can generalize this result by induction over all complex types. Note that we Δ is a restricted fragment of positive intuitionist logic. An even simpler intuitionistically sound argument suffices to prove: a ∨ b c ∨ d iff (a c or a d) and (b c or b d). Similar reasoning proves the last item in this fact. Accommodation says that if a term t must meet two type presuppositions α and β that are compatible, then t must meet their product type. This is guaranteed by the intuitionistically valid law: A → (B → (A ∧ B)). See Goldblatt (1979) and Luo (1994), among others.

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and bounded universals to finite conjunctions corresponding to finite coproducts and products that already exist within a CCC).37 Let me now turn to the internal semantics of TCL formulas. TCL makes use of the standard recursive type assignments from terms to types or objects in the categorial model given for the ordinary typed λ calculus, with type expressions denoting types. However, TCL has a special type π, which is a parameter of all terms derived from the lexicon. We must give some sort of semantic value to this type. π puts type constraints on argument positions; in effect π puts type constraints on the variables that end up occupying those positions. The effect of π semantically, then, is to make the assignment of types to variables not a single valued function from variables to types but a function from variables to a set of types. Thus, π’s semantic value is a function from variables to sets of types. The assignment of types to expressions should ideally obey the following constraint on coherent assignments. Definition 12 Coherent assignment: f is a coherent type assignment to the term λv1 . . . λvn λπφ iff ∀vi ⊥  f (π)(vi ) and f (λv1 . . . λvn λπφ) = β1 ⇒ (β2 ⇒ . . . ⇒ (βn ⇒ ( f (π) ⇒ α) . . .) only if ∀γ ∈ f (π)(vi ) βi γ. I assume that lexical entries are constructed in such a way that there is always a coherent assignment for the λ term for the entry that ensures that the λ term has a definite type. Presupposition type justification reduces the set of types assigned to each variable in f (π) to a singleton set or single valued function from variables to types, if it is successful. Lemma 1 A TCL term t has a type assignment iff there is a single valued type assignment to variables in t.38 Thus, TCL terms have a value in the internal semantics just in case the type presuppositions are all justified in the λ term reduction process. This is exactly what we need to make sense of semantic anomaly and semantic well37

38

To prove the correctness of the two subtyping rules for  finitely bound existentially quantified types, let us suppose that ∃x α A is understood  as β α {Aβ }. Clearly, there is the requisite morphism from each element of the family to β α {Aβ }, since each element is a subtype of itself. This suffices to prove the ∃ introduction rule. For the ∃ elimination rule, if there is a requisite morphism from A( xβ ) into B (this is what subtype guarantees), where β does not occur in B, then  there also exists a relevant morphism from each element of {Aβ } to B and hence from β α {Aβ } to B. Note that this also suffices to show Fact 2 as well as its generalization to all generalized property types. From left to right suppose that t: σ but that there is no single valued assignment to variables in t. Because of the mechanisms of type presupposition justification, this means that for every type assignment f there is some variable vi such that f (vi ) = {α, β} such that α β = ⊥. Computing the type of t recursively shows us that it cannot have a type. The right to left direction is proved by an induction on the complexity of TCL terms, using Fact 5.

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formedness. Since I have shown that the basic rules of TCL are sound in the model, we can now show: Fact 10 Any λ term t with a coherent assignment f in a structure A such that  for all variables vi in t, f (π)(vi )  ⊥, will have a definite type in A. To prove this fact, note that the lexical entries are assumed to have a definite  type by hypothesis. The condition that f (π)(vi )  ⊥ for all variables vi in t ensures that either Binding or Accommodation will reduce the type presuppositions assigned to the π parameters in t to singletons. Lemma 1 now entails the result. Another interesting model is the set of computable functions or computations over a countable domain closed under function space and cartesian product. This is another way of thinking of the proof theoretic model. I’ll call this model F . We can define an interpretation τ of types and of in F recursively: • • • • •

τ(⊥) = the empty function For a simple type α, τ(α) = {(a, 1): for some inhabitant a of α} τ(α ⇒ β) = { f : τ(α) → τ(β)} τ( ) = λx, y ∀ f ∈ y∃g ∈ x g ⊆ f Application is interpreted as the operation of applying a function to an argument. • Abstraction is interpreted as the operation of function construction.

Fact 11

The basic rules of TCL are valid in F .

In this section, I have set out an abstract, categorial model for types as proof objects which abstracts away from the details of the proof system. I’ve also gestured at a more concrete model F . This model ensures the soundness of the rules of TCL presented so far. In the next chapter, I take a look at a particular type constructor that also has a categorial interpretation.

5 The Complex Type •

An important question about TCL remains open from the last chapter: what sort of general type constructors do we need for investigating the meaning shifts evident in coercion, aspect selection, and copredication? The simply typed lambda calculus distinguishes between simple types and functional types, and TCL requires the existence of bounded quantified types. But we need other complex types to model aspect selection and copredications involving aspect selection, which are the subject of this chapter. A theory that invokes complex types has to say how these types get used in the analysis of the phenomena. And to do that, we need to provide a way of exploiting complex types and deciding what information flows from them— i.e., what are their effects on logical form and truth conditions. Underlying this system of type manipulation must be a semantics or model of what such complex types are. Given the nature of the type system and the separation between logical forms and types, two questions need to be answered: (1) How does the semantic conception of such complex types interact with the type hierarchy? And (2) how does information about types interact with logical form? In this chapter I investigate answers to these questions with respect to the complex type •, which has received the most scrutiny in Asher and Pustejovsky (2006) and Cooper (2005).

5.1 A type constructor for dual aspect nouns Do we need a separate type constructor to model the predication of properties to selected aspects, or to model the behavior of dual aspect nouns? I claim that we do, since aspect selection is a distinct form of predication from others. The intuition is that objects like books or lunches have two distinct aspects. Books, for instance, appear to have both a “physical” aspect and an “informational”

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aspect; which is selected in a predication depends on the type restrictions imposed by the predicate on its arguments. Books have a “dual” nature—two conceptualizations, if you will, that are equally “true of” or “faithful to” the object. Book is a paradigm example of a dual aspect noun. In one sense, almost every noun is a dual aspect noun. Consider (5.1) Ducks lay eggs and are common to most of Europe. (individual and kind) (5.2) Snow is common this time of year and all over my back yard. (kind and mass) (5.1) and (5.2) instantiate a general pattern—a copredication that involves predication on a kind as well as a predication on an individual or portion of matter. That is, while predicates like lay eggs apply to individuals, predicates like is common or is extinct apply, not to individuals, but to species or what linguists since Carlson (1977) have called kinds. In English the bare plural construction and the simple definite or definite DPs, a N or the N, where N is any noun, can, given a suitable predicate of which it is an argument, refer to a kind or provide some sort of quantifier over individuals. Note that kinds and individuals are incompatible aspects in the sense that they have different, even incompatible, metaphysical individuation and identity conditions. Besides the kind/individual duality, there are many examples of more specific, dual aspect nouns:1 (5.4) Mary picked up and mastered three books on mathematics. (physical object and informational content) (5.5) Je ne suis qu’un roseau mais je suis un roseau pensant. (Pascal) (physical object and thinking agent) (5.6) That is a lump of bronze but also Bernini’s most famous statue. (portion of matter and artifact) (5.7) Le prix Goncourt is 10000 euros and a great honor not accorded every year. (amount of money and prize and winner-person) (5.8) The lecture (interview, speech) lasted an hour and was very interesting. (event and information) (5.9) The bank (location) is just around the corner and specializes in sub prime loans. (physical object/ location and institution) 1

These copredications happen at the level of speech acts as well.

(5.3) Could you pass the salt please? Asher and Lascarides (2001) argue that please types its sentential argument as a request, while the syntax of (5.3) types the clause as a question, and this forces the discourse constituent introduced by (5.3) to have a complex type question • request. Asher and Reese (2005) and Reese and Asher (2007) extend this notion to biased questions which are both assertions and questions.

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(5.10) The canvas is immense, and, as an example of a bygone school, interesting to all art lovers. (adapted from an article of the New York Times, March 24, 1894) (physical object and informational object) (5.11) The promise was made but impossible to keep. (speech act (event) and proposition) (5.12) The belief that all non-Christians are immoral is false but persists in many parts of America. (informational content and state) (5.13) Mary’s testimony occurred yesterday and is riddled with factual errors and inconsistencies. (event and information content) (5.14) The very same thought that caused Mary to shudder amused Kim. (event and information content) (5.15) The house contains some lovely furniture and is just around the corner. (physical object and location) (5.16) Most cities that vote democratic passed anti-smoking legislation last year. (population and legislative entity) (5.17) Lunch was delicious but took forever. (food and event) We’ve seen evidence that such dual aspect nouns are ambiguous according to the contradiction test. However, dual aspect nouns and their aspects are certainly distinct from accidentally polysemous nouns like bank which don’t support copredication. Intuitively, dual aspect nouns should have a type in which both constituent types, the types of the aspects, are in some sense present. The copresence of these two types can lead to different individuation criteria for the same objects, as we saw in the last chapter and as we’ll see in the next. Nevertheless, postulating a new complex type for dual aspect nouns is controversial. The traditional story about the data involving dual aspect nouns is that there is a sort of coercion going on. For (5.4a) the story postulates that books are physical objects but that they have associated with them an informational content. Sometimes predicates apply to books properly speaking while some predicates apply to the associated informational content. These cases, many argue, are examples of sense transfer. Why should we need anything else? One objection to this proposal is that sense transfer functions don’t tell us that there’s anything special about lunches, books, and other elements that seem to have two constitutive types at once. Real books have to exist in some physical form (even e-books), but they also have to have some information content. The physical and informational aspects of books bear a very different relation to each other from the weak association that holds between me and a vehicle I am driving. If sense transfer functions capture the latter, they fail to

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explain what’s special about books, lunches, and other objects that I claim are of • type. In general it’s not easy to tell when a term is a dual aspect term. The linguistic test for such terms has two parts: first they must support copredications; secondly, predications that pick out a particular aspect of the objects associated with these terms must be able to affect the way such objects are counted or individuated. This second property of dual aspect terms is not shared by terms in standard coercive contexts. Consider the sentences below involving book, question, and newspaper, all of which are claimed to denote objects with multiple aspects in the literature. (5.18) a. The student mastered every math book in the library. b. The student carried off every math book in the library. (5.19) a. The teacher answered every student’s question. b. The teacher repeated every student’s question. (5.20) a. John bought two newspapers yesterday. (physical object reading) b. Rupert Murdoch bought two newspapers yesterday. (institution reading) The quantification over books in (5.18) is sensitive in one case to its informational aspect, and in the other to its physical aspect. In (5.18a), we simply quantify over all informationally distinct individuals without reference to the instantiations of these informational units; it is entirely unnecessary, for example, for the student to have read every distinct copy of every book in the library. In fact all of the math books in the library could have been mastered without the student’s ever having gone to the library at all—he could have read and mastered them via Google Books. In (5.18b), however, every physical individual must have been taken in order for the sentence to be true. Similar remarks hold for the distinction in (5.19b): one act of answering the same question posed on multiple occasions will count as an answer to each question; this is not the case with the act of repeating the question, however, since this refers to copying each individual speech act rather than providing the informational content of the answer. The same remarks apply to the pair in (5.20). Finally, the ability of almost any noun to refer to a kind or to a set of individuals entails that most nouns pass the quantification test with respect to their kind and individual aspects. Now contrast these examples with cases where coerced type shifts have been postulated, either of the Nunberg variety or the GL kind:

134 (5.21) b. (5.22) b.

The Complex Type • a. Everyone is waiting to go home. Everyone is parked out back. a. John enjoyed many cigarettes last night. John enjoyed no cigarettes last night.

In neither (5.21a) nor (5.21b) do we quantify over cars. (5.21b) says something like everyone came in a car that’s parked out back. And in (5.22a), while the object of enjoy is plausibly an event, it’s the event of smoking many cigarettes; the meaning of (5.22a) is not that John enjoyed many smokings of a cigarette or cigarettes. If coercions changed the denotation of the object noun phrase or DP, we would predict (5.22a) to be true when John enjoyed many smoking events that involved just a few cigarettes. The meaning or denotation of the object noun phrase or DP, and the things it quantifies over, don’t change; rather the argument of the verb becomes some meaning related to the noun phrase. Similar remarks apply to (5.22b). So the logical form associated with the object DP doesn’t shift when the coercion takes place. In contrast, the logical form of the whole noun phrase involving dual aspect nouns does seem to shift when we predicate a property of an aspect of an object. So predications involving aspects and predications involving coercions are not the same thing. One could attempt to use Nunberg’s transfer functions on different terms to simulate this behavior. To handle coercion, transfer functions could apply to arguments of verbs; whereas to handle aspect selection, transfer functions apply to the head noun phrase within the DP. This gets the counting arguments right, but copredication is going to be a problem. By either using the standard denotation or the shifted denotation of the head noun book on magic, we can satisfy the type requirements of one of the coordinated verbs. But we can’t satisfy them both. (5.23) John’s mother mastered and then burned the book on magic. If the use of transfer functions on terms does not look promising, then alternatively we can exploit standard approaches to ambiguity and postulate two senses of book: the basic physical book and “informational books” which have the following logical from: (5.24) i-books: λx: i∃y: p (book(y) ∧ info(y) = x) The ambiguity approach hypothesizes that book is ambiguous between i-book and p-book. Although standard tests for ambiguity indicate sometimes that dual aspect nouns are ambiguous, it would be wrong to treat all ambiguous terms in the same way. We’ve already seen that accidental and logical polysemy differ

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in several linguistically discernible ways—for instance, logically polysemous words pass the copredication and anaphora tests, while accidentally polysemous terms don’t. Furthermore, logical polysemy is not just a matter of association. The association between cars and individuals who drive them is at least as clear as the informational relation between physical objects and informational objects, but we can’t say (5.25) I’m parked out back and am an old Volvo anymore than we can say the bank specializes in IPOs and has eroded because of the recent floods. Nor is the peculiar predicative behavior of dual aspect terms versus other terms that undergo coercion a matter of semantic versus pragmatic ambiguities or type shifting. The Nunberg examples are often taken to be examples of a pragmatic shift, whereas the sort of type shifting or ambiguity postulated for books and lunches are taken to be semantic. Nevertheless, this can’t be the whole story, as there are semantically ambiguous words like bank that function equally badly with coopredication. Metaphysically, there is a big difference between objects that are of • type and objects that are not. Books are equally physical and informational objects, but there is no sense in which I am equally a person and my car. Such big metaphysical differences should be and are reflected in our conceptual scheme, that is, in the system of types. This is one principal motivation for introducing a complex type • for such objects. Let’s examine the ambiguity thesis and the postulated difference between ibooks and p-books a bit further. The distinction between i-books and p-books may seem a bit strange but it corresponds to a distinction between types and tokens. Barbara Partee has suggested (p.c.) that one might handle the quantificational ambiguity seen above with read and carry off by treating the entire phenomenon as an instance of the type/token distinction. According to this suggestion, (5.18a) makes reference to the type while (5.18b) refers to the token. There appear to be several problems with this solution. Simply reducing the above phenomenon to an ambiguity between i-books and p-books or types and tokens does not solve the problem of how the copredication works. In copredications involving the informational and physical aspects of books, we would have to say that one simultaneously accesses a type or i-book as well as a token or p-book. We would still in that case have to have a relational analysis of the token to the type and we would need to complicate our analysis of copredication in the same ways as we are presently envisioning for aspect selection. Furthermore, there are cases where reference seems to be made to more

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objects than are available under a simple type/token analysis. For example, in (5.26b), quantification is over informational tokens which are distinct from the actual physical object tokens that would be available. (5.26) a. John hid every Beethoven 5th Concerto score in the library. b. John mastered every Beethoven 5th Concerto score in the library. Hence, for an object of complex type, if there are type and token interpretations available for each component type of the dot, then the underlying typing is more complex than originally countenanced. I can sharpen this observation. Consider examples of real quantification over kinds. (5.27) a. John has stolen every book there is. b. Frances has grown every wildflower in Texas. While there are (improbable) interpretations exploiting the token reading of the quantified expression in each example above, the type interpretation is more felicitous. Let’s now see what a kind interpretation of the quantifiers in (5.18) would be. The verb carry off in (5.18b) would select for physical instantiations, or tokens, of kinds of books, while master would select for kinds of books. But then we will postulate a separate “book kind” for War and Peace, Anna Karenina, as well as for Pride and Prejudice. This stipulation, according to which we have a kind for each particular informational object, would get us what we want but this notion of kind is quite distinct from what we take to be kinds of books (novels, philosophical tracts, etc.), which group many particular informational objects together. The standard type reading of every book is distinct from the quantification over the particular informational content interpretation of the dot object in sentence (5.18a). Furthermore, making the dual aspect nouns ambiguous with respect to their type leads to real problems in the analysis of copredications. Which type for book do we choose in dealing with (5.4), for instance? Neither one on its own will do. So we must also postulate either sense transfers or ambiguities among the predicates of such objects. For instance, we could make master ambiguous between its normal sense where it requires an object of informational type and one where it takes an object of physical type as in (5.28): (5.28) p-master: λx: pλy∃z: i (master(y, z) ∧ info(x) = z) Alternatively, we could have an informational pick up.2 We now have multiplied ambiguities manyfold in the theory. We have i-books and normal books 2

This sense does indeed seem to exist, but it’s not the one at issue in the example (5.4).

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(or, vice versa, books as basic informational objects and p-books) and we also have normal mastering and p-mastering, or normal pickings-up and i-pickingup. In fact we can’t even choose between these alternatives; the dual nature of books says both are equally important. So we have in fact a “second-order” ambiguity in this analysis. A great many ad hoc decisions have to made now in lexical semantics. Moreover, books and lunches have only two constitutive aspects, but what about cities and other things that have more than two aspects? To be sure these are not “expensive” ambiguities; it’s easy to understand how one sense is related to the other, provided we spell out the functions linking them appropriately. But English and other natural languages don’t seem to exhibit these ambiguities. Worse, the account predicts that intuitively a priori false or odd sentences can be true. Consider, (5.29) John burned a book and then mastered it. Let’s suppose that the pronoun must find an antecedent with an i-book sense because master takes as a direct object something of informational type. This forces us to understand book in the “i-book” sense. Suppose now that burn requires as a direct object something of physical type. To handle the copredication, we must postulate a shift in the type of burn to an i-book sense: John burned some copy of an i-book. The trouble is many physical books may be instantiations of the same i-book. So this theory predicts a reading of (5.29) where John burns one copy of some i-book, and then masters a different one, which is perfectly possible. Perhaps there are other ways of resolving the ambiguities to get the right truth conditions for (5.29), but there are no principled ways of ruling out incorrect readings like the one I’ve just described. The traditional theory has to stipulate that books are ambiguous between physical books and i-books, and it also has to stipulate an ambiguity with respect to predicates of books; and this leads to problems. Thus, I will postulate • types for empirical and conceptual reasons.

5.2 Some not-so-good models of • types Now that I have motivated the need for a new complex type, let us return to the two questions with this chapter began: (1) How does the semantic conception of such complex types interact with the type hierarchy? And (2) how does information about types interact with logical form? Let’s see how these questions play out with the complex type •.

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5.2.1 • types as intersective types There is one model for • types already available to GL. Within the typed feature structure framework, types are modelled as propositions true at nodes; therefore, complex types have no natural interpretation, if the complex typeforming constructor has no apparent propositional operator correlate. The only possible propositional operator correlate is conjunction or (if one thinks of propositions in terms of sets of indices) intersection. This would mean that • types would in effect be intersective or conjunctive types (Coppo and Dezani 1978), with the following axiom governing conjunctive types of the form α β: • Conjunctive Types Axiom: x : α β iff x : α ∧ x : β This model of • adopts the following conjunctive types hypothesis (CTH). Let’s assume, as most work in lexical semantics does, that the type hierarchy forms a semi-lattice in which greatest lower bounds are assured to exist. Then: • Conjunctive Types Hypothesis: α • β := α β = glb{α, β}3 To see how CTH fares, let’s first see how the outlines of an account of copredication might go. Consider once again a classic copredication like (5.30): (5.30) The book is interesting but very heavy to lug around. Without going into too many details at this point, let’s imagine that the coordinated adjectival phrases place a conjunctive type requirement on the subject of the verb phrase—in this case, the subject must be both of type p and of type i. Now if we assume, again without going into details, that the type assigned to book, which is also by hypothesis p ∧ i, percolates up to the complete noun phrase as a whole, then the types match and the copredication succeeds. The problem CTH faces, however, is fundamental and stems from the basic axiom for conjunctive types. In particular, for certain dual aspect nouns that assign a variable x the type α • β, saying that x : α and x : β will entail that x can denote no object, because α and β may be inconsistent types. Two types α and β are inconsistent just in case assigning these types to any term t would make the denotation of t have incompatible properties. But hence according to CTH, α • β = glb{α, β} = ⊥, where ⊥ is the absurd type. If glb{α, β} = ⊥ in virtue of incompatible properties that objects of those types necessarily have, then no such complex type can have any inhabitants; that is, if x: ⊥, then the 3

This is of course how conjunction of types is also modelled in the internal semantics of types developed in chapter 4.

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interpretation of x relative to any points of evaluation is empty, because otherwise this semantic value or denotation of x would have to have incompatible properties that cannot be co-instantiated. ⊥ is a type with no inhabitants. Consider terms of • type like lunch or book. The term lunch supports copredications and is of the complex type that consists of a meal, an object with parts (courses, dishes), and an event, the eating of the meal. Events and physical objects have different, in fact incompatible, properties.4 Objects, from our commonsense point of view, perdure through time while events have a duration; objects are wholly present at each moment in time while events have temporal parts. So there is a strong reason to think that events and physical objects are disjoint types of entities. I’ve also already presented linguistic evidence that events and physical objects are distinct types. But that evidence points to a stronger conclusion—namely, that events and physical objects are incompatible types. Nothing is wholly an event and wholly an object, for they have incompatible essential properties: events have temporal parts, but objects don’t; objects perdure through time, but events do not; the whole event is not present at one of its temporal parts, whereas a ball or a person is fully present at each temporal instance in which it exists. These are straightforward truths of commonsense ontology. Thus, the intersection of the types event and physical object is empty. But if lunches, as I’ve argued, have the type physical object • event, then • is not to be understood as the intersection of two types or their meet, unless we want to say that there are no lunches. Similarly, one can argue that the population of a city is not at all identical to the legislative body that runs the city; a city like Austin has a million inhabitants but its legislative body consists of the mayor and the city council which has eight people in it. More generally, legislative bodies are a strict subset of the citizenry, unless we have a direct democracy where all the inhabitants constitute the legislative body. Books receive the type in TCL of physical-object • informational-object. Is it the case that physical-object informational-object = ⊥? The argument may be a bit harder to make here, but it would appear that informational objects and physical objects also have incompatible individuation properties. For instance, one important thing about informational objects is that they can have multiple concrete instantiations. Individual physical objects cannot have multiple concrete instantiations. This leads to different counting principles for informational objects and physical objects, which, as we shall see later, are at the heart of what Pustejovsky and I called “quantificational puzzles.” Moreover, these modal properties are incompatible and so it would appear that the 4

Though of course some philosophers, so-called four dimensionalists, don’t shrink from identifying all objects with event-like four-dimensional objects that course through time.

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intersection or meet of the two types is empty. Nevertheless, books are in some sense both informational and abstract objects. (5.31) a. Chris burned a controversial book. b. Chris read every book he burned. There are obviously books. So if we assume that books have a dual nature, CTH must be wrong. Intersective types don’t furnish the right model even when the two constituent types in a • type have a glb that is not ⊥, as in for example (5.32a,b). (5.32) a. The apple is red. b. The apple is juicy (is delicious). The apple as a physical object or as a food has a certain part, its skin of which one can make certain predications, viz., about its color, that don’t hold of the entire object. But if we think about types intersectively and as sets of objects, then the intersection of the type skin and food just gives us the skin of the food, and that’s not what tastes delicious or is juicy. So once again a simple analysis of a • type as an intersective or conjunctive type is incorrect. It just seems wrong to model objects of • types as instersections of types. Of course, this is an extensional construal of intersective types and I have assumed an intensional theory. But it stands to reason that such intensional intersective types should determine extensional meets in the type structure. I conclude that whatever • types are, they cannot be conjunctive types.

5.2.2 • types as pair types Another way to think of • types is as a pair or a product of types. We can consider the product construction to issue (as it does for the category of set) in a collection of pairs. • Pair Types Hypothesis: α • β := (α, β) If we think of objects of • type as inhabitants of such a collection of pairs, then each object of complex type would correspond to a pair, consisting of a component of each constituent type. This leads us to consider the Pair Types Hypothesis (PTH). Pair types can account for some of our intuitions about the different individiuation conditions of informational and physical objects.5 Suppose that 5

Thanks to Tim Fernando for this point.

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a. there are exactly two copies of War and Peace, two copies of Ulysses, and six copies of the Bible on a shelf. b. Pat has read War and Peace and Ulysses, and no other book. c. Sandy has read the Bible, and no other book. Now, consider the questions: • • • •

(Q1) How many books are there on the shelf? (Q2) How many books has Pat read? (Q3) How many books has Sandy read? (Q4) Who has read more books, Pat or Sandy?

My guess is that most people would answer: • • • •

ten to (Q1) two to (Q2) one to (Q3) Pat to (Q4)

The pairing approach gives us for the situation 10 pair objects consisting of a physical object represented by a number of the physical copy and an information content represented by the title of the book or an abbreviation thereof: • (1, W&P), (2, W&P), (3, U), (4, U), (5, Bible), (6, Bible), (7, Bible), (8, Bible), (9, Bible), (10, Bible) Given that the second element of these pairs are informational objects, it should be noted that, e.g., Bible stands for the same information content or informational object in the last six pairs. To answer (Q2)–(Q4), what matters are the second elements, the information contents, to counting how many books a person has read. Counting the first components or whole pairs is relevant for answering (Q1), but not for (Q2)–(Q4). Simple copredications are not the problem for this model. To handle copredication in the pair model, we proceed as with conjunctive types, with the difference that instead of requiring a conjunctively typed argument, the conjoined predicates of different types α, β require a pair typed argument (α, β), which is provided by a DP with a dual aspect noun in its head. But this model does have a problem, a big problem, with simple predications that require only one of the constituent types as in (3.44b), repeated below: (3.44b) The book weighs five pounds.

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The predicate weighs five pounds requires that its argument be of physical type, i.e., of type p. As p and i (the type of informational objects) are objects of a category, the product type p × i exists and furnishes a model of the complex type of book. Given categories p and i , the morphisms for the product type are guaranteed to exist. • π1 : p × i −→ p • π2 : p × i −→ i These morphisms appear to be what we need, to adjust the type book so that its type matches up with the type of the predicate in (3.44).6 Indeed, we need such a morphism or something like it to switch the type of the DP to get the right counts and the right answers to the questions I posed two paragraphs ago. It’s not enough just to be able to prove that the variable can have the appropriate type; it has to have the right type in order to quantify over objects of the right domain. Suppose, however, we use the projection π1 to isolate the p component of the book; thus, we retype the variable supplied by the DP the book as of type p. This manoeuvre coerces the term to pick out just one component of a complex object and leads to an analysis for (3.44). But it completely fails to give an adequate account of examples like those in (5.33) or (5.34). The problem is that when we use such a type projection, we end up losing the information that the variable introduced by the book is also of type i, information that we need to handle predications that occur in other clauses or with expressions that are anaphorically dependent upon the DP the book. (5.33) John’s Mom burned the book on magic before he could master it. (5.34) Mary bought a book that contradicts everything G¨odel ever said. It will not do to simply retype the term that had the • type with one of its constituent types. If we shift the book on magic to p so as to make the predication in the main clause of (5.33) succeed, then the anaphoric pronoun in the subordinate clause will not have an antecedent of an appropriate type. By being anaphorically linked to the book on magic, which now picks out an object of type p, it will also have the type p and this will lead to a type clash in the predication in the subordinate clause of (5.33). Alternatively, if we try to coerce the object of master back to an informational object, we get a typing conflict with the typing requirements of burn.7 6 7

Pustejovsky (1995) seems to have this in mind. One might think that an appeal to a sort of bridging inference could be made to rescue the view. That is, one might claim that the pronoun refers to the information content associated with the object of type p. Indeed, this seems plausible. But at the level of types, this will not

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Even worse, given the category theoretic notion of product, assigning book a product type entails that it also has the type both i and the type i. This lands us back with all the problems of intersective types. I conclude that PTH at least under such a construal cannot be right. The crucial insight needed to solve this difficulty is that the projections from complex types to the constituent types go with different terms, not the original term. So in addition to our projections on pair types, we must have function symbols f1 and f2 , which will give us new terms associated with t. We must modify the PTH to include the following axiom concerning these function symbols: • Separate Terms Axiom (STA): t : α • β iff f1 (t) : α ∧ f2 (t) : β With the relevant details now omitted, the analysis of (5.35) The book weighs five pounds and is an interesting story. under PTH and STA would have the following logical form: (5.35 ) ∃!x: p×i (book(x)∧ weighs five pounds( f1 (x))∧ interesting story( f2 (x))), where f1 (x) : π1 (p × i), f2 (x) : π2 (p × i) Unlike simple PTH, PTH + STA provides different terms for the complex object, its physical aspect and its informational aspect. No term has two incompatible types nor does any term have the intersection of the two constituent types of a • type. This approach is certainly an improvement over our previous attempts to understand •, but it complicates PTH; the projections associated with product types cannot be interpreted as providing sound typing information about a term t with a • type but rather provide information about other terms than t itself.

5.2.3 Relations between objects and their aspects Another problem remains with this analysis. STA assumes a functional relationship between an object of complex type and its aspects. This implies that there are always at least as many objects of complex types as aspects. The problem is, how do we count objects of complex types? We have seen that books as objects of complex type can be counted relative to the individuation criterion do. There simply is no way to recover the i type from the physical type projection. This would require that there be a map from arbitrary physical objects to i type objects and that does not exist. One would have to know that what one has is in effect a physical component of an object that has an informational component as well, but the formalism provided by PTH does not guarantee this. My proposal, discussed in the next section, will provide just this information.

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provided by the informational component type. They can also be counted relative to the criterion provided by the physical component type. Consider the relation between physical aspects a and a of a book b considered as a complex • type. It makes sense for us to say in the presence of a and a that: (5.36) a and a are copies of the same book b. We can have many physical copies of the same book (informationally speaking). So the individuation conditions for books in their informational aspect and in their physical aspect are different: if we count books with respect to their informational content, we get one number; if we count books with respect to their physical manifestations, we get a different number. Indeed, such individuation conditions offer yet another sort of argument for why we cannot consider physical object • information as an intersective type, as objects of informational type have different individuation conditions (and hence a different essential property) from objects of physical type. Can we count objects of complex type, where the constituent types provide two distinct criteria of individuation, using both criteria of individuation? The answer is “no,” since this yields absurd results. Consider once again a shelf of books, where there are three copies of the Bible and one copy of Jane Austen’s collected works, which contains Pride and Prejudice, Emma, Mansfield Park, Sense and Sensibility, Persuasion, Northanger Abbey and Lady Susan.8 In answering the question, How many books are there on the shelf? a person might say four counting physical volumes; but she might also answer eight, using the informational type to individuate the domain. Which of these will depend on context, certainly. But taking the pair types hypothesis as an ontological thesis—namely each pair of a distinct physical book and a distinct informational content constitutes a countable object—would yield the crazy count of 10 books. From the current perspective, that’s counting using two different principles of types, determined by two incompatible types. Thus, we can’t count objects of type α • β according to the combination of the principles of counting determined by the types α and β. We can only count according to one coherent principle of counting, one coherent individuation criterion. The cases where individuating criteria conflict calls into question the functionality of aspects presupposed by STA. Suppose we count books relative to the informational criterion. Then, as we’ve seen the relationship between books and physical aspects cannot be functional; one informational book may have several physical copies or aspects. On the other hand, if we consider the Jane Austen case again, then we see that books individuated phyiscally may have 8

Thanks to Julie Hunter and Laure Vieu for giving me this example.

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multiple informational aspects. Once again functionality fails to hold. STA presupposes something that cannot be satisfied in many cases. Further evidence for a nonfunctional relation between aspects and objects of complex type comes from the quantificational puzzle examples, which involve quantification over different aspects. I repeat the illustrative example (5.18) below. (5.18) a. The student mastered every math book in the library. b. The student carried off every math book in the library. The quantification over books in (5.18) is sensitive in one case to its informational aspect, and in the other to its physical aspect. In (5.18a), we simply quantify over all informationally distinct individuals without reference to the instantiations of these informational units. There are, typically, many different copies of certain math books in a library. It is not necessary, for example, for the student to have mastered the books by reading every distinct copy of every math book in the library—or even any physical copy of a book in the library in order for (5.18a) to be true. When we individuate books relative to their information content, there will be no functional relation between the variable ranging over books and the term ranging over the physical aspects of books. These considerations show that the relationship between objects of p • i and the physical books of type p is not a functional one, nor is it a relation of types to tokens. Instead we have two different types involved with two different counting principles, two ways of talking about tokens. STA has to be modified and complicated. In our model of • with product types, we have focussed on the set theoretic implementation of this idea. Like many others who have thought about this subject, I have implicitly assumed a naive isomorphism from the model of types to the ontological structure of objects along lines of figure 5.1. The problem is how to interpret the projection at the level of types from the complex type α • β to its component types α and β. Such projections are basic to the product construction, but we cannot interpret these projections as telling us that if x: α • β, then in virtue of the projections we have x: α or x: β. We must use STA. I have also shown that we cannot “read off” a similar projection over terms, let alone their denotations. The categorial product model of types, especially in its realization in the category of sets, helps us in understanding neither the typing of dual aspect nouns nor the nature of the objects that inhabit these types. However one might want to explicate the notion of pair object, the example of the shelf of books with Jane Austen’s collected works shows that understanding objects of • type as pairs of objects of the constituent types in the set theoretic sense yields absurd results when we attempt to count such

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π1

αO

Types:

x

βk

type

Terms:

type

f1 (t)

s

f2 (t) k

Objects:

o f

α•β

O

type

f1

denotes



π2

O

f2

t

denotes



o k

denotes



o

Figure 5.1 Naive isomorphism from types to objects

objects with the pair model. For according to the standard identity criteria for pairs, two books formalized in terms of their informational and physical aspects as a, b and c, d will be identical just in case a = c and b = d. In the Jane Austen example above, such individuation criteria would yield the unintuitive result that there are 10 books on the shelf.9 Even putting aside the arguments against having a functional relation between objects of complex type and their aspects, there is still the question, what is the interpretation of the pairing function at the level of truth conditional content and what is the analysis of the link between the semantic values of fi (t) and t? How do we spell out concretely the analogues of the morphisms π1 and π2 at the level of metaphysical constitution? We want fi (t) and t in the analysis of (5.30) to refer in some sense to the same book, but PTH + STA give us no clue by themselves. What is it to be an object of a pair type? If we knew the answer to this question, we could make progress on the relation between the values of fi (t) and t, where fi (t) is one of the terms for the projections postulated by STA. It’s not that the object is itself a set or a pair of objects! When I copredicate two properties of a book in (5.30), I refer to just one object, not two. Furthermore, that object is not a set; if it were, it could not be a physical object, which it is! The problem is that it’s in some sense once again both an informational and a physical object; we have not made much headway in understanding this puzzle. One suggestion is that we understand the dependence of the object of simple type on the object of complex type in terms of the parthood relation. In terms of 9

Of course, there is a dependence of identity criteria here: if book a is identical with book b on the physical individuation criterion, then of course they will have the same informational content as well.

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the notation of STA, fi (t) would be construed as a part of the semantic value of f .10 A lunch, for example, is part event and part food; it is the singular fusion of its parts. But normal parts of objects have names and can be referred to. This isn’t true of the inhabitants of • types like lunches. This should lead us to be suspicious of this view. Such suspicions point to a deeper problem. The presence of different individuation conditions for an object of • type like book militates against a simple mereological conception of objects of complex type as being the mereological sum of objects of the constituent types. For on such an account, such objects would have different identity conditions depending on which part one used to determine the criterion of individuation. We can formalize this situation using a relation of identity relative to some type providing a principle of individuation. Consider again two physical copies of the same book (individuated informationally). Letting = p stand for the identity relation relative to an individuation criterion appropriate for physical objects and letting =i stand for the identity relation relative to the criterion appropriate for informational objects, we have: (5.37) a. b1 =i b2 b. b1  p b2 In mereological terms, this implies that b1 and b2 have a common informational part but distinct physical parts. But then by the axioms of mereology, we have in terms of an absolute identity relation (in standard mereology two objects are equal just in case they have exactly the same parts): • b1  b2 This mereological approach predicts that in the example with the collected works of Jane Austen, we get once again the implausible count of 10 distinct objects! Thus, the mereological conception, like the pair conception of objects of complex type, seems fatally flawed. Another proposed analysis11 construes objects of complex type as collections like groups or orchestras. The latter suggestion certainly seems on the wrong track. Singular nouns that refer to groups can support plural anaphoric reference: (5.38) a. The orchestra got ready. Then they started to play the Bach suite. b. The battalion was in trouble. They were receiving heavy fire from the enemy. They called in for tactical air support. 10 11

Robin Cooper (p.c.), for instance, claims that lunches are composite objects with an event component and a food component. Also due to Robin Cooper, p.c.

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But plural anaphoric reference with lunch is crazy: (5.39) The lunchi for Chris Peacocke was very nice. Theyi pleased him too. Arguably, nice and please can take both events and meals as arguments, so we should be able to predicate this jointly of both the event and the meal. But the plural anaphora is semantically uninterpretable when the coreference is stipulated as above. Since a singular use of lunch cannot support plural anaphora, this strongly suggests that it does not refer to a plurality, in the way orchestra, battalion, and team arguably do. The alternative is to claim that an inhabitant of a • type is single but composite. Perhaps one can just bite the bullet about the counting argument given above. But mereological talk seems distinctly out of place when trying to make sense of objects like lunches. For ordinary objects like lunches, we have something like Benaceraff’s problem: an analysis of such objects in terms of a theoretical apparatus like mereology just seems wrong. (5.40) Part of the lunch is an event and part of the lunch is a meal. We readily make sense of a parthood relation among objects of the same type. Physical objects have physical parts; an apple has various physical parts, the core, the skin, the flesh, and so on. At least some events also have readily identifiable event parts. Such homogenous objects easily have causal interactions as well. A much vaguer notion of parthood must be invoked to explain the inhabitants of • objects on the mereological view. Unrestricted mereological composition aside, we normally do not think of objects as having parts of different types. Substance dualist positions concerning persons are an example of such, but much of the unintuitiveness of this view comes precisely from our inability to have a clear conception of how the various parts interact causally. Furthermore, in order to accommodate a mereological conception of the inhabitants of • types, we would have to suppose substance multi-ism. Not only would we have to worry about minds and bodies as being parts of the same person; we would have to worry about events and foodstuffs being parts of the same object, information and paper as being part of the same object, and so on. Dualism would be simple in comparison to the metaphysical view being proposed. In each case we would have to elaborate some sort of special causal or other relation telling us how changes in one part might affect another. But this seems crazy for inhabitants of • types. When I tear pages out of a book, an alteration in the physical part doesn’t cause a change in the informational part; my action changes both the physical and the information content together; it’s not that there are two parts—there just one object, the book, with two aspects. The part–whole model, at least where we understand such talk in its physical,

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spatio-temporal sense, is not the appropriate one for lunches or other objects of • type.12

5.3 The relational interpretation of • types In the previous section, we’ve seen problems for the PTH + STA hypothesis. We need to do away with the functio