Anaphora and Type Logical Grammar (Trends in Logic, Volume 24)

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ANAPHORA AND TYPE LOGICAL GRAMMAR

TRENDS IN LOGIC Studia Logica Library

VOLUME 24 Managing Editor Ryszard Wójcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Vincent F. Hendricks, Department of Philosophy and Science Studies, Roskilde University, Denmark Daniele Mundici, Department of Mathematics “Ulisse Dini”, University of Florence, Italy Ewa Or á owska, National Institute of Telecommunications, Warsaw, Poland Krister Segerberg, Department of Philosophy, Uppsala University, Sweden Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany

SCOPE OF THE SERIES

Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica – that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time.

Volume Editor Heinrich Wansing

The titles published in this series are listed at the end of this volume.

ANAPHORA AND TYPE LOGICAL GRAMMAR by

GERHARD JÄGER University of Bielefeld, Germany

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-3904-2 (HB) 978-1-4020-3904-1 (HB) 1-4020-3905-0 (e-book) 978-1-4020-3905-8 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springeronline.com

Printed on acid-free paper

All Rights Reserved © 2005 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

Contents

List of Tables Preface Acknowledgments

vii ix xiii

1. TYPE LOGICAL GRAMMAR: THE FRAMEWORK

1

1

Basic Categorial Grammar

1

2

Combinators and Type Logical Grammar

17

3

Historical and Bibliographical Remarks

65

2. THE PROBLEM OF ANAPHORA

69

1

Anaphora and Semantic Resource Sensitivity

69

2

Variables in TLG

72

3

Previous Categorial Approaches to Anaphora

76

4

Summary

116

3. LAMBEK CALCULUS WITH LIMITED CONTRACTION

119

1

The Agenda

119

2

Contraction?

120

3

The Logic LLC

121

4

Relation to Jacobson’s System

153

4. PRONOUNS AND QUANTIFICATION

157

1

Basic Cases

157

2

Binding by wh -operators

158

3

Binding by Quantifiers

159

4

Weak Crossover

169

5

Precedence Versus c-command

169

v

vi


6

Backward Binding and Reconstruction

174

5. VERB PHRASE ELLIPSIS 1 Introduction 2 VPE: The Basic Idea 3 Interaction with Pronominal Anaphora 4 Interaction of VPE and Quantification 5 VPE and Polymorphism 6 Parallelism Versus Source Ambiguity

183 183 186 187 195 201 206

6. INDEFINITES 1 Introduction 2 Dekker’s Predicate Logic with Anaphora 3 Bringing PLA into TLG 4 Donkey sentences 5 Indefinites and Scope 6 Sluicing 7 Summary and Desiderata

213 213 215 220 228 245 258 269

References

273

Index

283

List of Tables

1.1 1.2 2.1

The structural hierarchy Substructural Curry-Howard correspondences Categorial approaches to anaphora

vii

30 39 118

Preface

This book discusses how Type Logical Grammar can be modified in such a way that a systematic treatment of anaphora phenomena becomes possible without giving up the general architecture of this framework. By Type Logical Grammar, I mean the version of Categorial Grammar that arose out of the work of Lambek, 1958 and Lambek, 1961. There Categorial types are analyzed as formulae of a logical calculus. In particular, the Categorial slashes are interpreted as forms of constructive implication in the sense of Intuitionistic Logic. Such a theory of grammar is per se attractive for a formal linguist who is interested in the interplay between formal logic and the structure of language. What makes Lambek style Categorial Grammar even more exciting is the fact that (as van Benthem, 1983 points out) the Curry-Howard correspondence—a central part of mathematical proof theory which establishes a deep connection between constructive logics and the λ-calculus—supplies the type logical syntax with an extremely elegant and independently motivated interface to model-theoretic semantics. Prima facie, anaphora does not fit very well into the Categorial picture of the syntax-semantics interface. The Curry-Howard based composition of meaning operates in a local way, and meaning assembly is linear, i.e., every piece of lexical meaning is used exactly once. Anaphora, on the other hand, is in principle unbounded, and it involves by definition the multiple use of certain semantic resources. The latter problem has been tackled by several Categorial grammarians by assuming sufficiently complex lexical meanings for anaphoric expressions, but the locality problem is not easy to solve in a purely lexical way. The main purpose of this book is to develop an extension of Lambek style Type Logical Grammar that overcomes these difficulties and handles anaphora in a systematic fashion. The linguistic applications of the theoretical framework that is developed here focus on three classes of

ix

x


anaphora that are well-studied and well-understood as far as the empirical generalizations go. First and foremost, I will discuss the grammar of anaphoric third person singular pronouns, as illustrated by the following example. (1)

a. Johni invented a problem that hei could not solve. b. [Every student]i invented a problem that hei could not solve.

The second empirical domain that we are going to look at is verb phrase ellipsis, i.e., constructions like (2). (2)

a. John revised his paper, and Bill did too. b. John is happy with his job, but Bill isn’t.

As is well-known, VP ellipsis interacts with pronominal anaphora and quantification in complex ways. The logic of anaphora resolution that I will propose lends itself readily to a simple theory of this kind of ellipsis which covers the basic facts in an empirically adequate way. Finally, I will discuss a third class of anaphora, a version of ellipsis that has been called “sluicing” in the literature. Thereby I mean constructions in which a bare wh-phrase is interpreted as a (direct or indirect) question, as illustrated in (3). (3)

a. She’s reading something, but I can’t imagine what. b. A: She’s reading something. B: What?

The main goal of this work is not so much to develop a novel descriptive theory of anaphora but rather to demonstrate that anaphora can be integrated into Type Logical Grammar without giving up the attractive design of this theory of grammar. Nonetheless, the empirical predictions that we end up with do not always coincide with those of competing analyses, and I (naturally) try to argue that my analysis also gets the facts right in these cases. So the discussion might be interesting for non-Categorial grammarians who are interested in the analysis of anaphora as well. Also, the type logical analysis of donkey anaphora led to a partially novel account of the grammar of indefinites that diverges from established theories in several respects, both theoretically and empirically. The book does not expect prior knowledge of Categorial Grammar or any acquaintance with proof theory that goes beyond the level of some introductory logics course. I do assume though a working knowledge of set theory, first order logic and the typed λ-calculus. The technical level of the book should be easily accessible to anybody who has mastered

Preface

xi

some standard textbook on formal linguistics like Dowty et al., 1981 or Gamut, 1991. The structure of the book is as follows. Chapter 1 gives a selfcontained introduction to the framework of Type Logical Grammar. It does not make any reference to the issue of anaphora whatsoever. It can be used as an introductory text on its own. Readers that are already familiar with TLG, on the other hand, can safely skip this chapter. Chapter 2 discusses previous Categorial approaches to (pronominal) anaphora. This chapter too can be read on its own (or in combination with Chapter 1). The remainder of the book does not build on it in any significant way, so this chaper is not essential for understanding of the subsequent material. Chapter 3 is the core of the book. There I develop the novel type logical machinery that enables us to analyze anaphora resolution. In this chapter I focus on the proof theoretic properties of the resulting type logical calculus, i.e., I present the calculus in different proof theoretic formats, establish their equivalence, and prove essential meta-logical properties like Cut elimination, decidability, finite reading property, strong normalization, and completeness. The remaining chapters apply these theoretical tools to the mentioned empirical areas. Chapter 4 focuses on anaphoric pronouns and their interaction with quantification. Chapter 5 discusses two options for a type logical treatment of verb phrase ellipsis. In Chapter 6 I propose another extension of the underlying formalism to accommodate certain peculiarities of indefinite NPs. This has an impact on the issue of anaphora for two reasons. First, I believe that the problem of donkey anaphora is mainly a problem of indefiniteness, less a problem of anaphora. So an adequate treatment of donkey pronouns requires a theory of indefiniteness. Furthermore, sluicing is a form of ellipsis that interacts closely with the grammar of indefinites. I discuss these anaphora related aspects of indefiniteness, and I also consider some empirical issues pertaining to the grammar of indefinites as such, namely their peculiar scope taking behavior. Now that I have explained what the book is intended to be, a few words about what it is not. The intended audience are mainly formally inclined linguists and computational linguists with an interest in logic. I try to illustrate how a logical grammar can contribute to a both formally precise and empirically comprehensive linguistic theory. It is not my goal to give an introduction into the Lambek calculus and related substructural calculi for logicians. Therefore, issues that are important to logicians but of lesser relevance to the linguistic applications are not covered in great depth. This concerns especially model theory and the

xii


relation of type logics to modal logic and Linear Logic. Likewise, extensions of the Lambek calculus that are interesting from a logical point of view but without obvious linguistic applications—like negation or additive connectives—are not discussed. Neither could I deal with all facets of Type Logical Grammar that emerged within the last decade. The focus of the book is on the treatment of anaphora. The introductory Chapter 1 gives an overview over the “classical” version of Type Logical Grammar, but various new developments that have no immediate connection to anaphora are left out. This concerns especially non-associative Categorial calculi, multimodal extions of TLG, and the calculus of proof nets.

Acknowledgments

Most ideas that I present in this book were developed while I was a postdoc at the Institute for Research in Cognitive Science of UPenn in Philadelphia 1997 and 1998. It is no exaggeration to say that at that time, the IRCS was one of the best institutes in the world to conduct research on formal grammar. Thanks to Aravind Joshi for making the place what it is! I profited immensely from the discussions with my colleagues there. Many people gave me inspiration and feedback, but I feel the contacts to Robin Clark, Seth Kulick, Jeff Lidz, Mark Steedman and Yael Sharvit were especially important. Last but not least, Dick Oehrle’s occasional visits to Philadelphia were very rewarding. Natasha Kurtonina deserves a special mention. Due to a lucky coincidence, we came to the IRCS at the same time and wound up being office mates. She has an extraordinary gift for explaining things, and most of what I know about the “Logic” in “Type Logical Grammar”, I learned from her. She never tired of pointing out the flaws in my proofs and digging out literature that might be relevant for my research. Last but not least, I thank her for always being a good friend. When I left Philadelphia, my work on anaphora in TLG consisted of a couple of half finished papers and a lot of loose ideas. My time as a visitor at the Utrecht Institute of Linguistics in 2000 and 2001 gave me the opportunity to finally write everything down in a coherent way. It is hard to find a place with a higher concentration of excellent categorial grammarians than the OTS, and this created the right atmosphere to finish this work. Thanks to my Utrecht colleagues, especially to Michael Moortgat, for making the time in Utrecht a pleasant and productive one. While I was finally writing down the manuscript, Cornelia Endriss got me interested into the issue of indefiniteness again. We had a vivid intellectual exchange on this over several months. The last chapter of the book would have taken a different shape without this, and perhaps it

xiii

xiv


would not exist at all. Even though we finally drew different conclusions on what a correct approach to specificity should look like, many of the ideas and observations from this chapter are due to Neli. I had the opportunity to present material from this book at various talks at Berlin, UPenn, the MIT, Utrecht, D¨ usseldorf, Amsterdam and Leiden, and I am grateful for the comments I received at these occasions. I am also indebted to the students of my Categorial Grammar classes in Potsdam and in Utrecht for the feedback they gave me. I owe a lot to Raffaella Bernardi, Christian Ebert, Bryan Jurish, Manfred Krifka, Glyn Morrill, and Willemijn Vermaat for reading previous versions of the manuscript or parts of it, and making numerous helpful suggestions for improvement. All remaining errors are of course mine. Special thanks go to Bryan for spending a lot of effort correcting my English. This book is a revised version of my habilitation thesis that I defended at the Humboldt-University at Berlin in 2002. I would like to thank the committe members Marcus Kracht, Manfred Krifka and Michael Moortgat for their encouragement and support. It was also Marcus who suggested that I submit the manuscript to the “Trends in Logic”. Last but not least I thank the series editor Heinrich Wansing for the very good cooperation during the final preparations for publication, and the two anonymous reviewers for their suggestions and comments.

Chapter 1 TYPE LOGICAL GRAMMAR: THE FRAMEWORK

1. 1.1

Basic Categorial Grammar Informal Introduction

All versions of Categorial Grammar that have been developed in the past 30 years can be traced back to the pioneering work of Bar-Hillel, 1953. His system, despite its obvious limitations, in nuce contains most of the features that make the Categorial approach attractive to the present day. It is thus a natural starting point for a presentation of its more sophisticated descendant, Type Logical Grammar. It rests on a fundamental intuition about the structure of languages (both natural and formal ones) which says that linguistic signs may be complete or incomplete. Under this perspective, grammatical composition can be described as the process of completing incomplete linguistic signs. Basic Categorial Grammar (BCG henceforth) is probably the grammatical framework that expresses this intuition in its purest form. Consider the sentence (1)

Walter snores.

The name Walter has a simple semantic function; it just refers to the individual called “Walter”. We may thus consider the expression Walter to be complete, since its linguistic function does not depend on its linguistic context. Similarly, the sentence Walter snores is complete insofar as its denotation is a proposition with a truth value that depends on the extralinguistic context only. The verb snores, however, is incomplete in a sense. It serves to constitute a proposition, but it needs a subject to do so. Semantically it serves as a function that turns an individual into a proposition.

1

2


Following standard practice, I use the label np for names (and phrases that have a comparable distribution) and s for sentences. The verb snores is thus an incomplete expression that turns an np into an s. Using a notation from Linear Logic (Girard, 1987), this intuition could be expressed by snores : np−◦s In words this says that the expression snores has the category np−◦s. An important piece of information is missing here though. Consider the more complex sentence (2)

Walter knows Kevin.

Here the transitive verb knows is doubly incomplete; it requires two nps to turn it into an s. We may express this with knows : np−◦(np−◦s) However, for an adequate description of the linguistic facts we also need the information that one np occurs to the right and one to the left of the verb. Following Bar-Hillel, 1953, I therefore distinguish between two kinds of incomplete expressions: forward looking functors that have a category of the form A/B (pronounced: “A over B”) and expect the missing piece on their right, and backward looking functors that have a category of the form A\B (“A under B”) and expect the missing piece on their left. A more adequate description of the facts collected so far would thus be Walter, Kevin : np snores : np\s knows : (np\s)/np The derivation of the two example sentences can conveniently be expressed in tree format as s

s

np

np\s

Walter

snores

np\s

np Walter

(np\s)/np

np

knows

Kevin

3

Type Logical Grammar: The Framework

In the sequel, I will refer to the A in the types A/B and B\A as the goal category and to the B as the argument category. The rules that are used in this derivation are (where ab is the concatenation of the strings a and b): 1 If a has category A/B and b has category B, then ab has category A. 2 If a has category A and b has category A\B, then ab has category B. These derivation schemes are sometimes called cancellation rules since they bear an obvious analogy to the arithmetic law1 x ×y =x y

Complex categories. It should be added that both the goal category and the argument category of a complex category may be complex themselves. The category of transitive verbs—(np\s)/np—already provides an example. Manner adverbs like faintly illustrate this point further; they combine with an intransitive verb phrase (category np\s) to yield an intransitive verb phrase. Both the argument category and the goal category are complex here. (3)

a. faintly : (np\s)\(np\s) b. Kevin snores faintly s c. np Kevin

np\s np\s

(np\s)\(np\s)

snores

faintly

Recursion. The category of adverbs also illustrates that the argument category and the goal category may be identical. If a BCG assigns such categories to an expression, the described language will display a recursive structure (but note that recursion may be realized in other ways as well). Adjectives are another case in point. They are attached to a common noun phrase (category n) to produce an expression of exactly this category. Figure 1.1 on the following page illustrates this. 1 Bar-Hillel’s system is based on the work of Ajdukiewicz, 1935 where the analogy is even more striking since it does not distinguish between forward looking and backward looking functors.

4

ANAPHORA AND TYPE LOGICAL GRAMMAR s

np np/n The

np\s snores

n n

n/n old

n

n/n old

n

n/n old

Figure 1.1.

n/n

n

old

man

Recursion

Semantic composition. Categories in Categorial Grammar represent two kinds of information. They encode how a sign combines with other signs both syntactically and semantically. Incomplete signs denote functions, and syntactic composition is accompanied by function application in semantics. The structure of the category of a sign is mirrored in the type of the function that it denotes. If the goal category of a sign is complex, its denotation is a curried function (i.e., a function whose values are functions themselves). Categories with complex argument categories correspond to higher order functions—functions that take other functions as arguments. If the semantic component of signs is represented by terms of the typed λ-calculus, syntactic and semantic composition can be displayed simultaneously in a tree structure, as illustrated in Figure 1.2.

1.2

The Formal System

1.2.1 Syntax After this rather informal description of BCG, let us make these intuitions precise. I begin with a formal specification of the notion of category. A BCG comprises finitely many basic categories (also atomic categories). Most linguistic applications make do with very few—the set {s, np, n, pp} is sufficient in many cases, but this is not essential. Complex categories are formed from basic ones by means of the connectives “/” (forward looking slash) and “ \ ” (backward looking slash).

5

Type Logical Grammar: The Framework s faintly’(λx.call’(x, kevin’))walter’

np\s faintly’(λx.call’(x, kevin’))

np walter’ Walter

np\s λx.call’(x, kevin’)

Figure 1.2.

(np\s)/np λyx.call’(x, y)

np kevin’

called

Kevin

(np\s)\(np\s) faintly’ faintly

Semantic composition of Walter called Kevin faintly

Definition 1 (Categories) Let a finite set B of basic categories be given. CAT(B) is the smallest set such that 1 B ⊆ CAT(B) 2 If A, B ∈ CAT(B), then A/B ∈ CAT(B) 3 If A, B ∈ CAT(B), then A\B ∈ CAT(B)

Bracketing convention:. I assume that the forward slash associates to the left, i.e., A/B/C is shorthand for (A/B)/C. The backward slash associates to the right, i.e., A\B\C stands for A\(B\C). Furthermore, forward slash takes precedence over the backward slash; A\B/C means A\(B/C). Like any formal grammar, a BCG consists of a lexical and a syntactic component. Ignoring semantics for a moment, the lexicon of a BCG is a mapping that assigns finitely many categories to each element of some finite set of strings. A lexical unit may have more than one category since linguistic units may be lexically ambiguous (as for instance walk in English, which is both a common noun and an intransitive verb). Definition 2 ((Uninterpreted) Lexicon) Let an alphabet Σ and a finite set B of basic categories be given. A BCG-lexicon LEX is a finite relation between Σ+ (the set of non-empty strings over Σ) and CAT(B).

6


The syntactic component is identical for all BCGs. It consists of a series of axiom schemes and rule schemes that jointly constitute a deductive system. I choose sequent presentation as a convenient format for a description of a deductive system. A sequent consists of a sequence of formulae A1 , . . . , An (of some formal language)—the antecedent, and a single formula B, the succedent. Antecedent and succedent are connected by the deduction symbol ⇒. A1 , . . . , An ⇒ B This sequent expresses that the succedent B can be derived from the antecedent A1 , . . . , An . Applied to BCG, the formulae in the sequents are elements of CAT(B). Trivially, every category A can be derived from itself. This is expressed by the identity axiom scheme id. Here and henceforth, I use letters A, B, C, . . ., possibly augmented with indices, as variables over categories.

A⇒A

id

Furthermore, it is possible to use lemmas in a derivation. In other words, a preliminary result of a derivation can be plugged into another derivation. This is expressed by the Cut rule. Letters X, Y, Z, . . . are variables over (possibly empty) sequences of categories.2 X⇒A

Y, A, Z ⇒ B

Y, X, Z ⇒ B

Cut

Finally, the fraction cancellation schemes informally given above represent valid deductions:

A/B, B ⇒ A

A>

B, B\A ⇒ A

A

t, t\e/t ⇒ e/t

e/t, t ⇒ e

t, t\e/t, t ⇒ e

t/n, n, t\e/t, t ⇒ e | | | | 1 2 = 1 Figure 1.3.

A
Cut

Cut

Sequent derivation of “12 = 1”

sequent X ⇒ A, the succedent A is a subformula of some formula in X. (All axioms have this property, and it is preserved under Cut.) So we can do bottom up proof search by testing for a sequent in question whether (a) it is an instance of an axiom, or (b) it is the conclusion of some instance of Cut, and the two premises are derivable. The premises of a Cut rule always have a lower complexity (consist of fewer symbols) than its conclusion, so this procedure is bound to terminate after finitely many steps. If deduction is decidable, string recognition by a BCG grammar is decidable as well. Since both the lexicon and the set of designated categories is finite, recognition of a given string reduces to derivability of finitely many different sequents.

1.2.2 Relation to Context Free Grammars Basic Categorial Grammars are closely related to Context Free Grammars. These two grammar formats in fact (weakly) recognize the same class of languages. This equivalence was established in the seminal paper Bar-Hillel et al., 1960. I will sketch the basic idea of an equivalence proof. For an in-depth discussion of this and related issues the reader is referred to the excellent overview article Buszkowski, 1997. The notion of a derivation in BCG and CFG (Context Free Grammars) are highly similar to start with. Context free derivations can also be seen as deductions in a deductive system that contains the identity axiom and is closed under Cut. Instead of the two application schemes, however, CFGs have context free rules as additional axioms. To transform a BCG into an equivalent CFG, it is thus sufficient to demonstrate that only finitely many instances of the application schemes are used in actual derivations. These instances can then be reinterpreted as CFG rules. As was mentioned above, in every derivable BCG sequent, the succedent is a subformula of one element of the antecedent of this sequent (since this property holds of all axioms, and it is preserved under Cut).

10


Given this, it is straightforward to see that all categories that occur in a premise of a Cut rule are subformulae of categories that occur in the conclusion of this Cut. This in turn entails that in the derivation of a sequent X ⇒ A, only subformulae of X or of A are used. In particular, all application axioms used in the derivation consist of subformulae of the sequent to be derived. In the derivation of grammatical strings, only subformulae of lexical categories are used in the antecedents and only subformulae of designated categories as succedents. Since there are only finitely many such categories for a given BCG, in fact finitely many instances of application are sufficient. It follows from these considerations that every language that is recognized by a BCG is also recognized by some CFG. The inclusion in the other direction is much harder to establish—the proof is the central point of Bar-Hillel et al., 1960. It is easy to demonstrate though if we use the Greibach Normal Form Theorem (Greibach, 1965):

Theorem 1 (Greibach Normal Form Theorem) Every context free language L is recognized by some CFG G that only contains rules of the form A → aα (where “A” ranges over non-terminals, “a” over terminals, and “α” over (possibly empty) sequences of non-terminals). Let L be some context free language that is recognized by some CFG G in Greibach Normal Form. An equivalent BCG can easily be constructed. We identify the set of non-terminals of G with the set of basic categories. The CFG rules of G are transformed into lexical assignments by going from A→a to a;A and from A → aB1 . . . Bn to a ; A/Bn / · · · /B1 Finally we identify the start symbol of G as the only designated category of the BCG.


11

1.2.3 Semantics Semantic types. Despite this strong similarity between BCG and CFG, the former has at least one conceptual advantage over the latter pertaining to its natural connection to semantic interpretation. As I alluded to in the beginning, the Categorial architecture is actually strongly motivated by semantic considerations. In phrase structure grammars and related formalisms, the syntactic category of a sign on the one hand and its semantic type on the other hand are independent semiotic dimensions that have to be specified separately. In Categorial Grammar, these components are closely linked. In other words, the category of a linguistic sign codes two kinds of information: it determines the (syntactic) combinatory potential of this sign, and it specifies which type of denotation the sign has. Let us make this precise. I assume that the reader is more or less familiar with type theoretic interpretation and summarize the basic notions very briefly. Analogously to basic syntactic categories, there is a finite set of basic semantic types. The set of semantic types is the closure of the basic types under function space formation: Definition 6 (Semantic types) Let BTYPE be a finite set (of basic semantic types). The set TYPE of semantic types is the smallest set such that 1 BTYPE ⊆ TYPE 2 If a, b ∈ TYPE, then a, b ∈ TYPE. In linguistic applications, the basic types usually contain at least the types e (for “entity”) and t (for “truth value), but this is not part of the general format. Semantic types correspond to ontological domains. The semantic type of a sign indicates what kind of object the denotation of this sign will be. The space of semantic domains has a recursive structure analogous to the set of semantic types; a complex type a, b always corresponds to the set of total functions from the domain of type a into the domain of type b.

Definition 7 (Domains) The function Dom is a semantic domain function iff 1 The domain of Dom is TYPE, 2 for all A ∈ TYPE, Dom(A) is a non-empty set, and 3 Dom(a, b) = Dom(b)Dom(a)

12


This definition does not restrict the assignment of domains to basic types (beyond the requirement that domains are non-empty). Following the lead of Montague, 1974, the conventional basic types e and t are usually mapped to some set E of individuals and the set {0, 1} of truth values respectively, but again, this is not part of the general architecture. Given a correspondence between basic categories and semantic types, the semantic type of a sign of an arbitrary category can be predicted from the internal structure of this category. More formally put, the semantic type of a linguistic sign is the homomorphic image of its syntactic category.

Definition 8 (Category to type correspondence) Let τ be a function from CAT(B) to TYPE. τ is a correspondence function iff τ (A\B) = τ (B/A) = τ (A), τ (B) Compositional Interpretation. Semantic types serve a double function. Primarily, they restrict the possible denotations of linguistic signs. If an expression has the syntactic category A, its denotation will be an element of Dom(τ (A)). Semantic types simply identify categories that differ only syntactically but not semantically (like np/np and np\np). For practical purposes, we are rarely concerned with actual meanings (i.e., model theoretic entities) but we deal with meaning representations that are formulated in the language of the typed λ-calculus. So we are mainly interested in a compositional translation from natural language into the semantic representation language. Since the denotation of λ-expressions is unambiguous and well-understood, such a translation indirectly determines a compositional interpretation of the object language. Semantic types not only determine the range of possible interpretations of a linguistic sign, they also determine the syntactic properties of its translation. The semantic types that we use are the syntactic categories of the semantic representation language. The category-to-type correspondence restricts possible translations from natural language to the λ-calculus. The translation—and thus indirectly the meaning—of a basic expression is determined in the lexicon. So a Categorial lexicon for an interpreted language is a three place relation, relating the form of an expression, its syntactic category, and its translation into the λ-calculus. I revise the definition of a lexicon accordingly. EXPa is the set of expressions of the typed λ-calculus that have type a. Definition 9 ((Interpreted) Lexicon) Let an alphabet Σ, a finite set B of basic categories and a correspondence

13


function τ be given. An interpreted BCG-lexicon LEX is a finite subrelation of (Σ+ × {A} × EXPτ (A) ) A∈CAT(B)

The lexically determined form-category-meaning relation can be extended to all constituents that are recognized by the corresponding BCG. To this end, the axioms and rules of BCG have to be supplied with operations on the meanings of their operands (or, more precisely, with operations on their semantic representations). These are conveniently represented by means of labeled deduction. Antecedent formulae of sequents are labeled with variables and succedents with possibly complex terms of the λ-calculus. The actual translation of a constituent is obtained by replacing the free variables in the succedent term with the corresponding lexical translations of the formulae in the antecedent. The labeled BCG rules are given below. I write l : C for category C carrying the label l. First some remarks on notation are in order. I use lower case letters x, y, z, . . . as metavariables over variables of the λ-calculus and upper case letters M, N, O, . . . as metavariables over λterms. “N [M/x]” is the result of replacing all free occurrences of x in N by M . Variables (free or bound) that occur in different sequents are tacitly assumed to be different, so no variable clashes can arise. Finally, I omit brackets for function application; so the term “M N ” is the result of applying the functor M to the argument N (which is sometimes also written as “M (N )”). x:A⇒x:A

X⇒M :A

id

x : A/B, y : B ⇒ xy : A

Y, x : A, Z ⇒ N : B

Y, X, Z ⇒ N [M/x] : B A>

x : B, y : B\A ⇒ yx : A

Cut

A

A

to talk talk’ np\s

ask’b’ (np\s)/(np\s)

j’ np

f aintly

lex

faintly’ (np\s)\(np\s) faintly’talk’ np\s

ask’b’(faintly’talk’) np\s

lex A

A

talk’ np\s

ask’b’ (np\s)/(np\s) John

lex

to talk

ask’b’talk’ np\s

j’ np

lex f aintly

A>

faintly’ (np\s)\(np\s)

faintly’(ask’b’talk’) np\s

lex A

The derivation of the Right Node Raising construction (14) requires lifting of the two subjects to the category of quantifiers, followed by function composition of the lifted subjects with the transitive verbs. It is given in Figure 1.8 on page 24.

2.1.3 Left Node Raising Coordination of chunks that a phrase structure grammar (or BCG) would not analyze as constituents is pervasive in natural language. Right Node Raising constructions can informally be described as involving

22


deletion of right peripheral material in the first conjunct. The mirror image pattern exists as well. The following example, where clusters of arguments are conjoined, is a case in point. (16)

John introduced Bill to Sue and Harry to Sally.

Here apparently the sequence John introduced is missing in the second conjunct. A surface compositional derivation is possible in an extended Categorial Grammar if we make use of mirror images of the rules introduced above. Both backward type lifting T< and backward function composition B< are necessary here. Their semantics is identical to their forward oriented twins, and they differ from them just by the directionality of the slashes. (17)

a.

b.

X⇒M :A X ⇒ λx.xM : (B/A)\B X ⇒ M : A\B

T

lex

Figure 1.8.

lex

Derivations of (14) and (16)

A

likes like’ (np\s)/np

λy.like’ymary’ s/np

λy.think’(like’ymary’) (np\s)/np

lex B>

B>

λy.think’(like’ymary’)y np\s

john’ np

himself λRx.Rxx ((np\s)/np)\np\s

lex A

lex

know’ (np\s)/np λyz.know’(yz) (np\s)|np/np|np

G>

λz.know’z (np\s)|np

λv.know’vmary’ s|np Figure 2.12.

him λx.x np|np

lex A>

A>

Derivation for (41)

Binding of pronouns is achieved by identifying the anaphora slot that originates from the lexical entry of the pronoun with some np argument slot of a superordinate functor. This is implemented by means of the combinator Z. Since it operates on two-place functors, it comes in four directional variants.14 (45)

a.

b.

c.

14 Jacobson

X ⇒ M : A/B/C X ⇒ λxy.M (xy)y : A/B/C|B

Z> >

X ⇒ M : (B\A)/C X ⇒ λxy.M (xy)y : (B\A)/C|B X ⇒ M : C\A/B X ⇒ λxy.M (xy)y : C|B\A/B

Z< >

Z>

his mother mother’ np|np

λz.love’(mother’z)z np\s

lex A>

A>

every’man’(λz.love’(mother’z)z) s

In Jacobson’s “official” theory the formulation of Z is somewhat more complicated, but I skip over this here for ease of exposition. The purpose of the G-combinators is to pass unbound anaphora slots from subconstituents to superconstituents. As I have presented G up to now, this will only work for one single slot, but of course a 15 A

remark on terminology: I use the term “Skolem function” as synonymous to “function of type e, e” throughout this book, regardless whether or not an operation of “Skolemization” is involved.

103

The Problem of Anaphora

constituent may contain more than one unbound pronoun. Therefore a generalization of G is required as well. Jacobson assumes that there are infinitely many instances of G that are defined recursively. The definition given above represents the base case. The recursive rule takes the form of the following monotonicity rule (Jacobson assumes that the input to this inference scheme has to be obtained by applications of G> , G< , and G∗ only. I ignore this aspect for simplicity.) (47)

x:A⇒M :B y : A|C ⇒ λz.M [(yz)/x] : B|C

G∗

Written in tree format, this rule amounts to a form of hypothetical reasoning. To derive a conclusion B|C from a premise A|C, assume some hypothesis of type A, try to derive B from it, and discharge the hypothesis. The general scheme is given in Figure 2.13. y : A|C

1

yz : A .. . M :B λzM : B|C Figure 2.13.

G∗ , 1

G∗ in tree format

I illustrate the application of G∗ in example (48) below. To summarize this mechanism, suppose the argument in a functorargument structure contains an anaphora slot. Then either of two options apply: 1 The functor undergoes some version of G and the anaphora slot is thus projected to the superconstituent (as illustrated in Figure 2.12 on page 101). 2 The functor undergoes Z prior to applying it to its argument. As net effect, the anaphora slot in the argument is bound by some superordinate syntactic argument place of the functor (cf. (46)). As a consequence, Jacobson’s system agrees with Szabolcsi’s in the prediction that in a binding configuration, the binder always c-commands the pronoun.16 A welcome consequence of this is that the system han16 To

apply this view on binding for double object constructions, Jacobson also employs a wrapping operation in these cases, following basically the suggestions from Bach, 1979.

104 (48)


a. His mother loves his dog. b.

loves

lex

love’ (np\s)/np λrs.love’(rs) (np\s)|np/np|np his mother

λs.love’(dog’s) (np\s)|np

lex

mother’ np|np λx.xmother’ s|np/(np|np\s|np)

G>

dog’ np|np

G>

λuv.love’(dog’s)(uv) np|np\s|np λsuv.love’(dog’s)(uv) (np|np\s|np)|np

λzv.love’(dog’z)(mother’v) s|np|np

lex A>

1

love’(dog’s) np\s

T>

λyz.yzmother’ s|np|np/(np|np\s|np)|np

his dog

G< G∗ , 1 A>

dles basic cases of Weak Crossover correctly. Consider the contrast in (49). (49)

a. Every Englishmani loves hisi mother. b. *Hisi mother loves every Englishmani .

The binding in (49a) is achieved by applying Z to the verb loves before it is combined with the object. To get a similar binding effect in (49b), we would need a mirror image of Z, something like x : (A\B)/C ⇒ λyz.xz(yz) : (A|C\B)/C Incidentally, this is a directional version of Curry and Feys’ (1958) combinator S. Since, according to Jacobson, the grammar of English does not contain this combinator, the subject-object asymmetry observed in connection with crossover violations is correctly accounted for. Jacobson presents a considerable list of empirical arguments to show that the view “pronouns as identity maps” is in fact superior to both to the standard view using variables and to the Categorial treatments that locate the binding in the lexical entry of the pronoun. I will briefly review the most important arguments.


105

Functional questions. Consider a question like (50a): (50)

a. Who does no Englishman admire? b. Margaret Thatcher. c. His mother-in-law.

There is a general agreement in the literature (cf. Groenendijk and Stokhof, 1984, Engdahl, 1986, Chierchia, 1993) that such questions are ambiguous between an individual reading (which elicits answers like (b)) and a functional reading (where we expect answers like (c)). The two readings can be paraphrased as (51)

a. Which individual x is such that every Englishman y admires x? b. Which Skolem function f is such that every Englishman y admires f y?

As a consequence of this observation, it is inevitable to assume that a wh-phrase like who is semantically ambiguous, binding either an individual gap or a Skolem function gap in its sister clause. The advantage of the Jacobsonian approach is that no further apparatus is needed to handle the ambiguity. The functional gap is bound by exactly the same means as an ordinary pronoun, namely by employing Z. (This is in clear contrast to the mentioned alternative approaches, where considerable extra apparatus like internally structured traces is needed.) In a Jacobsonian approach, the interrogative pronoun receives the two lexical entries below. Here Q is the syntactic category of questions, and the formula ?xϕ is to be interpreted as the question “Which x is such that ϕ”. (I remain neutral with regard to the correct semantics of questions since this has no bearing on the issue discussed here.) (52)

a. who – λP ?x.P x : Q/(s/np) b. who – λP ?f.P f : Q/(s/np|np)

The functional reading is now easily derived (see Figure 2.14 on the next page), using Z and the lexical entry for who in (52b). (I give a simplified treatment of auxiliary inversion since this issue is inessential here.) Since the binding of functional gaps is treated analogously to the binding of pronouns here, the approach predicts that functional gaps are subject to Weak Crossover effects. This is in fact the case (as for instance discussed at length in Chierchia, 1993). A functional reading is missing if subject and object are reversed: (53)

a. Which woman admires no Englishman?

106

ANAPHORA AND TYPE LOGICAL GRAMMAR no λQR¬∃x(Qx ∧ Rx) s/(np\s)/n does

who

lex

lex

lex

Englishman englishman’ n

λR¬∃x(englishman’x ∧ Rx) s/(np\s)

A>

λy.¬∃x(englishman’x ∧ admire’(yx)x) s/np|np ?y.¬∃x(englishman’x ∧ admire’(yx)x) q

Figure 2.14.

lex

(np\s)/np admire’ λyz.admire’(yz)z (np\s)/np|np

λy.¬∃x(englishman’x ∧ admire’(yx)x) s/np|np

λx.x s/s

λP ?f.P f q/(s/np|np)

admire

lex

Z< > B>

B>

A>

Derivation of the functional reading of (50a)

b. Margaret Thatcher. c. *His mother-in-law. The argument in favor of the Jacobsonian treatment can be further strengthened if the meaning of a constituent question is identified with the set of (denotations of) its correct constituent answers (as for instance proposed in Hausser and Zaefferer, 1978, Zaefferer, 1984 and, more recently, in Krifka, 1999). Then the answer his mother-in-law to the question Who does no Englishman admire has to be interpreted as the Skolem function mapping each individual to his mother-in-law. Jacobson’s system provides this as the basic meaning anyway, while a variable-based account would need an extra type shifting device that λ-abstracts over the variable corresponding to the pronoun.17

Sloppy inferences. Free relative clauses display a similar polymorphism. This can be observed most clearly in connection with so-called “sloppy” inferences as in (54) (taken from Jacobson, 2000 who attributes it to Tanya Reinhart): (54)

a. John will buy whatever Bill buys. b. Billi will buy hisi favorite car. c. Therefore Johnj will buy hisj favorite car.

In its most prominent reading, (54a) is to be interpreted as ∀xe (buy’xbill’ → buy’xjohn’) 17 Needless

to say this argument does not apply if the meaning of a question is identified with a set of propositions, as in Karttunen, 1977 or in Groenendijk and Stokhof, 1984.


107

In this reading, the inference from (54a) and (b) to (c) is not valid. (54a) has another reading though that renders the argument valid. The critical reading can be represented as ∀fe,e (buy’(f bill’)bill’ → buy’(f john’)john’) A discussion of the semantics of free relatives would lead us too far afield here. The essential aspect of Jacobson’s analysis of the functional reading of free relatives is more or less parallel to her analysis of functional questions. In its basic meaning, a free relative pronoun like whatever binds an np gap inside the relative clause, and the free relative as a whole binds an np-position in the matrix clause. Jacobson assumes that whatever has a second reading where it binds an np|np-gap in the embedded clause and creates a free relative that binds such a position in the matrix clause. The meaning of whatever in this functional reading is basically a universal quantifier over Skolem functions. Again, binding is achieved by the same means as in the case of ordinary pronouns. In the example above, this means that the main verb both in the relative clause and in the matrix clause has to undergo Z. The same strategy can be applied if the object is sentential rather than nominal. So sentence (55a) is predicted to be ambiguous between the readings (b) and (c). (55)

a. Every Englishman believes whatever every Frenchman believes. b. ∀pt (∀x(french’x → bel’px) → ∀y(english’y → bel’py)) c. ∀Pe,t (∀x(french’x → bel’(P x)x) → ∀y(english’y → bel’(P x)y))

Under the second reading, the inference from (56a) and (b) to (c)— among others discussed by Chierchia, 1989 under the heading of “believe de se”—is correctly predicted to be valid. (56)

a. Every Englishman believes whatever every Frenchman believes. b. Every Frenchmani believes that hei should drink lots of red wine. c. Therefore, every Englishmanj believes that hej should drink lots of red wine.

Right Node Raising. In paragraph 2.1.2 (starting on page 21) I introduced a surface compositional Categorial treatment of Right Node Raising constructions like

108 (57)


John likes and Bill detests broccoli.

The analysis rests on the assumptions that the strings John likes and Bill detests form constituents that denote properties (the property to be liked by John and the property to be detested by Bill), and that the coordination particle and is polymorphic and denotes the join operation on properties (i.e., set intersection) in the construction above. Now consider a somewhat more complicated example where the object contains a bound pronoun: (58)

Every man loves but no man wants to marry his mother.

The sentence has a reading where the pronoun his is simultaneously bound by both quantifiers. It can be paraphrased by (59)

Every mani loves hisi mother but no manj wants to marry hisj mother.

If we analyze pronouns as variables and adopt the Categorial treatment of non-constituent coordination, this reading is underivable. The closest we can get at is the semantic representation (60)

λy(∀z(man’z → love’yz) ∧ ¬∃z(man’z ∧ wtm’yz))(mother’z)

Now λ-conversion would lead to the intended reading, but it is illicit without prior renaming of the quantified variables since y is not free for z here. With the renaming, we only obtain the reading where the pronoun is free. If one wants to maintain an analysis of pronouns as variables, one is forced to abandon the Categorial treatment of Right Node Raising. One has to adopt a reconstruction approach instead. So the input for the interpretation of (58) would be (61)

Every mani loves hisi mother but no mani wants to marry hisi mother.

This would give us the intended reading. However, a reconstruction approach without further constraints on the management of variable names leads to considerable overgeneration. For instance nothing prevents an interpretation of (62a) as (62b), where the pronoun is bound by the matrix subject in the first conjunct and by the local subject in the second one. Such a reading is impossible. (62)

a. Each boy believes that every man loves and no man marries his mother.

109


b. Each boyi believes that every mank loves (hisi mother) and no mani marries hisi mother. A variable free analysis of pronouns is compatible with the general Categorial treatment of coordination, and the resulting analysis avoids both the undergeneration of the Categorial approach and the overgeneration of the reconstruction approach that comes with the variable analysis. The critical reading (59) is derived if the conjunction operates on the category s/np|np, i.e., the two conjuncts are interpreted as properties of Skolem functions. Besides, it is possible to do coordination in the category s/np and to pass the anaphora slot up to the entire coordinated structure. This admits binding from outside. The two derivations are given in Figure 2.15. There are no further options, so non-parallel binding patterns as in (62b) are excluded. loves every man λP ∀x(man’x → P x) s/(np\s)

lex

wants to marry

lex

love’ (np\s)/np λyzlove’(yz)z (np\s)/np|np

λy∀x(man’x → love’(yx)x) s/np|np

Z B>

no man λP ¬∃x(man’x ∧ P x) s/(np\s)

wtm’ (np\s)/np

lex

λyzwtm’(yz)z (np\s)/np|np

λy¬∃x(man’x ∧ wtm’(yx)x) s/np|np

λy.∀x(man’x → love’(yx)x) ∧ ¬∃x(man’x ∧ wtm’(yx)x) s/np|np

lex Z B> his mother

Conj

mother’ np|np

∀x(man’x → love’(mother’x)x) ∧ ¬∃x(man’x ∧ wtm’(mother’x)x) s

every man λP ∀x(man’x → P x) s/(np\s)

lex

loves love’ (np\s)/np

λy∀x(man’x → love’yx) s/np

lex B>

no man λP ¬∃x(man’x ∧ P x) s/(np\s)

lex

wants to marry wtm’ (np\s)/np

λy¬∃x(man’x ∧ wtm’yx) s/np

λy.∀x(man’x → love’yx) ∧ ¬∃x(man’x ∧ wtm’yx) s/np λzw.∀x(man’x → love’(zw)x) ∧ ¬∃x(man’x ∧ wtm’(zw)x) s|np/np|np

A>

lex

B>

Conj

G

his mother

λw.∀x(man’x → love’(mother’w)x) ∧ ¬∃x(man’x ∧ wtm’(mother’w)x) s|np

Figure 2.15.

lex

mother’ np|np

lex A>

Bound and free reading of (58)

i-within-i effects. In the literature, this is the common name for the observation that a pronoun inside a complex definite NP cannot be coreferential with the matrix NP. So the following coindexations lead to ungrammaticality:

110


(63)

a. *[The wife of heri childhood sweetheart]i left. b. *[The wife of heri sister’s childhood sweetheart]i left. c. *[The wife of the author of heri biography]i left.

(64)

a. *[Heri childhood sweetheart’s wife]i came to the party. b. *[The author of heri biography’s wife]i came to the party.

Neither is it possible that a quantificational determiner binds a pronoun inside its (complex) complement noun. So the following structures are excluded as well. (65)

a. *[Every wife of heri childhood sweetheart]i left. b. *[Every wife of heri sister’s childhood sweetheart]i left. c. *[Every wife of the author of heri biography]i left.

(66)

*[Every author of heri biography’s wife]i came to the party.

If indices are taken to be part of the theory, this suggests a simple generalization which Chomsky, 1981:212 formulates as follows: (67)

“*[γ . . . δ . . .], where γ and δ bear the same index.”

However, there are systematic exceptions to this generalization (which are accommodated by Chomsky in a complication of the above rule given in a footnote). The indicated indexation becomes possible if the pronoun sits inside a relative clause that modifies the head noun of the matrix NP: (68)

a. [The woman whoi married {heri sister’s / heri } childhood sweetheart]i left. b. [The woman whoi married the author of heri biography]i left.

(69)

a. [Every woman whoi married {heri sister’s / heri } childhood sweetheart]i left. b. [Every woman whoi married the author of heri biography]i left.

Under the Jacobsonian view on pronoun binding, this pattern is in fact expected. Let us start with a good example like (69b). The anaphora slot originating from the pronoun her is inherited by the NP the author of her biography by repeated application of G. So this NP will receive category np|np. The verb married undergoes Z before it is combined with its object. As a consequence, the VP married the author of her biography receives the interpretation “λx.marry’(author’(biography’x))x”.


111

Starting from this VP meaning, the relative clause, the matrix NP and the matrix clause are assembled in the usual fashion, leading to the final meaning ∀x(woman’x ∧ marry’(author’(biography’x))x → leave’x) which corresponds to the coindexation in (69b). The same mechanism works ceteris paribus for all other good examples. So why is it impossible to assign the same meaning to (65c)? As far as the semantics goes, nothing prevents this. To get the reading in question, the 2-place predicate wife of has to undergo Z, just like married does in (69c). This is impossible though because the preconditions for the application of Z are defined in terms of syntactic categories rather than semantic types. While both married and wife of are semantically of type e, e, t, the former has category (np\s)/np and the latter category n/np. Only the former meets the preconditions for Z.

Paycheck pronouns. There is a class of pronoun occurrences that can neither be accommodated under “binding” nor under “coreference”. The name “paycheck pronouns” comes from the following example (from Karttunen, 1969). (70)

a. The man who gave his paycheck to his wife was wiser than the man who gave it to his mistress. b. The man whoi gave hisi paycheck to hisi wife was wiser than the man whoj gave hisj paycheck to hisj mistress.

Sentence (70a) has a reading which is synonymous with (70b). The example is problematic because the pronouns it does not have a coreferential/binding antecedent, even though it is evidently anaphorically related to the NP his paycheck in the first conjunct. There are two possible strategies to analyze this kind of anaphor in the literature. One may consider the critical pronoun as an E-type pronoun in the sense of Evans, 1977, i.e., as shorthand for a definite description. The paraphrase given in (70b) would thus be the main part of the analysis. This kind of analysis is incompatible with the program of surface compositionality since a certain syntactic copy mechanism has to be evoked before interpretation can proceed. Alternatively, one may assume that the paycheck pronoun it retrieves two meaning components from the context by means of anaphora resolution, namely a Skolem function and an individual, and it denotes the result of applying the function to this individual. In the above example, these components are the function f mapping individuals to their

112


paychecks, while the individual slot is bound by the relative pronoun who. Analyses along these lines were among others proposed in Cooper, 1979 and Engdahl, 1986. The latter kind of analysis has the advantage of being compositional. It is faced with three problems: 1 How can a pronoun take several antecedents simultaneously? 2 How can the NP his paycheck evoke a Skolem function as value of a subsequent paycheck pronoun? 3 How exactly does anaphora resolution of the two anaphoric components of paycheck pronouns proceed? Obviously, the second question receives an immediate answer if we assume Jacobson’s analysis—the meaning of his paycheck is the paycheckfunction. Let us turn attention to the first question. Syntactically, the paycheck pronoun in the example above takes his paycheck as one antecedent, and its second anaphoric slot is bound by a relative pronoun. The category of his paycheck is np|np, and the relative pronoun binds a gap of category np. The category of the paycheck pronoun should thus be (np|np)|(np|np)—an anaphor that takes first an np|np and second an np as antecedent and returns an np. This category is derivable from the basic pronoun category np|np by a variant of the well-established Geach rule: x : A|B ⇒G λyz.x(yz) : (A|C)|(B|C) | Applying this rule to the lexical entry of it gives us the derived sign (71)

it – λf.f : (np|np)|(np|np)

(Note that λf x.f x = λf.f due to the extensionality of functions.) So the first question is answered by assuming G| as a general type shifting rule. Given this, no extra apparatus is needed to answer the third question. The value for the Skolem function is retrieved by means of accidental coreference, while the individual component is bound by Z. A sample derivation for the critical clause in the simplified example in (72) is given in Figure 2.16 on the next page. The category of the clause is s|(np|np), i.e., it denotes a function from Skolem functions to propositions. The Skolem function slot is filled by the denotation of the antecedent phrase his paycheck. (72)

Every man spent his paycheck. Mary kept it.

113

The Problem of Anaphora kept M ary

lex

mary’ np λP.P mary’ s/(np\s)

lex

keep’ (np\s)/np λyz.keep’(yz)z (np\s)/np|np

T>

λuv.uvmary’ s|(np|np)/(np\s)|(np|np)

G>

λrsz.keep’(rsz)z (np\s)|(np|np)/(np|np)|(np|np)

G>

λsz.keep’(sz)z (np\s)|(np|np)

λv.keep’(vmary’)mary’ s|(np|np)

Figure 2.16.

it

Z< >

lex

λx.x np|np λf.f (np|np)|(np|np)

G| A>

A>

Derivation for (72)

Bach-Peters sentences. The above analysis of paycheck pronouns leads to a straightforward account of Bach-Peters sentences, i.e., sentences with two complex NPs each containing a pronoun that is coindexed with the other NP. A classical example is (73)

[The man who deserves iti ]k gets [the prize hek wants]i .

Even though this coindexation pattern leads us to expect a kind of circular reference (and thus pragmatic deviance), the construction is perfectly intelligible. The question is how the interpretation is to be derived in a compositional way. Let us first note some asymmetries in these constructions. To start with, the first NP is unrestricted with regard to the form of its determiner. It may be indefinite, definite (as above) or quantified: (74)

a. [A man who deserved iti ]k got [the prize hek wanted]i . b. [Men who deserved iti ]k got [the prizes theyk wanted]i . c. [Every man who deserved iti ]k got [the prize hek wanted]i .

On the other hand, the second matrix NP must be definite (or specific if indefinite). (75)

a. ??? [The man who deserves iti ]k gets [a prize hek wants]i . b. *[The man who deserves iti ]k gets [every prize hek wants]i .

This pattern is not really surprising; the attempted backward bindings in (75) are ruled out as mundane Weak Crossover violations. Definites and specific indefinites are known to be exempted from Weak Crossover.

114


Furthermore, the first pronoun seems to be subject to Weak Crossover as well. Exchanging the pronoun and the gap corresponding to the relative pronoun in the first NP results in deviance. (76)

*[The man whom iti always evaded]k finally got [the prize hek wanted]i .

The second pronoun can occur both in subject position and in object position though. (77)

[The man who deserves iti ]k finally got [the prize that always evaded himk ]i .

The kind of subject-object asymmetry displayed by the first pronoun is also characteristic for paycheck pronouns: (78)

a. The man who sees [his brother]f regularly is better off than the man whoi never visits himf i . b. *The man who sees [his brother]f regularly is better off than the man whomi hef i is never visited by.

This asymmetry is predicted by Jacobson’s account; the relative pronoun binds the np slot of the paycheck pronoun just like an ordinary pronoun, and an object relative pronoun cannot bind an anaphora slot in the subject. Finally it should be observed that paycheck pronouns can precede their functional antecedent. This is not surprising either, given that accidental coreference does not involve binding and thus does not evoke Weak Crossover. (79)

The man whoi sees himf i regularly is better off than the man who never visits [his brother]f .

Putting these pieces together, we have evidence that the first pronoun in a Bach-Peters sentence is a paycheck pronoun, while the second one is an ordinary bound pronoun. This is exactly the analysis Jacobson proposes: In a sentence like (80), it is analyzed as a paycheck pronoun. Its Skolem function slot remains free on the level of sentence semantics— the category of the whole sentence is thus s|(np|np), while its np slot is bound by the relative pronoun (i.e., deserves undergoes Z). The pronoun him is treated as an ordinary pronoun that gets bound by the matrix subject (i.e., gets undergoes Z as well). The full derivation is given in Figure 2.17 on the facing page. (80)

Every man who deserves it gets the prize that pleases him.

lex

Z< >

lex

T>

n|(np|np)/(n\n)|(np|np)

the

(s/np\s)\s

np\s

G>

lex

T

lex

A>

lex

Figure 2.17.

Derivation of (80)

λf.∀x(man’x ∧ deserve’(f x)x → get’(ιy.prize’y ∧ please’xy)x) s|(np|np)

λrs.rs(λx.get’(ιy.prize’y ∧ please’xy)x) (s/np\s)|(np|np)\s|(np|np)

λf Q∀x(man’x ∧ deserve’(f x)x → Qx) (s/(np\s))|(np|np)

A>

G>

A

pleases please’ (np\s)/np

G>

(np\s)/np|np

Z< >

lex

him

A>

λx.x np|np

A>

lex

A>

G>

(np\s)|(np|np)

(np\s)|(np|np)/(np|np)|(np|np)

(n\n)|(np|np)

(np\s)|np/np|np

A>

(n\n)|np

G>

lex

n|(np|np)

(n\n)|np/(np\s)|np

n|np

lex

(n\n)|(np|np)/(np\s)|(np|np)

λP Qx.Qx ∧ P x (n\n)/(np\s)

that

G>

λP Qx.Qx ∧ P x (n\n)/(np\s)

who

...

G

n/(n\n)

lex

λf Q∀x(man’x ∧ deserve’(f x)x → Qx) (s/(np\s))|(np|np)

(s/(np\s))|(np|np)/n|(np|np)

λP Q∀x.(P x → Qx) s/(np\s)/n

every

man’ n

man

deserves deserve’ (np\s)/np

it lex

A>

(np|np)|(np|np)

λx.x np|np

A>

G|


115

116


The syntax-semantics interface supplies the meaning (81)

λf.∀x(man’x ∧ deserve’(f x)x → get’(ιy.prize’y ∧ please’xy)x)

The paycheck pronoun it is still unresolved, so the meaning is a function from Skolem functions to propositions. In the Bach-Peters reading, it is accidentally coreferent with the prize that pleases him, which denotes the Skolem function (82)

λxιy.prize’y ∧ please’xy

So the final interpretation is obtained by applying (81) to (82), which yields the desired (83)

∀x(man’x ∧ deserve’(ιy.prize’y ∧ please’xy)x → get’(ιy.prize’y ∧ please’xy)x)

Let us briefly wrap up the discussion of Jacobson’s approach. Her crucial assumptions are that the meaning of pronouns is the identity map, and that the binding of pronouns is achieved by means of a syntactic operation, namely Z. The fact that these simple assumptions suffice to explain a considerable range of quite diverse data is strong evidence that her theory is on the right track. Still, some critical remarks can be made. From a theoretical point of view, the collection of combinators that are necessary to make the system work seems ad hoc. The instances of Z that were discussed here only deal with constructions where the binder is the subject. Other configurations require other versions of Z. An empirically adequate modeling of the inheritance of pronoun slots requires an even larger proliferation of combinators; we need infinitely many instances of G. So it seems that some generalization has been missed here. In the ideal case, all these combinators should be theorems of a more general deductive system. Besides, Jacobson’s system has certain empirical shortcomings. It assumes that c-command is a structural precondition for pronoun binding. As discussed above, this is inadequate in many cases. This problem becomes more severe if we strive for a unified treatment of pronominal anaphora and ellipsis. In ellipsis construction, c-command of the ellipsis site by the antecedent is the exception rather than the rule. In the next chapter, I will thus develop a type logical version of a Jacobson style treatment of anaphora that avoids these problems.

4.

Summary

Categorial grammars generally employ a version of the Curry-Howard correspondence for meaning assembly. This entails a variable-free con-


117

ception of the syntax-semantics interface. In other words, under the Categorial perspective on meaning composition, the grammar does not include variable binding operations. Furthermore, both Lambek Categorial Grammars and most versions of Combinatory Categorial Grammar are in a sense subsystems of Linear Logic. This means that every meaning of a lexical item that occurs in a complex construction must be used exactly once in the composition of the complex meaning. This seems to be at odds with the empirical facts in connection with anaphora phenomena like pronominal anaphora and ellipsis. By definition, anaphora involves the re-use of semantic resources. If the overall variable-free design is to be maintained, there are two basic strategies to accommodate anaphora into the general picture: 1 Anaphora is triggered by certain lexical items, and the recycling of semantic resources is due to the interpretation of these lexical items. Accordingly, anaphoric lexical items have semantic representations where a λ-operator binds more than one variable occurrence. This strategy is pursued among others by Szabolcsi, 1989, Szabolcsi, 1992 in a Combinatory and by Moortgat, 1996a, and Morrill, 2000 in a Type Logical setting. 2 Anaphora resolution is handled in syntax. This means that the grammar contains operations specifically designed for this purpose. Lexical items typically have Linear meaning representations here, while the grammatical operations go beyond the resource management of Linear Logic. This approach was first explored by Mark Hepple (Hepple, 1990, Hepple, 1992), who uses a version of Type Logical Grammar. Drawing on his insights, Pauline Jacobson reformulated this idea within the framework of CCG (Jacobson, 1999, Jacobson, 2000). Mixed approaches are possible. Jacobson, for instance, follows the general Categorial consensus in treating coordination ellipsis in the lexicon while pronominal anaphora is dealt with in syntax. All mentioned approaches, including the second group, assume that anaphora is somehow lexically triggered. So they will not easily lend themselves to an analysis of ellipsis phenomena like stripping that are apparently not lexically triggered. (84)

Most people want to be millionaires, but not John.

The landscape of Categorial approaches to anaphora is schematically summarized in Table 2.1 on the next page.

118


Author

Locus of resource multiplication

Szabolcsi

lexicon

Moortgat

lexicon

Morrill

lexicon

Hepple

syntax

Jacobson

syntax

Table 2.1.

Lexical entry of he

Non-standard operations

λxyz.y(xz)z (s/np)\((np\s)/s)\(np\s) λxy.xyy q(np, np\s, np\s) λxyz.y(xz)y (((s ↑ np) ↑2 s) ↓2 (s ↑ np))/(np\s) λx.x np/ np λx.x np|np

none none none BIR Z, G

Categorial approaches to anaphora

There is an obvious tradeoff between a fairly complex lexicon in the first three approaches and a complication of the grammatical machinery in the last two ones. Given that Jacobson gives empirical arguments a) that the meaning of a pronoun is in fact the identity function and b) something like her Z-operation is needed also in the absence of anaphora (for instance in functional questions), the second class of approaches seems to be superior. Furthermore they do without a lexical ambiguity between bound, coreferential and free pronouns, while this complication is virtually inevitable in the first group of theories. On the other hand, both Hepple and Jacobson assume some version of a c-command constraint on anaphora resolution. As argued above, this is empirically inadequate in certain cases of pronominal anaphora, and it practically blocks the extrapolation of their anaphora machinery to ellipsis. Furthermore, Hepple’s system displays certain unpleasant formal properties (like the failure of Cut elimination), and the (infinite!) collection of combinators needed to make Jacobson’s system work seems to be ad hoc. In the following chapter, I will develop a simple extension of the Lambek calculus which enables us to derive all relevant instances of Jacobson’s Combinatory approach as theorems. The system is proof theoretically well-behaved, and it is straightforwardly applicable to several kinds of ellipsis phenomena in natural language.

Chapter 3 LAMBEK CALCULUS WITH LIMITED CONTRACTION

1.

The Agenda

After having reviewed the main Categorial approaches to anaphora from the literature, in this chapter I will develop a new proposal. My aim is to extend the Lambek style core of Type Logical Grammar in such a way that a comprehensive treatment of anaphora phenomena becomes possible. The discussion from the previous chapter leads to the following agenda: Resource multiplication should be done in syntax (as in Hepple’s 1992 and Jacobson’s 1999, 2000 systems) rather than in the lexicon. There are three main reasons for taking this decision: 1. Doing resource multiplication in the lexicon means we have to stipulate ambiguity between bound and free pronouns, 2. binding-in-syntax lends itself more naturally to an extension to the discourse level than binding-in-lexicon, and 3. Jacobson supplies convincing empirical evidence that the meaning of a pronoun is in fact the identity function. This leads to the second desideratum: The meaning of a pronoun should come out as the identity function on individuals. The general topic of the present investigation is the analysis of anaphora in TLG, hence: The analysis should be formulated in an extension of the Lambek calculus.

119

120


I thus want to improve on Hepple’s proposal: The system should be proof-theoretically well-behaved, i.e., the logic should enjoy Cut elimination, decidability, the subformula property and the finite reading property. Furthermore there should be a natural Curry-Howard correspondence as syntaxsemantics interface. Finally, I want my analysis to incorporate the insights from Morrill’s (2000) system. Neither the structural positions of anaphors nor the positions of antecedents should be limited in an empirically unjustified way. The anaphora resolution mechanism should do without c-command restriction. The latter point is certainly controversial among linguists, and I will discuss the empirical aspects of this decision at length in the next chapter.

2.

Contraction?

Under a type logical perspective, doing resource multiplication in syntax means that the logic of grammatical composition derives Curry-Howard terms where one λ-operator binds more than one occurrence of a variable. According to the Curry-Howard correspondences of substructural logics, this amounts to the assumption that the structural rule of Contraction is part of the logic of grammar in one way or another. The canonical version of this rule is repeated here for convenience: X, x : A, y : A, Y ⇒ M : B X, x : A, Y ⇒ M [x/y] : B

C

Looking at this rule under a bottom-up proof search perspective, it says that antecedent formulae can be multiplied at will. It is easy to see that the proof search space becomes infinite as soon as we incorporate this rule, since we can apply Contraction to the premise of this rule again (still in the bottom-up direction) etc. and thus run into an infinite regress. While logics using Contraction might still be decidable (Intuitionistic Logic and some versions of Relevant Logic are), we nevertheless lose the finite reading property.1 Such a logic would thus be a priori too 1 The

simplest illustration of this point is the identity theorem x:A→A⇒M :A→A

where the Curry-Howard term M can be any λx.f n x for arbitrary n in Intuitionistic or Relevant Logic.

Lambek Calculus with Limited Contraction

121

powerful as a logic of grammatical composition. Contraction thus has to be limited in a suitable way to avoid this collapse. One may try to do this by employing multimodal techniques along the lines of Hepple’s (1990) work. I will pursue another strategy though, which is inspired by Jacobson’s work. I will extend the Lambek calculus with a third version of implication, and I will compile a limited version of Contraction directly into the logical rules for this new connective. This will allow us to keep the power of Contraction under strict logical control.

3. 3.1

The Logic LLC Vocabulary

In this section I will introduce a conservative extension of L called Lambek Calculus with Limited Contraction (abbreviated LLC), where a limited version of the structural rule of Contraction is compiled into the logical rules of a logical connective. Starting from the Lambek calculus L, I extend the inventory of category forming connectives by a third kind of implication (written as |). So the set of categories F over a collection of atomic categories A is given by

Definition 37 (Categories) F ::= A, F\F, F • F, F/F, F|F As in Jacobson’s system, the vertical slash creates categories of anaphoric items. A sign has category A|B iff it needs an antecedent of category B and, provided it finds one, behaves like an item of category A. Pronouns will thus come out as np|np. As in L, the product is interpreted as Cartesian product and the implications as function space formation. So the category-to-type correspondence for LLC is given by

Definition 38 (Category to type correspondence) τ (A • B) = τ (A) ∧ τ (B) τ (A\B) = τ (B/A) = τ (B|A) = τ (A), τ (B)

3.2

Sequent Presentation

The Gentzen style sequent formulation extends the corresponding presentation of L by a left rule and a right rule for the new implication slash. It is given in Figure 3.1 on the following page. Let us have a closer look at the two new rules. If the left premise of the rule of use |L is instantiated with an identity axiom, we obtain the

122


x:A⇒x:A

id

X⇒M :A

Y, x : A, Z ⇒ N : B

Y, X, Z ⇒ N [M/x] : B X⇒M :A

Y ⇒N :B

X, Y ⇒ M, N : A • B

Cut

•R

X, x : A, y : B, Y ⇒ M : C X, z : A • B, Y ⇒ M [(z)0 /x][(z)1 /y] : C X, x : A ⇒ M : B X ⇒ λxM : B/A X⇒M :A

/R Y, x : B, Z ⇒ N : C

Y, y : B/A, X, Z ⇒ N [(yM )/x] : C x : A, X ⇒ M : B X ⇒ λxM : A\B X⇒M :A

•L

/L

\R Y, x : B, Z ⇒ N : C

Y, X, y : A\B, Z ⇒ N [(yM )/x] : C

\L

X, x1 : A1 , Y1 , . . . , xn : An , Yn ⇒ M : B X, y1 : A1 |C, Y1 , . . . , yn : An |C, Yn ⇒ λz.M [(y1 z)/x1 ] · · · [(yn z)/xn ] : B|C

|R

n>0 Y ⇒M :B

X, x : B, Z, y : A, W ⇒ N : C

X, Y, Z, z : A|B, W ⇒ N [M/x][(zM )/y] : C Figure 3.1.

|L

Labeled sequent presentation of LLC

simplified formulation below (from which the original formulation can be recovered via Cut): X, x : B, Z, y : A, W ⇒ N : C X, x : B, Z, z : A|B, W ⇒ N [(zx)/y] : C Intuitively this rule says: If an anaphoric resource of category A|B is preceded by an antecedent of category B, it may be resolved and thus be replaced by a resource of category A. The meaning of the resolved anaphor is obtained by applying the meaning of the unresolved anaphor to the

123


meaning of its antecedent. (Typically, the meaning of the anaphor is just the identity function, so the resolved meaning of the anaphor winds up being identical to the meaning of the antecedent in these cases.) Note that the metavariable Z ranges over sequences of categories, including the empty sequence, so antecedent and anaphor may or may not be adjacent. The same intuition is possibly expressed more transparently by the two axioms below, which are jointly equivalent to the sequent rule above (i.e., extending L with the two axioms has the same effect as extending L with |L): x : A, y : B|A ⇒ x, yx : A • B x : A, y : B, z : C|A ⇒ x, y, zx : A • B • C The Curry-Howard labeling of |L reveals that this operation corresponds to three Intuitionistic operations: 1. Contraction (because M occurs twice in the proof term of the succedent), 2. the rule of Modus Ponens (corresponding to function application of z to one copy of M ), and Cut (corresponding to replacing x by M and y by zM ). In fact, if all three implications of LLC are mapped to the Intuitionistic implication and the product of LLC to Intuitionistic conjunction, |L becomes a derivable rule of Intuitionistic Logic, as can be seen from the natural deduction derivation in Figure 3.2. This translation into Intuitionistic Logic justifies the Curry-Howard labeling used here. z:B→A⇒z:B→A

id

w:B⇒w:B

z : B → A, w : B ⇒ zw : A

id →E

X, x : B, Z, y : A, W ⇒ N : C

X, x : B, Z, z : B → A, w : B, W ⇒ N [(zw)/y] : C .. . X, x : B, w : B, Z, z : B → A, W ⇒ N [(zw)/y] : C Y ⇒M :B

X, x : B, Z, z : B → A, W ⇒ N [(zx)/y] : C X, Y, Z, z : B → A, W ⇒ N [M/x][(zM )/y] : C

Figure 3.2.

Cut

P P C

Cut

Intuitionistic derivation of |L

The rule of proof |R expresses the insights that anaphora slots can percolate up inside larger constituents, and that they can be merged. The first fact is relevant for instance in answers to functional questions; we want to be able to assign the phrase (1b) the category pp|np: (1)

a. In which town is every Englishman happy?

124


b. In his hometown. This percolation mechanism can be covered by two axioms and one inference rule. An anaphora slot can percolate up to a superconstituent from either of its subconstituents. This corresponds to the two axioms x : A|C, y : B ⇒ λz.xz, y : (A • B)|C x : A, y : B|C ⇒ λz.x, yz : (A • B)|C Furthermore, anaphora slots are preserved under unary derivations. The corresponding inference rule is x:A⇒M :B y : A|C ⇒ λz.M [(yz)/x] : B|C Second, a syntagma may contain several anaphoric expressions that are understood as being co-anaphoric (i.e., depending on the same antecedent) even though no antecedent is present. A case in point is the adjective local which arguably has category (n/n)|np (i.e., it requires an np-antecedent to be an attributive adjective). In (2b) the two occurrences of local are preferably interpreted as co-anaphoric: (2)

a. What happened in three cities last year? b. The local press accused the local politicians of corruption.

The axiom covering this merge operation is x : A|C, y : B|C ⇒ λz.xz, yz : (A • B)|C These axioms and rules taken together are equivalent to the rule of proof |R given in fig 3.1 on page 122.2 As for the rule of use, the labeling is justified by the fact that the rule is Intuitionistically derivable (cf. Figure 3.3 on the next page).

3.3

Cut Elimination

Despite the fact that they incorporate Contraction, the two logical rules for the new implication have the subformula property, just as all logical rules of L. Hence the premise of any inference rule has a lower complexity 2 In J¨ ager, 2001 I used a rule of proof that is slightly stronger than the one given here. The present version is a generalization of a proposal from Glyn Morrill (p.c.). His rule is the special case of the one given here for n = 1.

125

Lambek Calculus with Limited Contraction yi : C → A i ⇒ y i : C → A i

id

zi : C ⇒ z i : C

yi : C → Ai , zi : C ⇒ yi zi : Ai

id →E

X, x1 : A1 , Y1 , . . . , xn : An , Yn ⇒ M : B

X, y1 : C → A1 , z1 : C, Y1 , . . . , yn : C → An , zn : C, Yn ⇒ M [(yi zi )/xi ] : B .. .

P

X, y1 : C → A1 , Y1 , . . . , yn : C → An , Yn , z1 : C, . . . , zn : C ⇒ M [(yi zi )/xi ] : B X, y1 : C → A1 , Y1 , . . . , yn : C → An , Yn , z : C ⇒ M [(yi z)/xi ] : B X, y1 : C → A1 , Y1 , . . . , yn : C → An , Yn ⇒ λz.M [(yi z)/xi ] : C → B

Figure 3.3.

Cutn

P C n−1

→I

Intuitionistic derivation of |R

than its conclusion, and bottom-up proof search reduces complexity. Again this does not hold for the Cut rule. As in L, however, Cut is admissible in the Cut free sequent presentation of LLC.

Theorem 9 (Cut Elimination) If LLC X ⇒ A, then there is a Cut-free sequent proof of X ⇒ A. Sketch of proof: The proof is essentially identical to Lambek’s Cut elimination proof for L (as sketched from page 51 onwards in Chapter 1), except for the fact that we have two more cases to consider for principal Cut. These are the configurations where the Cut formula is the active formula in both premises. Since the rule |R also introduces anaphora-implications on the left hand side of the sequent, there are two new configurations for principal Cut: The left premise of the Cut is a conclusion of |L and the right premise is a conclusion of |R, or both premises are conclusions of |R. The Cut elimination steps for these configurations are schematically given in Figures 3.4 and 3.5 on the following page. (For the second configuration it is assumed that 1 ≤ i ≤ n.) In either case, the principal Cut is replaced by a Cut of lower degree. Lambek’s Cut elimination algorithm is thus guaranteed to terminate. Because every rule in the Cut free sequent presentation of LLC has the subformula property, the bottom-up proof search space is finite. As in L, this leads to the following consequences:

Theorem 10 (Decidability) Derivability in LLC is decidable. Proof: Identical to the corresponding proof for L.

126


U, D1 , V1 , . . . , Dn , Vn ⇒ A U, D1 |B, V1 , . . . , Dn |B, Vn ⇒ A|B

Y ⇒B

|R

X, B, Z, A, W ⇒ C

X, Y, Z, A|B, W ⇒ C

X, Y, Z, U, D1 , V1 , . . . , Dn |B, Vn , W ⇒ C

|L

Cut

;

B⇒B

U, D1 , V1 , . . . , Dn , Vn ⇒ A

id

X, B, Z, A, W ⇒ C

X, B, Z, U, D1 , V1 , . . . , Dn , Vn , W ⇒ C

X, B, Z, U, D1 |B, V1 , D2 , V2 , . . . , Dn , Vn , W ⇒ C .. . Y ⇒B

X, B, Z, U, D1 |B, . . . , Dn−1 |B, Vn−1 , Dn , Vn ⇒ C X, Y, Z, U, D1 , V1 , . . . , Dn |B, Vn , W ⇒ C

Cut

|L

|L |L |L

Principal Cut for |, first configuration

Figure 3.4.

X, A1 , Y1 , . . . , An , Yn ⇒ Bi X, A1 |C, Y1 , . . . , An |C, Yn ⇒ Bi |C

Z, B1 , W1 , . . . , Bm , Wm ⇒ D

|R

Z, B1 |C, W1 , . . . , Bm |C, Wm ⇒ D|C

Z, B1 |C, W1 , . . . , X, A1 |C, Y1 , . . . , An |C, Yn , Wi , . . . , Bm |C, Wm ⇒ D|C

|R Cut

; X, A1 , Y1 , . . . , An , Yn ⇒ Bi

Z, B1 , W1 , . . . , Bm , Wm ⇒ D

Z, B1 , W1 , . . . , X, A1 , Y1 , . . . , An , Yn , . . . , Bm , Wm ⇒ D

Cut

Z, B1 |C, W1 , . . . , X, A1 |C, Y1 , . . . , An |C, Yn , Wi , . . . , Bm |C, Wm ⇒ D|C Figure 3.5.

|R

Principal Cut for |, second configuration

Corollary 2 (Finite reading property) For a given unlabeled LLC-sequent, there are at most finitely many Curry-Howard labelings. Proof: Identical to the corresponding proof for L.


3.4

127

Natural Deduction Presentation

During the discussion of L we saw that the sequent system is indispensable since it guarantees decidability, but for practical purposes it is rather awkward. A presentation in natural deduction format is better suited for concrete derivations. Besides, it has an appealing allusion to the tree format linguists are used to. I start with a sequent style presentation of the natural deduction system (Figure 3.6 on the next page). Next to the identity rule and the Cut rule (which are identical to the corresponding rules in the sequent system), we have an introduction rule and an elimination rule for each connective. The rules for the Lambek connectives are identical to the corresponding rules in the natural deduction presentation of L. Additionally, we have an introduction rule and an elimination rule for the anaphora slash. The |-introduction rule is a combination of the rule of proof for | in the sequent system and Cut, and thus requires no further elaboration. The elimination rule is straightforwardly derivable from the rule of use from the sequent system (and vice versa). The derivations are given in Figure 3.7 on the following page and 3.8 on page 129.

Cut elimination for sequent style natural deduction. As in the sequent system, Cut is an admissible rule in the natural deduction system in tree format. Here Cut elimination does not even affect the CurryHoward term of the proof. Theorem 11 (Cut Elimination) If LLC X ⇒ M : A, then there is a Cut-free natural deduction proof of X ⇒ M : A. Proof: The proof follows the same strategy as the corresponding proof for the sequent system. There are two notable differences: The degree of a Cut application is measured by the complexity of the Curry-Howard term of the conclusion of the Cut. This guarantees that every Cut elimination step reduces the degree of the Cut, also if Cut is permuted with an elimination rule. Second, since there are no left rules in the natural deduction calculus, the configuration for principal Cut never arises. Since the elimination of a principal Cut is the only configuration where Cut elimination leads to a change in the Curry-Howard term, Cut elimination in the natural deduction calculus preserves Curry-Howard term assignment.

128


x:A⇒x:A

id

X⇒M :A

Y, x : A, Z ⇒ N : B

Y, X, Z ⇒ N [M/x] : B X⇒M :A

Y ⇒N :B

X, Y ⇒ M, N : A • B X ⇒M :A•B

Cut

•I

Y, x : A, y : B, Z ⇒ N : C

Y, X, Z ⇒ N [(M )0 /x][(M )1 /y] : C X, x : A ⇒ M : B X ⇒ λxM : B/A

/I

X ⇒ M : A/B

Y ⇒N :B

X, Y ⇒ M N : A x : A, X ⇒ M : B X ⇒ λxM : A\B X⇒M :A

/E

\I Y ⇒ N : A\B

X, Y ⇒ N M : B for 1 ≤ i ≤ n : Zi ⇒ Ni : Ai |C

•E

\E

X, x1 : A1 , Y1 , . . . , xn : An , Yn ⇒ M : B

X, Z1 , Y1 , . . . , Zn , Yn ⇒ λz.M [(Ni z)/xi ] : B|C X⇒M :A

Y ⇒ N : B|A

|I

Z, x : A, W, y : B, U ⇒ O : C

Z, X, W, Y, U ⇒ O[M/x][(N M )/y] : C Figure 3.6.

(Labeled) Natural Deduction presentation of LLC

X⇒A Y ⇒ B|A

Z, A, W, B, U ⇒ C

Z, X, W, B|A, U ⇒ C Z, X, W, Y, U ⇒ C Figure 3.7.

Derivation |L ; |E

Cut

|L

|E

129


A|B ⇒ A|B

id

Y ⇒B

X, B, Z, A, W ⇒ C

X, Y, Z, A|B, W ⇒ C Figure 3.8.

|E

Derivation |E ; |L

3.4.1 Natural Deduction in Tree Format Natural deduction proofs are more concisely presented in tree format. To give a natural representation for |-elimination, this format has to be extended somewhat in comparison to the tree format for L. Strictly speaking, a natural deduction proof tree is not necessarily a tree, but rather a sequence of finite directed acyclic graphs (DAGs) with labeled nodes. I will continue to use the term “proof tree” nevertheless. As in conventional syntax trees, the nodes in a proof tree are partially ordered by two relations, immediate dominance D and precedence 1. Then, by induction hypothesis, σ(A2 |B, Y2 , . . . , An |B, Yn ) → σ(A2 , Y2 , . . . , An , Yn )|B Via the identity axiom, associativity and monotonicity of the product, and the third axiom, this gives us σ(X, A1 |B, Y1 , . . . , An |B, Yn ) → σ(X, A1 , Y1 , . . . , An , Yn )|B Now we can turn to the actual proof of the lemma. It is obvious that the axioms above are derivable in the sequent system, and the monotonicity rule is a direct consequence of |R. Hence the if -direction is straightforward. We prove the only-if -direction via induction over sequent derivations. The lemma obviously holds for identity sequents. For the rules of L, the induction step was established in the proof of Theorem 6 on page 55 in Chapter 1. Suppose the following sequent is derivable in the sequent system: X, A1 , Y1 , . . . , An , Yn ⇒ C Then the following arrow is derivable in the axiomatic system: σ(X, A1 , Y1 , . . . , An , Yn ) → C Due to the monotonicity rule above, this gives us σ(X, A1 , Y1 , . . . , An , Yn )|B → C|B Together with the result we just proved and Cut, we have σ(X, A1 |B, Y1 , . . . , An |B, Yn ) → C|B

150


So if the lemma holds for the premise of an application of |R in the sequent system, it also holds for the conclusion. It remains to be shown that the truth of the lemma is preserved by |L. As we just showed, it is derivable in the axiomatic system that σ(Z, A|B) → σ(Z, A)|B Together with the fourth axiom plus monotonicity and associativity of the product, we get σ(B, Z, A|B) → σ(B, Z, A) Suppose it is derivable in the sequent system that Y ⇒B and X, B, Z, A, W ⇒ C Then via induction hypothesis, the previous result, Cut, and associativity and monotonicity of the product, we get σ(X, Y, Z, A|B, W ) → σ(C)

This completes the proof.

Theorem 16 (Soundness) For each LLC-sequent X ⇒ A, if LLC X ⇒ A then for all models M, XM ⊆ AM Proof: I will prove soundess of the axiomatic version. Together with the previous lemma, this gives soundness of the sequent system as well. The soundess of the axioms and rules of the axiomatic version of L carries over from Theorem 7 on page 58 since each model for LLC is based on an associative frame. There is a straightforward correspondence between the postulates in the definition of LLC-models and the axioms in the axiomatic presentation. We start with the first axiom A • B|C → (A • B)|C


151

Suppose x ∈ A • B|C (for some model M which I leave implicit henceforth). Then there are y ∈ A and z ∈ B|C such that Rxyz. Furthermore, there is a w ∈ B such that Szwg(C). Due to the first postulate, there is a v such that Sxvg(C) and Rvyw. Hence v ∈ A • B, and thus x ∈ (A • B)|C. A|B • C → (A • C)|B Suppose x ∈ A|B • C. Then there are y ∈ A|B and z ∈ C, and thus there is a w ∈ A such that Sywg(B). The second postulate entails that there is a v such that Sxvg(B) and Rvwz. Hence v ∈ A • C, and thus x ∈ (A • C)|B. A|C • B|C → (A • B)|C Suppose x ∈ A|C • B|C. Then there are y ∈ A|C and z ∈ B|C such that Rxyz. Therefore there are w ∈ A and v ∈ B such that Sywg(C) and Szug(C). According to the third postulate, there is an r with Sxrg(C) and Rrwv. Hence r ∈ A • B, and thus x ∈ (A • B)|C. A • B|A → A • B Suppose x ∈ A • B|A. Then there are y ∈ A and z ∈ B|A with Rxyz. Therefore there is a w ∈ B with Szwg(A). According to the fifth postulate, y ∼ g(A), and thus Rxyw due to the fourth postulate. Hence x ∈ A • B. A → B A|C → B|C Finally, suppose that AM ⊆ BM in all models M , and suppose that x ∈ A|C. Then there is a y ∈ A such that Sxyg(C). By assumption, y ∈ B, and hence x ∈ B|C. The completeness proof also follows closely the analogous proof for L.

Theorem 17 (Completeness) For all sequents X ⇒ A, if for all LLC-models M XM ⊆ AM then LLC X ⇒ A

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Proof: We start with the construction of a canonical model. The set W is simply the set of LLC-categories. For all atomic categories p, f (p) = {A| A ⇒ p} The relation R is defined as RABC iff A ⇒ B • C Likewise, S is defined as SABC iff A ⇒ B|C Finally, A ∼ B iff A ⇒ B, and g(A) = A for all categories A and B. The fact that W, R is an associative frame follows from the associativity of the product (see the completeness proof for L on page 58). It is straightforward to show that the first four additional postulates are fulfilled by this model: 1 If x ⇒ y • z and z ⇒ w|u, then x ⇒ y • w|u due to the monotonicity of the product, and thus x ⇒ (y • w)|u. So y • w has the properties that are required for v. 2 Likewise, w • z would be a witness for the required v for the second postulate. 3 For the third postulate, w • v is the required witness for r. 4 Suppose x ⇒ y • z, and z ⇒ w|y. Due to monotonicity of the product, we have x ⇒ y • w|y, and with the fourth axiom and Cut this entails that x ⇒ y • w, hence Rxyw. The fifth postulate requires that A ∈ B entails that A ⇒ B. We will show something stronger, namely

Lemma 5 (Truth lemma) In the canonical model CM, it holds for all categories A, B that A ∈ BCM iff A ⇒ B We prove this by induction over the complexity of B. If B is atomic, the claim follows from the way CM is constructed. If the main connective of B is one of the three Lambek connectives, the proof of the induction step is identical to the corresponding step in the proof of Theorem 7 on page 58. It remains to be shown that the induction step also goes through for the anaphora slash.


153

→ Suppose that B = C|D, and suppose A ∈ C|D. (Unless otherwise stated, interpretation is with respect to the canonical model.) This means that there is an E ∈ C such that SAEg(D). By the model construction and the induction hypothesis, it follows that E ⇒ C and A ⇒ E|D. Due to the monotonicity rule for the anaphora slash and Cut, it follows that A ⇒ C|D. ← Now suppose that A ⇒ C|D. By the construction of the model, we thus have SACD, and thus SACg(D). By induction hypothesis C ∈ C, and hence A ∈ C|D. This completes the proof of the truth lemma. The fifth postulate for LLC-models follows as a corollary, so the canonical modelis in fact a model for LLC. Now suppose that LLC X ⇒ A. Since XCM = σ(X)CM , it follows from the truth lemma that σ(X) ∈ XCM , but σ(X) ∈ ACM . Hence XCM ⊆ ACM , and hence X ⇒ A is not valid. By contraposition, every valid sequent must be derivable. This completes the completeness proof. Before we look at the linguistic applications of LLC, I will close the chapter with a brief review of the formal similarities and differences between Jacobson’s system and LLC.

4.

Relation to Jacobson’s System

The central innovation of Jacobson’s system is the combinator Z. There are four directional instances of Z (if we ignore wrapping, which Jacobson needs to analyze double object constructions properly). However, of these four axioms, only one is used in linguistic analyses, namely x : (A\B)/C ⇒ λyz.x(yz)z : (A\B)/C|A It is easy to show that this instance of Z is a theorem of LLC. The derivation is given in Figure 3.22 on the following page. Despite this kinship, there is a crucial conceptual difference between Z and |E. Z is essentially based on a notion like c-command. Binding is an operation that connects argument places of an operator, and it is determined by the argument structure hierarchy which argument place can bind which other argument place. |E, on the other hand, is purely precedence based. The only structural constraint on binding is the requirement that the binder precedes the bound element. This difference has certain empirical consequences when it comes to the linguistic applications.

154


y : C|A

[z : A]i

2

x : (A\B)/C

yz : C

x(yz) : A\B x(yz)z : B

λz.x(yz)z : A\B

|E, i /E

\E

\I, 2

λyz.x(yz)z : (A\B)/C|A Figure 3.22.

1

/I, 1

Derivation of Z in LLC

Z roughly corresponds to |E, and there is a similar correspondence between the different instances of G in Jacobson’s system and |I in LLC. The two directional instances of G are repeated here for convenience. X ⇒ M : A/B X ⇒ λxy.M (xy) : A|C/B|C X ⇒ M : B\A X ⇒ λxy.M (xy) : B|C\A|C

G>

G
∀] *[∀ > ∃] b. Every girl will be happy if some movie is shown. [∃ > ∀][∀ > ∃]

Fodor and Sag, 1982 suggest that indefinites are ambiguous between a quantificational and a referential (= specific) reading. This predicts though that indefinites take either local or global scope. The existence of intermediate readings has been established by several authors, however, notably by Farkas, 1981 and by Abusch, 1994. The following two examples are taken from Kratzer, 1998 (they are slight modifications of examples from Abusch, 1994): (44)

a. Every professor rewarded every student who read some book he had recommended. ∀ > ∃ > ∀ b. Every one of them moved to Stuttgart because some woman lived there. ∀ > ∃ > because

The conclusion that has to be drawn from the investigations of the authors mentioned and others is that the scope of indefinites is structurally unrestricted, even if local and global scope readings might be preferred pragmatically. The sharp contrast between indefinites and other quantifiers suggests that different mechanisms are at work here. Existential closure in the sense of DRT is an obvious candidate for a mechanism to assign scope to indefinites. It leads to mispredictions though if the indefinite has a non-trivial descriptive content. The following example from Reinhart, 1995 illustrates this point. (45)

a. If we invite some philosopher, Max will be offended. b. ∃x((philosopher’x ∧ invite’xwe’) → offended’max’)

Analysing (45a) in a DRT-style way without employing any further scoping mechanisms leads to a semantic representation like (45b) for the specific reading of (a), where the existential impact of the indefinite some philosopher takes wide scope, while the descriptive content remains in the antecedent of the conditional. As already observed in Heim, 1982 for a parallel example, (45b) does not represent the truth conditions of the specific reading of (45a). The former is true if there is one nonphilosopher, while (45a) in the wide-scope reading requires the existence of a philosopher x with the property that Max will be offended if we invite x. Since the existence of the non-philosopher Donald Duck is sufficient to verify (45b) but not (45a), this problem is sometimes called the Donald Duck problem in the literature.

247

Indefinites

To overcome this and related problems, several authors have proposed to employ choice functions for the analysis of indefinites (see for instance Reinhart, 1992, Reinhart, 1995, Reinhart, 1997, Kratzer, 1998, Winter, 1997). To cut a long story short, according to these theories, the semantic counterpart of an indefinite determiner is a variable over a choice function, i.e., a function that maps non-empty sets to one of their elements. This variable is subject to existential closure in a way akin to the treatment of free individual variables in DRT. (45a) would therefore come out as (46)

∃f (CH(f ) ∧ (invite’f (philosopher’)we’ → offended’max’))

The extension of the predicate constant CH is the set of choice functions of type e, t, e, i.e., ∀f (CH(f ) ↔ ∀P (∃xP x → P (f P ))) (46) in fact represents the truth conditions of (45a) in an adequate way. Generally speaking, the choice function approach solves two problems in one stroke. Since it uses unselective binding to assign scope to indefinites, it covers the fact that the scope of indefinites is structurally unrestricted. Second, the choice function mechanism makes sure that the existential impact of an indefinite is not unduly divorced from its descriptive content. On the other hand, the choice function approach faces at least two serious problems. First, what happens if the extension of the descriptive content of an indefinite is empty? Consider a slight variation of (45a): (47)

If we invite some Polish friend of mine, Max will be offended.

If the indefinite some Polish friend of mine receives a specific reading, the sentence can be paraphrased as There is a certain Polish friend of mine, and if we invite him, Max will be offended. Suppose I don’t have any Polish friends. In this scenario the sentence is false in the relevant reading. The choice function approach as such does not supply clear truth conditions in this case, since the argument of the choice function f in the term f (polish friend of mine’) denotes the empty set. Both Reinhart and Winter suggest mechanisms that make the smallest clause containing such a term false. This works fine for simple sentences such as (48)

We invited some Polish friend of mine.

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This sentence is in fact false if there are no Polish friends of mine. This very fact would make (47) true though, while the sentence should come out as false. Let us call this problem the empty set problem. The second problem arises if the descriptive content of an indefinite contains a pronoun that is bound by some superordinate quantifier. The following example (from Abusch, 1994) can serve to elaborate this point. (49)

Every professori rewarded every student who read some book hei had recommended.

According to the choice function approach, the sentence should have a reading which can be represented as (50)

∃f (CH(f ) ∧ ∀x(professor’x → ∀y(student’y ∧ read’(f (λz.book’z ∧ recommend’zx)) → reward’yx)))

Suppose two professors, a and b, happened to recommend exactly the same books to their students. Then the expressions λz.book’z ∧ recommend’za and λz.book’z ∧ recommend’zb denote the same set, and thus the terms f (λz.book’z ∧ recommend’za) and f (λz.book’z ∧ recommend’zb) denote the same individual. So the reading that is described in (50) can be paraphrased as follows: Every professor has a favorite book. He recommends this book (possibly along with other books), and he awards students that read his favorite book. Furthermore, if two professors recommend the same books, they have the same favorite book. The last condition is entirely unnatural, and the sentence has no such reading. This problem is discussed among others by Kratzer, 1998 (who attributes the observation to Kai von Fintel and P. Casalegno) and Winter, 1997. Again, the literature contains several proposals regarding how to circumvent this kind of overgeneration, but so far no suggested solution is really satisfactory. I will call this problem the bound pronoun problem bound pronoun problem.

Indefinites

249

These and a variety of other problems for the choice function approach have been pointed out by several authors, see for instance Reniers, 1997, Geurts, 2000 and Endriss, 2001. The works mentioned also contain discussions of the possible solutions of these problems, and arguments why they are not fully satisfactory. The conclusion that has to be drawn from the discussion in the literature so far is that the choice function approach is not a viable alternative to a quantificational treatment of indefinites. So an adequate theory of indefinites should avoid the problems that were discussed in this section so far. At the same time, such a theory should take the fact into account that the scope taking behavior of indefinites is virtually unrestricted, and it should be able to deal with the peculiar pronoun binding abilities of indefinites which were the subject of the previous two sections. The theory I sketched there already meets the first two requirements. The scope of indefinites is syntactically handled by the rule ∧ , and its applicability is determined by the scope of static closure. In other words, an indefinite is predicted to take scope either over an entire sentence or over a constituent that is subject to static closure. If we assume that clause embedding operators like verbs of propositional attitude etc. apply static closure to their arguments, this generalization is empirically correct. The scope of ordinary quantifiers on the other hand is syntactically modeled by means of the rule qE in our type logical framework. While I only gave a formulation of this rule that does not take domains of its applicability into account, the clause boundedness of quantifier scope can easily be modeled by means of multimodal techniques, as for instance Morrill, 1994 demonstrates. While multimodality goes beyond the scope of this book, it is important to point out that the scoping mechanisms for indefinites and for quantifiers are independent from each other in our framework, and it is thus not surprising that they are subject to different constraints. In the previous two sections, I demonstrated how the “dynamic” binding abilities of indefinites can be modeled in TLG. It remains to be shown how this framework can take the descriptive content of indefinites into account, thereby avoiding the Donald Duck problem, the empty set problem, and the bound pronoun problem. The basic idea underlying my proposal can be sketched as follows. Recall that I assumed that an indefinite like something denotes the identity function over individuals. Let us extend this idea to other indefinites. For some philosopher, I assume that it denotes the identity function as well, but the domain of this function is confined to the set of philosophers. The application of this function to non-philosophers is not defined. This function combines with its linguistic environment in the

250


manner discussed above for indefinites with trivial descriptive content. A sentence like (51)

John invited some philosopher.

will denote a partial function f from individuals to truth values. Applying f to a philosopher that was invited by John yields the value 1. Applying f to a philosopher that was not invited by John yields the value 0. The application of f to non-philosophers is not defined. The sentence is true with respect to a sequence e iff there are individuals c such that applying f to c yields the value 1. This is the case iff there is at least one philosopher that was invited by John. Likewise, the static closure of the denotation of (51) is 1 iff there is a c such that f c = 1 and 0 otherwise. So the descriptive content of an indefinite is interpreted as a domain restriction for the argument place that corresponds to this indefinite. Existentially closing such an argument place has the effect of asserting the existence of an element of this domain. This makes sure that the descriptive content of an indefinite always has the same scope as its existential impact. Thus we avoid the Donald Duck problem. If this domain happens to be empty, both static closure and the global truth definition lead to falsehood, so there is no empty set problem either. Finally, since the descriptive content of an indefinite always has the same scope as its existential impact, a wide scope reading for the indefinite in (49) is excluded, since then the pronoun could not be bound by the quantifier. So the bound pronoun problem does not arise either.7 To formalize this idea, I extend the term language. ε-abstraction now optionally comes with an explicit domain of the function that is created. I thus add the following clauses to the syntax and the semantics of the term language respectively:

Definition 64 1 If x is a variable of type A, M is a term of type t, and N is a term of type B, then εxM N is a term of type B A . 2 εxM N M,g = {c, N M,g[x→c] |c ∈ Dom(A) ∧ M M,g[x→c] = 1} 7 A very similar analysis could probably be carried out within an unselective bindingframework if the descriptive content of indefinites is analyzed as a restriction on the corresponding variables, and if restricted variables are assumed to be undefined if their value does not obey their restriction. Farkas, 1999 points out that the Donald Duck problem can be avoided by using restricted variables, but she does not develop a semantics with a partial interpretation function.

Indefinites

251

So the denotation of εxM N is similar to the denotation of λxN , except that the domain of this function is restricted to the extension of λxM . The truth definition and the definition of semantic closure on denotations have to be adjusted accordingly. Basically, while existential closure of ε-arguments involves existential quantification over the whole domain of the type of the bound variable, we now only quantify over the extension of the restriction.

Definition 65 (Truth) 1 e |= α : t iff α = 1 2 e |= α : A|e iff e − 1 |= αe1 : A 3 e |= α : Ae iff ∃c ∈ Dom(α) : e |= αc : A

Definition 66 (Static closure of sentential denotations) 1 ↓ (α : t) = α : t 2 ↓ (α : S|e) = λc. ↓ (αc) :↓ S|e

3 ↓ (α : S e ) = {(↓ (αc) :↓ S)|c ∈ Dom(α)} There are some noteworthy facts concerning the behavior of restricted abstraction.

Fact 5 1 e |= εx1,M1 · · · εxn,Mn N : t iff e |= ∃x1 .M1 ∧· · ·∧∃xn .Mn ∧N 2 (↓ εxM N ) : t = ∃x(M ∧ ↓ N ) 3 εxM [((εzN O)x)/y] = εxN [x/z] M [(O[x/z])/y] provided x is free for z in O and N , and no variable in N that is bound on the left hand side is free on the right hand side Proof: Immediate from the definitions.

The first two parts simply state that existential closure of restricted εabstraction turns it into restricted existential quantification. The third part is basically a restricted version of β-reduction for ε-abstraction. If we apply an ε-abstract εzN O to a variable x that is itself ε-bound, we may perform β-reduction (subject to the usual restrictions) and thus simplify the function application to O[x/z], but the restriction N on z has to be passed on to the ε-operator that binds x. The side condition ensures that no variable in N may become unbound by this operation.

252


Indefinites are still analyzed as identity functions, but these are created by restricted ε-abstraction, and the common noun phrase of an indefinite NP supplies the restriction on the abstract. So the lexical entry for the indefinite determiner some comes out as in (52). (The indefinite article a is treated analogously.) (52)

some – λP εxP x x : npnp /n

In the remainder of this section I will demonstrate that this treatment of the descriptive contents of indefinites adequately extends the treatment of indefinites of the previous two sections to the general case, and that it avoids the problems of the unselective binding approach and the choice function approach that were discussed above. Let us start the discussion with a simple example like (53)

John invited some philosopher.

The syntactic derivation does not differ from previous examples where the descriptive content of the indefinite was empty. It is given in Figure 6.13 for completeness. some

philosopher

lex

λP εxP x x npnp /n invited John john’ np

lex

lex

invite’ (np\s)/np

philosopher’ n

εxphilosopher’x x npnp (εxphilosopher’x x)y np

invite’((εxphilosopher’x x)y) np\s invite’((εxphilosopher’x x)y)john’ s

lex /E

i /E

\E

∧, I

εy.invite’((εxphilosopher’x x)y)john’ snp Figure 6.13.

Derivation of (53)

The semantic representation of the sentence is (54a). According to the third part of Fact 5, this is equivalent to (54b), which in turn has the same truth conditions as (54c). (54)

a. εy.invite’((εxphilospher’x x)y)john’

253

Indefinites

b. εyphilospher’y .invite’yjohn’ c. ∃y(philospher’y ∧ invite’yjohn’) I continue with another look at the interaction between indefinites and negation. Example (55) is analogous to (17) apart from the fact that someone has been changed to some farmer. (55)

Some farmer doesn’t beat his donkey.

The syntactic derivations of (55) are structurally identical to the four derivations of (17) (which are given in the figures 6.5 – 6.8). They lead to the four semantic representations in (56). (56)

a. ∼ πwεv.beat’(donkey of’w)((εxfarmer’x x)v) b. εv ∼ πw.beat’(donkey of’w)((εxfarmer’x x)v) c. ∼ εv.beat’(donkey of’((εxfarmer’x x)v)) ((εxfarmer’x x)v) d. εv ∼ beat’(donkey of’((εxfarmer’x x)v)) ((εxfarmer’x x)v)

According to the third part of Fact 5, these terms can be rewritten by the equivalent (57)

a. b. c. d.

∼ πwεvfarmer’v .beat’(donkey of’w)v εvfarmer’v ∼ πw.beat’(donkey of’w)v ∼ εvfarmer’v .beat’(donkey of’v)v εvfarmer’v ∼ beat’(donkey of’v)v

Some further elementary transformations render these representations truth conditionally equivalent to8 (58)

a. b. c. d.

πw¬∃v(farmer’v ∧ beat’(donkey of’w)v) πw∃v(farmer’v ∧ ¬beat’(donkey of’w)v) ¬∃v(farmer’v ∧ beat’(donkey of’v)v) ∃v(farmer’v ∧ ¬beat’(donkey of’v)v)

So in all four readings, the restriction on ε-abstraction is always turned into a restriction of an existentially quantified variable. This fact accounts for the absence of a Donald Duck problem in the present account. Reconsider the critical example (45), which is repeated here in 8 The

calculation for (b) makes use of the fact that εxM πyN and πyεxM N are truth conditionally equivalent if y is not free in M , which follows directly from the definitions.

254


a slightly modified form as (59a). Giving the indefinite wide scope over the conditional leads to the semantic representation (59b). (59)

a. If John invites some philosopher, Max will be offended. b. εx(invite’((εyphilosopher’y y)x)john’ → offended’max’)

Transferring the restriction on the inner ε to the outer ε leads to (60a). Expanding the abbreviational convention for → gives us (b), and employing the correspondence between the Dekker-connectives and the classical first order connectives makes this equivalent to (c). This in turn is truth conditionally equivalent to (d). (60)

a. b. c. d.

εxphilosopher’x (invite’xjohn’ → offended’max’) εxphilosopher’x ∼ (invite’xjohn’ & ∼ offended’max’) εxphilosopher’x ¬(invite’xjohn’ ∧ ¬offended’max’) ∃x(philosopher’x ∧ (invite’xjohn’ → offended’max’))

Note that the truth conditional equivalence between (59b) and (60d) holds for all models, including those where philosopher’ has an empty extension. If there are no philosophers, the sentence is predicted to be false. So the present account avoids the empty set problem. The semantic reason for this is the fact that the denotation of (59a) is a function from philosophers to truth values. If there are no philosophers, this is the empty function. A truth valued function is true according to our truth definition iff there are arguments for which the function returns the value 1. The empty function never returns any value, therefore the sentence is false in such a model. Let us now turn our attention to a run-of-the-mill conditional donkey sentence like the classical (61)

If a farmer owns a donkey, he beats it.

The syntactic derivation of this sentence is analogous to the one for (21). It leads to the semantic representation (62)

εxfarmer’x εydonkey’y own’yx → πzπwbeat’wz

Expanding the definition of → leads to (63a). Dynamic binding makes this equivalent to (b). Employing the interaction between the Dekker connectives, static closure and the classical connectives allows us to rewrite (b) as (c). Using the second part of Fact 5 twice, we get (d), and this is first-order equivalent to (e).

255

Indefinites

(63)

a. b. c. d. e.

∼ (εxfarmer’x εydonkey’y own’yx & ∼ πzπwbeat’wz) ∼ εxfarmer’x εydonkey’y (own’yx & ∼ beat’yx) ¬ ↓ εxfarmer’x εydonkey’y (own’yx ∧ ¬beat’yx) ¬∃x(farmer’x ∧ ∃y(donkey’y ∧ own’yx ∧ ¬beat’yx)) ∀x(farmer’x → ∀y(donkey’y ∧ own’yx → beat’yx))

Quantificational donkey sentences are analyzed in a similar way. Apart from the form of the indefinite, the example in (64) is analogous to (30), which was discussed in the previous section. Again, the syntactic derivation is analogous (cf. Figure 6.12 on page 244), and we end up with the semantic representation in (64b,c) for the weak and the strong readings, respectively. (64)

a. Every farmer who owns a donkey beats it. b. ∀z(↓ εx(farmer’z ∧ own’((εydonkey’y y)x)z) → ↓ (εx(farmer’z ∧ own’((εydonkey’y y)x)z) & πu.beat’uz)) c. ∀z(↓ εx(farmer’z ∧ own’((εydonkey’y y)x)z) → ↓ (εx(farmer’z ∧ own’((εydonkey’y y)x)z) → πu.beat’uz))

β-reduction leads to (65)

a. ∀z(↓ εxdonkey’x (farmer’z ∧ own’xz) → ↓ (εxdonkey’x (farmer’z ∧ own’xz) & πu.beat’uz)) b. ∀z(↓ εxdonkey’x (farmer’z ∧ own’xz) → ↓ (εxdonkey’x (farmer’z ∧ own’xz) → πu.beat’uz))

Expanding the definition for → and performing dynamic binding (together with some minor routine manipulations) leads to the reformulations (66)

a. ∀z(↓ εxdonkey’x (farmer’z ∧ own’xz) → ↓ εxdonkey’x .(farmer’z ∧ own’xz) & beat’xz) b. ∀z(↓ εxdonkey’x (farmer’z ∧ own’xz) → ∼↓ εxdonkey’x .(farmer’z ∧ own’xz) & ∼ beat’xz)

Crucially, all ε-operators in these representations are immediately preceded by ↓. Due to Fact 5, this amounts to existential quantification over the corresponding argument places, i.e., we get (67)

a. ∀z(↓ ∃x(donkey’x ∧ farmer’z ∧ own’xz) → ↓ ∃x(donkey’x ∧ farmer’z ∧ own’xz) & beat’xz)

256


b. ∀z(↓ ∃x(donkey’x ∧ farmer’z ∧ own’xz) → ∼↓ ∃x(donkey’x ∧ farmer’z ∧ own’xz) & ∼ beat’xz) This in turn is equivalent to (68)

a. ∀z(∃x(donkey’x ∧ farmer’z ∧ own’xz) → ∃x(donkey’x ∧ own’xz ∧ beat’xz)) b. ∀z(farmer’z → ∀x(donkey’x ∧ own’xz → beat’xz))

So to sum up this point, the domain restriction on ε-bound variables is always turned into a restriction on the corresponding existential quantifier when this ε-slot is existentially bound. This avoids the Donald Duck problem, and the treatment of donkey constructions that was proposed in the previous section carries over to indefinites with non-trivial restrictions without problems. It remains to be shown how the present system handles cases where the descriptive part of an indefinite contains a bound pronoun. A simple example is (69)

Every girli visited some boy that shei fancied.

In the indicated binding configuration, the subject quantifier must take scope over the indefinite object because otherwise the corresponding proof tree would not be well-formed (cf. the discussion of this issue on page 167 in Chapter 4). So the only derivation of (69) is the one that is sketched in Figure 6.14 on the next page.9 The semantic representation of (69) is thus (70)

∀x(↓ girl’{x} →↓ (girl’{x} & (λuεv.visit’((εyboy’y∧fancy’yu y)v)u){x}))

According to the laws of structural function application,this is equivalent to (71)

∀x(↓ girl’x →↓ (girl’x &εv.visit’((εyboy’y∧fancy’yx y)v)x))

β-reduction leads to (72)

∀x(↓ girl’x →↓ (girl’x &εvboy’v∧fancy’vx .visit’vx))

Some elementary transformations lead to the equivalent first order formula 9 I use only the weak reading of every here since the strong reading leads to an equivalent result.

(73) i

lex

∧, k

Figure 6.14.

Derivation of (69)

∀x(↓ girl’{x} →↓ (girl’{x} &(λuεv.visit’((εyboy’y∧fancy’yu y)v)u){x})) s

εv.visit’((εyboy’y∧fancy’yu y)v)u snp

visit’((εyboy’y∧fancy’yu y)v)u s

\E

qE, i

(εyboy’y∧fancy’yu y)v np

/E

k

λw.boy’w ∧ fancy’wu n

εyboy’y∧fancy’yu y npnp

lex

visit’((εyboy’y∧fancy’yu y)v) np\s

visit’ (np\s)/np

λQ.∀x(↓ girl’{x} →↓ (girl’{x} & Q{x})) q(np, S, s) [u]j np

visited

every girl

some λP εyP y y npnp /n

boy that she f ancied πzλw.boy’w ∧ fancy’wz n|np

/E

|E, i

Indefinites

257

∀x(girl’x → ∃v(boy’v ∧ fancy’vx ∧ visit’vx))

According to these truth conditions, the sentence could also be true in a situation where two girls fancy the same boys but visit different boys. This is in line with the semantic intuitions. As discussed above, the

258


choice function approach furthermore predicts a non-existent reading where girls that fancy the same boys must visit the same boy to make the sentence true. It might be argued that this reading is actually there but hard to detect, because it is logically stronger than the ordinary narrow-scope reading (73). This is not the case anymore though if we use a downward monotonic quantifier in subject position, as in (74)

At most three girls visited a boy that they fancied.

According to the choice function approach, this sentence should have the reading given in (75a), which is truth-conditionally equivalent to (75b). (75)

a. ∃f.CH(f )∧|λx.girl’x∧visit’(f (λy.boy’y∧fancy’yx))x| ≤ 3 b. |λx.girl’x ∧ ∀y((∃z(boy’z ∧ fancy’zx) → boy’y ∧ fancy’yx) → visit’yx)| ≤ 3

Under the assumption that every girl fancies some boy, the prediction is that the sentence has a reading that is synonymous to At most three girls visited every boy that they fancied. Intuitions are fairly solid here that such a reading does not exist. Intuitively, this bound pronoun problem in connection with the choice function approach is similar to the Donald Duck problem of unselective binding: In both approaches, the interpretation of the descriptive content of an indefinite is divorced from its existential impact, while these two semantic components of indefinites always occur in tandem. Modelling the scope of indefinites by means of existential closure over partial functions covers this fact. It deserves to be mentioned that the bound pronoun problem of the choice function approach has been taken as evidence by Geurts, 2000 and by Endriss, 2001 that the scope of indefinites is assigned by means of some form of syntactic movement. The present solution proves that this conclusion is not inevitable. The scope of indefinites is assigned in an entirely surface compositional way here, without making reference to transformations between syntactic representations. (Recall that the manipulations of the semantic representation that I used in the discussion above are meaning preserving reformulations in the semantic representation language without any significance for the meanings that the theory assigns to natural language expressions.)

6.

Sluicing

Donkey anaphora is an empirical domain where the grammar of indefinites is intricately linked with the grammar of anaphora. The same

Indefinites

259

holds for the phenomenon of sluicing. After a brief recapitulation of the basic issues that arise in connection with this form of ellipsis, I will demonstrate that the LLC-treatment of anaphora in combination with the analysis of indefinites that was developed in the previous sections can easily be combined in a natural approach to sluicing. Briefly put, sluicing is a version of ellipsis where under certain contextual conditions, a bare wh-phrase stands proxy for an entire (embedded or matrix) question. The phenomenon was first systematically described in Ross, 1969, where also the name is coined. Typical examples are (76)

a. She’s reading something, but I don’t know what. b. Some guy knows how to get in here. Do you have any idea who? c. They hired a new system administrator. Guess who!

As with VP ellipsis, sluicing involves a source clause and a target clause. The source clause is typically a declarative clause which contains an indefinite NP. The target clause is (interpreted as) the question that is obtained if this indefinite is replaced by a wh-phrase. At the surface structure, everything but this wh-phrase is deleted. So on the face of it, the examples above are related (via deletion, reconstruction or whatever) to the non-elliptical counterparts (77)

a. She’s reading something, but I don’t know what she’s reading. b. Some guy knows how to get in here. Do you have any idea who knows how to get in here? c. They hired a new system administrator. Guess who they hired!

Interestingly, sluicing constructions remain grammatical in cases where the non-elliptical counterpart involves an island violation of the whphrase. Consider the following example (like some of the subsequent examples, it is taken from Merchant, 1999).10 (78)

10 I

a. They wanted to hire somebody who speaks a Balkan language, but I don’t know which.

follow the standard assumption that the deletion of the common noun phrase inside the wh-phrase (i.e., which instead of which Balkan language) is independent of sluicing, and I will ignore this kind of ellipsis.

260


b. *They wanted to hire somebody who speaks a Balkan language, but I don’t know which Balkan language they wanted to hire somebody who speaks. In the non-elliptical version (78b), the wh-phrase which Balkan language binds a gap inside a relative clause island. Therefore the example is ungrammatical. Nonetheless, the corresponding sluicing construction (78a) is impeccable. The same point can be made with regard to a whole range of syntactic island constraints. The following list is not meant to be exhaustive.

Adjunct islands. (79)

a. Ben will be mad if Abby talks to one of the teachers, but she couldn’t remember which. b. *Ben will be mad if Abby talks to one of the teachers, but she couldn’t remember which of the teachers Ben will be mad if Abby talks to. (from Merchant, 1999)

Complex NP islands. (80)

a. The administration has issued a statement that it is willing to meet with one of the student groups, but I’m not sure which one. b. *The administration has issued a statement that it is willing to meet with one of the student groups, but I’m not sure which one the administration has issued a statement that it is willing to meet with. (from Chung et al., 1995)

Sentential subject islands. (81)

a. That certain countries would vote against the resolution has been widely reported, but I’m not sure which ones. b. *That certain countries would vote against the resolution has been widely reported, but I’m not sure which ones that would vote against the resolution has been widely reported. (from Chung et al., 1995)

Embedded question islands. (82)

a. Sandy was trying to work out which students would be able to solve a certain problem, but she wouldn’t tell us which one.

Indefinites

261

b. *Sandy was trying to work out which students would be able to solve a certain problem, but she wouldn’t tell us which one Sandy was trying to work out which students would be able to solve. (from Chung et al., 1995)

Coordinate structure constraint. (83)

a. Bob ate dinner and saw a movie that night, but he didn’t say which. b. *Bob ate dinner and saw a movie that night, but he didn’t say which movie Bob ate dinner and saw that night. (from Merchant, 1999)

These facts suggest that sluicing does not involve syntactic operations like reconstruction or deletion. Rather, an approach that requires some form of semantic correspondence between source clause and target clause seems viable. On the other hand, the morphological form of the remnant wh-phrase is not arbitrary. In languages with overt case marking, the wh-phrase has to have the same case as the indefinite in the source. The following German example is from Ross, 1969, where this effect (as well the island insensitivity of sluicing) was first observed. (84)

a. Er will jemandem schmeicheln, aber sie wissen nicht {wem / *wen}. He wants someoneDAT flatter but they know not {whoDAT / *whoACC } b. Er will jemandem schmeicheln, aber sie wissen nicht, {wem / *wen} er schmeicheln will. he wants someoneDAT flatter but they know not {whoDAT / *whoACC } he flatter wants ‘He wants to flatter someone, but they don’t know who (he wants to flatter)’

(85)

a. Er will jemanden loben, aber sie wissen nicht {*wem / wen}. He wants someoneACC praise but they know not {*whoDAT / whoACC } b. Er will jemanden loben, aber sie wissen nicht, {*wem / wen} er loben will. he wants someoneACC flatter but they know not {*whoDAT / whoACC } he praise wants ‘He wants to praise someone, but they don’t know who (he wants to praise)’

262


The German verbs schmeicheln (‘to flatter’) and loben (‘to praise’) govern dative case and accusative case respectively on their object. The sluiced wh-phrases in the (a)-examples have to have the same case marking as the corresponding indefinites in the source clause. In other words, they must have the same case marking that they would have in the corresponding non-elliptical constructions. Under a pure identity-of-meaning approach, this morphological correspondence would seem mysterious. A third peculiarity of sluicing is the fact that the wh-phrase in the target clause and the corresponding indefinite in the source clause must have parallel scope. (This has been pointed out by Chung et al., 1995.) Recall that in (43b)—repeated here as (86a)—the indefinite some movie may have narrow scope or wide scope relative to the quantifier every girl. If this sentence is used as a source clause in sluicing, only the wide scope reading is possible. (86)

a. Every girl will be happy if some movie is shown. [∃ > ∀][∀ > ∃] b. Every girl will be happy if some movie is shown, but I don’t know which movie. [∃ > ∀] *[∀ > ∃]

As I will try to demonstrate in the remainder of this section, the theory of indefinites that was developed in the previous sections lends itself naturally to a TLG account of sluicing that is based on LLC and covers the empirical generalizations just discussed. Let us consider a simple example like (87)

John invited someone, but it is unclear who John invited.

The details of the semantics of questions that one adopts are of minor importance for the subsequent discussion. Therefore I remain neutral in this respect and represent the semantics of the sluiced question who John invited as (88a). The interrogative pronoun who has the lexical entry in (88b). So the missing piece of meaning that is required to interpret the ellipsis in (87) is (88c). (88)

a. ?x.invite’xjohn’ b. who – λP ?xP x : Q/(s/np) c. λx.invite’xjohn’

The denotation of the term in (88c) is identical to the denotation of the source clause John invited someone according to the semantics of indefinites given above. So the adequate reading can easily be derived via anaphora resolution if we assign the interrogative pronoun who the additional lexical entry

263

Indefinites

(89)

who – λP ?xP x : Q|(snp )

The semantics of the two readings of who is identical. In the sluicing version, who is an anaphor that needs a declarative clause containing an indefinite as antecedent to yield a question. This is just a formal reformulation of the informal description of sluicing patterns given above. Note that the only difference between the two lexical entries for who lies in the fact that they use different substructural versions of Intuitionistic implications. This is similar to the relation between the ordinary English auxiliaries and their VPE-counterparts. They too have pairwise identical meanings, and their categories differ with regard to the implication they use (vp/vp versus vp|vp). An analogous lexical ambiguity has to be assumed for all interrogative pronouns and interrogative determiners. What happens if the descriptive content of the indefinite is nontrivial, as in (90)? (90)

John invited some philosopher, but it is unclear which philosopher.

Here, the missing piece of meaning is also λx.invite’xjohn’, but the meaning of the source clause is the partial function εxphilosopher’x .invite’xjohn’ We can transform this partial function into a total function by means of an operation tot’, which is defined as 1 iff f c = 1 ∀f, c.tot’f c = 0 else So for the sluicing version of the interrogative determiner which we have to assume the lexical entry11 (91)

which – λP λR?x.P x ∧ tot’Rx : Q|(snp )/n

The (simplified) syntactic derivation for (90) is thus as in given in Figure 6.15 on the following page. The semantic representation which corresponds to this derivation is (92a), which is truth conditionally equivalent to (92b). (I treat but as synonymous with and.) 11 The

correct analysis of the semantic contribution of the restrictor of which is an intricate issue since it gives rise to a de re/de dicto ambiguity (cf. the discussion in Groenendijk and Stokhof, 1984, pp 89). I think that an analysis using Boolean conjunction as in the entry in (91) can be maintained if we admit free world indexing of common nouns in the restrictor of operators, which is arguably necessary anyway. The issue is orthogonal to our present concerns, so I omit further discussion.

np

John

lex

s

[snp ]j

s

k

∧, i

(np\s)/np

invited

np\s

lex

\E

npnp /n

some

np

npnp

lex

s ∧, k

/E

lex

Figure 6.15.

snp

/E

i

n

philosopher

lex s\s

\E

s

/E

|E, j

n

/e

philosopher

/E

Q

s/Q

Derivation for (90)

(s\s)/s

but

Q|(snp )

lex

it is unclear

Q|(snp )/n

which

lex

264 ANAPHORA AND TYPE LOGICAL GRAMMAR

Indefinites

(92)

265

a. εx.(εyphilosopher’y invite’yjohn’)x∧ unclear’?z.philosopher’z∧ tot’(εyphilosopher’y invite’yjohn’)z b. ∃x.philosopher’x ∧ invite’xjohn’∧ unclear’?z.philosopher’z ∧ invite’zjohn’

Let us now change the example slightly to (93)

John invited some philosopher, but it is unclear who.

Here, the descriptive content of the indefinite in the source clause does not coincide with the restrictor of the wh-phrase in the target. Nevertheless, (93) is synonymous to (90). To derive this fact, I have to revise the lexical entry of who slightly. The example (93) demonstrates that the antecedents of who-sluices may be partial functions. Therefore, the totalizing function tot’ has to be incorporated into the semantics of sluicing-who as well. The modified entry is thus12 (94)

who – λP ?xtot’P x : Q|(snp )

Given this, the interpretation of (93) comes out as in (95a), which is truth-conditionally equivalent to (92b) as well. (95)

εx.(εyphilosopher’y invite’yjohn’)x∧ unclear’?z.tot’(εyphilosopher’y invite’yjohn’)z

The fact that the descriptive content of the source indefinite always serves as an additional restriction of the remnant wh-phrase in the sluiced question was first observed (and accounted for) in Chung et al., 1995. It provides a major stumbling block for any theory that analyzes sluicing via syntactic copying or deletion. The prime obstacle for a purely syntactic approach to sluicing is of course the lack of island sensitivity of sluiced constructions. Ross, 1969 suggested that syntactic island constraints only apply to phonetically non-empty structures. Versions of this idea recur at several places in the relevant literature, most recently—in a qualified form—in Merchant, 1999. A full discussion of how this problem is dealt with in TLG would of course require a discussion of islandhood within this framework. Like the problem of restrictions on quantifier scope, this issue goes beyond the 12 We

may as well assume that ordinary who has the same semantics and thus maintain the synonymy between the two readings, since the argument of who in ordinary questions is always a total function and the presence of tot’ does not make any difference.

266


scope of this work, and the interested reader is referred to the relevant discussion in Morrill, 1994. However, quite independently of the precise type logical analysis of islandhood that we adopt, it should be clear that sluicing is predicted to be insensitive to it. Consider again a relevant minimal pair such as (96)

a. They wanted to hire somebody who speaks a Balkan language, but I don’t know which one. b. *They wanted to hire somebody who speaks a Balkan language, but I don’t know which Balkan language they wanted to hire somebody who speaks.

The wh-phrase which one in (96a) has the syntactic category Q|(snp ). So it acts as a question as soon as the linguistic context supplies an antecedent of category snp —a clause containing a wide scope indefinite. The source clause in (96a) has this category, provided the indefinite a Balkan language is given wide scope there. The (ungrammatical) nonelliptical question in (96b) plays no role in the analysis of (96a), so no matter how we exclude (96b), this analysis will not affect the analysis of (96a). Instead, we predict that the locality constraints in sluicing exactly mirror the locality constraints for the scope of indefinites. Since the latter is in principle unbounded, so is sluicing. The discussion in the previous paragraph readily suggests an explanation of one half of the scope parallelism facts mentioned above. Note that the source clause in (96a), taken in isolation, is ambigous between a narrow scope reading and a wide scope reading of the indefinite a Balkan language. According to the present theory of indefiniteness, this semantic difference is reflected in the syntactic category of this clause. If the indefinite has narrow scope, the matrix clause has the category s. Wide scope of the indefinite corresponds to the category snp for the matrix clause, and only in this category may it serve as antecedent for the sluice in the second conjunct. These considerations derive one half of the scope parallelism constraint: the indefinite in the source clause that licenses sluicing must at least have scope over the entire source clause. We predict that it may have wider scope though. The derivation in Figure 6.15 on page 264 already provides an example. If cross-clausal binding of pronouns by indefinites is analyzed in the way done here, i.e., by assuming that the binding indefinite takes scope over the entire construction, this conclusion is desired and even inevitable. Nonetheless in examples like (97), only the reading where the indefinite and the wh-phrase take exactly parallel scope (i.e., some girl takes narrow scope with respect to knows) is possible.

Indefinites

(97)

267

Everybody knows that John wants to marry some girl, but John’s mother still doesn’t know which one (John wants to marry).

This might be due though to the fact that constituent questions trigger existential presuppositions. So the sluiced question in (97) triggers the presupposition John wants to marry some girl (with a girl having wide scope). If the indefinite takes wider scope than the wh-phrase, this presupposition has to be bound via bridging, while it can be directly bound if the scopes are parallel. Let us now turn to the third empirical generalization discussed above, the morphological parallelism between the licensing indefinite in the source clause and the remnant wh-phrase in the target. I repeat Ross’ example. (98)

a. Er will jemandem schmeicheln, aber sie wissen nicht {wem / *wen}. He wants someoneDAT flatter but they know not {whoDAT / *whoACC } ‘He wants to flatter someone, but they don’t know whom’ b. Er will jemanden loben, aber sie wissen nicht {*wem / wen}. He wants someoneACC praise but they know not {*whoDAT / whoACC } ‘He wants to praise someone, but they don’t know who (he wants to praise)

For a detailed discussion of the treatment of morphology in TLG, I have to refer the reader once again to Morrill, 1994, but for the present purposes a sketch will do. Suffice it to say that basic categories in a morphologically informed version of TLG are not unstructured atoms but (atomic) first order formulae, i.e., they consist of a predicate (unary predicates suffice) which takes complex terms as arguments. Morphological feature structures can be coded as first order terms. Underspecified aspects of the morphological structure can be represented as universally quantified individual variables. Morphological feature structures can thus be incorporated into a first order version of LLC+∧ . In such a first order version of TLG, the category of a dative NP in German will be an atomic formula of the form np(...dat...), where dat is a (possibly complex) term representing the case information “dative”. Let us abbreviate this category with np(dat). Likewise, the category of accusative NPs shall be sketched as np(acc). The German interrogative pronouns wem (dative) and wen (accusative) thus have the syntactic categories Q/(s/np(dat)) and Q/(s/np(acc)) respectively, i.e., they bind an np-position with the matching case information in the interrogative clause.

268


The case features of an indefinite NP appear at two places: at the argument and the result of the substructural implication in its syntactic category. An indefinite in dative case has the category np(dat)np(dat) , and likewise for other cases. A clause containing a wide-scope indefinite in dative thus has the category snp(dat) . The sluicing version of the dative interrogative pronoun has the category Q|(snp(dat) ), i.e., it requires a clause as antecedent that contains a wide scope indefinite with dative case. The ungrammatical versions of (98) are excluded because the case features of the anaphoric wh-phrase do not match with the corresponding feature in the antecedent. Among the analyses of sluicing from the literature, the present one is probably closest to the one from Chung et al., 1995. These authors adopt a DRT style unselective binding analysis of indefinites. According to them, sluicing invokes the copying of the LF of the IP of the source clause into the target clause. So after this copying operation, our previous example (99a) would receive approximately an LF as (99b). (99)

a. John invited some philosopher, but it is unclear who. b. ∃x[IP John invited some philosopherx ], but it is unclear whox [IP John invited some philosopherx ].

So the indefinite some philosopher introduces a free variable both in the source clause and the target clause. This variable is bound by unselective existential closure in the source and by the wh-operator in the target. If the source did not contain a free variable (i.e., an indefinite), vacuous binding in the target and thus ungrammaticality would ensue. Furthermore, the copying mechanism ensures that the descriptive content of the indefinite contributes to the interpretation of the target question. Finally, the connection between the wh-operator in the target and the variable that it binds is not established via movement and thus not predicted to be sensitive to island constraints. The main problem of Chung et al.’s (1995) approach is inherited from the unselective binding approach as such—it is susceptible to the Donald Duck problem. The example (100a) will receive the LF (100b). (100)

a. Max will be offended if we invite some philosopher, but it is unclear who. b. Max will be offended if we invite some philosopher, but it is unclear whox [IP Max will be offended if we invite some philosopherx ].

So the question part of this sentence can be paraphrased as which x is such that Max will be offended if x is a philosopher that we invite. Given

Indefinites

269

that Donald Duck is not a philosopher, “Donald Duck” should be a good answer to this question, but it isn’t. To sum up the discussion of sluicing, it can be said that the present theory covers the core facts of this kind of ellipsis in a simple and adequate way. However, our theory is essentially an identity-of-meaning theory, and the literature contains quite a few instances of sluicing that prima facie do not lend themselves easily to such an analysis. The most problematic cases are those where sluicing is not licensed by an overt indefinite; implicit existentially quantified arguments in the source clause can do that job as well. Chung et al., 1995 call this version of sluicing “sprouting”. The following illustrate this phenomenon (101)

a. She served the soup, but I don’t know to whom. (from Chung et al., 1995) b. She was reading, but I couldn’t make out what. (from Chung et al., 1995) c. He’s writing, but you can’t imagine where/why/how fast. (from Ross, 1969)

While it might be suggestive to assume that here, the licensing indefinite is somehow incorporated into the verb, such an analysis won’t work in examples like the following (also from Chung et al., 1995). (102)

Joan ate dinner but I don’t know with whom.

Here, the source clause does not entail that Joan ate dinner with someone, so the elided material in the target clause is not present in its entirety in the source clause, no matter what identity criterion we assume. So a plain identity-of-meaning theory like the present one has nothing to say about these cases. One might argue that these cases seem to involve some version of bridging, a phenomenon that is well attested in all classes of anaphora.

7.

Summary and Desiderata

The main purpose of this chapter was to demonstrate that the LLCanalysis of anaphoric pronouns can be extended to donkey pronouns. However, the difficult part of the analysis of donkey anaphora is not how to analyze pronouns but how to analyze indefinites, so most space was devoted to this problem. The chapter consisted of three parts. In the first part, I introduced Dekker’s Predicate Logic with Anaphora, and I showed how the PLA-analysis of indefinites can be combined with the LLC-treatment of pronouns. I thereby extended LLC to the Categorial logic LLC+∧ . Basically, indefinites are treated analogously to pronouns,

270


with the crucial difference that indefinites cannot be resolved. Another way to look at it is to say that I gave a type logical reformulation of Heim-style DRT, where free variables are replaced by identity functions. The Novelty Condition for indefinites is reconstructed as the absence of resolution rules. By translating Dekker’s analyses of the standard logical connectives of conjunction and negation into the term language accompanying LLC+∧ , we were able to reproduce the core of the DRT analysis of donkey anaphora within TLG. The second part focused on the issue of how the descriptive content of indefinites is to be analyzed. I suggested that indefinites in general denote (possibly partial) identity functions over individuals, and that the descriptive content of an indefinite supplies the domain of this function. These functions function-compose with their linguistic environment, and the descriptive content of an indefinite is thus inherited by the denotations of its super-constituents. I showed that this mechanism, paired with an operation of existential closure of argument slots, circumvents certain problems which plague other current theories of the scoping of indefinites. The last part of the chapter applied these findings to the problem of sluicing. I showed that the functional semantics of indefinites, paired with the LLC-mechanism of anaphora, lends itself naturally to a simple identity-of-meaning theory for sluicing. The basic empirical generalizations about this kind of ellipsis fall out immediately. Each of these three topics is of considerable complexity, and a host of issues has to remain untouched, let alone resolved. As for our account of donkey anaphora, this analysis basically reformulates “classical” Dynamic Semantics (i.e., Dynamic Predicate Logic in the sense of Groenendijk and Stokhof, 1991b), even though the philosophical underpinning is different. This of course means that the empirical weaknesses of DPL are inherited. Our account of the scoping of indefinites is confined to singular NPs. The issue becomes considerably more intricate if plural NPs are taken into account. Weak quantifiers like three men are as unrestricted in their scope taking behavior as singular indefinites, so one would expect the same mechanisms to be at work. However, with plural indefinites, two scoping mechanisms are involved. For instance, sentence (103a) (taken from Winter, 1997 who attributes it to Ruys, 1995) has a reading that can be paraphrased as (103b). (103)

a. If three relatives of mine die, I’ll inherit a fortune. b. There are three relatives of mine, and if each of them dies, I’ll inherit a fortune.

Indefinites

271

The specific reading of three relatives of mine thus actually involves two quantifiations, a wide scope existential quantification over sets of relatives of mine with the cardinality three, and a narrow scope universal quantification over elements of this set. It seems that the former is as unrestricted as the existential impact of singular indefinites, while the universal quantification is confined to the local clause (i.e., obeys the same constraints as other non-indefinite quantifiers). Reniers, 1997 gives a TLG analysis of these facts using two versions of Moortgat’s in situ binder. A reformulation into the present framework, where locally restricted quantification is handled by qE and unrestricted existential quantification by ∧ is easy to provide. However, a lot of issues remain open in in connection with this issue, such as the question of which quantifiers exactly can be subject to a double scope interpretation, and what properties qualify a determiner to belong to that class. These issues have been discussed in different theoretical frameworks in Szabolcsi, 1997 and Endriss, 2001. It remains to be seen whether the findings of these authors are compatible with the present theoretical setting. Finally, the analysis of sluicing presented here remains somewhat sketchy for two reasons. First, a detailed semantics of questions has to take intensionality into account. While there is no fundamental obstacle against a Curry-Howard style intensional semantics,13 the syntaxsemantics interface becomes considerably more complex and less transparent if relativization to possible worlds is added. Therefore, this issue was left out throughout this book. Second, I believe that an adequate treatment of sluicing requires a theory of presupposition resolution and a theory of bridging, and it has to take the effects of information structure into account. While there is no lack of formal approaches to these phenomena, they are completely independent of the type logical aspects of anaphora. The discussion was therefore confined to those aspects that have a direct bearing on the topic of the book as a whole.

13 See

for instance Morrill, 1994 for a fully worked out formalization.

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About the Author

Gerhard J¨ ager is professor of linguistics at the University of Bielefeld.

281

Index

214, 219, 222, 223, 225–227, 232, 238, 256, 260, 267–269 deep a., 181 donkey a., x, xi, 256, 267, 268 surface a., 181 Anderson, 28 Andreka, 61 antecedent, 6, 8–10, 13, 14, 23, 25, 26, 28–31, 37, 39, 41, 49, 52, 55, 58, 59, 61, 63, 64, 70, 73, 83, 84, 88, 91, 93, 95, 97, 109, 110, 112, 114, 118–122, 137, 138, 142–145, 155, 156, 165, 167, 172, 175–178, 181, 183– 193, 196, 198, 200, 202–205, 209, 210, 222, 244, 260, 263, 264, 266 application (function a.), 4, 6, 8–10, 13, 17, 23, 25, 31, 41, 42, 54, 56, 75, 80, 100, 108, 121, 125, 133, 135–137, 139, 140, 148, 162, 163, 176, 185, 186, 190, 238, 247–249 structural function a., 240, 254 argument category, 2–4, 45 arrow, 31, 54, 55, 59, 146, 147 assignment function, 32, 71, 73, 206, 215, 225, 228, 229 associative, 31, 54, 57–59, 61, 65, 83, 86, 144, 148, 150, 155 auxiliary, 33, 69, 103, 173, 182, 184, 185, 198, 199, 201, 204, 209, 210, 227, 229, 261 auxiliary inversion, 103, 173 axiom, 6–10, 13, 23, 25, 27, 42, 54–56, 59, 92, 93, 122, 146–148, 150 -atic presentation, 54, 146, 148 -atic system, 54–56, 59, 147, 148

α-equivalence, 33, 141 αβη-equivalence, 33, 141 β-redex, 135, 137, 139 β-reduction, 33, 36, 131, 135, 140, 163, 249, 253, 254 ε-operator, 225, 249 η-redex, 135, 136, 139, 140, 162 η-reduction, 33, 36, 43, 45 λ-abstraction, 31, 71, 81, 136, 137 λ-calculus, ix, x, 4, 12, 13, 16, 31, 33, 34, 36–38, 43, 67, 70, 73, 135, 141, 221, 224, 225 λ-conversion, 33, 106 A, 6, 9, 13, 14, 99–101, 103, 107, 110 A’-movement, 45 Abusch, 244, 246 accessibility relation, 57 accidental coreference, 110, 112, 156, 172, 176, 177 accusative, 76, 78, 86, 88, 158, 259, 265 Ades, 65 adjective, 3, 122 adverb, 3, 158, 212 Ajdukiewicz, 3, 65 alphabet, 5, 7, 12, 42 ambiguity, 15, 74, 103, 117, 134, 154, 158, 176, 185–187, 191, 194, 195, 200, 204, 207, 212, 261 lexical a., 15, 53, 116, 261 scope a., 163, 166 spurious a., 15, 53, 207, 208, 229 structural a., 14, 15, 53, 74, 143, 237 anaphora, ix–xii, 30, 67, 69, 70, 74, 79, 86, 91–93, 96, 98, 100, 101, 106, 108, 109, 112, 114–118, 121, 122, 125, 137, 142, 143, 145, 150, 151, 155, 165, 167, 168, 170–173, 177–179, 181, 183– 187, 193, 198, 204, 211, 213,

283

284

ANAPHORA AND TYPE LOGICAL GRAMMAR identity a., 6, 9, 15, 17, 26, 41, 49, 51, 54, 58, 62, 119, 130, 131, 137, 146, 147 scheme, 6, 25, 26, 49, 54, 62

B, 77, 103, 107 Bach, 83, 102, 171, 172 Bach-Peters sentence, 110, 112, 154, 176 Bar-Hillel, 1, 2, 9, 10, 61, 65 Barry, 39, 92 Barss, 168 Barwise, 212 Basic Categorial Grammar, 1, 3–11, 13, 15–17, 19–21, 23, 25, 26, 40– 43, 46, 49, 56, 61, 62, 65 BCG, see Basic Categorial Grammar Belnap, 28 Bernardi, 165 binary reduction lemma, 63, 64, 73 binding, 45, 69, 70, 74–79, 81, 82, 84, 91– 93, 98, 99, 102–104, 106, 108, 109, 112, 114, 151, 154, 156, 157, 159–161, 165–172, 175– 179, 203, 206, 209, 214, 226, 232, 238, 243, 247, 254, 264, 266 backward b., 111, 171, 172, 175–178, 214 dynamic b., 211, 217, 234, 243, 247, 252, 253 Interpretation Rule, 94–96 Theory, 75, 237 Principle A, 78 Principle B, 78, 88, 91, 209 Principle C, 75 Bresnan, 168, 176 bridging, 265, 267, 269 implicational b., 203 Buszkowski, 9, 62 c-command, 16, 75, 76, 88, 91, 114, 116, 118, 151, 167–172, 178 canonical model, 60, 150, 151 Carpenter, 45, 66, 163, 165 Cartesian product, 119 Casalegno, 246 cataphora, 172, 176, 177 Categorial Grammar, ix, x, 1, 4, 11, 16, 17, 22, 23, 25, 46, 61, 65–67, 92, 96, 114, 171, 200 CFG, see context free grammar CG, see Categorial Grammar Chierchia, 102, 103, 105 choice function, 244, 245 approach, 245–247, 250, 254, 256 Chomsky, 43, 61, 65, 75, 108, 167 conjecture, 61 Chung, 258, 260, 263, 266, 267

Church-Rosser property, 36, 163 classical logic, 26, 27, 29, 30, 38 Cohen, 61 combinator, 17, 21–23, 46–49, 73, 75–77, 81, 96, 98, 102, 114, 116, 151, 160, 178 Combinatory Categorial Grammar, 96 combinatory logic, 22, 74 common noun, 3, 5, 239, 241, 249, 257, 261 complete, 58, 60, 61, 142, 144 completeness, xi, 59, 61, 149–151, 250 complexity, 9, 34, 50, 60, 61, 65, 122, 123, 125, 130, 141, 150, 224, 268 compositional interpretation, 12, 16 compositionality, 73, 143 principle of c., 67, 72, 219 surface c., 109 concatenation, 3, 83, 85, 86 confluent, 141, 142 constituent, 13, 14, 16, 18, 20, 21, 30, 46, 71, 76, 81, 83–85, 91, 94, 96–98, 100, 104, 105, 121, 122, 168, 170, 171, 183, 202, 213, 247, 264, 268 constructive logic, 28 context free grammar, 9–11, 61, 62, 64–66 Contraction, 26, 27, 29, 30, 49, 74, 79, 92, 93, 95, 117–119, 121, 122, 137 Cooper, 81, 109, 163 coordinate structure constraint, 258 coordination, 17–22, 43, 48, 68, 74, 78, 105, 106, 115, 158, 182, 183, 199, 205, 210 Boolean c., 18, 19 non-constituent c., 46, 106 scheme, 18–20 Cormack, 200 curried, 4, 239 Curry, 21, 22, 35, 49, 102, 178 Curry-Howard correspondence, 31, 34, 35, 38, 48, 65, 114, 118 Curry-Howard isomorphism, 35 Cut, xi, 6–10, 13–17, 25–27, 36, 41, 49–54, 56, 58, 62–64, 95, 96, 116, 118, 120–126, 128, 130–132, 142, 147, 148, 150–152, 160, 162, 222 elimination, xi, 49, 51, 54, 95, 96, 116, 118, 122, 123, 125, 130, 142, 160–162, 222 Dahl, 191, 205 Dalrymple, 69, 189, 190, 202, 205, 206 dative, 259, 265, 266 de Groote, 66 de-accenting, 192, 206

Index decidability, xi, 49, 50, 53, 96, 118, 123, 125, 160, 161, 175, 176, 222, 224 decidable, 8, 9, 53, 95, 118, 123, 142 Dekker, 212–214, 216, 218, 219, 222, 224– 226, 228, 229, 232–234, 243, 252, 267 denotation, 1, 4, 11, 12, 20, 56, 71, 110, 156, 173, 176, 185, 215, 218– 220, 224, 225, 228, 239, 248, 252, 260 designated category, 7–10, 42, 61, 64, 65 determiner, 43, 107, 111, 164, 211, 212, 240, 241, 243, 245, 249, 269 interrogative d., 175, 261 discontinuity, 79, 83 Discourse Representation Structure, 16, 216 Discourse Representation Theory, 16, 74, 212, 213, 216, 218, 222, 225, 239, 244, 245, 266, 267 disjunction, 26, 35 Doˇsen, 58, 61 dominance, 16, 127, 128 immediate d., 127 Donald Duck problem, 244, 247, 248, 251, 254, 256, 266 donkey anaphora, see anaphora donkey sentence, 211, 212, 217, 218, 226, 234, 238–240, 243, 252, 253 double object construction, 76, 102, 151, 158, 170, 171, 182 Dowty, x DRS, see Discourse Representation Structure DRT, see Discourse Representation Theory Dunn, 28 dynamic binding, see binding Dynamic Predicate Logic, 218, 226, 227, 234, 268 Dynamic Semantics, 212–216, 239, 268 E-type pronoun, 109 ellipsis, x, xi, 19, 69, 79, 114–116, 154, 177, 178, 181–196, 200, 202, 204–207, 209, 210, 224, 256, 257, 260, 266, 268 antecedent contained deletion, 183, 184 cascaded e., 187, 188, 194, 199 coordination e., 68, 115, 183 gapping, 177, 182, 183 sluicing, x, xi, 183, 184, 213, 256, 257, 259–261, 263, 264, 266– 269 stripping, 115, 182, 183

285 verb phrase e., x, xi, 69, 181, 182, 184, 185, 193, 194, 196, 198– 203, 205–210, 257 empty abstraction, 37, 38 empty category, 16, 171 empty set problem, 245, 247, 248, 252 Endriss, 247, 256, 269 Engdahl, 102, 109 Evans, 109 existential closure, 216, 218, 228, 244, 245, 249, 256, 266, 268 extensionality, 33, 110 f-command, 172 Faltz, 68, 69 Familiarity Condition, 222 Farkas, 244, 248 Feys, 21, 22, 35, 49, 102, 178 Fiengo, 185, 191, 199, 202, 203 finite reading property, xi, 53, 95, 96, 118, 153, 160, 161, 222, 224 Fodor, 243, 244 Fox, 191 frame, 57, 60, 61 associative f., 57–59, 61, 144, 148, 150 conditions, 60 language f., 61 relational f., 60 ternary f., 56–58, 60, 61 free relative, 104 Frege, 65 function composition, 21, 22, 46, 47, 49, 77, 81, 96 function space formation, 11, 119, 221 functional gap, 103 functional question, 102, 104, 116, 121, 154, 175, 179 functional reading, 102–104 G, 98–101, 107, 108, 110, 114, 115, 152– 155 Gaifman, 9, 10, 61, 65 Gamut, x Gardent, 206 Gawron, 78, 168, 178, 188–190, 194, 195, 199 Gazdar, 73 Geach, 47 Geach rule, 47, 73, 76, 96, 98, 110, 153 gender, 97, 237 generative capacity, 65, 66, 73 Gentzen, 26 Gentzen style sequent presentation, see sequent presentation Gentzen style sequent system, see sequent system Geurts, 246, 256

286


Girard, 2, 29, 66 glue language, 67, 73, 74 goal category, 2 Greibach, 10 Greibach Normal Form, 10, 61 Groenendijk, 102, 104, 212, 218, 261, 268 Hankamer, 181 Hardt, 185, 192, 202, 203 Hausser, 104 Heim, 16, 70, 212, 244 Hepple, 39, 66, 92–97, 115–119, 142 Hirschb¨ uhler, 199–201 Howard, 35 Husserl, 65 hyperintensional, 142 hyperintensionality, 145 hypothetical reasoning, 25, 43, 45, 86, 100, 156, 157, 165, 172, 178, 188 i-within-i effects, 106, 154 in situ binder, see q indefinite, x, xi, 111, 164, 184, 196, 211– 214, 218–224, 226, 227, 229, 232, 233, 236–241, 243–251, 253, 254, 256, 257, 259–261, 263–268 intension, 142 interpolation, 62, 63 interpretation function, 32, 57, 59, 215, 224, 248 Intuitionistic Logic, ix, 25, 27–31, 33–35, 38, 92, 118, 121, 135 island, 46, 257–259, 263, 266 adjunct i., 258 complex NP i., 258 embedded question i., 258 sentential subject i., 258 J¨ ager, 122 Jacobson, 74, 92, 96, 97, 100, 102–104, 108, 109, 112, 114–117, 119, 151–155, 167, 175, 176, 183 Janssen, 17 Kamp, 16, 74, 212, 218 Kanazawa, 212 Kandulski, 66 Karttunen, 104, 109 Kayne, 168, 171 Keenan, 68, 69 Kehler, 191 Kempson, 200 Kleene star, 7 Klein, 73 Kratzer, 16, 70, 244–246

Krifka, 104 Kurtonina, 60, 61 L, xi, 17, 23, 25, 26, 29–31, 39–43, 45, 46, 48–51, 53–66, 68, 72, 73, 80, 83, 84, 86, 92, 94–96, 117, 119, 121–125, 127, 132, 141, 142, 145–150, 155, 158–160, 163, 201 labeled deduction, 13, 34 Ladusaw, 258, 260, 263, 266, 267 Lakoff, 206 Lamarche, 66 Lambek, ix, 17, 23, 25, 31, 39, 49, 50, 53– 55, 65, 160 Lambek calculus, see L Lambek Calculus with Limited Contraction, see LLC Lamping, 69 Larson, 171 Lasnik, 168 left node raising, 21 left rule, 49, 119 Leslie, 39, 92 Lewis, 212 lexicon, 5, 7–9, 12, 14, 17, 70, 72, 74, 91, 95, 100, 115, 117, 154, 178, 184, 199 LF, see Logical Form lifting, 20–22, 46, 48, 53, 54, 73, 76, 81, 96, 98, 160 Linear Logic, xi, 2, 29, 30, 37–39, 49, 68, 70, 92, 114, 115 Link, 173 LLC, 117, 119–121, 123–126, 128, 130, 135, 137, 138, 141–144, 146, 148–155, 157, 160–163, 171, 172, 178, 179, 181, 184, 188, 189, 193, 198, 201, 206, 210, 218–222, 232, 256, 260, 265, 267, 268 LLC+∧ , 221, 222, 232, 265, 267 Logical Form, 16, 70, 266 logical rule, 26–29, 41, 43, 49, 51–53, 56, 86, 119, 122, 160, 222 m-command, 16 many pronoun puzzle, 199 May, 163, 185, 191, 199, 202, 203 McCloskey, 258, 260, 263, 266, 267 meaning multiplication, 70, 181, 183, 184 Merchant, 183, 257–259, 263 Merenciano, 83 Mikulas, 61 modal logic, xi, 57 model, xi, 12, 16, 56–60, 71, 92, 142, 144, 145, 148–151, 157, 181, 215, 220, 225, 232, 252

Index theory, xi, 56, 142 Modus Ponens, 25, 53, 121, 155, 185 monostratal, 16, 172 Monotonicity, 26–28, 30, 68, 143 Montague, 12, 20, 65, 69, 163, 211, 213 Montague Grammar, 211 Moortgat, xiii, xiv, 46, 49, 66, 79–84, 87, 88, 115, 157, 159, 165, 172, 195, 268 Moot, 165 Morrill, xiv, 39, 45, 46, 66, 79, 82–84, 86, 91, 92, 115, 118, 122, 154, 166, 176, 184, 247, 263, 265, 269 multimodal, xii, 46, 49, 66, 83, 96, 119, 165, 247 Multimodal Type Logical Grammar, 66 natural deduction, 25, 26, 34, 40, 66, 71– 73, 80, 85, 86, 88, 94, 121, 125, 127, 130, 132, 134, 135, 142, 155, 160–162, 221–223 negation, xii, 27, 214, 216, 226–229, 232, 235, 250, 267 nominative, 76, 78, 88 non-associative, xii, 31, 60, 65, 66, 83 non-subject sloppy reading, 203, 209 normalization, 36, 68, 137, 138, 141, 162 β-n., 36, 135, 136, 138–140, 162, 163 η-n., 36, 135, 137, 140, 162, 163, 187 proof n., 36, 135, 141, 142, 163 strong n., xi, 36, 137, 222 term n., 36, 163 Novelty Condition, 222, 267 NP-complete, 65 number (morph. category), 97 pair formation, 32, 33, 37, 83 Pankrat’ev, 61 parallelism, 65, 181, 202–208, 264, 265 Partee, 47, 172 Peirce’s Law, 27 Pentus, 61, 62, 65, 66, 73 Pereira, 69, 165, 189, 190, 202, 205, 206 Permutation, 26, 27, 29, 30, 46, 49, 92, 95, 194 Pesetsky, 168, 171 Peters, x, 78, 168, 178, 188–190, 194, 195, 199 pied piping, 80 pied-piped, 78, 175 PLA, see Predicate Logic with Anaphora plural, 173, 268 polymorphic, 17, 18, 78, 105, 158, 175, 204, 210, 227, 228, 235, 241 polymorphism, 104, 199, 224, 227, 235 possessive, 100, 172 possible world semantics, 57

287 Postal, 167 pre-order, 54, 55, 61 precedence, 5, 97, 127, 128, 151, 167–172, 179 Predicate Logic with Anaphora, 213–219, 224, 227, 232, 234, 235, 239, 243, 267 presupposition, 264, 265, 269 product, 25, 28, 29, 37, 38, 43, 49, 54– 56, 59, 60, 62, 83, 84, 86, 119, 121, 130, 131, 134, 144, 146– 148, 150, 163 elimination, 37, 131 pronoun, 69, 70, 77–79, 84, 86–88, 91, 93, 96, 98–100, 102–112, 114, 116, 117, 143, 154–156, 166– 173, 175–179, 185–188, 191, 192, 195, 196, 199, 202, 203, 205, 208–211, 213–222, 224, 226, 229, 232, 233, 236–239, 246–248 anaphoric p., 74 bound p., 70, 77–79, 81, 82, 105, 112, 165–167, 175, 176, 211, 254, 256 problem, 246–248, 256 donkey p., xi, 235, 267 interrogative p., 103, 175, 260, 261, 265, 266 paycheck p., 109, 110, 112, 177 reflexive p., 68, 74–78, 81, 82, 84, 86, 88, 94, 127, 145, 171, 172 relative p., 43, 45, 46, 104, 109–112, 156, 167, 184 proof nets, xii, 66 proof search, 9, 50, 53, 96, 118, 123 proof tree, 13, 14, 40, 41, 72, 127–135, 137, 138, 140, 141, 155, 161– 163, 165, 166, 190, 193, 194, 196, 198, 222, 254 proposition, 1, 23, 27, 29, 35, 97, 104, 110, 112, 206 prosodic labeling, 85 Pullum, 65, 73 q, 80–84, 115, 157, 159–166, 169, 174, 193–198, 201, 226, 229–231, 241–243, 247, 255, 268, 269 Quantifer Raising, 163 quantification, x, xi, 155, 157, 159, 160, 193, 195, 212, 214–217, 232, 249, 253, 268, 269 quantifier, 19–21, 70, 79–81, 84, 91, 99, 104, 105, 157–160, 163–169, 172, 175, 178, 179, 193–196, 198, 200, 201, 207, 210, 211, 214, 216, 219, 224, 226, 232,

288


238, 241, 243, 244, 246–248, 254, 260, 263, 268, 269 downward monotonic q., 256 existential q., 212, 215, 217, 225, 226, 254 interrogative q., 173 Quantifying In, 163 recipe, 39, 43, 66, 68, 73 reconstruction, 106, 170, 172, 173, 175– 177, 179, 185, 213, 257, 259 recursion, 3, 72, 227 reflexive, see pronoun Reinhart, 78, 79, 167–169, 202, 244, 245 relative clause, 43, 70, 71, 91, 104, 105, 108, 167, 183, 198, 257 Relevant Logic, 28–30, 37, 38, 49, 118 Reniers, 246, 268 residuation, 56, 83, 84 resource conscious logic, 28 resource sensitivity, 67 Restall, 30 Retoré, 66 Reyle, 16, 74, 212 right node raising, 21, 105, 106, 154, 182, 183 right rule, 49, 119 Roorda, 62, 63, 66 Rooth, 47, 202, 203, 205, 212 Ross, 257, 259, 263, 267 rule of proof, 49, 121, 122, 125 rule of use, 49, 119, 122, 125 Russell, 65 Ruys, 268 Sag, 73, 181, 184, 193, 199, 205, 243, 244 Saraswat, 69 satisfaction, 213–216, 224 scope, xi, 45, 46, 79, 81, 83, 91, 160, 163–167, 190, 193–196, 198– 202, 207, 208, 210–212, 214, 216, 225–227, 229, 232, 235– 238, 240, 241, 243–245, 247, 248, 251, 254, 256, 260, 263– 266, 268 double s., 269 inversion, 201 parallelism, 264 secondary wrap, see wrapping Sem, 191 semantic composition, 4, 65, 68, 70, 74, 79, 86, 93, 97, 185, 188, 219 sentential category, 223, 224 sequent, 6–10, 13, 15, 25, 31, 34, 36, 37, 39, 41, 49–51, 53–56, 58, 60– 65, 72, 73, 80, 94, 95, 99, 119,

123–125, 130–132, 134, 142– 144, 147–149, 151, 153, 159, 160, 189, 221 calculus, 53 derivation, 13, 63, 66, 147, 198 format, 40, 41, 131, 132, 160, 166, 198 presentation, 6, 39, 41, 49–51, 55, 120, 123, 142, 146, 161 proof, 49–51, 123, 134 rule, 23, 50, 53, 73, 95, 121, 131, 132, 160, 161 system, 49, 50, 125, 142, 147, 148 Shamir, 9, 10, 61, 65 Shieber, 69, 189, 190, 202, 205, 206 Skolem function, 100, 102–104, 106, 109, 110, 112, 154, 173, 177, 186, 191, 220 slash, ix, 4, 5, 22, 23, 25, 65, 83, 86, 95–97, 119, 125, 142, 143, 150, 151 elimination, 25, 41, 46, 55, 56, 62, 86, 95, 143 introduction, 25, 41, 43, 46, 47, 49, 55, 56, 62, 86, 95 sloppy, 104, 154, 177, 178, 185, 187–192, 195, 196, 198, 202, 203, 205, 207–210 Solias, 83, 184 sound, 39, 58, 60, 142, 144 soundness, 58, 146, 148 specific, 111, 200, 235, 241, 244, 245, 268 sprouting, 267 Stalnaker, 213, 222 static closure, 227–229, 232, 233, 241, 247, 248, 252 Staudacher, 212 Steedman, 17, 22, 49, 65, 184 Stokhof, 102, 104, 212, 218, 261, 268 strict, 79, 97, 119, 185–189, 191, 192, 195, 196, 201, 205, 207–209, 224 string recognition, 8, 9, 43, 61, 64 structural hierarchy, 17, 26, 29, 30 structural rule, 26–30, 37, 38, 46, 49, 60, 68, 74, 92, 118, 119, 137, 194 subformula, 9, 49, 50, 53 property, 50, 96, 118, 122, 123, 142, 161 substructural logic, 65, 198 succedent, 6, 9, 13, 14, 25, 28, 38, 39, 49, 52, 54, 58, 73, 121, 142 surface structure, 16, 17, 257 syntactic composition, 4, 145 Szabolcsi, 68, 69, 74–79, 81, 82, 87, 88, 102, 115, 213, 269 S, 49 T, 20–22, 75, 99, 101, 110 Tarski, 218

Index

289

Thrainsson, 172 Tiede, 66 TLG, see Type Logical Grammar topicalization, 45, 175 trace, 16, 43, 45, 103 transformation, 16, 17, 41, 51, 135, 256 truth, 1, 11, 12, 18, 27, 28, 58, 59, 71, 142, 148, 150, 151, 212, 214, 216, 219, 220, 224, 225, 228, 233, 235, 248, 251, 252, 261 conditions, 207, 213, 214, 216, 217, 219, 224, 225, 227, 229, 237, 244, 245, 250, 254 Turing, 66 Type Logical Grammar, ix–xii, 1, 17, 23, 35, 43, 45, 46, 49, 66, 67, 70– 73, 79, 91, 115, 117, 157, 166, 172, 194, 198, 201, 213, 218, 219, 224, 225, 234, 235, 239, 247, 260, 263, 265, 267, 268

validity, 58, 59, 142, 146 van Benthem, ix, 61, 65 variable free, 67, 70, 73, 74, 106, 203, 213, 224 Versmissen, 83 von Fintel, 246 VPE, see ellipsis, verb phrase

underspecification, 69 unselective binding, 240, 245, 248, 250, 256, 266

Z, 98–103, 105, 107, 108, 110, 112, 114– 116, 151, 152, 155 Zaefferer, 104

Wall, x Wansing, 38 weak crossover, 102, 103, 111, 112, 154, 167, 177, 178 Wescoat, 202 Williams, 176, 178, 191 Winter, 245, 246, 268 wrapping, 76, 81, 83, 84, 88, 91, 92, 102, 151, 171 secondary wrap, 83–86

TRENDS IN LOGIC 1.

G. Schurz: The Is-Ought Problem. An Investigation in Philosophical Logic. 1997 ISBN 0-7923-4410-3

2.

E. Ejerhed and S. Lindström (eds.): Logic, Action and Cognition. Essays in Philosophical Logic. 1997 ISBN 0-7923-4560-6

3.

H. Wansing: Displaying Modal Logic. 1998

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4.

P. Hájek: Metamathematics of Fuzzy Logic. 1998

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H.J. Ohlbach and U. Reyle (eds.): Logic, Language and Reasoning. Essays in Honour of Dov Gabbay. 1999 ISBN 0-7923-5687-X

6.

K. Do˘sen: Cut Elimination in Categories. 2000

7.

R.L.O. Cignoli, I.M.L. D’Ottaviano and D. Mundici: Algebraic Foundations of manyvalued Reasoning. 2000 ISBN 0-7923-6009-5

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E.P. Klement, R. Mesiar and E. Pap: Triangular Norms. 2000

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ISBN 0-7923-6416-3 9.

V.F. Hendricks: The Convergence of Scientific Knowledge. A View From the Limit. 2001 ISBN 0-7923-6929-7

10.

J. Czelakowski: Protoalgebraic Logics. 2001

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G. Gerla: Fuzzy Logic. Mathematical Tools for Approximate Reasoning. 2001 ISBN 0-7923-6941-6

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M. Fitting: Types, Tableaus, and Gödel’s God. 2002

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F. Paoli: Substructural Logics: A Primer. 2002

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S. Ghilardi and M. Zawadowki: Sheaves, Games, and Model Completions. A Categorical Approach to Nonclassical Propositional Logics. 2002 ISBN 1-4020-0660-8

15.

G. Coletti and R. Scozzafava: Probabilistic Logic in a Coherent Setting. 2002 ISBN 1-4020-0917-8; Pb: 1-4020-0970-4

16.

P. Kawalec: Structural Reliabilism. Inductive Logic as a Theory of Justification. 2002 ISBN 1-4020-1013-3

17.

B. Löwe, W. Malzkorn and T. Räsch (eds.): Foundations of the Formal Sciences II. Applications of Mathematical Logic in Philosophy and Linguistics, Papers of a conference held in Bonn, November 10-13, 2000. 2003 ISBN 1-4020-1154-7

18.

R.J.G.B. de Queiroz (ed.): Logic for Concurrency and Synchronisation. 2003 ISBN 1-4020-1270-5

19.

A. Marcja and C. Toffalori: A Guide to Classical and Modern Model Theory. 2003 ISBN 1-4020-1330-2; Pb 1-4020-1331-0

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S.E. Rodabaugh and E.P. Klement (eds.): Topological and Algebraic Structures in Fuzzy Sets. A Handbook of Recent Developments in the Mathematics of Fuzzy Sets. 2003 ISBN 1-4020-1515-1; Pb 1-4020-1516-X

21.

V.F. Hendricks and J. Malinowski: Trends in Logic. 50 Years Studia Logica. 2003 ISBN 1-4020-1601-8

22.

M. Dalla Chiara, R. Giuntini and R Greechie: Reasoning in Quantum Theory. Sharp and Unsharp Quantum Logics. 2004 ISBN 1-4020-1978-5

23.

B. Löwe, B. Piwinger and T. Räsch (eds.): Classical and New Paradigms of Computation and their Complexity Hierarchies. Papers of the conference “Foundations of the Formal Sciences III” held in Vienna, September 21–24, 2001. 2004 ISBN 1-4020-2775-3

24.

G. Jäger: Anaphora and Type Logical Grammar. 2005

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