Structures and Categories for the Representation of Meaning

Structures and categories for the representation of meaning develops a way of representing the meanings of linguistic expressions which is independent of any particular language, allowing the expressions to be manipulated in accordance with rules related to their meanings which could be implemented on a computer. This requires a new two-dimensional notation, different from that of modern logic. The book begins with a survey of the contributions of linguistics, logic and computer science to the problem of representation, linking each with a particular type of formal grammar. Taking Frege as his guide, the author then presents a system of graphs organized by scope relations in which linguistic constituents are sub-graphs whose configuration is determined by their categories. In developing this system, he extends the notion of scope and argues that anaphoric and relative pronouns are structural signs not linguistic constituents. Certain count nouns are made the basis of this system and a new account of proper names, relating to count nouns, is given.

Structures and categories for the representation of meaning

Structures and categories for the representation of meaning Timothy C. Potts Formerly Senior Lecturer in the Department of Philosophy, University of Leeds

CAMBRIDGE

UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521434812 © Timothy C. Potts 1994 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1994 Reprinted 1996 This digitally printed version 2007 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Potts. Timothy C. Structures and categories for the representation of meaning / Timothy C. Potts p. cm. Includes bibliographical references and index. ISBN 0 521 43481 5 1. Grammar, comparative and general. 2. Categorization (linguistics). 3. Semantics. 4. Language and logic. 5. Computational linguistics. I. Title P151.P66 1993 40T.43-dc20 93-788 CIP ISBN 978-0-521-43481-2 hardback ISBN 978-0-521-04250-5 paperback

to

Peter Thomas Geach whose lectures on the theory of syntax at the University of Leeds in 1967 originally inspired this work

Die stillschweigenden Abmachungen zum Verstandnis der Umgangssprache sind enorm kompliziert. (Wittgenstein, 1921, 4.002) Quia quaesisti a me, quomodo oportet incedere in thesauro scientiae acquirendo, tale a me tibi super hoc traditur consilium: ut per rivulos, et non statim in mare, eligas introire; quia per facilia ad difflcilia oportet devenire. (Aquinas, letter De modo studiendi)

Contents

Preface Acknowledgments

page

xi xv

1 1.1 1.2 1.3 1.4 1.5

Linguistics: strings String grammars Semantic roles Passives Pronouns and relative clauses The semantic component

1 9 15 24 33

2 2.1 2.2 2.3 2.4 2.5

Logic: trees Tree grammars Logic and meaning Operator and operand Categorial grammar Quantification

46 48 54 64 75

3 3.1 3.2 3.3 3.4

Computer science: graphs Graphs and graph grammars Semantic networks Conceptual dependency Frames

94 96 111 122

4 4.1 4.2 4.3 4.4 4.5

Categorial graphs Scopeways Converging scope: (1) pronouns Converging scope: (2) relative clauses Scope at higher levels Sub-categorization

131 137 148 162 168

5 5.1

Basic categories: count nouns Frege's categorization of count nouns

176

ix

x

Contents

5.2 5.3 5.4

Difficulties in Frege's view Count nouns as a basic category Generic propositions

182 193 201

6 6.1 6.2 6.3

Basic categories: pointers The category of pointers Proper names The relationship of proper names to count nouns

211 222 232

7 7.1 7.2 7.3 7.4

Quantifiers, pronouns and identity Quantification revisited Anaphoric pronouns and relative clauses revisited Plurative and numerical quantifiers Identity

239 246 259 270

Epilogue

278

Bibliography Index

289 300

Preface

This work addresses the representation problem - to use the jargon of computer scientists. To be sure, they speak of the representation of knowledge, but that is a misnomer, reflecting their intentions rather than the nature of the problem. What counts as knowledge must be true, yet any notation in which we can express what is true must equally allow us to express what is false. The problem, therefore, is how best to represent the meanings of linguistic expressions so that they may be manipulated in accordance with rules, such as rules of inference or of translation. One might call this the 'semantic form' of expressions, by analogy with 'logical form'. My interest is restricted to expressions of everyday language. This is not a synonym for 'natural language'. The implied contrast is with technical language, for example the language of mathematics, which might also qualify as natural language. I also assume that, in the case of expressions which are accounted either true or false (propositions1), the central core of their meanings will be given by specifying the circumstances under which they would be true, so that semantic form or structure will relate to and should facilitate these specifications. Identifying the structure is, indeed, the very first step in such a specification, for the meaning of an expression is determined by the meanings of its parts and the manner of their combination; that much is implicit in the possibility of learning a language (see Davidson, 1965). Yet there seems to be a remarkable reluctance on the part of those concerned with the study of meaning to discuss structural issues. Time and time again one finds that an author is simply taking a certain structure for granted, in order to press on as quickly as possible to offer an account of the truth conditions of propositions containing the types of expression in 1

This is the traditional sense of 'proposition', and the sense in which it will be used throughout this book. It should not be confused with a more recent sense, deriving from Russell, in which a proposition is an abstract entity constituting the meaning, or perhaps denotation, or perhaps reference of a proposition in my sense.

xi

xii

Preface

which he or she is currently interested. Structures are, moreover, usually assumed to be of very simple kinds, even at the cost of very complex specifications of truth conditions. This prejudice against structural investigation is especially remarkable in view of the manifest aptitude of the human mind for grasping structure, by contrast, for example, with its poor showing at computation. Our delight in music is one evidence of this and most of us, indeed, enjoy these complex sound patterns without any theoretical understanding of them, just as we can speak and write a language without any theoretical knowledge of linguistic structures. It would be more controversial to claim that appreciation of painting or sculpture turned upon apprehension of structure, but for architecture the case requires no argument; one has only to recall the enormous importance always accorded to proportion by architectural theorists, from ancient Greece through Renaissance figures like Alberti and Palladio to modern masters such as le Corbusier. Without our apprehension of structure there would not even be any computation, for the subject-matter of pure mathematics, upon which applied mathematics depends, is, precisely, structure. So an account of meaning which emphasizes structure is a priori far more credible than one which stresses computation. The one really great success story of modern logic should also have been a warning against this lack of interest in structure. The failure of logicians in the late middle ages to give a correct account of generality specifically, of the logical relationships of propositions containing more than one general term, such as 'every', 'few', 'some', etc. - was overcome by Frege in the late nineteenth century thanks to a new structural description of general propositions (to be explained in detail in the sequel). And whereas medieval logicians, relying on over-simple structures, tried to compensate with more and more complex specifications of the conditions for valid inferences, Frege, operating with more complex structures, was able to offer a simple account of validity. Taking Frege as my guide, then, I have tried to develop aspects of his structural analysis of language with respect to meaning which remain implicit within his work. This stage is reached in chapter 4, which is the pivot of the work. It is preceded by three chapters in which I survey the contributions made by linguistics, logic and computer science respectively to the representation of meaning. Conveniently, although perhaps slightly artificially, I have linked each of these disciplines with a particular type of formal grammar: linguistics with string grammars, logic with tree grammars and computer science with graph grammars. These grammars proceed in order of complexity, which has determined the order in which the contributions of the three disciplines are presented.

Preface

xiii

I have not, of course, attempted a general survey of each discipline's contribution to the study of meaning in general, but have singled out what each has to offer by way of structural analysis, as that alone is germane to my purpose. In the remaining three chapters, I diverge from Frege, calling into question, first, his treatment of count nouns as disguised intransitive verbs (chapter 5) and, second, his use of proper names as the basic category of his system (chapter 6). I propose an alternative categorization for both count nouns and proper names, so a final chapter re-works the ground covered by chapter 4 in order to make the necessary modifications, with some extensions. It is a matter of considerable regret to me that I have been unable to include a treatment of temporal expressions, which occur in the majority of contingent propositions. This lack consequently inhibits practical application of the system of representation proposed here; I hope to remedy it at a later date. Meanwhile, I have given a brief taste in the Epilogue of how categorial graphs might be used to handle a longrecognized stumbling block for Frege's ideography, adverbial modification. Computer scientists who look for a notation which can be implemented immediately on a machine will also be disappointed to find that, while I argue for distinct, though related, structural analyses with respect to meaning and with respect to the accepted forms of expression in a particular language, I restrict myself entirely to the former, thus leaving one side of the representation problem untouched. This is properly a task for linguists but, if the ideography which I develop here is on the right lines, their current proposals would demand substantial modification. With the exception of the first section of each of chapters 1-3, argument and notation proceed hand in hand throughout this book. This is essential to my purpose, since a notation is, precisely, a means of representing structures of a certain kind. A discussion of structure must, therefore, involve an ongoing discussion of notation. To set out the final notation at the beginning would be tantamount to assuming from the outset everything that the book sets out to justify. Some readers may find the structures which I discuss difficult to grasp: in my experience, some people find it much easier to apprehend auditory structures, others visual structures. I myself am in the latter group, and so my representations are primarily visual. This may present an obstacle to those who need a notation which they can pronounce. In addition, with one exception (in section 7.1), I am only concerned with structure, so that the reader who looks for full specifications of truth conditions will be disappointed. Although structural analysis must

xiv

Preface

constantly advert to questions about truth and inference, it does not require a full specification of truth conditions; the structural analysis is, rather, a precondition for the latter. Moreover, a system for representing meaning has many uses for which a detailed specification of truth conditions may be unnecessary, for instance machine translation, expert systems (see Galton, 1988). This is fortunate, since experience shows that spelling out the truth conditions of propositions of everyday language is an enormously difficult task. To do so for the range of constructions considered in this book would be a totally unreasonable demand; I hope, by the end, to convince the reader that structural analysis with respect to meaning is both demanding and worth-while in its own right. N O T E ON N U M B E R I N G OF E X A M P L E S Examples are numbered consecutively, beginning anew with each chapter. Analyses of examples are given the same number as the example itself, followed by a letter: P for a phrase marker, L for a representation in linear notation (but F if the representation is based on Frege's principles), LF for the 'logical form' of transformational grammar, and S for shallow structure. Graphs are numbered in their own sequence (Gl) etc.

Acknowledgments

My thanks are due to Professor David Holdcroft for encouraging me in this enterprise and for reading an earlier version; to Mrs Stella Whiting for useful criticisms, one of which persuaded me to change substantially my account of demonstratives; to Mr Brandon Bennett for comments and information which have helped me especially in the chapter on computer science; and finally to three publishers' readers, whose names I do not know, for criticisms which led to a major revision of an earlier version. I am also much indebted to a publisher's reader of the penultimate version, who submitted a very detailed report which has prompted several excisions and a few additions to the main text as well as many footnotes.

xv

1

Linguistics: strings

1.1 STRING GRAMMARS Throughout this book, I shall be using 'grammar' in the sense given to it by mathematicians. That calls for a word of explanation, for the terms 'grammar' and 'syntax' are in process of interchanging their meanings and the new usage has not yet crystallized, with consequent occasions of confusion. The way in which we classify linguistic expressions and the way in which which we analyse them structurally depends upon our purposes. People's jobs, for example, can be distinguished into manual versus 'white-collar' or, alternatively, into those concerned with production and those concerned with providing services. Sometimes we need one classification, sometimes the other. Language is no exception. The branches of learning whose subject-matter is language already exhibit at least three distinct (though not unrelated) purposes. Oldest, perhaps, is the study of meaning and of validity in argument. Then came grammar (now, more often, called 'syntax'), the description, roughly at the level of words and phrases, of those combinations which are used in a given language. More recently, phonology and phonetics have sought to classify sounds and their combinations, from several points of view, including, for example, the relation of sound production to the physiology of the throat and mouth. There is no a priori reason to suppose that the same structural analysis will be apposite to all of these purposes. Indeed, quite the contrary, although, to the extent to which the purposes are related, it should be possible to inter-relate the corresponding structural systems. The contrast which is of prime concern here is that between the study of meaning, on the one hand, and that of the accepted forms of expression in a particular language, on the other. Etymologically, 'syntax' simply means 'order', 'arrangement', 'organization', so that it is precisely the study of structure. Consequently, if we distinguish more than one structural system in language, we shall correspondingly have more than one syntax, for example a syntax 1

2

Linguistics: strings

relating to meaning and another relating to the forms of expression which are accepted in a particular language. But recently it has become common to contrast syntax with semantics. In the mouths of linguists, this is a contrast with the study of meaning, and syntax is roughly what grammar used to be, a classification of the accepted forms of expression for a particular language, though largely omitting the morphology of individual words. Meanwhile, grammar has been used by some philosophers for the combination of expressions in relation to their meanings, as in 'philosophical grammar'. Moreover, its application has been extended very considerably by mathematicians to embrace almost any system of rules for the combination of elements, not necessarily linguistic, into structures. An example is a (graph) grammar to describe the development of epidermal cell layers (Lindenmayer and Rosenberg, 1979); another is a (context-free) grammar for string descriptions of submedian and telocentric chromosomes (Ledley et al., 1965). In the mathematician's usage, a grammar is always a formal system, whereas philosophical grammar is usually understood to be implicit, the unwritten rules for the combination of expressions with respect to their meanings; the philosopher's task is then to make as much of it explicit as his immediate purposes may require. This slight ambiguity is unavoidable when we are dealing with everyday language, for we are not free to specify what rules we please: we are constrained by language which is already present; yet, at the same time, we want to be as explicit as possible. It seems too late to protest successfully against this reversal of the terminology, but it is with regret that I acquiesce in it. Most important, however, is to have a clear and stable usage. We do not have this at present, as the reversal is not yet complete, so that 'grammar' and 'syntax' are still sometimes used interchangeably, for instance 'universal grammar' instead of 'universal syntax'. We do need two distinct terms here, one for the study of linguistic structures in general, of whatever type, and another for the study of forms of expression which are accepted in a particular language. So, bowing to the new custom, I shall reserve 'grammar' for the former and 'syntax' for the latter. To the extent that linguists have concerned themselves with specifying grammars formally, most of the grammars which they have proposed for everyday language have been among those known to mathematicians as string grammars. String grammars, naturally, generate strings, that is, symbols concatenated in a line, a well-ordering, so that we could identify each as the first, second, third, etc., symbol in the string. Alternatively, a string is an ordered list. It is also possible for a member of a string itself

String grammars

3

to be a string, so that we obtain a structure of nested strings. So, if the grammar is used as a basis for representing meaning, there is an implicit claim that meaning can be represented adequately by string structures, that nothing more complicated is needed. A string grammar G s consists of an ordered set (N,£,P,S), where N is a finite set of non-terminals, I of terminals, P of productions or rewriting rules, and S is the starting symbol. Intuitively, terminals are expressions of the language which the grammar generates, non-terminals are category symbols, of which the starting symbol S is one. V, the union of N and £, is the alphabet of the grammar, while V* is the closure of V, that is, the denumerably infinite set of all finite strings composed of members of V, but including the empty string {excluding the empty string, it is V4"). In general, productions take the form a => /?, meaning that a may be re-written as /?, where a is in V + and /} is in V*. Linguists have largely confined themselves to string grammars, of which a wide variety has now been proposed. Their interest, however, has primarily been in syntax, so we need only be concerned with these grammars to the extent that they have been expected to sustain an account of meaning. To this end, some exposition of formal syntax is unavoidable; yet, at the same time, a comprehensive survey of every theory, even from this point of view alone, would call for a book to itself. I propose, therefore, to concentrate upon the most famous and the most fully developed string grammar for everyday language, transformational grammar, due originally to Chomsky (1957), which from the start has sought to encompass an account of meaning as well.1 I shall not, however, discuss the syntactic arguments used to support the proposed structural analyses. Moreover, linguists who favour a different theory of syntax will have to ask for themselves whether the points of criticism which I raise carry over to their preferred theory. Transformational grammar grew out of constituent-structure analysis (Bloomfield, 1933; Wells, 1947; Harris, 1951; Postal, 1964). Sentences were first divided into their immediate constituents, typically phrases, 1

My exposition is based on Borsley (1991) and Radford (1988), supplemented from Radford (1981), Jacobsen (1986) and Chomsky (1977, 1982a, 1982b, 1986a and 1986b). For the most recent version of the semantic component, May (1985) is the central text. To be exact, there have been three transformational grammars, the second dating from Chomsky (1965) and the third from Chomsky (1981). In each case continuity of notation has masked fundamental changes to the theory. Thus from 1965 to 1981 transformations were held to be meaning-preserving, with the semantic component operating upon deep structures, while since 1981 the semantic component has been attached to shallow structures, which are almost surface structures, and transformation rules merely move items within a structure derived from the phrase-structure rules. The revisions have been prompted in large measure by challenges from logic.

4

Linguistics: strings

and then the latter were progressively divided until their ultimate constituents, typically single words, were reached. The criterion upon which the divisions were based was initially intuitive, but later a substitution test was introduced: if, for an expression occurring in a sentence others could be substituted whilst preserving grammaticality and if, further, the same substitutions were possible in any other sentence in which the expression could occur, then that expression was accounted a constituent of any sentence in which it occurred. The test was thus also a method of classifying linguistic expressions, the members of each group belonging to the same syntactic category.2 By using a symbol for each category, it became possible to describe a series of sentence-patterns. The categories (constituting N, the finite set of non-terminals) used in most formal theories today are derived from those of traditional syntax. While there remain some differences, there is also a broad measure of agreement, starting from Noun (TV), Verb (F), Adjective (^4) and Preposition (P). These are known as lexical categories; investigations of constituent structure revealed a need to classify corresponding phrases, so four phrasal categories were introduced as well. Subsequently arguments were put forward for intermediate categories to cater for expressions which were more than single words or morphemes, yet smaller than the phrases already recognized. The intermediate categories are commonly indicated by a prime, the phrasal categories by a double prime, for example N\ N" (or NP for the latter). The original start symbol was S (Sentence) and there was no phrasal category corresponding to it. Subsequently it was itself recognized as a phrasal category, the head of such phrases being an Inflexion (/) catering for variations of tense, aspect and modality, so that 5, accordingly, was replaced by /'. A further category C (Complementizer) was introduced later to provide for subordinate and relative clauses as well as for mood; thus, examples of complementizers are relative pronouns and 'that' introducing a subordinate clause. As a result of these developments, the start symbol in the latest version of the theory is C" but, as I shall only be concerned with indicative sentences, I shall omit the first few steps in each derivation and begin with /". This will make it easier to see the essential features of derivations for the purpose in hand. Another development which can be largely ignored here is to break down the original categories into sets of features, each of which has a value. Thus verbs and prepositions share common behaviour which 2

This test for syntactic constituents has now been supplemented by several more, which are clearly set out in Radford (1988, p. 90); see also Borsley (1991, ch. 2).

String grammars

5

nouns and adjectives lack, such as being able to combine with a noun phrase, so are assumed to incorporate a feature, rather confusingly dubbed 'V; they are + F, whereas nouns and adjectives are — V. Similarly, a feature 'N' is credited to nouns and prepositions but not to verbs and adjectives; while a further feature BAR (derived from an earlier notation) with 0, 1 and 2 as values can be used to differentiate lexical, intermediate and full phrasal categories respectively. The theoretical interest of this feature theory is as a basis for justifying the system of categories; but, so long as it is understood that these are syntactic categories, they are not our concern. The form of production rules for a string grammar cited above, according to which a category symbol may be re-written as a string, combines two elements. First, there is the replacement of the category symbol by a list of symbols; second, there is the ordering of that list. So long as we use a linear notation to represent the structure of a string, these two elements are difficult to disentangle. Take, as an example, the sentence (1)

Every doctor visited at least one patient.

Let us suppose that 'every doctor' and 'at least one patient' are noun phrases, composed of an expression of a new category, Det (Determiner) ('every' and 'at least one') and a noun ('doctor', 'patient'), while 'visited at least one patient' is a verb phrase, composed of a verb 'visited' and the noun phrase 'at least one patient' already identified. We can then represent a structure for (1) in a linear notation by (1L)

(Det N) (I (V (Det N))).

This is a string whose first member is a string of two members and whose second member is also a string of two members, with its second member being in turn a string of two members, and so once more. However, it omits any information about the categories of the sub-strings. We could supply this by placing sub-scripts on the closing brackets, as follows (intermediate categories are omitted in the interest of simplicity): (1L')

((Det NV (I (V (Det N)N.)v«))r.

This is known as a labelled bracketing, but it is a much less clear representation than the following tree, known as a phrase-marker (to which I have added the terminal symbols for (1)):

6

Linguistics: strings

(IP)

I"

I

N"

I

Det

N

I

every

doctor

e[past]

I

V"

V visit

N"

J_ Det

at least one

I

N patient

The symbol e under the / node indicates that it is empty; tense, however, is regarded as a feature and is therefore shown in square brackets. As we shall be dealing with trees a great deal in the sequel, a few terms will be useful. A tree consists of nodes, joined by edges. Both nodes and edges may be labelled; here only the nodes are labelled and are shown by their labels. The node at the top of the tree is called its root, whose label is always the starting symbol. The nodes at the bottom of the tree are called its leaves\ here they are all terminal symbols. Phrase-markers are ordered from left to right and from top to bottom. The left-to-right ordering gives rise to a relationship of precedence between nodes: for any two distinct nodes X and Y, X precedes Y just in case X occurs to the left of Y. This relationship remains invariant in any given string, because edges in a phrase-marker may not cross. The top-to-bottom ordering produces a dominance relationship: ^dominates Fjust in case there is a path down the tree from X to Y; X immediately dominates Y just in case it dominates Y and no node occurs between them. Symbols which are immediately dominated by the same symbol are called sisters. Other relationships important in transformational grammar can wait until they are needed. The first phrase-structure rules formulated by transformational grammarians combined immediate dominance and precedence, for example the following, in accordance with which we could obtain (IP): S => N"V" N" => DetN V" => VN" Since then, they have undergone continuous change and refinement. First, immediate dominance (ID) and (linear) precedence (LP) have been separated; the motivation for this is that it is more economical when making cross-comparisons between different languages, which may differ

String grammars

7

in their word-order conventions while nevertheless breaking down phrases into the same elements. Second, the ID rules have now been reduced to three types, which can, accordingly, be stated as rule-schemas, that is to say, rule-patterns which use symbols for which the category names given above may be substituted in order to yield particular rules. I shall use Greek minuscules as such symbols; on the right-hand side of the rule they will be separated by commas, to indicate that their order is indifferent. Each of the four types of rule relates to a type of modifier, what is modified being the head of the construction (which I write first), and the four modifiers belong to the same group as the traditional terms subject, object, indirect object. The first is that of specifier, which takes us from a full phrasal category to an intermediate category which is the head of the new string; the ruleschema is: (S)

<x" => a', (/?")•

The parentheses indicate that /?" is an optional element, so that a" may simply be re-written by a'. If /?" is present, however, it is the specifier of a' and the latter its head. It is not possible to substitute just any category name for the Greek letters and still always obtain a syntactically correct rule, but some substitutions which work out all right for English are the following: (SC) (SI) (SV) (SN) (SP) (SA)

C" I" V" N" P" A"

(N") (N") =» V', (N") =• N ' , Det => F, Det =• A', Det

=> c, =*• r,

The parentheses indicate that the symbol enclosed in them is optional. Thus C" may simply be re-written as C , without any branching, and similarly for F and V". In such cases I shall often omit the intermediate category, in the interest of keeping phrase-markers as simple as possible. So far as linear precedence is concerned, in English the head always follows the specifier in applications of these rules. The second type of modifier is a complement; the rule-schema for introducing complements takes us from an intermediate to a lexical category, the latter again the head of the new construction, and is: (C)

a' =» a, OS"), ()'"), (• • .)•

Thus a rule may introduce more than one complement. Some examples are:

8 (CC) (ci) (CV) (CN)

Linguistics: strings C r V N'

=> => =* =>

C, I" i, v" V, (N"), (A"), (P"), (I") N, (P") / (I")

(The slash in this rule indicates that P" and I" are exclusive alternatives). (CP) (CA)

P' =* P,(N") A' =* A, (N"), (P")

Thus this type of rule allows us to re-introduce phrasal categories. The LP rules for English require that a lexical category or a noun phrase precedes any phrasal category which is its sister, and that a sentence (/") follows all its sisters. By using a specifier rule for a given category followed by a complement rule, we may descend from the double-barred category to the corresponding unbarred one, the derivation proceeding thus:

I

I

specifier

a'

i

i

a

complement |

1 complement

The third type of modifier is an adjunct, which is simply added to an intermediate category; thus the rule-schema is: (A)

a'

=> a', p"

The following are examples: (AV) (AN) (AP) (XA)

V' N' P' A'

=» =» => =*

V, A" / N" / P" N\ N" / A" / P" / I" P', A"/P" A',A"/P"

These rules are optional. The arguments for adjuncts from specifiers and complements are mainly syntactic and need not concern us here. There is one type of construction for which the three rule-schemas described above do not provide, namely, that involving coordinating conjunctions such as 'and' and 'or\ According to transformational grammarians, coordinating conjunctions can be used to join together any two expressions of the same category; moreover, they can be used to form lists, so, if their category be K, we need a rule-schema on the following lines:

Semantic roles (K)

9

a =• K, a b a2, (a3, • • ., an.)

with the LP rule for English that K must occur in the penultimate position. A typical example is when a sentence is re-written as a disjunction of two sentences, i.e. I" =* I" or I", and another occurs when a complex subject is formed, such as 'Jack and Jill*, which requires: N" =» N" and N". (I have ignored, here, the complication which arises when the conjunction as shown is the second part of a two-part expression, as in 'either . . . or' and 'both . . . and'.) 1.2 S E M A N T I C ROLES So far, these rules will yield phrase-markers whose leaves are lexical category symbols, but they do not provide for the introduction of terminal symbols; hence the resulting structure is known as a pre-terminal string. In order to obtain a deep structure from a pre-terminal string, the non-terminals must be replaced by terminals. This is effected by lexical insertion, the substitution of words, phrases or other signs having a fixed meaning for the category symbols (including an empty or null symbol, 0). At its simplest, this could be done by providing, for each category symbol, a list of linguistic expressions which might be substituted for it. In practice, a more complicated arrangement is needed. Suppose, for example, that we had the pre-terminal string (1L'); clearly, a transitive verb must be substituted for V, because of the second (Det N). By contrast, if the pre-terminal string had been (Det N) (I V) instead, an intransitive verb would have to be substituted for V. In view of this and many similar cases with other categories, linguistic expressions were sub-categorized, an idea due originally to Matthews and Chomsky (see Chomsky, 1965, p. 79), which was effected by showing what kinds of string they may fit into. These strings, in turn, were specified by recourse to the phrase-structure rules, since the pre-terminal strings may contain more detail (often variable) than is needed for the purpose. Thus a transitive verb like visit could be listed as V, +[— N"], where the brackets enclose a sub-categorization frame into which the expression would fit at the horizontal line, and the plus-sign indicates that the specified frame must be present.

10

Linguistics: strings

A subsequent development, however, has now made it possible to eliminate sub-categorization in the lexicon (though some linguists prefer to retain it). This is case grammar,3 which is concerned with the semantic relationships between verbs and their subjects, direct and indirect objects, etc. Borrowing from mathematics via logic, Fillmore calls these the arguments of a verb or predicate. Neither he nor other transformational grammarians, however, use the term precisely in its mathematico-logical sense, so it is better to define it, as they do, in terms of phrase-markers: an argument is any N" which is dominated either by another N" or by an F. These relationships between verbs and their arguments are expressed in everyday language, according to case-grammar, by cases, prepositions or postpositions, and may be characterized as role-types: human languages are constrained in such a way that the relations between arguments and predicates fall into a small number of types . . . these role types can be identified with certain quite elementary judgments about the things that go on around us: judgments about who does something, who experiences something, who benefits from something, where something happens, what it is that changes, what it is that moves, where it starts out, and where it ends up. (1968b, p. 38) Fillmore eventually settled for nine of these roles (1971, pp. 42, 50-1). He does not suppose a one-to-one correspondence between roles and (in English) prepositions; a single role can be indicated, in different sentences, by more than one preposition and the same preposition may be used for more than one role. Typical examples of prepositions for each role, however, are: Agent Experiencer Object

'by' 'by' -

Source Goal Instrument

'from' 'to' 'with'

Location Path Time

'in' 'along', 'through' 'at', 'during'

There is no preposition for the object-role in English.4 These roles are most easily understood from examples. It would be difficult to construct a single example exhibiting all nine, but (2)

Margot opened the cupboard outside her sister's bedroom with her key at 9:15 p.m. on her way from the kitchen to the attic along the landing

crams in eight of them. Margot is the Agent, the cupboard is the Object, outside her sister's bedroom is the Location, her key is the Instrument, 3 4

Due originally to Gruber (1965) but largely developed by Fillmore (1966, 1968a, 1968b, 1971, 1975a) and Anderson (1971, 1977). A more recent addition to this list is Benefactive, the role of something which benefits from an action. Indeed, some authors posit as many as twenty-five distinct roles, whereas others restrict them to four.

Semantic roles

11

9:15 p.m. is the Time, the kitchen is the Source, the attic the Goal and the landing the Path. For an example of Experiencer, (3)

Margot felt a sharp pain in her left foot

will do; although 'Margot' is the subject of both (2) and (3), Margot is not represented as an agent in the latter, but rather as undergoing something, experiencing it. Her left foot is another example of a Location. Fillmore's term 'awe-roles' is unfortunate, for it uses a syntactic phenomenon to characterize a semantic feature. These roles are to be found, indeed, just as much in uninflected languages like English which use prepositions where inflected languages use cases. Moreover, in speaking of cases we are referring to inflexions of nouns, whereas the roles belong to what is described or named by the nouns: proposition (2) presents Margot the woman, not her name, as Agent, the cupboard and not the expression 'the cupboard' as Object, etc. All the same, it is the proposition which presents Margot, the cupboard, etc., in these roles, which they are not committed to playing perpetually (Margot is Agent in (2) but Experiencer in (3), while in another proposition the cupboard might be Agent, for instance if it were represented as falling upon somebody). So, instead of 'case-roles', it will be more appropriate to speak of semantic roles.5 These semantic roles are properly to be regarded as going with verbs, in spite of their original introduction via inflexions of nouns. For each verb carries a selection of semantic roles with it and excludes others: thus giving involves a Source (the giver) and a Goal (the recipient). Verbs might even be classified by the roles which they carry; thus 'open' might be an Agent/Object/Instrument verb and 'give' an Object/Source/Goal verb. This idea has been incorporated into transformational grammar by including a semantic grid in the lexicon entry for each verb, for example [Agent, Object] for 'visit', [Experiencer, Object, Location] for 'feel' and [Source, Object, Goal] for 'give'. One semantic role is underlined in each of these lists to indicate that it relates to the subject of the verb, called its external argument.6 The remaining, internal arguments of the verb are its complements, so, given the semantic grid, we no longer need a separate sub-categorization frame, provided that we know how each role is to be expressed (for example the Instrument role by a P" whose P is 'with'). It 5 6

I first suggested this in (1978a). Chomsky adopts the same terminology in (1986a, p. 60), but most transformational grammarians use thematic roles (0-roles) instead. Chomsky has argued that the semantic roles of external arguments are determined by the whole verb phrase (1986b, pp. 60-2; cf. Radford, 1988, pp. 386-8).

12

Linguistics: strings

has also been found that certain other syntactic information can be predicted from semantic grids. Yet there are problems in taking over this account of semantic roles as it stands. First and foremost is the question how each role is to be identified and characterized, and how many there are. Fillmore's list of nine has not met with universal acceptance. No wonder, for its justification is far from clear: he describes the roles in very general terms and in formulations which do not stand up to close scrutiny, whereas the roles are presented to us in a language by its verbs. Thus his methodology is inverted; it is not that we first have a set of abstract role concepts and then proceed to recognize instances of them as being presented in propositions. Rather, it is through our understanding of the meanings of propositions that we come to formulate descriptions of roles presented by them. 7 Hence the correct method of identifying the roles would be by studying groups of verbs. A start has been made on this by a computer scientist, Roger Schank, whose work will be described in section 3.3. Meanwhile, any list of semantic roles must be regarded as tentative. A second problem for semantic roles is raised in a point made by Kenny a generation ago. If the meaning of a verb is partly characterized by the number and type of semantic roles which it introduces, then we should expect a fixed number of semantic roles to go with each verb, unless the verb had more than one meaning. But Kenny pointed out that, from a proposition like (4)

Brutus killed Caesar with a knife in Pompey's Theatre on the Ides of March

we can validly infer any proposition derived from it by deleting one or more of its prepositional phrases (1963, pp. 156-62). So if we were to classify 'kill' in (4) as, say, an [Agent, Object, Instrument, Location, Time] verb, how would we classify it in, for example 'Brutus killed Caesar on the Ides of March'? If as an [Agent, Object, Time] verb, and the meaning of a verb is partly determined by its associated semantic roles, then 'kill' cannot have exactly the same meaning in the two propositions. But not only would that be a totally arbitrary conclusion, without any independent evidence to support it; the validity of the inference would also be called into question by the equivocation in the meaning of 'kill'. Davidson (1967a) proposed to overcome this difficulty by analysing propositions like (4) so that each semantic role is introduced by a

7

This criticism is argued in detail in Potts (1978a).

Semantic roles

13

conjunction, thus: There is a killing and it is by Brutus and it is of Caesar and it is in Pompey's Theatre and it was on the Ides of March'. One of the simplest of all logical inferences is that from 'p and q" to 'p' (or to V), s o it is easy to see that any of the conjoined clauses may be dropped. This analysis certainly achieves its intended object, but at a heavy price. It destroys the category distinction between the verb and the names or descriptions of the players of the various semantic roles. Moreover, it is unclear what would be represented if we were to drop the very first (verbal) clause from the conjunction, as we are entitled to do; or, again, if we were to drop all but the first and second conjuncts, since, if 'killed' is a transitive verb, *'Brutus killed' is not a sentence. In one case, that of Location, the analysis is also inadequate. For the most part, the various actors in an action are in the same place, but not always. On Davidson's analysis, however, they must be, since the place is given as an attribute of the action, not of the individual actors. Communication provides common counter-examples to this assumption, such as (5)

Andrew telephoned from Leeds to Philip in New York.

If there was, then, a telephoning, was it in Leeds, in New York, in the transatlantic cable or now, perhaps, in the satellite and pathways through space? Clearly, all we can safely say is that Andrew telephoned to Philip and Andrew was in Leeds and Philip was in New York. Now this does, indeed, give us three conjuncts, so perhaps Davidson was on the right lines in that aspect of his analysis. Only we are left with an irreducible sentence 'Brutus killed Caesar', since neither *'Brutus killed' nor *'killed Caesar' is a sentence. However, this tallies with the intuition that it is part of the meaning of 'kill' that in any killing there must be both a killer and a killed (not necessarily distinct, though). We should then have to say that 'Caesar was killed', in which there is no explicit mention of the agent, has the same meaning as 'Something killed Caesar', but that we do not need to add 'by something' to the former precisely because it is implied in the meaning of the verb. The conclusion to be drawn from this discussion of Kenny's problem is that Fillmore's list of semantic roles would have to be pruned so long as we insist that a semantic role must be associated with a verb. Transformational grammar can handle this, however, by its distinction between complements and adjuncts. Thus in (4) 'Caesar' is a complement of 'kill' but 'with a knife', 'in Pompey's Theatre' and 'on the Ides of March' are all adjuncts. So the semantic grid for 'kill' would be just [Agent, Object]. We could then have semantic grids for prepositions as well, such as [Instrument] for 'with'. And a semantic role which was

14

Linguistics: strings

introduced by a prepositional adjunct to one verb might be introduced as a complement to another; thus it might be held that 'unscrew' demands the semantic grid [Agent, Object, Instrument]. A third and final difficulty remains: Fillmore requires that the same semantic role shall not occur more than once in the same sentence. As it stands, this requirement clearly will not do: where a proposition contains more than one verb, through the use of connectives, relative clauses, psychological verbs and so on, the same semantic role may be introduced by each of two different verbs and with respect to a different participant. So the requirement would at least have to be re-phrased to the effect that no verb shall have the same semantic role associated with it more than once. This is how it has been put in the Theta Criterion of recent transformational grammar, with the complementary stipulation that not more than one semantic role shall be assigned to each argument of a verb: each argument bears just one semantic role and each semantic role is assigned to just one argument (see Chomsky, 1981, p. 36). Even then, however, a justification for this restriction is still lacking. Were the restriction not present, there would often be room for dispute about which semantic role a participant was playing. Thus, in stating a semantic grid for 'give', I assumed that the giver and recipient of a gift play the roles of Source and Goal respectively, with the gift itself as Object. But why not, following the hint offered by the syntactic distinction between direct and indirect object, say that the giver is Agent, the gift the first Object and the recipient the second Object? Only because duplication of semantic roles has been excluded. The requirement is clearly convenient] but convenience hardly amounts to a justification. Moreover, it is prima facie counter-intuitive to insist that the giver of a gift is not, in respect of his action, an agent, whatever we may say about gift and recipient. This example brings out a further important point. The original application of Source and Goal was to movements of bodies (in the sense of Newtonian mechanics), the Source being their starting-place and the Goal their end-place. Now of course it is perfectly legitimate to look at giving as analogous to this: from the point of view of the gift, it 'moves' from being in the ownership of the giver to being in the ownership of the recipient. However, this is not the original sense of 'Source' and 'Goal', but a new, analogous one, which arises precisely from looking at giving in this way. And the 'movement' of ownership may not be accompanied by any physical movement of the gift itself, for example the gift of a house, so this is also movement in an analogous sense. Yet it is surely also legitimate to regard giving as an action in which the giver is the agent and the recipient the object, with the gift either as a further object or perhaps

Passives

15

as analogous to the instrument of a causal action, that by means of which a giving is effected. In spite of the uncertainty which still surrounds both the number and distribution of semantic roles, the notion promises to be of great importance in grammar, both syntactic and semantic. It will surface several times throughout this book, and one must hope that more intensive study will be devoted to it in the near future. Its immediate application, however, is to lexical insertion. When a verb is inserted under a V node, its complements will, where they are its internal arguments, simultaneously be assigned semantic roles and the whole verb-phrase verb will assign a semantic role to its subject, where that is an external argument. Clearly, then, it will only be possible to re-write in a context where the correct number of complements, to match the number of semantic roles, is available; hence sub-categorization frames are no longer needed. Strictly speaking, semantic grids are only assigned to syntactic structures at the point at which they are semantically interpreted. Nevertheless, semantic roles interact with syntactic operations which take place earlier, so it will be convenient to show semantic grids in phrasemarkers. They will be written underneath the lexical category symbols to which they pertain, though abbreviating each semantic role to a single letter. I shall use only three semantic roles, A (Agent), P (Patient) and / (Instrument). Thus the semantic grid for 'visit' will be [A,PJ. Correspondingly, the role assigned to each argument will be shown under its category symbol; the argument is then said to be semantically marked. So, for instance, if a prepositional phrase contains a noun phrase which, in context, describes an instrument, I shall write [I] under PP. Sub-categorization has been retained in head-driven phrase-structure grammar (Pollard and Sag, 1988), where it plays a large role. In this grammar, sub-categorization is introduced as a feature of lexical categories, with a list of phrasal categories as its value; these are the categories of its complements. In the case of verbs, if the list is empty, we have an intransitive verb, whereas for 'visit' we should have

V[SUBCAT:(NP)J and for 'give' V[SUBCAT:(NP,PP)], etc. This is less informative than a semantic grid, because it tells us nothing of the semantic roles associated with the complements, and nothing about the external argument. 1.3 PASSIVES A construction which is of especial interest with respect to meaning is the passive. The reason for this is as follows. If we compare the active

16

Linguistics: strings

sentence 'Dr Patel visited Mrs Wilson' and the corresponding passive, 'Mrs Wilson was visited by Dr Patel', we notice that they will be true under just the same circumstances. In order to understand an indicative sentence, however, we must at least know in what circumstances it would be true; that is a part, indeed a large part, of its meaning. Consequently, if we ask what is the structure of each of the above sentences with respect to its meaning (its semantic structure), we should expect the two structures at least to have much in common, even if they were not exactly the same. But now consider the passive sentence corresponding to (1), (6)

At least one patient was visited by every doctor.

A possible interpretation of (6) is that there is at least one patient such that every doctor visited him or her. This would be strained as an interpretation of (1), whose more likely meaning is that, taking each doctor in turn, you will find at least one patient whom he or she visited. That meaning could also be given to (6), especially in a context where the first suggested interpretation is unlikely to be true. For although any circumstances which satisfy the first interpretation will also satisfy the second, the converse does not hold. If the doctors in question were Dr Patel, Dr Hawbrook and Dr Farbridge, it would be enough to make the second true that Dr Patel visited Mrs Wilson, Dr Hawbrook Mr Oddy and Dr Farbridge Miss Verity, but not enough to make the first true; for that, all three doctors must have visited the same patient. From the point of view of truth conditions, therefore, propositions like (1) and (6) have a double complexity: first, they incorporate a distinction between visitor and visited and, second, a distinction between relating every doctor to some patient or other, perhaps different for each, and relating one particular patient to every doctor, the same for each. Moreover, the second type of complexity is absent from pairs like 'Dr Patel visited Mrs Wilson' and 'Mrs Wilson was visited by Dr Patel'. The two interpretations thus have different truth conditions and, hence, should be represented by different semantic structures. However, there is no difference upon one point: both sentences and both interpretations present the doctors as the visitors and the patients as the visited. English syntax gives priority to this point; according to the voice of the verb, the positions in which we place the noun phrases relative to it determines which represents the visitors and which the visited. How, then, can we account for the two interpretations with regard to the relationship between every doctor and at least one patient? For the grammar so far expounded there is a prior problem, since it will not even provide a structural analysis of (6), never mind relate it to

Passives

17

(1). The difficulty lies with the semantic grid for 'visit', for, if we try to use a modified version of (IP), we shall get:

? at least one patient

I—[—I I

V"

I

I

past

V | be visited [A,P]

I P" [A] | by every doctor

The semantic grid under ' V requires the external argument of 'was visited' to be associated with the Agent role, and the internal argument with the Patient role, but here they are the wrong way round. This is but one example of a much more general problem: there are many other constructions, too, for which the grammar will not provide. Now string grammars are classified into four types, known as the Chomsky hierarchy. These differ in the kinds of productions which are allowed. The first, type 0, is unrestricted. The second, type 1, is contextsensitive. Its productions must take the form yAd => y\xd, where strings of mixed terminals and non-terminals may be substituted for the Greek letters but only non-terminals for the Latin capital; y and 5 may be empty but JX may not. These grammars are called 'context-sensitive' for the obvious reason that A can be re-written as JJ, only when it occurs in the context of y and 8. By contrast, type 2 grammars are context-free, and have productions of the form A => fi. Finally, type 3 grammars are regular and have productions of the form A => bC or A => b, where only terminals may be substituted for the small Latin letters. Each of these grammars is a special case of the type one lower than it, but the languages which it generates are labelled as being of the type of the most restricted grammar which generates them. 8 The grammar as so far presented is context-free; yet 'there is little reason to doubt that all natural languages can be . . . generated by context-sensitive phrase-structure grammars' 9 (Chomsky, 1966, p. 29). So 8

9

For further details of string grammars, see Gonzalez and Thomason (1978), Gross (1972), Aho and Ullman (1972), Hopcroft and Ullman (1969), Marcus (1967), Chomsky and Miller (1956) and Chomsky (1956). A detailed attempt on these lines has since been made by Gazdar et al. (1985).

18

Linguistics: strings

why not use a context-sensitive grammar instead? The answer originally given was that grammar as a whole can . . . be regarded as a device for pairing phonetically represented signals with semantic interpretations, this pairing being mediated through a system of abstract structures generated by the syntactic component. Thus the syntactic component must provide for . . . each interpretation of each sentence . . . a semantically interpretable deep structure and a phonetically interpretable surface structure, and, in the event that these are distinct, a statement of the relation between the two structures. (Chomsky, 1962, p. 52) The motivation for transformation rules is thus a belief that the structure determining the meaning of a sentence will normally differ from that determining its form of expression in a given language and, hence, that the syntactic part of a linguistic theory must consist of two components, one generating deep structures (the phrase-structure rules) and the other relating them to surface structures (the transformation rules). The latter are so called, accordingly, because they allow us to transform one derivation of a structure into another. The original version of transformational grammar contained a Passive Transformation rule which derived passive sentences from the deep structures which, unaltered, would have yielded the corresponding active ones, by swapping the positions of the two noun phrases and passivizing the verb. A rough equivalent in terms of the syntax outlined above is: C N"i I V N"2

=*

C N"2 I (be + en) V by N"i.

If we apply this rule to the example above, the output will be: At least one patient past (be + en) visit by every doctor, and subsequent rules inverted the order of past be to yield be past and then was, as also of en visit, which becomes visit en and then visited. Thus the Passive Transformation rule took a string which would otherwise yield the sentence (1) and gave us a string which would yield the sentence (6). According to the theory, there should be no difference of meaning between these two sentences. In that case, we must say that both are ambiguous as between the two interpretations and, moreover, that the ambiguity cannot be resolved in terms of deep structures. Yet that is hardly satisfactory, for the second interpretation is unlikely for (1), so how does it become so much more probable for (6)? Moreover, if we start with an active sentence like 'Dr Patel visited Mrs Wilson', the rule gives us 'Mrs Wilson was visited by Dr Patel'; now these do have the same meaning (at least, to the extent that they will be true under just the same conditions), so how is it that the rule preserves meaning in this case but not in the transformation of (1) into (6)?

Passives

19

The Passive Transformation rule also has its drawbacks from a purely syntactic point of view. It does not offer any explanation of the structural changes which it introduces, so that we do not know why the object of the active verb is moved into subject position, nor why this cannot happen to the subject of an active verb, nor why the changes are just what the rule says they are. It is, moreover, heavily biased towards English, while even in English some passive forms are found for which it will not account, and there are many more such cases in other languages (see Keenan, 1979; Chomsky, 1981a, pp. 117-27). Transformational grammarians now prefer, accordingly, to decompose passive formation into a number of steps each of which has a much wider range of instances and is, hence, syntactically more fundamental. Within this framework, transformation rules have become essentially movement rules, allowing a constituent of a given category to be moved to an empty position of the same category (substitution rules)10 or allowing a constituent to be adjoined to another to form a constituent of the same category as that to which it is adjoined (adjunction rules). But this alone would be too permissive, allowing ungrammatical expressions to be generated, so it is tempered by various restrictions. One of these has already been mentioned, that no argument may have more than one semantic role, but most of the restrictions are expressed in terms of government, an application of a traditional syntactic concept (according to which verbs, prepositions, etc., govern different cases) to nodes of phrase-markers. The basic idea is then that an expression may only move to a position which governs it. The definition of government is based upon that of ccommanding: X c-commands Y just in case X does not dominate Y and every Z that dominates X dominates Y. Thus a node will c-command its sisters and everything which they dominate. If Z in this definition be restricted to full phrasal categories, then X is said to m-command Y, so that if one node c-commands another it m-commands it, but not conversely. M-commanding is the weaker notion; for example, a head introduced by the complement rule-schema will m-command, but not c-command, the intermediate category node which immediately dominates it. Government is the same as m-commanding, except that there may be barriers to it. These barriers are always full phrasal category nodes. The basic case in which such a node is a barrier to government is when it is not semantically marked, that is, has not been assigned any of the 10

The F-movement rule is an exception to this, because it moves a verb from being under a F-node to being under an /-node.

20

Linguistics: strings

semantic roles; another way of putting this is that a full phrasal category is a barrier unless it is V" or the complement of a verb or adjective. This has an immediate application to passive constructions, for, if we suppose that the 'by'-phrase of a passive sentence originates in deep structure as a complement of the verb, it cannot be semantically marked, because the semantic role which should be associated with it is the external argument of the active verb. So the P" node above the 'by'-phrase would be a barrier to government, for example by the head of the verb phrase. This account of barriers is supplemented by a provision for barrierhood to be inherited by a full phrasal node higher up the tree from one which it dominates; but I shall not go further into this complication here. The notion of barrier is used to formulate a further restriction upon transformational movements, the Subjacency Condition: a movement may not cross more than one barrier. Reverting, now, to (6), one proposal for generating passive constructions is to categorize 'visited' as an adjective, with 'at least one patient' and 'by every doctor' as its complements, the former being moved into an empty N" position generated by the (SI) rule (Jacobsen, 1986, p. 158). The problem with this is that adjectives do not carry semantic roles, so it leaves 'at least one patient' and 'every doctor' with no roles assigned to them. Yet it is evident that these expressions do have semantic roles in (6). Most transformational grammarians, however, draw a distinction between syntactic and lexical passives. The latter, which do not concern us here, are illustrated in English by words like 'untaught' and 'unknown'; they are simply treated as adjectives and provided as such by the lexicon for insertion into appropriate deep-structure phrasemarkers. In (6), by contrast, 'visited' is a syntactic passive, and the supposition is that, in the deep structure, it carries with it the direct object of its active form, which is subsequently moved into an empty subject position to become the subject of the eventual passive sentence. The latter thus derives from a different deep structure from the active one. According to another proposal, then, 'be' is the main verb of a passive sentence and the passive participle is the verb of a subordinate clause. Let us consider how this might apply to (6). If'visited by every doctor' is the remnant (after deletion of empty positions) of a subordinate clause, what is the latter's category? On this account, it is a small clause. Small clauses are effectively sentences lacking an inflexion and sometimes, also, lacking a verb, for instance the italicized phrase in 'Romeo expected Juliet {to come) before noon'; but they always have a subject. Opinions differ as to their structure; one view, in which they are compatible with the three ruleschemas, is to assign them to a full phrasal category X", where Xmay be

Passives

21

/, V, A or P. We then have X" => N" X' by rule-schema (S) and X' then decomposes into a head and complements in the normal way.11 For a passive sentence we should want the structure of the small clause to be N" V. The suggestion is then that the object of the (active) verb in V is moved first to the (empty) N" position, and thence outside the small clause altogether to become the new subject of the whole sentence (see Radford, 1988, p. 445). Thus, for 'Mrs Wilson was visited' we might have the deep structure: I" 1 N" V"

r

I

e[past] V

be

I

I" I N"

I

I

V

I visited [A,P]

V" 1

N" [P] Mrs Wilson

A transformation is also required in order to inflect the verbs; this is effected by moving the verb under the /-node whenever the latter is empty. 12 It does not concern us here and I shall take it for granted in future examples. This solution only caters for passives which lack a 'by'-phrase, and it is obvious that it will run into difficulties if we supply, for example, 'Dr Patel' as the subject of the small clause; for then the movement of 'Mrs Wilson' to that position is blocked, while there is nowhere for 'Dr Patel' to go in order to yield 'visited by Dr Patel'. The most recent proposal is therefore that the passive participle be analysed into two constituents, the first the verb root, which carries with it the semantic roles of its internal 11 12

See Stowell (1981). This condition is subsequently relaxed to allow movement of the verb under the /-node when the latter is occupied by the past participle suffix -en (see below).

22

Linguistics: strings

arguments, and the second the suffix -en which carries the semantic role of the verb's external argument. Where, however, a 'by'-phrase is also present in the sentence, the semantic role of the external argument of the verb is assigned to it by the passive suffix; this is achieved by providing an optional sub-categorization for the latter with the frame / by N"] (Jaeggli, 1986). This idea has been further developed by Baker, Johnson and Roberts (1989). Instead of splitting past participles under a Fnode into verb root and suffix, they propose that the latter be considered an inflexion in deep structure, to be combined with the verb root later by a transformation. 'Be' in the passive voice is treated as a separate verb, which thus requires its own inflexion for tense, etc. The resulting deep structure for (6) would then be given by the phrase-marker (6P). (6P)

I" I N" e

1 r l

r

; past

1

1 r

V

1

1

be

1

1 r 1

I I

I

I

V"

i

i

-en [A]

V I I

l_

r

i

P" I

N" [P]

I

I

visit by every doctor | [A>p] at least one patient Where, as here, a 'by'-phrase is present, the external semantic role assigned by the verb is considered to be transferred down the tree from the passive suffix to the noun phrase within the prepositional phrase. The relevant transformation is movement of 'at least one patient' to the empty TV" position. However, a trace is left in the vacated positions, and this is to be regarded as filling the place which it occupies, so that the latter is no longer available to be filled by a subsequent transformation movement. Moreover, the moved constituents are indexed and the same

Passives

23

index is attached to the trace marking their original positions (or intermediate positions, in the case of successive moves); constituent and traces are then said to be co-indexed. Using 't' for traces, and numeric sub-scripts for co-indexing, (6P) will yield the following shallow structure: (6S)

at least one patienti was2 t2 visited3 t3 by every doctor t\.

From a semantic point of view, there is a serious difficulty in this proposal which its authors do not even consider: what sense can we attach to the notion of a semantic role being associated with the passive suffix? It makes sense to associate a semantic role with, for example, one of the arguments of a verb, because the noun phrase occurring in that argument describes whatever plays that role. In the case of the passive suffix, however, nothing is described which could play the role of for instance, Agent. Consequently we cannot attach a meaning to the proposal, despite its conformity with the technicalities of transformational grammar; the language in which it is set out has simply 'gone on holiday', to use Wittgenstein's phrase. The essence of the current analysis of passives by transformational grammarians is to see them as containing a subordinate clause (on which the small clause is a variant). The analogy is with sentences like Thilip reckons the computer (to be) good value', which can be paraphrased by Thilip reckons that the computer is good value', in which the subordinate clause is explicit. The intended parallel is then: Philip reckons the computer (to be) good value Mrs Wilson was Dr Patel -visited in which the Agent of the passive sentence is the analogue of the subject of the subordinate clause in the first example; I have tried to bring this out by forming a composite predicate 'Dr Patel-visited' which is something that Mrs Wilson can be. But some transformational grammarians remain unhappy with small clauses (Radford, 1988, pp. 515-20; cf. Williams, 1983, and Postal and Pullum, 1988), while another has written of their supposed analogues: 'It is one of the enduring embarrassments of the field that this apparently routine question of constituency has proven so difficult to resolve' (McCloskey, 1988, pp. 556). Head-driven phrase-structure grammar also adopts a two-verb analysis of passives, but avoids transformations by sub-categorization which makes the grammar context-sensitive. The central provision is to sub-categorize 'be' in passives so that its subject is the same as the

24

Linguistics: strings

(missing) subject of the verb phrase which follows it; this is done by introducing a variable as the value of the feature 'subject', as follows: V [SUBCAT,(VP [ +PASSIVE; SUBJ,Y]>; SUBJ,Y]. There is then no need to generate a structure with empty N" positions, but this complication is avoided only at the price of further, extensive subcategorization instead. The detail need not concern us, because the basic structural analysis is the same as in transformational grammar.

1.4 P R O N O U N S AND RELATIVE CLAUSES Two other types of construction which put a theory of meaning to the test are propositions containing anaphoric pronouns (those which have antecedents within the proposition) and relative clauses. Transformational grammar treats pronouns as noun phrases, that is, insertions directly under N'\ but distinguishes them into anaphors and pronominals. Anaphors comprise reflexive pronouns and expressions like 'each other'; pronominals include personal pronouns, whether anaphoric or not. In deep and shallow structures, the antecedent of an anaphoric expression is shown by co-indexing, which must, accordingly, be regulated. This is done by principles of binding, which is defined in terms of ccommanding: X is bound | 1 3 X is an argument and for some argument Y: Y c-commands X and Y is co-indexed with X. Arguments which are not bound are said to btfree. Since both X and Y in this definition have to be arguments, they will both be noun phrases. There are three principles of binding, which relate it immediately to the treatment of pronouns: (Bl) (B2) (B3)

an anaphor is bound in its governing category; a pronominal is free in its governing category; other noun phrases are free everywhere.

This means that an anaphor must be co-indexed with an antecedent related to it in a particular way, determined by the notion of governing category, which is defined as follows: X is a governing category for Y J X is the minimal I" or N" containing both Y and a governor of Y. 13

Following Frege, I use 'I' as a sign for a definition.

Pronouns and relative clauses

25

The effect of the binding principles on the treatment of pronouns may best be illustrated by examples. Consider, then, the deep structure required in order to generate (7)

Octavian defeated Antony and he killed himself.

The phrase-marker, with co-indexing shown, as before, by sub-script numerals, would be as in (7P). (7P)

I"

F

I

I

I" I

I

N"[A] Octavian

I

and

I e[past] V

I

I" 1

r

N"[A]

V"

he, N" [P]

defeat | [A, P] Antony2

I

r

I

V"

e[past] V I

N" []

kill [A, P] himself,

If we look at the branch of this phrase-marker to the right of'and', we see that 'kill' governs [N\V]. The minimal T' or N" dominating both of these is the / ' to the right of 'and', so this is the governing category for [Nf\V], We have inserted an anaphor under that node, so by (Bl) it must be co-indexed with an argument dominated by the same /" node and which c-commands it. The only candidate is [N"J']y under which we have inserted a pronominal. The latter, accordingly, must be co-indexed with the anaphor. However, by (B2), the pronominal is free with respect to the F node. We are, though, free to co-index the pronominal with either of the nouns in the branch of the tree to the left of the 'and'. Here it is shown co-indexed with 'Antony', giving the sense that Antony killed himself, the most natural interpretation of (7). But we could have coindexed it with 'Octavian' instead, and then the sense would be that Octavian defeated Antony and Octavian killed himself - a possible but unlikely reading of (7). Finally, if the pronominal were not co-indexed with either of the preceding nouns, an alternative for which the binding rules allow, the sense would be that Octavian defeated Antony and

26

Linguistics: strings

someone else (exactly who being grasped, presumably, from the context) killed himself. In treating relative clauses, transformational grammarians distinguish between restrictive and appositive ones. The former are sometimes also called defining relative clauses, as in this characterization by Fowler: A defining relative clause is one that identifies the person or thing meant by delimiting the denotation of the antecedent: Each made a list of books that had influenced him; not books generally, but books as defined by the that-clause. Contrast with that: / always buy his books, which have influenced me greatly; the clause does not limit his books, which needs no limitation; it gives a reason (= for they have), or adds a new fact (= & they have). (1926, entry that, rel. pron.) This is a semantic explanation of the difference between the two types of clause; transformational grammarians would no doubt prefer a more syntactic characterization. Thus restrictive clauses may be introduced by a relative pronoun inserted under an TV" node, by the complementizer 'that' (finite clauses only), or may have no overt introducing expression; they can sometimes be separated from their antecedents, but they cannot modify proper nouns. Appositive clauses, by contrast, are always introduced by an overt relative pronoun inserted under an N' node, cannot be separated from their antecedents but can modify proper nouns; they are by way of parenthetical comments or afterthoughts, and this is often indicated by intonation or punctuation (commas, hyphens, parentheses) (Radford, 1988, pp. 480-1). Sometimes the intonation or punctuation is essential to mark the difference; thus the relative clause in (8)

Eskimos, who live in igloos, have lots of fun

is appositive, but in (9)

Eskimos who live in igloos have lots of fun

is restrictive. On other occasions a sentence containing a relative clause may be ambiguous: (10)

Serious works on Russia from Polish sources, which are not intended as merely propagandist pamphlets, are a valuable contribution towards a better understanding of that country

on which Fowler comments: 'If the clause is non-defining, . . . none of these serious works are propagandist, and all are valuable. The real meaning is that some of them are free of propaganda, and are therefore valuable' (1926, entry which) (that) (who, 3).

Pronouns and relative clauses

27

Fowler, indeed, proposes to mark this difference of meaning by reserving 'who' and 'which' for appositive relative clauses, and 'that' for restrictive ones: The two kinds of relative clause, to one of which that and to the other of which which is appropriate, are the defining and non-defining; and if writers would agree to regard that as the defining relative pronoun, & which as the non-defining, there would be much gain both in lucidity & in ease. Some there are who follow this principle now; but it would be idle to pretend that it is the practice either of most or of the best writers. (1926, entry that, rel.pron.)

He gives a long list of examples to show that the practice which he recommends would aid in clarity; but contrary usage both before and since he wrote makes its general adoption, however advantageous, highly improbable. Yet it can still be a useful convention in the presentation of linguistic examples and I shall follow it where possible. From the present point of view the importance of this distinction is that it sometimes carries with it a difference in the circumstances under which a sentence is true and, hence, a difference of meaning. Thus (8) says of eskimos in general both that they live in igloos and that they have lots of fun, whereas (9) says only of eskimos that live in igloos that they have lots of fun. Similarly for the two readings of (10). Our question must therefore be whether this difference of meaning can be accounted for by a structural difference. Jackendoff (1977, section 7.2) notes that where a noun is qualified both by a restrictive and by an appositive relative clause, the former must come first and the latter after it, thus: The man that came to dinner, who was drunk, fainted'. Moreover, restrictive relative clauses can be concatenated, for instance, 'The man that came to dinner that gobbled his soup fainted', whereas appositive ones must be joined with an 'and', for example 'Eskimos, who live in igloos and who hunt seals, have lots of fun'. Again, when the noun modified by the relative clause is in object position, the sentence can be negated if the clause is restricted but not if it is appositive. Thus we can have 'Baldwin did not greet the man that was sitting opposite him' but not *'Baldwin did not greet the man, who was sitting opposite him'. From all this, Jackendoff concludes that restrictive relative clauses are more intimately tied to the noun which they qualify than are appositive ones. His version of X-bar syntax allows for triple-barred categories, so he accounts for these differences by generating appositive clauses as complements of N" (immediately dominated by triple-bar N), but restrictive clauses as complements of N' (immediately dominated by

28

Linguistics: strings

N").14 This resort is not available in a system restricted to double-barred categories but, since part of the argument for distinguishing adjuncts from complements is that adjuncts always follow complements, it is open to us to generate restrictive relative clauses as complements of N, and appositive ones as adjuncts of N'. Since a head can have more than one complement, this allows for concatenated restrictive clauses; but as a head can only have one adjunct (at a time), multiple appositive clauses at the same level would have to be formed by conjunction. This yields the phrase-marker (9P) for (9). (9P)

I" I

I

I

N" [A]

I

I

N"

I

I

eskimos

I

C"

I

N" I

I

I C I

have lots of fun [A]

I

C 1

I

I I" I

I

N" r [A] I live in igloos [A] whThe reason for the empty N" here is that, where the verb of the relative clause is transitive, its object may give rise to the relative pronoun, as in 'someone whom I met'. In that case, 'wh-' will be inserted under the N" node under /' (not shown here) and subsequently moved by a transformation rule to the empty N" position. In order to produce Fowler's canonical form of (9), 'Eskimos that live in igloos have lots of fun', we insert 'that' as complementizer under the C node; 'wh-' is also inserted as above, and moved into the empty N" position, being deleted in the phonetic component. It is therefore supposed that, even in the present case, 'wh-' is also moved to the empty N" position. 14

Van Benthem (1988, p. 40) asserts that there is strong syntactic evidence in many languages for the structure '(Determiner Noun) Relative-Clause' rather than 'Determiner (Noun Relative-Clause)', but he presents no evidence and transformational grammarians evidently disagree with him.

Pronouns and relative clauses

29

The phrase-marker for (8) is only slightly different from (9P), but places the relative clause at one further remove from the modified noun. (8P)

I" I N" [A] have lots of fun [A]

1

N' I C

N' N

1

1*

imos

e;

1

C 1

1

I"

C

r N" [A]

1

live in igloos [A]

wh-

This makes no difference to the transformation and traces which it leaves, so that the shallow structure for both (8) and (9) is: (8/9S) eskimos wh-i e tx live2 t2 in igloos have3 /3 lots of fun, with 'that' replacing 'e' as an option for (9S). I shall now consider how transformational grammar would handle three examples which combine reflexive, personal and relative pronouns and which are especially challenging when we ask how to represent their meanings. In the first example, the relative clauses are restrictive (according to the criteria given by transformational grammarians) and are concatenated: (11)

Anyone that hurts anyone that hurts him hurts himself.

I assume that 'hurts' is an [Agent, Patient] verb. The two restrictive relative clauses here are nested, not concatenated: the clause 'that hurts him' qualifies the second 'anyone', not the first. So, using a broken line for omitted steps unimportant to the present example and which the reader can supply for himself, the phrase-marker of the deep structure should be as shown in (IIP).

30

Linguistics: strings

(IIP)

I

N"[A] I

Det

N'

I any

I

r N

I

I

C"

I

one! N"

hurt [A,P]

1

I

N"[P]

V

c

that

V I

himselfj

I

N"[AJ V

whI V

I N"[P]

_J

hurt Det [A,P] I any

N'

r N one

I N"

C" I

r

c

that

e l

r N"[A] wh-

H

V

V N"[P]

hurt hinij [A,P]

The anaphor at the end of the sentence is compulsorily co-indexed with the first occurrence of 'anyone', by the same reasoning as in (7P). The pronominal 'him' is free in its governing category, which is the first /"

Pronouns and relative clauses

31

above it, so we are permitted to co-index it also with the first occurrenee of 'anyone', as the most natural sense of (11) requires. By way of transformation, each occurrence of 'wh-' will be moved to the nearest empty N" position to its left, leaving, of course, a trace. Assuming also movements of the verbs to their respective / nodes (omitted above), so that they can be inflected for tense, we shall then have as the shallow structure: (1 IS) anyonei wh-2 that t2 hurts3 /3 anyone wh-4 that /4 hurts515 himi hurts6 t6 himself!, which will yield (11) by elimination of 'that' and the traces, together with appropriate inflexion of 'wh-' (as 'that', rather than 'who', under the Fowler convention). My second example is a Bach-Peters sentence, which is interesting because the subject noun phrase contains a pronominal which relates to the object noun phrase, so that the pronominal has a 'postcedent' rather (12P)

Det

fool [A,P]

love [A,P]

hini2

32

Linguistics: strings

than an antecedent, while the object noun phrase also contains a pronominal whose antecedent is in the subject noun phrase: (12)

A boy that was fooling her kissed a girl that loved him.

For the present, I shall assume that these relative clauses are also restrictive, on the ground that they, too, satisfy the criteria described above; but this will be called into question in chapter 4. The phrasemarker for the deep structure will then be (12P). Following the same transformational and co-indexing procedures as in the previous example, we shall obtain from this the shallow structure (12S) A boy! wh-2 that t2 was fooling her3 kissed4 /4 a girl3 wh-5 that t5 loved6 t6 My final example comes from the genre of 'donkey'-sentences, so-called (13P)

I" I

I

N"[A] I

Det

I

N' I

every N

onei

I

shall [past]

C"

r

I

I c

N"[P]

I

N"

i that

I

I

N"[A]

I

return [A,P]

it2

F

r

I

V"

wh-i I

r V

V 1

I I borrow Det [A,P]

N"[P] I

I

N' N

book2

The semantic component

33

after the medieval example 'Every man who owns a donkey beats it'. For a change, I offer the following variant: (13)

Everyone that borrows a book should return it.

The full interest of this example will only emerge in chapter 4, but as a foretaste we may note that an attempt to form a corresponding passive sentence, *'It should be returned by everyone that borrows a book' is inadmissible given that the pronoun is to be anaphoric. The phrasemarker (13P) for the deep structure presents no special difficulties, however. With the usual transformations again, the shallow structure is: (SI3) Every onej wh-2 that /2 borrows3 t$ a. book4 should return it4. Transformational grammar is thus able to generate all of these examples; in head-driven and generalized phrase-structure grammar, they are again catered for by sub-categorization. In the case of relative clauses, a feature SLASH is introduced which has categories as its values. Rules then provide for introducing this feature at the / node of the relative clause and passing it down to each subsequent node on a path which eventually leads to e. There is thus no fundamental difference between the structures posited by these grammars and the shallow structures of transformational grammar described above. Our next question, therefore, must be whether it can interpret them satisfactorily. In order to answer that, we have to look at the semantic component of the grammar.

1.5 THE SEMANTIC COMPONENT In the original version of transformational grammar, semantic interpretation was applied to deep structures, as an appendage to the system of syntactic features which governed lexical insertion. Interpretative semantics, as it was called, aimed to explain how one or more meanings may be attached to a sentence given one or more meanings for each of its constituents (Katz and Fodor, 1963; Katz and Postal, 1964). The method adopted was, first, to specify a form for dictionary entries of lexical items which distinguished different meanings of each item by a tree, each meaning corresponding to one path along the tree; and, second, to specify rules telling us how such paths may be combined. This approach was criticized by Weinreich on the ground that it tried to combine a recognitional theory of meaning with a generative syntax, whereas 'Semantic theories should be formulated so as to guarantee that deep structures (including their lexical components) are specified as

34

Linguistics: strings

unambiguous in the first place . . . and proceed from there to account for the interpretation of a complex expression from the known meanings of its components' (1966, pp. 398-9) Instead of modifying the semantic component on the lines suggested by Weinreich, however, transformational grammarians now propose a completely different account. To begin with, semantic interpretation is applied to shallow structures, 15 that is, the output of the transformation rules, and no longer to deep ones; the most important reason for this change is that the interpretation requires the traces introduced by the transformation rules, since it is assumed that the assignment of semantic roles is left undisturbed by the latter. Next, semantic interpretation is separated into two stages. The first maps a shallow structure onto LF ('logical form' 16 ) which is restricted to 'those aspects of semantic representation that are strictly determined by grammar, abstracted from other cognitive systems' (Chomsky, 1977, p. 5) and whose status and properties are empirical matters, 'not to be settled in terms of considerations of valid inference and the like' (Chomsky, 1986b, pp. 67, 205n.). The second stage brings to bear such extra-grammatical knowledge as is required to complete an explanation of the meaning of the expression concerned. The semantic component of the new transformational grammar deals only with the first stage, and is thus considerably less ambitious than the earlier version. The basic rule for deriving LF from S-structure, termed by May (1985, p. 5) QR, allows us to move a noun phrase occurring under an S node and adjoin it to the left of the latter. It leaves a trace, with which it is coindexed. May sees QR as analogous to the rule for forming questions (or relative clauses) which moves 'wh-' to the C position under S' in deriving surface from deep structures. Thus we may compare: (14)

John saw who?

yielding (14LF) ((whoOc (John saw (/,)N0s)s with (15)

John saw everyone

yielding 15

16

Alternatively, shallow structures can be fed to a phonetic-form component which yields surface structures proper, that is, actual sentences. This component will not concern us here. This turns out to be different enough from what logicians would call 'logical form' to warrant retention of the scare quotes whenever the term is used.

The semantic component

35

(15LF) ((everyoneON- (John saw (/i)N-)s)s This, of course, pre-dates the latest modifications to transformational grammar. The movement of 'wh-' is now presumed to be to a leftmost node under C" which may be any of N\ P" or A" (see Radford, 1988, section 9.10, especially p. 504). Thus (14LF) becomes: (14LP)

«who,) N . (John saw (/,)N")e)c".

If the analogy is to be preserved, (15LF) would have to be altered to: (15LF)

((everyonei)N« (John saw (/i) N ")c)c.

the place for 'everyone' being provided by the specifier rule (SC) and no longer by an adjunction rule. We could then formulate the QR rule as follows: (QR)

(. . .(X)N-. . .)c => ((Xi)N"(. • .(/s)N-. • .)c)c-.

May imposes certain restrictions on QR. One is that Xj c-commands t\\ this is already built into the formulation above. Another is that QR may not be applied to proper nouns, only to a noun phrase consisting of a determiner and count noun. May's initial approach to multiple quantification is to allow repeated applications of QR, the first application yielding the leftmost noun phrase. Now the shallow structure deriving from (IP) will be (IS)

(every doctor)N» (visited\ t\ (at least one patient)N")r,

where t is the trace left after moving 'visit' under the empty /-node. With one application of QR, this would become: (every doctor2)N» (h visited i t\ (at least one patient)N»)r and then, by a second application: (lLFa) (every doctor2)N» (at least one patient3)N» (/2 visited! t\ ty)v But it would be equally legitimate to apply QR to 'at least one patient' first, giving the result: (lLFb) (at least one patient3)N» (every doctor2)N» (h visitedi t\ t^r It would thus be possible to obtain two distinct LF representations from (IS), corresponding to the two interpretations which its syntactic form leaves open. (In (lLFa) 'every doctor' c-commands 'at least one patient' but, in (lLFb), conversely.) Exactly the same applies to (6); from (6S) we can obtain either (6LFa) (at least one patient3)N» (every doctor2)N» (/2 was4 /4 visitedi by t\ ty)v

36

Linguistics: strings

or (6LFb) (every doctor2)N- (at least one patient3)N» (t2 was4 t4 visited i by t\ /3)i', with no explanation why the latter is much less probable than the former. May allows that in a particular case only one interpretation might be possible but, if so, then the other will be ruled out on grounds which are not purely syntactic. He does acknowledge a difficulty, though, with Bach-Peters sentences like (12). Beginning with a simplification corresponding to (16)

A boy kissed a girl that loved him,

he temporarily accepts the LF representation (16LF) (a boyi)N" (a girl that loved him2)N» ('i kissed /2)r but rejects (a girl that loved him2)N' (a boyi)N» (t\ kissed t2)v on the ground that the pronoun 'him' is not c-commanded by 'a boy' in the latter, whereas it is in (16LF). When we try to apply QR to (12S), however, there is no way in which we can so represent its LF that 'him' is c-commanded by 'a boy that was fooling her' and 'her' c-commanded by 'a girl that loves him', for the only possibilities are (a boy that was fooling heri)N» (a girl that loved him2)N» (/i kissed /2)r, in which 'him' is c-commanded by 'a boy that was fooling her' but 'her' is not c-commanded by 'a girl that loved him', and (a girl that loved him2)N» (a boy that was fooling heri)N« (t\ kissed /2)i', in which the converse obtains. His solution is to propose an Absorption of two noun phrases, one of which c-commands the other, into 'something like a conjoined constituent' (1985, p. 21) in which each c-commands the other. This requires a new expedient in indexing, in which an index is attached to the conjoined constituent: (12LF) ((a boy that was fooling heri)N» (a girl that loved him2)N»)N"2 (ti kissed r2)r. Donkey-sentences are treated in the same manner (pp. 74, 152). He rejects the analogue of a booki ((everyone that borrows

/I)N"2

(h should return i

The semantic component

37

for (13) on the ground that the 'wh-' movement would produce *'Which book should everyone that borrows return it?', and proposes instead: (13LF) ((a booki)N- (everyone that borrows /I)N")N-2 (f2 should return iti)r. This appears to be more of an ad hoc than a principled solution, for it assumes a grammatical rule quite foreign to every version of transformational grammar, N" =» N" N", which is crucially unlike a conjoined constituent in having no conjunction. Moreover, so far as I can see, there is no way in which Absorption might be integrated into the latest version of transformational grammar as a case of attribution or of adjunction, for that would have to take place at the N' level, at which the quantifying phrase has been stripped of its determiner. Absorption would also apply, of course, to examples like (1) and (6), so that we finally have (lLFc) (every doctori (at least one patient2)N')N»2 ('i visited /2)r and (6LFc) (at least one patient3 (every doctor2)N»)N"2 (h was4 t4 visitedi by tx h)v But (lLFc) would be susceptible both of the interpretation that it is true of every doctor that he or she visited at least one patient, and of the interpretation that it is true of at least one patient that every doctor visited him or her, and a similar ambiguity would be intrinsic to (6LFc). So there would be no way of distinguishing these two meanings. May now discovers a difficulty in his representations of LF for sentences involving multiple quantification, prompted by the analogy which he sees between quantifiers and interrogative or relative pronouns. In order to rule out *'What did who admire?' (in contrast to 'Who admired what?'), transformational grammarians have proposed an Empty Category Principle (ECP) to the effect that an empty category must be governed by the expression to which it is co-indexed. This means that when the empty category occurs in a subject position, the co-indexed phrase must be adjacent to it. Example (lLFb) fulfils this condition, but (lLFa) and (lLFc) do not, so if the ECP applies at LF, the only LF corresponding to (1) would be (lLFb). How, then, are we to provide for the two interpretations of (1)? May's answer is to modify the definition of 'c-command' so that the two quantifying phrases mutually govern each other and then to provide alternative interpretations of (lLFa). This is to claim, in effect, that the difference between them is not structurally

38

Linguistics: strings

determined by syntax. Yet previously he thought that it was so determined. What caused him to change his mind? Not, it seems, any clearer view of what aspects of meaning are determined by syntax, but the presumed analogy with interrogative pronouns and their treatment in the syntactic theory. The modification also raises a further difficulty, which is brought out by the example: (17)

Some student admires every professor, but John doesn't.

May assumes that 'but' conjoins two sentences here. Consequently, (every professori (some student2 (/2 admires /i)r))r but (John doesn't (admire /i)v")r must be rejected, because the second occurrence of t\ is not c-commanded by 'every professor'. Instead, he proposes that 'every professor' be adjoined to the verb phrase, being repeated on the latter's second occurrence: (some student2 (f2 (every professori (admires fi)y)r)r)r but (John doesn't (every professor3 (admire f3)v")r)r« May finds further support for this proposal in examples like (16), where (16LF) would commit him, as one possible interpretation, to *'there is at least one girl who loved him such that a boy kissed her'; in order to avoid this, he proposes to adjoin 'a girl that loved him' to the verb phrase. In the most recent version of transformational grammar, 'every professor' and 'a girl that loved him' would have to be attributes of F \ with the effect that / occurred to the left of it. So now we have two positions to which QR may move a noun phrase: it may become an attribute of C" or of V'\ and QR would, accordingly, require amendment. May acknowledges that quantifying phrases as adjuncts of verb phrases are awkward, in that a quantifying phrase is customarily regarded as qualifying a sentence frame (frame, because it will contain at least one empty category trace). Although he argues that quantifying phrases adjoined to verb phrases do also c-command the traces in subject position immediately preceding them, he subsequently discovers further exceptions to the Empty Category Principle when extended to quantification (1985, pp. 115-17). This leads him to replace it by a Path Containment Condition (PCC). A path is here understood to be 'a set of occurrences of successively immediately dominating categorial nodes' in a phrase-marker from an empty category trace to the quantifying phrase with which it is co-indexed (p. 118). The condition is then that paths must not cross over each other, though they may share common segments

The semantic component

39

(parts). This still rules out (lLFa), since paths 1 and 2 would cross, but allows (lLFb), in which they do not. Similarly, it allows for verb-phrase adjunction, since the paths do not then intersect at all. So far, paths have been defined as starting with a co-indexed empty category symbol, but May then broadens the definition to allow them to begin with co-indexed pronouns. With this extension, the PCC differentiates between LFs for examples like (16) and (17) in which the quantifier phrase is adjoined to the verb phrase, which conform to it, and those in which it is adjoined to a sentence-node, which do not. Moreover, while allowing (13), it excludes the corresponding passive *4It should be returned by everyone that borrows a book' in which the pronoun is to be understood as anaphoric. He also claims that the PCC legitimizes (12LF), for the path from t2 to wa girl . . .' does not cross that from t\ to 'a boy . . .', while the paths from the two pronouns to their related quantifier phrases are internal to the combined noun phrase. However, as he does not analyse the restrictive relative clauses in each of the quantifier phrases, we need to look at this more closely, applying QR to (12S). The result would be: (12LF')

(a boy } wh- 2 that t2 was fooling her3)N» (a girl3 wh-4 that

loved

N'^N"! (/ 3 kissed t\)\>.

Now in (12LF) 'a boy' and 'a girl' are not indexed, but according to the rules for co-indexing pronouns, they must be so when the phrasemarkers for the quantifying phrases are spelled out in full. Paths 2 and 4 are duly nested within paths 3 and 1 respectively and pose no problem. But there is no way of avoiding a crossing of paths 1 and 3 at the N" node which dominates 'a girl that loved him', and that is a categorial node. This will be clear from inspection of a simplified phrase-marker corresponding to (12LF'): C" I

i

Det

N"

N"

N"

N'

D

N' N

C"

boy!

that was fooling her 3

a

r N girl3

I

I'

7 kissed t\

C" that loved hi

40

Linguistics: strings

Path 1 (from 'him') = , but path 3 (from t3) = and the two paths must cross either at N"2 or at some point below it on the 'a girl that loved him' branch. Thus May has no way of representing an LF for this example which conforms to his own rules. Other transformational grammarians treat pronouns in LF rather differently from May. If we consider again the first example of section 1.4, the shallow structure is the same as the deep structure, so far as the pronouns are concerned: (7S)

(Octavian (defeated2 t2 Antonyi))i» and (hei (killed3 t3 himselfi))r'.

Here the antecedent of the pronouns is a proper noun. It seems from examples (see Chomsky, 1977, p. 195; Jacobsen, 1986, p. 346) that such an antecedent is considered on a par with a quantifying phrase and preposed in the form 'for x = Antony'. The pronouns themselves can then be interpreted in accordance with the anaphora rule, to the effect that a pronoun whose antecedent is a quantifier phrase is to be re-written as the letter which is added to the latter when it is preposed. The numerical sub-scripts then become redundant and the LF for (7S), omitting the traces of the verb movements, will be: (7LF) for x = Antony [Octavian defeated x and x killed x]. It is unclear whether 'Octavian' should be treated in the same manner; examples in the literature containing proper nouns which are not the antecedents of pronouns are left unchanged. There is some dispute whether relative pronouns should be represented on a par with interrogative ones, but Chomsky prefers to treat a relative clause as a predicate of its head; thus, given a shallow structure the mani [wh-2 hisi mother loved /2 best], a rule of predication identifies T and '2' to yield: the mani [wh-! t\'s mother loved t\ best] (1982, pp. 13 and n. 11, 92-5). This would simplify slightly the structures proposed by May, for example by identifying sub-script 2 with sub-script 1 in (12LF'), and sub-script 4 with sub-script 3, but would make no essential difference. Chomsky has subsequently extended this treatment to clauses from which the relative pronoun has been omitted; the empty category sign 0 is then replaced by a letter and 'wh-' by an operator 'O' (1986b, p. 84). The difficulty with (12) also remains under Chomsky's proposals. The anaphora rule only allows replacement of a pronoun by a co-indexed

The semantic component

41

trace when the pronoun occurs after the noun phrase (its antecedent) to which its meaning relates (see Chomsky, 1977, p. 202). In (12), 'her' relates to 'a girl' and precedes it (its postczdznt). Lasnik (1976) tried to tackle this problem, but Evans (1980) has shown that the solution which he proposes will not work. Evans himself, however, proposes a rule having the effect that a pronominal may be co-indexed with a noun phrase provided that it does not both precede the latter and c-command it (1985, p. 244). This covers (12), since 'her', though preceding 'a girl', does not c-command it {vide (12P)). None of these proposals for the treatment of relative clauses in LF will serve to distinguish the meanings of (8) and (9), for, as we concluded in section 1.4, both have the same shallow structure: (8/9S) eskimos wh-i e t\ live2 t2 in igloos have3 /3 lots of fun, which would yield (8/9LF)(eskimos4 (t4 wh-i e t\ live2 t2 in igloos have3 /3 lots of fun)) on May's principles, with sub-script 1 changed to 4 for Chomsky. Yet, by analogy with May's representation of other examples of restrictive relative clauses, it seems that he would want the LF for (9) to be: (9LF) (eskimosi wh-2 / t2 Hve3 /3 in igloos)N» (t\ have4 t4 lots of fun)r. But this would mean that the distinction between restrictive and appositive relative clauses was introduced at LF, rather than different LF representations being 'triggered' by a difference in shallow structures. Finally, we have to consider how (11) would be represented at LF. The shallow structure was (1 IS)

anyonei wh-2 that t2 hurts3 /3 anyone wh-4 that t4 hurts5 t5 himj hurtS6 t6 himself i.

The literature offers no guidance on how to deal with an example like this, so we shall have to improvise. Supposing the relative clauses to be restrictive, the second appears to be embedded within the first, so that a representation consonant with May's general approach to restrictive relative clauses would be: (anyonei wh-2 that t2 hurts3 /3 (anyone wh-4 that /4 hurts5 t5 himi)N»)N" (ti hurts6 But this is not yet in LF, because the second occurrence of 'anyone', although a quantifying phrase, is not co-indexed. The problem, however, is with what to co-index it, since the subject position of the second occurrence of 'hurts' is already filled by the co-indexed trace f3. The best

42

Linguistics: strings

we can do is probably to give it the same index, 4, even though there is no rule to justify this: (11LF) (anyonei wh-2 that t2 hurts3 /3 (anyone4 wh-4 that /4 hurts5 t5 himi)N»)N" (t{ hurtS6 /6 himselfi)i' At least it is clear that none of the paths will then cross. A final judgment upon this new semantic component of transformational grammar must be postponed until we have considered the structural analyses of these examples which logicians can offer, for transformational grammar leaves it to other disciplines to tell us how to proceed from this point onwards in assigning a meaning to the structures which it delivers. But it should be clear already that the semantic component is still very inchoate and is beset with many unanswered problems. Part of the trouble undoubtedly lies in the methods of transformational grammarians: they are constantly modifying their notation and rules (which is perfectly legitimate and, indeed, necessary to progress) but do not then go back to re-work earlier examples (which is not). This often happens even within a single work (as illustrated several times in this chapter). The result is that the pieces do not fit together and anyone who tries to extract a coherent whole becomes extremely frustrated. But there are also problems of substance. Thus the analogy which transformational grammarians see between the treatment of relative clauses in shallow structures and of quantifying phrases in LF may be questioned. So far as LF is concerned, the sub-script letter attached to 'wh-' appears redundant, in that it does no work. This can be seen if we replace sub-scripted 'wh-'s by 'such', for example in (12LF"): (12LF") ((a boyi such that t\ was fooling her3)N» (a girl3 such that /3 loved himi)N»)N"i (/3 kissed /i) r .

The phrase 'such that' - a logicians' device - is perhaps pedantic, but clear enough, and shows that nothing is lost to the meaning by eliminating the sub-script letter. Semantics is undoubtedly the Achilles heel of transformational grammar. This is now its second attempt at adding a semantic component to the syntactic theory, and it seems destined to be no more succesful than the first. Transformational grammar has now been around for over thirty years, so this deficiency can hardly be attributed to growing pains. It must surely prompt us to ask whether the theory is not mistaken about the relationship of syntax to semantics. In retrospect, the relegation of semantics as an appendage of syntax, a feature common to most theories due to linguists, has historical rather than theoretical justification:

The semantic component

43

modern synchronic linguistics began in the nineteenth century with phonology, and only discovered formal syntax in the thirties of this century, while its excursions into semantics remained at a pre-formal stage until very recently. Moreover, linguists were concerned to claim a certain autonomy for syntax. In transformational grammar this takes the form of a principle to the effect that no syntactic rule can refer to pragmatic, phonological or semantic information (see Chomsky, 1977, pp. 36-59).17 But that does not exclude an appeal to semantic arguments in favour of a particular syntactic rule, which are sometimes given quite explicitly by transformational grammarians. Moreover, the principle itself presupposes that we can always distinguish pragmatic, phonological and semantic information from syntactic. Yet it is admitted that the native speaker, upon whose judgments of acceptability syntactic theory is ultimately based, is for the most part unable to tell whether a deviant expression is meaningless or asyntactic18 or merely false (see Chomsky, 1977, p. 4; Radford, 1988, pp. 13-16). What settles this question is a theory: if an example violates the rules of my semantic theory, I say it is meaningless, but if the rules of my syntactic theory, then I say it is syntactically ill-formed. Indeed, this very expression presupposes a system of rules by the implied contrast with well-formed, that is, formed in accordance with certain rules. But is not this position viciously circular, with theory dependent upon judgments which in turn are dependent upon the same theory? It could be saved from this charge if, instead of being for the most part unable to tell whether a deviant expression were meaningless or asyntactic, we were for the most part able to do so. Then the theory would be built on the agreed judgments and used only to settle borderline cases. A good case can be made out for constructing a theory of meaning on this basis. There is a long tradition of philosophical criticism which takes the form of arguing that what an opponent has written is meaningless. This builds upon a type of protest which any language speaker quite naturally makes from time to time: 'It doesn't make sense'. In both cases, of course, the critic must be prepared, if challenged, to support the charge 17

18

It is to observe that transformational grammar does not itself entirely conform to this principle, since it embraces the assignment of semantic roles at the point of lexical insertion to yield deep structures, and subsequently appeals to semantic roles in defining government. This is not a term used by transformational grammarians, but it is cited in the OED in just the sense required here, with a quotation from Mark Pattison. There is no corresponding term 'asemantic', doubtless because 'asyntactic' transliterates a Greek word meaning 'disordered' and thus pertaining to the old sense of 'syntax'.

44

Linguistics: strings

of meaninglessness with argument, and may turn out in the end to have been wrong: such judgments are not infallible. It is much less plausible to try to construct a theory of syntax upon some notion of syntactic ill-formedness. The reason for this is that, whereas we are familiar with meaningless expressions that cannot be faulted on the ground that their authors do not know how to speak or write their language, we are very largely quite unfamiliar with syntactically ill-formed expressions whose sense cannot be impugned. One can give enough examples, perhaps, to show that there is a difference between semantic and syntactic ungrammaticality, but syntactic illformedness does not have to go far before it begins to impair intelligibility, so that, without a theory for a guide, one soon becomes uncertain whether the deviance is semantic or syntactic in origin. The ground of this asymmetry between syntax and semantics is that syntax is the handmaid of semantics. This may be unpalatable to many linguists brought up in neglect of semantics, but the primary purpose of knowing how to express oneself in English, Spanish or German, say, according to the generally accepted forms in those languages, is to communicate with others who share that knowledge. We study the syntax of a foreign language primarily because we want to be understood in it, to be able to convey our meaning to others. This does not exclude other purposes for syntax which have nothing to do with meaning, nor does it in any way deny that syntactic rules should for the most part be formulated in terms of syntactic structure. It does, however, explain why attempts to explain syntactic ill-formedness as a notion upon which to build a syntactic theory are doomed to failure, and one only has to look at any of these attempts to see how feeble they are. 19 It is possible, nevertheless, to extract structures for sentences containing multiple quantifying phrases upon which a correct account of meaning can be based from those generated in transformational grammar. A method for doing this has been developed by computer scientists and will be described in section 3.2. But the complexity is considerable even with a restricted vocabulary and quantification is only one aspect of a wider problem, to be explained in the next chapter under the heading of scope ordering. A serious attempt to extend the method to a wider range of linguistic structures would almost certainly turn out to be messy and enormously complicated. The basic reason for this is that in seeking to transform syntactic into semantic representations we are moving from simpler to more complex structures. There is therefore a 19

See, for example Radford, 1988, pp. 7-17, which explains well enough what grammaticality is not, but says almost nothing about what it is.

The semantic component

45

strong case for working the other way round instead, starting with the more complex semantic structures and projecting them onto simpler syntactic ones. We need, therefore, a different model of the relationship of syntax to semantics. The centrepiece of the model will be a theory of semantic structures, that is, an explanation of the meanings of expressions in so far as that can be given by reference to structures assigned to them. The task of a theory of syntax will then be to show how these semantic structures can be mapped onto forms which are generally acceptable in various languages. Since the syntactic theory will carry no burden of semantic interpretation as an appendage, it should be simpler than current transformational grammar. It is an open question whether the syntax would have to be specified by a tree grammar, or whether a string grammar would suffice. Transformational syntax cries out for a tree grammar, but that may be because of its (partly hidden) semantic content. It is idle to speculate further about syntax, however, until we have a much clearer idea of semantic structures, which brings us to the contribution of logic to our problem. Meanwhile, thirty years' work in transformational grammar, together with the rivals which it has engendered, has made linguists think of language in formal terms as never before and produced a wealth of syntactic insights and observations which will surely be incorporated into a syntax related to semantics in the way just outlined, even though many generalizations may demand modification to fit a different theoretical framework. The work on LF in transformational grammar described above makes much more sense as a contribution to the problem of parsing, that is, of obtaining a semantic structure or structures from syntactic ones - the converse of generation. For it sets out to extract from shallow structures a system of structures upon which an account of meaning can be based, which is, precisely, semantic parsing. In this perspective, the theory of LF has still not taken the measure of Weinreich's criticism of interpretative semantics: it, too, is still working the wrong way round, trying to attach a theory of semantic parsing to one of syntactic generation. But as part of a wider theory of syntactic and semantic parsing, it could have a future. As Wittgenstein said of his Tractatus, it is not like a box of junk (say, parts of old clocks) but, rather, like a clock that tells the wrong time.

Logic: trees

2.1 TREE GRAMMARS In logic, what corresponds to grammar is the set of formation rules for the symbols, that is, the rules which define a formula. Although I shall eventually argue that some of the structures exemplified in the formulas of modern logic are more complex than trees, many of them certainly are trees and probably the majority of logicians has assumed that all of them are. So it is fitting to take tree grammars as typical of logic in the way that string grammars have been taken as typical of linguistics. A tree is a finite set of one or more nodes which fulfil two conditions: first, that just one node is called the root and, second, that the remaining nodes are partitioned into disjoint trees, called sub-trees of the whole. Each root is connected to the roots of each of its sub-trees by a line and, intuitively, the resulting diagrams look like upside-down trees, with the root at the top and the branches spreading out downwards. The bottommost nodes are called the leaves of the tree and the set of leaves constitutes the tree frontier. As with a string grammar, a distinction is drawn between nonterminals (category symbols) and terminals but, in addition, each of these symbols is assigned a degree (sometimes also called a 'rank'), which is the number of sub-trees which issue from it. A tree grammar Gt consists of an ordered set (V,r,P,S), where V is the alphabet (the union of nonterminals C and terminals E) and r a set of integers giving the degree of each member of V. P is again the set of productions, which are of the form Ti => Tj, where Tx and Tj are trees. Finally, S is a finite set of starting trees, each of whose nodes must be labelled with an item from the alphabet. Thus, instead of allowing us to re-write one string by another string, a tree grammar allows us to re-write one tree by another tree. The rules of the simplest tree grammars - which will be enough for our needs - are in expansive form, viz. 46

Tree grammars Co => I C|

T I ....

47

I Cn

where Co, . . ., Cw are category symbols and T is a terminal symbol of degree n. Co, . . ., C,, may be empty; application of such a rule will terminate a branch of a tree. In order to facilitate comparison and contrast with string grammars, let us see how a string grammar might be used to generate the sentence used as an example in chapter 1, 'Every doctor visited at least one patient', with the absolute minimum of changes to the category and terminal symbols. The process by which the sentence is generated will consist of a series of trees and, as it is possible to express trees in a linear notation, to do so will make for a more compact exposition. Thus the general expansive form given above can be written: Co =» T(C,,. . .,Cn) and, as rules of this form, we may lay down the following, using the categories of transformational grammar: (Rl) (R2) (R3) (R4)

I" V" N" N

=> Past(V") =» visit (N", N") =» every (N), at least one (N) => doctor, patient.

Thus 'visit' is given a degree of 2, 'Past', 'every' and 'at least one' a degree of 1, and 'doctor' and 'patient' a degree of 0. Here, then, is the derivation of 'Every doctor visited at least one patient', using I" (sentence) as the start symbol: 1 I" 2 Past (V") by (Rl) 3 Past (visit (N",N")) by (R2) 4 Past (visit (every (N), at least one (N)) by (R3) 5 Past (visit (every (doctor), at least one (patient))) by (R4) In the original notation, the final tree will be: Past visit

r

I

every

at least one

doctor

patient

48

Logic: trees

The difference between this tree and the phrase-marker for the corresponding string is patent. Unlike the latter, it contains no category symbols and terminal symbols are not only to be found as leaves, but also at all the other nodes, including the root. There the sentence consisted of the tree frontier only, the remainder of the tree showing how it was derived, whereas here the sentence itself is displayed as having a tree structure, and the history of its derivation is given by a series of trees. 2.2 LOGIC AND M E A N I N G Even if we grant, however, that logic demands a tree grammar, by what right do we suppose that logic has anything to contribute to the representation of meaning! Traditionally, logic is the study of arguments (or of types of argument) in order to determine whether they are valid, and in practice it has concentrated upon theoretical deductive arguments. The attraction of using logical methods in order to represent everyday language is that they are geared to deduction, making it easier to determine what follows from a given premiss or premisses. But argument is not the only use of language, not even the principal one, so we need some guarantee that logical form has wider import or, at the very least, that it is capable of supplementation to cater for other uses. The way forward is shown by a method of establishing that an argument is invalid which is in such common use that we seldom reflect upon its implications. When we do not know independently whether the conclusion of an argument with which we are presented is true, we look for a parallel, a similar argument. If we suspect that the original is invalid, then we try to find a parallel the truth of whose premisses and the falsity of whose conclusion will not be contested. But even this does not settle the matter, for it may not be accepted that our new argument is parallel to the original, so this forces us to consider more carefully what we mean by 'parallel' in this context. By what process did we obtain the second argument from the first? The answer, clearly, is by changing some parts of the original but leaving others intact and this, in turn, implies that we saw the first argument as having some pattern or structure, which is preserved in the second. In some cases, producing the parallel argument may involve changing entire propositions among those constituting the original, but far more commonly we only change parts of its constituent propositions, thus recognizing the latter as having an internal structure which is relevant to the validity of the argument. So already we find ourselves committed in logic to a structural analysis of propositions. However, supposing that our second argument can be related to the first by a series of changes in

Logic and meaning

49

the latter, so that it cannot be denied that both have the structure which we saw in the first, it may still be objected that we have not shown the first argument to be invalid. The objection may be correct; if so, our mistake will have been that we changed too much in the original, that we treated something as mere content which should have been recognized as belonging to the structure or pattern. Of course, if we have found but one example of the structure which we saw in the original argument which has true premisses but a false conclusion, we have shown that that pattern of argument is invalid; but that does not establish that every argument in which the pattern can be seen is also invalid, for it may also have another structure of which there is no such example. One need only instance the pattern consisting of any two premisses and a conclusion, of which invalid examples can be thought up in a moment. Yet almost all of the valid forms of argument catalogued by Aristotle in his Prior Analytics have this pattern, too. Finding invalid patterns of argument is therefore also of limited use, and will never yield conclusive proof of the invalidity of any particular argument. But in practice it is often effective in convincing someone to withdraw an argument, by throwing upon him the burden of finding a valid pattern in it, which he may not be able to do. The logician, accordingly, is primarily interested in cataloguing valid patterns of argument. Yet he cannot do this simply by going through examples of each pattern. True, the more examples he tries out, failing on each occasion to find a counter-example, the more confident he can be that the pattern is a valid one, but at no stage does he have a guarantee that he has not simply overlooked an example which would show the pattern to be invalid. In this resort, he has two alternatives. The first consists in breaking down the argument into a series of minimal steps, and then justifying each step. The justification proceeds in two stages. First, each of the minimal steps is represented as a simple argument pattern. A representation of an argument pattern is known as a schema, the Greek word for 'pattern', but in order to avoid any confusion, it will be convenient here to distinguish between an argument schema and an argument pattern, the latter being the structure which the argument is seen as having and the former a representation of that structure. Logicians customarily use a notation for schemas in which the parts of the argument whose structure is being represented and which are replaceable by others without affecting the structure are indicated by letters or perhaps other symbols; these are, accordingly, called schematic symbols} An actual argument may therefore be obtained from a schema 1

I find this term much preferable to 'variable', an unfortunate loan-word from mathematics which has caused some confusion in logic.

50

Logic: trees

by substituting linguistic expressions for the schematic symbols which it contains. Yet we cannot allow any linguistic expression we please to be substituted for a schematic symbol, for the result of doing so might not make sense. Consequently, a schema must always be accompanied by a key to its schematic symbols stating what kind of expression may be substituted for each. The key gives us the directions for using the schema in order to obtain actual arguments from it; without the key, it has no use and, hence, no meaning. So, together with the logician's commitment to structural analysis goes a commitment to classifying linguistic expressions, that is, assigning them to categories. Since a theoretical deductive argument consists of propositions, the proposition will be the first of these categories and, in order to illustrate the method presently under consideration, it will be enough to introduce the schematic symbols T \ 'Q' and ' R \ for each of which a proposition may be substituted. Ex hypothesis a minimal step in argument will consist solely of one or more premisses and a conclusion. If there were any intermediate propositions, then it would not be a minimal step. Let us, then, use the following convention for argument schemas which represent minimal steps and, more widely, for patterns of argument from which any intermediate steps are omitted. The premisses will be separated by commas, and the premisses from the conclusion by 'f\ As an example of an argument schema, we may then cite: (51)

if P then Q, P f Q

Each schematic symbol occurs twice in this schema. That it does so is part of the pattern represented. So, in order to do justice to this feature of the pattern, a further restriction is imposed upon substitutions for the schematic symbols; not only must a proposition be substituted for each, but the same proposition must be substituted for every occurrence of the same schematic symbol. We can claim that (SI) represents a minimal step in argument because there is evidently no way of breaking it down into still simpler steps. So now we can proceed to the second stage, in which its validity is justified. One might well urge that no justification is needed or even possible in this case, that the schema is self-evidently valid. Anyone with experience of teaching logic, however, will know that it is perilous to rely upon what people consider to be self-evident. The schema (52)

if P then Q, Q \ P

is invalid, but to many a student beginning logic, this has been far from self-evident. Aristotle, too, would hardly have gone to the trouble of

Logic and meaning

51

baptizing the move in argument represented by this schema as the fallacy of asserting the antecedent (of a conditional) if its invalidity were so universally self-evident. A justification of the validity of (SI) is therefore required. Yet since ex hypothesi it represents a minimal step in argument, it cannot be justified by further argument. We have to ask, therefore, what is lacking in anyone who fails to appreciate that it is valid. The only answer left is that he does not understand the meaning of 'if. For it is already assumed that he understands the key to the schematic symbols and, since any proposition may be substituted for each of them, we cannot appeal to any particular choice of substitutions. Only 'if then remains, though taken in the context of the whole pattern represented by the schema, and not in isolation, as the invalidity of (S2) shows. Thus the pursuit of logic leads rapidly into the study of meaning, and this, in turn, demands that the structures represented by schemas be relevant to meaning. These argument schemas can be regarded as rules of inference, in accordance with which valid arguments are constructed. We can thus define a method of proof in terms of them which allows us to show that many other argument patterns are valid. Given any argument schema consisting just of premisses and conclusion, we see whether the latter can be derived from the former by a series of steps each of which consists in an application of one of the rules.2 The second method of determining the validity of a pattern of argument ignores any intermediate steps, concentrating upon the premisses and the conclusion. This is a defect vis-a-vis linguistic arguments, because someone might well reach a conclusion which follows validly from his premisses, yet do so by invalid steps. In other words, it does not test the validity of a chain of reasoning. It makes essential use of the notion of truth conditions, the truth conditions of a proposition being the circumstances under which it is true. This is extended also to apply to proposition schemas. The latter, of course, are neither true nor false and so, strictly, have no truth conditions. However, where we have an argument schema, it is possible to specify the truth conditions for the premisses and for the conclusion relative to one another, although not absolutely, and that is enough for the purpose of investigating validity. For the schema will be valid just in case there are no substitutions for its schematic symbols (legitimized by the key to 2

In practice, the basic argument schemas have to be supplemented, for this purpose, with some further rules of inference of which the premisses and the conclusion are themselves argument schemas. An example is the rule that at least one of a set of premisses from which a contradiction is derivable must be false. Such rules, following the Stoics, are called argument themas.

52

Logic: trees

them) which will yield true premisses but a false conclusion. Hence, if we can find a way of classifying the possible substitutions for the schematic symbols by which each type of substitution will yield a true conclusion providing that it yields true premisses, we shall have excluded the possibility of a counter-example and so have shown that the schema is valid. It is easiest to explain how the method is applied by way of an example, for which the following schema will do: (S3)

If either P or Q then R h If Q then R

For each of the schematic symbols here, a proposition may be substituted. But, ex hypothesis propositions are either true or false. So a simple way of classifying the possible substitutions in the schema is immediately to hand, according to whether the propositions substituted are true or are false. Since the schema contains three schematic symbols, there will be eight types of substitution, as there are two types of substitution for each. But there is no need to consider all of these, for, if the schema is invalid, there will be at least one which yields a false conclusion from a true premiss. We therefore suppose that the conclusion is false and see where this leads. Now, if we can relate the truth or falsity of substitutions for 4 Q' and 'R' in the proposition schema 'If Q then R' to the truth or falsity of the resulting proposition, it will be possible to infer from the falsity of the conclusion of the schema to the truth or falsity of the substitutions in it. In the present example, this is customarily done by stipulating that a proposition whose structure is representable by 'If Q then R' will be false if the proposition substituted for 'Q' is true and that substituted for 'R' is false, but will otherwise be true. Thus, on our supposition with regard to (S3), we only need to consider the cases in which a true proposition is substituted for 'Q' and a false one for *R\ The next move is to consider whether the premiss will be true or false for this substitution. Here we are looking at the premiss in relation to the conclusion, because the schematic symbols which occur in the latter also occur in the former and we are only considering the type of substitution for them which will yield a false conclusion. Thus, so far as the premiss is concerned, we have only two cases to consider: in both, a true proposition will be substituted for 'Q' and a false one for 'R' but, in the first, a true proposition will be substituted for T ' and, in the second, a false one. The problem is then to specify whether the premiss as a whole will be true or false in each of these two cases; if it is true in either case, then we have our counter-example to the argument schema and it is invalid.

Logic and meaning

53

The premiss has the same over-all structure as the conclusion but, instead of the schematic symbol 'Q\ has the (subordinate) schema 'either P or Q'. Since we are only considering the case in which a false proposition is substituted for 'R', it follows from our previous stipulation of the truth conditions of propositions having the structure represented by 'If Q then R\ that we have now to consider whether a substitution in the (subordinate) schema 'either P or Q' which is consonant with our substitution of a true proposition for 'Q' in the conclusion could yield a false proposition from 'either P or Q\ For, if the latter is true, the whole premiss will be false, which is not a counter-example to the schema. The next step, therefore, is to stipulate the truth conditions of propositions having the structure represented by 'either P or Q\ So we lay down that a proposition having the structure represented by 'either P or Q' is to be accounted false if a false proposition is substituted for T' and a false proposition for 'Q\ but is otherwise to be accounted true. In the example under test, it is given that the substitution for 'Q' is a true proposition; consequently the result of substituting in 'either P or Q' will be true whether the proposition substituted for 'P' is true or is false. Hence, every substitution in the schema which yields a false conclusion will also yield a false premiss, and so there is no counter-example to it: it is valid. Looking back at this method in the light of the example, we see that it involves structural analysis just as much as the first method. The structural analysis is, of course, partially accomplished already in the argument schema, but a further step is taken in the course of applying the method: the premiss 'If either P or Q then R' is treated as being constructed from two proposition schemas, by substituting 'either P or Q' for 'P' in 'If P then R\ The depth to which the analysis is pursued is also the same in both methods. It terminates when we reach the simplest proposition schemas which can be given for each piece of language which is regarded as reflecting the structure of the argument - minimal schemas, as I shall call them. Rule-schemas and themas prescribe how proposition schemas having the same over-all structure may be introduced or eliminated in the course of argument: thus (SI) can be regarded as a rule for eliminating 'if in the context 'If P then Q\ A proof, therefore, which only uses rule-schemas will take the form of breaking down the premisses into these minimal units and then building up the conclusion from the latter; the picture is overlaid, and so more difficult to discern, when the proof also involves the use of mle-themas. In an exactly parallel way, the method of truth conditions constructs an account of the truth conditions of the conclusion and then of the premisses from those

54

Logic: trees

which have been stipulated for the minimal units of which they are composed. If validity as defined by the method of truth conditions is to answer to validity as we should recognize it in linguistic arguments, the stipulations of truth conditions for the minimal schemas cannot, of course, be arbitrary. Hence these, like rule-schemas, must be justified by appeal to the meanings in everyday language of the 'structural' words in the minimal schemas. So this method, too, leads us straight into considerations about meaning and, accordingly, also demands a structural analysis which relates to meaning. Whichever method we prefer, therefore, the study of validity, that is, logic, is central to the analysis of meaning, and valid inferences will afford us some of the most important clues to that structure.

2.3 OPERATOR AND OPERAND Modern logic was inaugurated by Frege and the definitive advance which he achieved depended essentially upon a new structural analysis of propositions. Frege himself was quite clear about this from the start and made explicit to his readers the importance which he attached to it: A distinction between subject and predicate does not occur in my way of presenting a j u d g m e n t . . . In this I follow strictly the ideography of mathematics in which, also, one can distinguish subject and predicate only by doing violence. (1879, para. 3) I hope that logicians, if they do not allow themselves to be frightened off by the first impression of unfamiliarity, will not refuse their assent to the innovations to which I have been driven by a necessity inherent in the subject-matter itself... In particular, I believe that the replacement of the concepts of subject and predicate by argument and function will prove itself in the long run. (1879, Preface, p. xiii)

Yet this manifesto does not explain what difference to semantic structures will be made by the replacement, and indeed there has been much confusion over the question since. The first task must therefore be to elucidate the structures proposed by Frege, in order to compare and contrast them with those of traditional logic and syntax, and to understand just how they prepared the ground for a new era in the history of logic. If argument and function are to replace subject and predicate in an account of semantic structure, then we should expect them to play a comparable role. Now, in the first place, subject and predicate are always linguistic expressions, so argument and function should both be linguistic expressions, too. Second, though, neither subjects nor predicates are

Operator and operand

55

kinds of expression, in the way that, for example, nouns and verbs, adjectives and adverbs are. Sometimes the same expression can occur as a subject and as a predicate or, at least, as part of a predicate; for instance, 'smoking' in 'Smoking can damage your health' and 'Elizabeth is smoking'. 'Subject' and 'predicate' are relative terms; given a sentence, we can distinguish its subject and its predicate. Similarly, then, function and argument should not be categories but, rather, be distinguishable in a given proposition or, perhaps, other expression. Originally, Frege quite explicitly defined argument and function as expressions, but his discussions of examples do not entirely tally with this; thus, he says, if we imagine 'Cato' in the sentence 'Cato killed Cato' as being replaceable by some other expression at both of its occurrences, then 'killing oneself is the function (1879, section 9). Yet the expression 'killing oneself does not even occur in the sentence chosen as example. Writing on the same topic twelve years later, Frege had sorted out this muddle. He begins with the distinction between a numeral and a number: 'One could imagine the introduction some day of quite new numerals, just as, for example, the Arabic numeral superseded the Roman. Nobody is seriously going to suppose that in this way we should get quite new numbers . . . with properties still to be investigated' (1891, p. 4). Numerals, he says, are signs which signify numbers; thus 'VIII' and '8', being different signs, are different numerals, but both signify the same number. Now, if we take a simple example of what Frege calls a 'calculating expression' (Rechnungsausdruck), like '3 + 4', it indicates that the numbers 3 and 4 are to be added together - not the numerals, because the value of the expression is the number 7 and not the numeral '34'. If we regard the calculating expression as a command, then it will be executed by doing something with the two numbers, namely, adding them, and 'III + IV' would express the same command in a different notation. In mathematical terminology, addition is a function which is applied to at least two arguments when an appropriate calculating expression is executed, the arguments being the numbers which are added. The expressions from which the whole calculating expression is built up are, however, the two numerals '3' and '4' and the sign ' + '. Consequently, if Frege wanted to replace the terms 'subject' and 'predicate' by some terminology which would correspond to the mathematician's use of 'function' and 'argument', he needed a distinction which would contrast the numerals in a calculating expression like '3 + 4' with the remaining expression. Although he eventually recognized this, he never introduced such a terminology. He did, indeed, distinguish between proper names and

56

Logic: trees

function names, but that is a categorial distinction; in some contexts, a function name can signify an argument and even a proper name a function. Perhaps Frege's failure to supply this lack was responsible for his subsequent reversion to the subject/predicate terminology, though using it, now, in a totally new way. At any rate, it has since become quite commonplace to speak of 'logical predicates' and 'logical subjects'. This witnesses to a need for an appropriate pair of terms, but it can only be confusing to impress terms from linguistics and then to endow them with different senses in logic (for example a sentence may have more than one logical subject). It also lacks the generality required; thus it goes against the grain to call ' 3 ' and '4' the logical subjects of '3 + 4' and ' + ' its logical predicate, because '3 + 4' is not a sentence, not even a mathematical one. Yet '3 + 4' expresses the composition of function and argument(s) which is Frege's paradigm. Fortunately, there is an alternative to hand which, though not very widespread, has gained some currency: operator and operand? The etymology of 'operator' and 'operand' also suggests, very nicely, the intuitive content of Frege's basic assumption about semantic structure. An operator is something which works upon something else, while an operand is its correlative, something which is worked upon. The seed of this idea is already present in traditional grammar, which tells us that adjectives modify the meanings of nouns while adverbs modify the meanings of verbs, that is, that adjectives and adverbs work upon the meanings of nouns and verbs respectively, so that, in a phrase consisting of an adjective and a noun, the adjective is the operator and the noun its operand, and similarly for a phrase consisting of an adverb and a verb. In that case, then, the phrase is ordered, with respect to its meaning, by the notion works upon. This ordering, moreover, does not always coincide with the order of the words in speech or writing; thus, in English, an adverb usually comes after the verb which it qualifies, so that, in such a case, the works upon relation goes from right to left across the written page (the adverb is a suffix operator). And, in other languages, adjectives are placed after the nouns which they qualify. Frege extended this structural principle to all complex linguistic expressions, giving the following prescription for analysing them: Suppose that a word or phrase occurs in one or more places in an expression. If 3

The only jarring note in previous uses of this pair of terms is to be found in Wittgenstein (1922, 5.21-5.251), who tried to contrast operations with functions, though, on closer inspection, this turns out to be a sub-division of functions in the accepted sense.

Operator and operand

57

we imagine this word or phrase as replaceable by another (the same one each time) at one or more of its occurrences, then the part of the expression that shows itself invariant under such replacement is called the function; and the replaceable part, the argument of the function. (1879, section 9) Here we should substitute 'operator' for 'function' and 'operand' for 'argument'. 'Replaceable' means 'replaceable salva congruitate\ that is, supposing the original expression to be meaningful, the new expression obtained by the replacement will also be meaningful, though it will not necessarily have the same meaning as the original one, nor, if the expression in question happens to be a proposition, will the replacement necessarily preserve the truth or falsity of the original. In short, it is a replacement which preserves semantic coherence. It should be distinguished from a replacement which preserves syntactic coherence but may yield an expression to which no meaning has been given. Frege's prescription will not yield any determinate results, however, for the simple reason that almost any part of a linguistic expression is replaceable salva congruitate. This can easily be seen from Frege's own examples. Thus he suggests that, first, we regard 'hydrogen' in (1)

Hydrogen is lighter than carbon dioxide

as replaceable by 'oxygen' or 'nitrogen'. Then 'is lighter than carbon dioxide' will be the operator and 'hydrogen' its operand. Second, we can regard 'carbon dioxide' as replaceable by 'hydrogen chloride' or by 'ammonia'. 'Hydrogen is lighter than' will then be the operator and 'carbon dioxide' its operand. Finally, we could regard both 'hydrogen' and 'carbon dioxide' as replaceable, in which case 'is lighter than' will be the operator, which will now have two operands, 'hydrogen' and 'carbon dioxide' (1879, section 9). That these three alternatives should be available is part of Frege's intention. But there are also other possibilities. Thus 'light' can be imagined as replaceable by 'heavy' (allowing ourselves the morphological change from 'heavyer' to 'heavier'), and this immediately creates a problem. 'Hydrogen' and 'carbon dioxide' are rather conveniently placed at the ends of the sentence, so that what remains when either or both of them are removed is still a single phrase. If 'light' is imagined as replaceable by 'heavy', though, we are left with two pieces when we try to say what the operator is, 'Hydrogen is' and 'er than carbon dioxide'. Well, perhaps we could get round that difficulty by allowing dots of omission to feature in operators, so that 'Hydrogen is . . .er than carbon dioxide' could be accounted the operator. But the possibilities do not end there. We could imagine 'is' as replaceable by 'weighs' or even 'looks' and, although 'than' cannot be replaced salva congruitate, '. . .er than'

58

Logic: trees

could be replaced by 'and so is' or by 'as . . . as'. Already, then, we have been able to imagine each part of the sentence in turn as replaceable by another expression, so it would appear that virtually any word or phrase can be regarded as an operand and similarly, therefore, any word or phrase as an operator. This offers an extreme contrast to the subject/predicate distinction, which allows us only one way of dividing each sentence. It is, indeed, too great a contrast. The analysis of an expression into operator and operands is intended to elucidate its meaning and it is, on the face of it, unlikely that any consistent account of meaning could be built upon an absolutely free choice of operator. At the same time, Frege had good reason for allowing alternative analyses, so we do not want the operator/ operand distinction to be as rigid as the subject/predicate one. Intuitively, certain alternatives are compatible, whereas others are not. Thus, if one takes 'is lighter than carbon dioxide' as the operator in (1) and 'hydrogen' as its operand, it is open to us to analyse the operator, in turn, as itself consisting of an operator 'is lighter than', with 'carbon dioxide' as operand. It is not implausible to regard this result as equivalent to an initial analysis of 'is lighter than' as operator, with two operands, 'hydrogen' and 'carbon dioxide'. Again, if 'hydrogen is lighter than' is taken as the operator, with 'carbon dioxide' as operand, this operator can be further analysed, with the same result as before. Thus there is a sense in which all of these analyses are mutually compatible. By contrast, if we were to take 'hydrogen' as the operator and 'is lighter than carbon dioxide' as the operand, that would not be compatible with the previous series of analyses, because, in positing 'hydrogen' as operand, they supposed that its meaning is worked upon, either by 'is lighter than' or by 'is lighter than carbon dioxide'. To call 'hydrogen' the operator is, however, to posit that its meaning works upon that of 'is lighter than carbon dioxide'. But if, as it seems, working upon is an asymmetric relation, then either of these analyses will exclude the other. So our aim should be to characterize the operator/operand distinction in a way which excludes incompatible analyses yet finds room for compatible alternatives. This would be feasible if we had at our disposal a classification of different kinds of expression, or system of categories, for, given a combination of expressions of known categories, we could state which was the operator and which its operands. Indeed, a much more modest apparatus will suffice, because any operator can be characterized by stating the kinds of expression which it requires as operands and the kind of expression which it forms by combining with them. Thus all that we need to begin with are some expressions which can only occur as operands, and a category or categories for them. I shall call the categories to which

Operator and operand

59

these expressions are assigned basic categories. But a warning must be issued that this carries no implication that expressions belonging to basic categories are not susceptible of further structural analysis. Indeed propositions themselves will be expressions of a basic category, for it is propositions whose meanings semantics must elucidate in the first instance, while, as may be seen from propositional logic, they always occur as operands when set in the context of larger propositions. Yet if we were to deny structural complexity to propositions, our whole enterprise would be cut off at the root. To offer an example, then, if propositions form a basic category and also names of substances like 'hydrogen' and 'carbon dioxide', say the categories P and S respectively, then we should no longer be free to regard 'hydrogen' as the operator in (1) and 'is lighter than carbon dioxide' as its operand. Yet this move does not deprive us of the compatible alternative analyses described above. If 'hydrogen' and 'carbon dioxide' are both designated as expressions of the basic category 5, then it will follow that 'is lighter than' is an operator which will form a proposition from two expressions of that category. Since we can characterize operators in terms of the categories of their operands and the categories of the expressions which they form, this yields a notation for category names by writing the latter first, followed by a bracketed list of the former: in the present case, therefore, P(S,S). But if 'is lighter than' be combined with just one expression of category S - whether preceding or following it does not matter - then we shall have an expression which will combine with one expression of category S to form a proposition, that is, an operator of category P(S). Thus it still remains open to us to analyse (1) into 'hydrogen' as operand and 'is lighter than carbon dioxide' as operator, or into 'hydrogen is lighter than' as operator and 'carbon dioxide' as operand. Now Frege came very close to this solution. Indeed he simplified it even further, by having only a single basic category, that of proper name (Eigenname). The price of this simplification was that the category of proper name had to be very hospitable, including propositions as well as most proper nouns. Inevitably this has been controversial, for it is counter-intuitive to suppose that propositions relate to meaning in the same way as proper nouns. But, for the moment, the particular choice of a basic category or categories does not concern us. The essential point is that at least one category should be designated as basic, in the sense that expressions of that category may only occur as operands.4

4

Frege's system does conform to this requirement, but there is a complication. We cannot say without further ado that proper names can only occur as operands, because this is

60

Logic: trees

The structures to which the notion works upon gives rise are most clearly represented as trees. Trees can be read in two directions, horizontally and vertically, each of which corresponds to a distinct type of ordering. Frege partly recognized this, adopting a planar notation in order to represent 'if. . ., then . . .', 'both . . . and . . .' and 'either . . . or . . .', but otherwise retaining a linear notation. Each of these expressions forms a proposition from two (subordinate) propositions. If we go along with Frege for the time being in assigning propositions to the category of proper names and calling that category E (for Eigenname), then each of the three expressions will be an operator of category E(E,E). Now, supposing 'A' and 'B' to be propositions, Frege's representation for 'if A, then B' is

and that is a tree, although branching from left to right instead of from top to bottom, the more usual convention. In Frege's notation, the horizontal dimension represents the notion works upon from left to right, while the vertical dimension from bottom to top represents the 'direction' of'if A, then B': the latter orders 'A' and ' B \ since 'if B, then A' does not mean the same, although the meanings of both 'A' and 'B' are worked upon by 'if. . ., then . . .'. It is, of course, also possible to represent this structure in the linear notation. The usual convention is to write the operator first (as a prefix operator) with its operands enclosed in parentheses, for example 'if (A,B)\ The way in which the order of the operands relates to the meaning of the operator is then fixed by a convention. The convention which I

ambiguous. By 'proper name' we can understand a certain expression, after which a category is named, or an expression qua member of a category, in virtue of which membership it is called a proper name. Thus Frege held that a proper name can sometimes occur as an operator (1893, section 22); though, in that case, he no longer assigned it to the category of proper names, but to a different category. The difficulties presented by this view will be considered later. For the moment, the point is that an expression may sometimes be assignable to more than one category. In such cases, we really need a neutral way of describing it, but, because it is relatively rare (even in Frege's view), for a proper name to occur as an expression of a category other than that of proper name, it is called after the category to which it usually belongs. In spite of this, however, it remains true for Frege that an expression which occurs as a proper name, that is, as a member of that category, can only occur as an operand. So we can disregard this complication and proceed on the understanding that Frege has a basic category in the sense explained above.

Operator and operand

61

shall follow when using a linear notation is that the order of the operands will be the same as in English, except that, where in English we have an infix operator (that is written between the first operand and the remainder), its first operand will be written at the end. Thus 'if is infix in 'B if A' and so I write 'if (A,B)', but it is prefix in if A, B', so again we have 'if (A,B)\ Similarly, (1) becomes: 'is lighter than (carbon dioxide, hydrogen)'. The reader should note that this is not the most common convention for this purpose among logicians. Frege represented 'both . . . and . . .' and 'either . . . or . . .' by combining his representation for 'if . . ., then . . .' with a sign for 'not'. This was a small vertical line placed under the horizontal one, as it were dividing the horizontal line into two halves. He understood 'if A, then B' to be true in every case except that in which 'A' is true and 'B' false, so that 'both A and B', for example, could be represented as 'not if A then not B'. Now 'not' is also an operator here, whose operand Frege takes to be a single proposition, so that it belongs to his category E(E). In linear notation, therefore, we can express 'both A and B' as 'not (if (A, not (B)))'. But it is not very easy, visually, to disentangle the works upon relationship (expressed by the parentheses) from the semantic direction of 'if in relation to its operands (expressed by the ordering of items between commas). A planar notation is much clearer, whether Frege's own or a tree with its root at the top. I place the two versions side by side for comparison, setting out the second as a labelled tree: 1

1

-B A

— •

works upon

not

I

if

I

works upon

Although Frege used a linear notation to represent semantic structures for examples like (1), it is evident that they, too, can be shown in the planar notation. The virtues of a planar as against a linear notation can be argued on both sides, but there can be little to be said for a mixed notation which is partly linear and partly planar. The verdict of history has so far gone against Frege's effort to sustain a partly two-dimensional notation. Yet although a linear notation may be much more convenient for working in, for parsing and synthesizing actual sentences and for setting out proofs, a planar one may be far better for theoretical

62

Logic: trees

purposes, when it is of primary importance to obtain a clear view of the structures with which we are dealing or, perhaps, are positing. So it is that, here, the planar notation for Fregean structures is more apposite to our purposes than the linear one. It will be evident that it can be extended, where Frege failed to extend it, to the style of analysis which he proposed for (1), for which we simply have: (IF)

is lighter than

I hydrogen carbon dioxide Moreover, at the bottom of each branch of the tree, as its leaves, we shall always have expressions of a basic category, that is, in Frege's system, always proper names. In order to generate this type of structure, we shall evidently need a tree grammar, for example, one with the rules: E =» I E

is lighter than 1 1 E

E =>

hydrogen, carbon dioxide

The principle of vertical ordering introduced by a tree grammar is nowadays called scope. An expression is said to fall or lie within the scope of another, just in case the latter lies above it on the path from it to the root of the tree. Thus in the tree representation of 'both A and B' given earlier, 'B' lies within the scope of 'if but not within the scope of 'A'. In the linear notation, everything which is enclosed in the parentheses following an operator lies within its scope, but items separated by commas do not fall within each others' scope. I shall find it useful to distinguish one expression as falling within the immediate scope of another when the latter is at the next node above it in the tree; lying within the immediate scope of an expression is then the same as being worked upon by it. Frege's replacement of subject and predicate by argument and function may thus be interpreted as putting trees in the place of strings, ordering linguistic structures by scope instead of by concatenation. But this is to confine attention to linguistic structures without any thought about how those structures are to be interpreted. Although Frege seems to have begun by thinking of argument and function as expressions, he had certainly abandoned this view by 1891, and thereafter he held them to be the respective Bedeutungen - meanings - of what I have called operands and operators. The Bedeutungen of proper names were christened objects (in the case of proper nouns, these were to be their bearers, in the case of numerals, the corresponding numbers and, in the case of propositions,

Operator and operand

63

truth and falsity). Among operators, those which form propositions were singled out as having a special kind of function, concepts, for their Bedeutungen, the values of concepts being always truth or falsity. This allowed Frege to provide an explanation of the truth or falsity of propositions exactly parallel to that of the values of calculating expressions: only, instead of the functions mapping numbers onto numbers, as in the simplest kind of calculating expression, in the simplest kind of proposition, concepts map objects onto truth or falsity. This aspect of Frege's appeal to argument and function in linguistic analysis was certainly original but, although it has been widely influential, it remains controversial. However it can hardly be regarded as a replacement for subject and predicate, since the latter are indubitably expressions, whereas function and argument in Frege's mature sense are explicitly distinguished from the expressions which signify them. Operator and operand replace predicate and subject, not function and argument and, although the replacement was necessary so that Frege could apply the function/argument distinction to everyday language, there is no complementary necessity to interpret a tree grammar in terms of functions and their arguments, however widespread the custom. Frege's theory of meaning was put forward as a package deal, but we are not compelled to take it or leave it as a whole. It seems, then, that at the time when Frege made such large claims for the advantage of replacing subject and predicate by argument and function, he was by no means clear about what that replacement comprised. If, indeed, he thought of it as more than a structural innovation, we must nevertheless distinguish sharply between the structural analysis and the functional account of truth conditions built upon it. It is then evident that the latter does not come up for assessment until we are satisfied with the former. Yet if all that we are concerned with initially is the suggestion that, in asking for the meaning of a proposition, we must see it as having a tree structure, it is dubious whether that marks any advance upon or even any major innovation to previous logic. Perhaps Frege was the first logician to make the idea explicit, but it has surely been implicit at least since the Stoics inaugurated propositional logic. The only really new feature is that, where an operator has more than one operand, no distinction is made between the operands corresponding to the syntactic distinction between subject and object, or between direct and indirect object. But that is just a more thorough-going application of the idea of a tree structure, so the principal contrast lies not between Frege and previous logicians, but rather between those who think that a string grammar is enough to sustain a theory of meaning and those, including Frege, who think that it is not.

64

Logic: trees

2.4 C A T E G O R I A L G R A M M A R The first attempt to construct a tree grammar inspired by Frege's replacement of subject and predicate by operator and operand was made by Ajdukiewicz (1935). Ajdukiewicz began by proposing a system of category names, which I have already anticipated, though using a different notation; whereas I have used linear tree-notation, he used a horizontal line on the model of division in arithmetic, for example |

for

E(E),

i

for

E(E,E),

etc.

Ajdukiewicz differed from Frege in distinguishing two basic categories, sentence and name. 'Sentence', of course, translates 'Satz' in the German version of Ajdukiewicz's article; but in spite of the title, which uses 'syntactical' in the logician's sense, he is certainly concerned with meaning. Both sentence and name, for example, are described as basic semantic categories; so we need to amend 'sentence' to 'proposition'. As to names, he avers that at least two semantic categories must be distinguished under this head in everyday language, 'the singular names of individuals' and 'general names . . . as names of universals'. But he nevertheless proposes to follow Lesniewski in having only a single category of names. From the sole linguistic example which he gives, (2)

Lilac smells very powerfully and roses bloom,

in which 'lilac' and 'roses' are assigned to the category of names, it appears that he is including count nouns in this category. Frege would certainly have demurred at this assimilation but, as Ajdukiewicz himself points out, the idea of characterizing semantic coherence by means of a system of categories is independent of any particular choice of basic categories. The choice of basic categories will be considered later; meanwhile, it will be convenient to follow Ajdukiewicz while expounding his theory. If Co . . . Cn are categories, then the general form of a category name is C0(Cj,. . .,C,J, where n is the degree (or rank) of expressions belonging to that category (for basic categories, n = 0) and, for each of Co . . . Cn any category name may be substituted. The category names of the two basic categories are S (Satz, proposition) and N (Name, name). Taking his example, Ajdukiewicz assigned 'lilac' and 'roses' to category N9 'smells' and 'bloom' to category S(N), 'and' to category S(S,S), while the adverb 'powerfully' is assigned to category S(N)(S(N)) and 'very' to category S(N)(S(N))(S(N)(S(N))). The grammar then consisted of a single rule-schema,

Categorial grammar (A)

65

C0(C,,. . .,(?„) C, . . . Cn » Co,

If this be written out in Ajdukiewicz's own notation, it is analogous to the multiplication of fractions in arithmetic, the 'numerator' operands cancelling out with the 'denominators' of the operator; hence his choice of that notation. I have used ' » ' instead of '=>' in stating this rule because it is the converse of a production for a tree grammar, a combination rule instead of a re-write rule. It works, moreover, the opposite way round from the rules considered in section 2.1, that is, it is & parsing rather than a generative rule. In order to apply it, Ajdukiewicz assumes that the expression concerned has first been re-written so that each operator precedes its operands; thus, for his example, and S(S,S)

(very S(N)(S(N))(S(N)(S(N)))

(powerfully S(N)(S(N))

(smells(lilac))),bloom(roses)) S(N) N S(N) N

Then, for the first application of the rule-schema, we take

C0=Cj = S(N)(S(N)) (Al)

to yield the rule

S(N)(S(N))(S(N)(S(N)))

S(N)(S(N))

»

S(N)(S(N))

which allows us to combine 'very' and 'powerfully' to give 'very powerfully' as an expression of category S(N)(S(N)). In a second application, we take COZ=CI = S(N), yielding the rule (A2)

S(N)(S(N)) S(N) » S(N),

so that we can combine 'very powerfully' with 'smells', giving S(N) as the category of 'very powerfully smells'. Next, with Co = S and Cy = N, we get the rule (A3)

S(N) N » S,

which can be applied twice: first, to license combining 'very powerfully smells' with 'lilac' to form a proposition; second, to yield another proposition by combining 'bloom' with 'roses'. Finally, by taking Co=Ci = C2— S in the rule-schema, we have the rule (A4)

S(S,S) S S » S,

which allows us to combine 'and' with both 'very powerfully smells lilac' and 'bloom roses' to obtain the result that (2) is a proposition. Unfortunately Ajdukiewicz never spelled out any procedure for rearranging the parts of a proposition into the correct order for applying the combination rule-schema. Instead, bar-Hillel (1953) proposed a modification to the notation for category names which would allow the combination procedure to be carried out on a proposition with its words

66

Logic: trees

in the correct order for the language concerned. This was to distinguish operands occurring before (on the left of) an operator from those occurring after (on the right of) it. Thus Dj. . .D m \C 0 /Ci. . .C m is the category name of an operator which will combine with m operands preceding it and n operands following it. If m = 0, then Co is a prefix operator; if n = 0, it is a suffix operator; and if both m > 0 and n > 0, then it is an infix operator. A corresponding modification of the rule-schema is required, to: (B)

D , . . . D m D 1 . ..D m \C 0 /C,. ..C n C,. ..C n

»

Co

We can then write category names in the new notation under the words of (2) just as they stand and apply the new rule-schema with the same result as before: Lilac N

smells N\S

very powerfully and roses bloom ((N\S)\(N\S))/((N\S)\(N\S)) (N\S)\(N\S) S\S/S N N\S very powerfully (N\S)\(N\S) smells very powerfully N\S Lilac smells very powerfully roses bloom S S

There are, however, quite common examples for which even this modification does not cater. Bar-Hillel himself mentions one of them, of which an example (though not his) is the placement of 'not' in English immediately before a verb, as in (3)

The cat did not move.

From the point of view of meaning, 'not' is best regarded as an operator upon a proposition, that is, as of category S(S), for it says that something is not the case, is not true. So the standard logical analysis of (3) is 'not (the cat moved)', with 'not' as the operator and 'the cat moved' as its operand. But neither an assignment to category S/S nor an assignment to category S\S will allow us to write the category names under the words in (3) in the order in which they stand and so that we can show that it is a proposition by using schema (B). The other three operators of propositional logic provide a further exception, for, if we take the forms 'both . . . and . . .', 'either . . . or . . .' and 'if . . ., then . . .', each operator consists of two separated parts; under which, then, do we write the category name and what do we write under the other? Many subsequent writers on categorial grammar have pursued this attempt to make it fit the forms of expression of a particular language, usually English (for example Lambek, 1958; Montague, 1974, ch. 8; Ades

Categorial grammar

67

and Steedman, 1982; Bach, 1984; Huck, 1988). We see, here, the obverse of transformational grammar, in which a syntactically inspired structural analysis was extended to embrace meaning. Now a semantically inspired analysis is being extended to embrace syntax. Of course it would be convenient, and economic, too, if a single structural analysis would simultaneously serve for both syntax and semantics, but the more one goes into detail the more the difficulties pile up. Those which arise for an account of meaning affixed to a syntactic analysis have been discussed in the previous chapter, while the ultimate source of difficulty in affixing syntax to a semantic analysis is that, whereas semantic structures may be expected to be relatively independent of the vagaries of any particular language, syntax will differ from one language to another. Have we, in any case, any right to expect that semantic structures will be the same as syntactic ones? 'Every doctor visited at least one patient' has already been used to show that a proposition or sentence can be seen as having a string structure, but can also be seen as having a tree structure. When we consider it as generated by a string grammar, we see it as having a string structure; when we consider it as generated by a tree grammar, we see it as having a tree structure. It is no use asking which of these structures it 'really' has. The way in which we see anything complex as being organized - what we count as its parts and how they are interrelated - will depend upon our purposes. Giving an explanation of the meaning of a proposition is a different purpose from explaining why it is a correct form of expression in a given language. Hence there is no a priori reason to suppose that we need to see it as having the same type of structure qua proposition and qua sentence. On the contrary, our expectation should be that semantic structures will be different from syntactic, and we should require cogent arguments to persuade us otherwise. That the divergence of purpose between semantics and syntax should give rise to two different ways of discerning a structure within an expression, each legitimate in its own context, but immediately productive of confusion when used for the wrong purpose, is a very ancient idea, which is already quite explicit in Aristotle: 'In fallacies arising from the form of expression, the cause is the similarity of the expression. For it is difficult to distinguish kinds of thing expressed similarly and kinds of things expressed differently (indeed he who can do this is very near the contemplation of the truth and is especially wise in judgment)' (De sophisticis elenchis 7, 169a29-36). It has, indeed, been a persistent theme throughout the history of philosophy, even though the emphasis which it has received has varied; in our time it has again come to the fore: thus Wittgenstein echoes Aristotle when he says: 'Our

68

Logic: trees

investigation . . . is a grammatical one. Such an investigation sheds light on our problems by clearing misunderstandings away. Misunderstandings concerning the use of words, caused, among other things, by certain analogies between the forms of expression in different regions of language . . . Philosophy is a battle against the bewitchment of our intelligence by means of language' (1953, 1.90,109). The phenomena of paraphrase and ambiguity lend further support to the supposition that two different kinds of structure are discernible in expressions, each of which is necessary to a full account of language. Indeed, paraphrase and ambiguity then appear as converses. One expression is a paraphrase of another if both have the same meaning; hence if there is one structure relevant to meaning and another relevant to the particular form of expression employed, we might expect to discern in two expressions which paraphrase each other the same semantic structure but different syntactic structures. By contrast, when a single expression has two different meanings, we should be able to find in it two semantic structures although there is only one syntactic structure. The distinction between semantic and syntactic structure will appear more clearly if we extend the notion of paraphrase to include translation. For one expression to be a translation of another, in a different language, is also for both to have the same meaning. Although two such expressions, because they have the same meanings, will have the same semantic structure, their syntactic structures might differ so considerably that a man might be able to understand the one without being able to understand the other, even though he had the requisite vocabulary. This often happens with a foreign language which we do not know well: we know the translation of each word in the expression but cannot see precisely how their meanings fit together. How could this occur if semantic and syntactic structure were the same? Ambiguity is the other side of the coin: not the ambiguity which occurs when one element in a sentence has more than one meaning, but structural ambiguity, when it is possible to see the sentence as being constructed in more than one way. Here is an example from a modern hymn: 'Though storm-clouds threaten the day he will set me free'. None of the individual words in this example is ambiguous; it is a question of how we read the structure, as a complete sentence with an implicit comma after 'the day', or as a clause in which 'he will set me free' qualifies 'the day'. Another example is the following notice: 'No parking in front of these gates at all times'; yet another, an advertisement for a newspaper: 'Everything you need to know about sport every Monday'. On the face of it, we have a single syntactic structure in each of these examples, for I do not know what syntactic structure is if two equiform expressions of the

Categorial grammar

69

same language do not ipso facto have the same syntactic structure; but ex hypothesi we have two semantic structures, since the expression has two meanings. Moreover, for centuries it has been a staple of philosophical criticism to argue that an author has degenerated into writing nonsense, yet without in the least implying that he was ignorant of the syntax of his native language and unable to write correctly formed expressions in it. Typically, though not invariably, the criticism is directed to expressions which make liberal use of abstract nouns ('pain* and 'pleasure' are classical examples). So what is often being contended is that certain items of vocabulary may be combined in ways which are syntactically coherent, but still do not yield, in combination, an over-all meaning. The implication of this must be that an expression can have a syntactic structure without having any semantic structure; and, hence, that the elements of which semantic structures are composed are not always the same as those of which syntactic structures are composed. Many linguists, too, are committed by their principles to support philosophical tradition on this point, for they have been at pains to develop arguments for certain types of syntactic structure which are independent of any appeal to meaning (see Akmajian and Heny, 1975, passim). To argue that semantic and syntactic structures are distinct is not to say that they are unrelated. The contrary, indeed, follows if we understand an expression through its syntactic structure; for then the latter will represent the meaning of the expression too, but by the use of certain conventions which are peculiar to the language concerned and to which other languages offer alternative conventions. So there must be some systematic means by which syntactic structures can be related to semantic ones, and these will eventually have to be spelled out. Here, however, it is my purpose to elucidate semantic structures and so I shall assume that categorial grammar, when used in the service of a theory of meaning-representation, need not concern itself with syntax. The categorial grammar which I have expounded so far is that of Ajdukiewicz, but although Ajdukiewiz was to some degree inspired by the logic of Frege, it contains a feature, to which bar-Hillel has drawn attention, which contrasts with Frege's analysis: 'It makes a considerable difference . . . whether we treat loves (say) as an operator which out of a left N John and a right N Mary forms a sentence, John loves Mary, IN ONE COMPLEX STEP, or as an operator which out of a right N Mary forms an operator, loves Mary, which out of a left N John forms a sentence IN TWO SIMPLE STEPS' (1964, p. 70). The distinction between left and right combination here is an unnecessary complication and can be ignored. The point at issue remains if we take the simpler system in which

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Logic: trees

operators are assumed to precede their operands. The difference is then between assigning 'loves' to category S(N,N), which is the category of expressions requiring two names to form a proposition, and thus of degree 2, and assigning it to category S( N) (N), the category of expressions requiring one name to form an expression of category S(N)9 and thus of degree 1. These are distinct categories in Ajdukiewicz's grammar; in Frege's system, they are not. The reason for this is that Frege classifies proposition schemas, not operators. In the previous section, I observed that it was convenient for Frege's purposes that 'hydrogen' and 'carbon dioxide' occur at the ends of (1), so that when both of them are removed we are still left with a single phrase, 'is lighter than'; for if 'light' be removed instead, the remainder of the proposition falls apart into two pieces, 'hydrogen is' and 4-er than carbon dioxide'. To overcome this difficulty, I suggested that dots of omission might be allowed to occur in operators, so that we could write, for example 'Hydrogen is . . .er than carbon dioxide' as an operator. But this is only a makeshift, which will not serve in the extreme case, illustrated by the power notation in mathematics, when nothing remains if all of the operands are removed. That notation consists in writing the power to which the number is to be raised as a super-script to its right, so that we have two numerals, either or both of which can be considered operands. The pattern in which they are arranged has been given a determinate meaning by mathematicians. There is, however, a low limit to the number of distinct patterns in which operands alone can be arranged. The same goes for speech: a sound pattern is a sequence. The example of music shows that, potentially, sound patterns can be much more than mere sequences of sounds, but these potentialities are largely undeveloped in spoken language, perhaps because it is important to us to be able to write a language as well as to speak it, and writing is strictly sequential. So, in order to be able to construct distinct patterns freely, we incorporate signs as parts of a pattern. Thus a common computer notation for powers is illustrated by '3 A 2\ The pattern has not been eliminated here in favour of the new sign, because the numerals still have to be related to the sign in a particular way: one must immediately precede it, the other immediately follow it, and the latter indicates the power to which the number designated by the former is to be raised. Consequently '2*3' arranges '2' and ' 3 ' in a different way from '3*2', and each combination of signs has a different meaning. Pocket calculators which embody a power function overcome the problem of having no special sign for it in standard mathematical notation by writing two letters, each of which relates to a working register

Categorial grammar

71

on the calculator, in the same pattern as the numerals should be written. This representation of a pattern of numerals is clearly a schema in the sense introduced in section 2.2, although not a proposition schema: the user is invited to substitute numerals for the letters written on the key, by entering them into the appropriate registers, so the letters are also schematic symbols. This shows us that what is essential to a schema is no more than schematic symbols arranged in a determinate pattern; it need not contain any words or other special signs. The solution to the problem raised by treating 'light' as the operand in proposition (1) is, of course, to use schemas instead of operators, for example 'Hydrogen is 9er than carbon dioxide', together with a specification of what expressions may be substituted for '9'. From 1891 onwards, Frege adopted this course. He distinguished between complete and incomplete expressions and by 1893 was using 'proper name' versus 'function name' to mark the same distinction. A Fregean incomplete expression or function name is the same as a schema; it always contains at least one schematic symbol. Frege used Greek letters for schematic symbols, because he did not want them to be confused either with the link letters of his quantifier notation or with the letters used in algebra. As this convention is now familiar, I shall follow it. In section 2.2 I introduced a distinction between schema and pattern according to which a schema is a representation of a pattern. Now Frege remarked that the essential contrast to be discovered in a (complete) expression lies between what is present in a schema 'over and above' its symbols, and the expressions which are substituted for them (1891, p. 6; 1893, section 1). Wittgenstein subsequently drew out this hint that the distinction should be between pattern and operands, rather than between operator and operands: what signifies that the meanings of the operands are worked upon in a certain way is that they are arranged in a certain pattern (see 1922, 3.1431-2). Looking at an expression as being structured in this kind of way, therefore, implies that its only parts are its operands. A schema from which the expression may be obtained (by substitution for its schematic symbols) is not a part of the resulting expression; it is merely a representation of the arrangement of the parts, whether or not it contains any words or signs in addition to the schematic symbols. Notwithstanding this conclusion, what we have hitherto called the operator is normally part of a schema. It will be convenient to retain 'operator' in this sense, for we can ask about the meaning of a word or phrase in an expression whether it is an operand or an operator in that context. Moreover, although we may conceive of a proposition as having been constructed by substitution in a schema, it is at the very least dubious whether we can consider the schema as occurring in the

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proposition, for a schema by definition contains schematic symbols, whereas a proposition does not. The operator of the schema, however, certainly does occur in the proposition. Linguistic description would be crippled without any means of referring to certain words or phrases in an expression, so it will still be valuable to have a term which indicates that the role of some words with respect to the meaning of an expression is fundamentally different from that of others. Yet, if the contrast to be found in a proposition lies between its operands and the pattern in which they are arranged, is it not perverse to represent operators as well as operands as nodes on trees? Surely this is to invite misrepresentation, for a structure consists of inter-related parts, so that, where it is represented by a tree, what could be more natural than to suppose that the nodes represent the parts and the edges (the lines between nodes) their inter-relationships? In that case, ought not the operands to be the only nodes, and everything else to be shown by the way in which the nodes are inter-connected by the edges? In theory, this sounds absolutely right, but in practice it cannot be carried through. Some examples will show why. First, we could no longer represent linguistic structures by trees of any kind. Consider the proposition: (4)

Plato taught Aristotle and Aristotle admired Plato.

This contains two subordinate propositions, each of which in turn contains two proper names. Hence each is constructed from a schema: the first from % taught £', and the second from '( admired
taught

Aristotle

Aristotle

admired

Plato

These are trees of a sort, in that they are strings (ignoring the labels of the edges) and strings are a limiting case of trees. Yet they are not trees of the kind we require, since we do not want to represent 'Plato' as working upon the meaning of 'Aristotle' or conversely but, rather, 'taught' as working upon the meaning of both. Moreover, if we now go on to consider how the whole proposition (4) might be represented, 'and', being an operator, will also have to label an edge, which must, presumably, join the two representations above. The result

Categorial grammar Plato

73

Aristotle and

taught Aristotle

admired Plato

is not a tree at all, because the 'and' edge does not join the other edges at a node. So much the worse for trees, someone will doubtless say; why not develop, instead, the notation just sketched out? Well, this new notation depends upon an absolute distinction between operator (or schema) and operand, since operands only label nodes and operators only label edges. When considering the replacement of 'subject' and 'predicate' by 'operand' and 'operator', I observed that both pairs are relative terms, which implies that an expression which is an operator in one context may be an operand in another. No examples have yet been cited, but they will make their appearance shortly, and this tells decisively against any method of representation which is based upon the schema/operand distinction rather than upon category differences. Nevertheless, we can give due weight to the argument for drawing the primary distinction between schema and operand by insisting that what corresponds, in a tree, to a schema is not just an operator labelling a node, but the operator together with the edges associated with it in virtue of its degree and also the category symbols (which are schematic symbols) at the other nodes of those edges. In other words, the sign for a schema in tree notation is everything on the right-hand side of a production rule in expansive form which introduces a terminal symbol of degree > 0. The operator is a part of this sign just as it is a part of a schema. To return, now, to bar-Hillel's point about the difference between assigning 'loves' to category S(N,N) and to category S(N)(N)\ if, following Frege, we regard 'loves' as the operand of a schema '( loves £' of degree 2, then we have three alternative ways of using it. First, we may simultaneously, 'in one complex step', as bar-Hillel puts it, substitute 'John' for '(' and 'Mary' for '£' to obtain the proposition 'John loves Mary'. That corresponds to Ajdukiewicz's category S(N,N). Second, we may simply substitute 'Mary' for '£', to obtain the schema '£ loves Mary', of degree 1. That corresponds to bar-Hillel's first 'simple step' when 'loves' is assigned to category S(N)(N), which combines with one expression of category TV to yield another operator, of category S(N). Third, we may just substitute 'John' for '(', obtaining the schema 'John loves
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Logic: trees

But, for Frege, it is the same schema which has been used in all three cases, so, for him, there could be no distinction between category S(N,N) and category S(N)(N). Moreover, the flexibility offered by Frege's account is essential to logic, because there may be some argument schemas in which '( loves Mary' occurs but others in which 'John loves £' occurs, but no further analysis of either is needed. Thus 'Everyone loves Mary; ergo John loves Mary' is an instance of the schema: 'Everyone (/>; ergo John ', whereas 'John loves everyone; ergo John loves Mary' is an instance of the schema ' everyone; ergo (j) Mary'. In neither schema does the substitution for (j> have to have the inner structure exemplified by 'loves Mary' and 'John loves' respectively. 'Yawns' could be substituted for (f) in the first schema and 'There is a devil in' for <> / in the second. This flexibility can be secured for categorial grammar by two modifications. The first is a simplification: in a category name, only basic category names may be substituted for Co. This corresponds to the result of substitution for each of the schematic symbols in a schema, which will always be an expression of a basic category, precisely because it no longer contains any schematic symbols. This rules out Ajdukiewicz's category S(N)(N), leaving only S(N,N). But now, second, we must modify the rule of combination to allow for a partial combination, that is, as well as S(N,N) N N »

S,

we must allow S(N,N) N »

S(N).

The obvious way to do this is to break up the first combination into two steps, allowing only one operand to be combined with an operator at a time, viz. (C) C 0 (C b . . .,Civ . .,Cn) Q

»

C0(C,,. . .,Q-i,C l+1> . . .,Cn).

I have called the system which results from these modifications Fregean categorial5 grammar in order to distinguish it from Ajdukiewicz's variety 5

Actually, the term 'categorial' is unfortunate in that every grammar is necessarily categorial in the sense of employing categories. Ajdukiewicz used 'functor' as equivalent to my 'operator', so 'functorial grammar' would have been better. But 'categorial grammar' is now the accepted usage. At any rate, I shall count as categorial any grammar which embodies the idea that expressions have an operator/operand structure. Martin (1984) has argued that Frege's Grundgesetze system is not a categorial grammar, but his case rests upon an idiosyncratic notion of the latter. He claims, first, that Frege does not set out a grammar 'in the sense of generating all well formed expressions from a limited lexicon and a restricted set of formation rules' (p. 151). But nearly all of what has been called 'grammar' throughout history would fail this criterion, while the demand for

Quantification

75

(Potts, 1978b), but I shall call (C) 'Ajdukiewicz's rule' because it is the rule-schema which corresponds to (A) in Fregean grammar. 2.5 Q U A N T I F I C A T I O N The divergence between Ajdukiewicz and Frege is even more marked over generality, that is, the analysis of propositions containing one or more quantifying phrases - typically, a count noun and a qualifying numerical adjective, like 'three lorries'. But their interest centred upon two unspecific numerical adjectives, 'every' and 'some' or 'a' (in the sense of 'at least one') and, in Frege's case, the 'dummy' count noun 'thing', giving us the combinations 'everything' and 'something' respectively, the universal and existential quantifiers.6 Ajdukiewicz assigned the quantifiers and other quantifying phrases to category S(S), the same as that of 'not'. On the face of it, this will not work at all. Take a very simple example like (5)

A dog barked.

We can argue as follows that the category of 'barked' is S(N). Given that 'Fido' is the name of a dog (category N), then 'barked' in (6)

Fido barked

must be of category S(N), since (6) is a proposition (category S). But there is a valid inference from (6) to (5), so 'barked' must have the same meaning in both, or the inference would be invalid by reason of ambiguity. Now our categories are semantic, that is, categories relative to meaning, so a difference of category automatically carries with it a difference of meaning. Hence the category of 'barked' in (5) must also be S(N). So far, Ajdukiewicz would have agreed. Yet there is no way in

6

explicit formation rules is blind to the importance of notation, to which Frege always devoted great attention. The point of Frege's ideography is that it should be impossible to express in it a structure which could represent nonsense and, with such a notation, explicit formation rules are redundant. Martin also appears to hold that an account of the truth conditions of propositions in terms of the Bedeutungen of their parts is integral to categorial grammar, for he writes 'The explicit versions of the categorial interpretation all agree that the operators, both the connectives and the variable binding operators, are to be read as referring expressions standing for semantic operations on references' (p. 147). It will be clear from the end of section 2.3 that my use of 'categorial grammar' carries no such implication; at most it supposes that the meaning of an expression is to be explained in terms of the meanings of its parts (operands) and the manner of their combination (the schema), but leaves quite open just how the explanation should proceed. The term 'quantifier' relates to the question 'How many?' and not, as its affinity to 'quantity' might suggest, to the question 'How much?' As to Frege's representation of the specific count nouns in quantifying phrases, that will engage us in a later chapter.

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which 'S(S) S(N)9 can be combined by rule (A), (B) or (C) to yield a proposition. So how could Ajdukiewicz have thought that the category of quantifiers was S(S)1 The answer is that Ajdukiewicz based his assignment on Frege's notation for generality but, like many others, misunderstood it. This will become clear by explaining Frege's notation. 7 To begin, if we are justified in assuming that 'barked' has the same category in (5) as in (6), namely S(N), it seems that we must take it to be the operand in (5), although it is the operator in (6). For the expression 'a dog' is not a proper name, whereas the category of 'barked' shows that, if it occurs as an operator, it requires a proper name as its operand. Hence, the operator in (5) must be 'a dog' and, since (5) as a whole is a proposition (category S), while the operand 'barked' is of category S(N)> it follows that the category of 'a dog' will be S(S(N)). In that case, the structure of (5) should be simply: a dog (barked). But this is inadequate, and it is extremely important to see why. Wittgenstein observed, of such a notation, that 'we should not know what was generalized' (1922, 4.0411). We need a more complex example than (5) in order to bring out the force of this criticism, such as example (6) of section 1.3, viz.: (7)

At least one patient was visited by every doctor.

By analogy with our approach to the analysis of (5), we begin by categorizing the verb 'visit', which can be done by showing how it will combine with proper names to form a proposition, as, for instance, in (8)

Dr Patel visited Mrs Wilson.

This is comparable to example (1). Since 'visit' requires two proper names to form a proposition, its category will be S(N,N). So a semantic structure for (8) can be displayed in linear notation as (8A)

visited (Mrs Wilson, Dr Patel).

Now we have already decided that 'a dog' should be assigned to category S(S(N)). So presumably the same will go for other quantifying phrases, of which 'at least one patient' and 'every doctor' will be examples. For in each case we can use them to construct propositions which are strictly 7

A parallel account will be found in Dummett (1973, chapter 2, pp. 8-33), though with category assignments remaining implicit. Lewis (1970, p. 193) assigns quantifiers to category S(N,S), a possibility which Ajdukiewicz (1935, p. 17) discusses and rejects. But this shows how necessary it still is to expound Frege's notation.

Quantification

77

comparable to 'a dog barked', for example 'at least one patient died' and 'every doctor was incompetent'. If we now compare (7) with (5), we see that the structure which was assigned to the latter is also to be found in the former, with 'at least one patient' corresponding to 'a dog' and 'was visited by every doctor' to 'barked'. Hence, if 'at least one patient' belongs to category S(S(N)) then 'was visited by every doctor' will belong to category S(N) and be the operand of 'at least one patient'. However, 'was visited by every doctor' has a complexity which 'barked' lacks, and we have already decided that 'visited' belongs to category S(N,N) while 'every doctor' belongs to category S(S(N)). Of course, 'was visited by every doctor' is not a simple combination of 'visited' and 'every doctor'; the verb is in the passive voice. But the whole expression 'was visited by' can also be assigned to category S(N,N)9 since we can have, for example, 'Mrs Wilson was visited by Dr Patel'. So it seems that an expression of category S(S(N)) ('every doctor') must be able to combine with one of category S(N,N) ('was visited by') to form an expression of category S(N) ('was visited by every doctor'), even though we cannot yet offer a justification for this result. If this combination is to be approved, it still remains to decide which expression is the operator and which the operand. Analogy suggests that the quantifying phrase is the operator and the verb its operand, for although 'was visited by every doctor' is not a proposition, it is composed, like (5), of a quantifying phrase and a verb, only now a transitive instead of an intransitive verb. This would mean that the works upon relationship goes from right to left, but that is a phenomenon which we have already encountered: the quantifying phrase is a suffix operator. In any case, (7) could be rearranged in the same way: At least one patient, every doctor visited is a little unusual, but not out of the question in English syntax. At the same time, it puts the verb back into the active voice, so if we took our cue from this, we might display a semantic structure for (7) by: (7')

at least one patient (every doctor (visited)).

This brings us at last to Wittgenstein's criticism. Visiting requires a visitor and something visited. In this case both are generalized, and our analysis forces us to write both of the general expressions in front of the verb. Normally, however, we can tell which is the visitor and which the visited by the positions of the corresponding expressions relative to the verb: when the verb is in the active voice, the name of the visitor is written before it and the name of what is visited after it, and the opposite

78

Logic: trees

when the verb is in the passive voice. This indication of the meaning is lost when operators are always written before operands, so how do we know from our structure for (7) that the latter is about doctors visiting patients rather than conversely? Are we talking, now, of doctors visiting patients or patients visiting doctors? There appears to be a quite straightforward reply to this difficulty. Our representation of (8) used a convention to determine which of Dr Patel and Mrs Wilson is visitor and visited, so why not lay down a corresponding convention for representations of (7) and similar propositions? Thus we could agree that the immediate operator upon 'visited' is to describe the visitor and the operator upon that, the visited. But this will not work. Looking at (7) again, there is no reason why we should not analyse it, in the first instance, into 'every doctor' as operator and 'at least one patient was visited by' as operand; if we then go on to analyse the latter into 'at least one patient' as operator and 'was visited by' as operand, we obtain the structure: (7")

every doctor (at least one patient (was visited by)).

Now we agreed that the immediate operator upon the active verb should describe the visitor, so the corresponding convention for the passive verb would be that its immediate operator should describe the visited. That is exactly what we have here, with the more remote operator describing the visitor, as we should also expect. It was shown in section 1.3, however, that (9)

Every doctor visited at least one patient

and (7) will then have different truth conditions and that, in consequence, we have to account for a double complexity in these propositions: we have to show the semantic relationship of each quantifying expression to the verb and also that between each of the quantifying expressions. Now the analyses (7A) and (7B) show that it is not enough merely to distinguish between the categories of proper names and of quantifying expressions in order to give a correct structural account of generality. But it is an essential first move, because it permits us to exhibit the scope relationships of the two quantifying expressions. A price, however, has been paid; by contrast with the original propositions, the analyses no longer show how the quantifying expressions relate to the meaning of the verb (for that, we had to invoke a supplementary convention). So the next step must be to restore an explicit indication of this.

Quantification

79

Frege's solution was to write the verb as it is represented in (8A), but to link8 the positions occupied there by operands to the quantifying expressions. In the case of the simple example (5) with which we began, there is just one such link, so that we have: a dog (barked ( I ))

I

Instead of (8B), however, we obtain: every doctor (at least one patient (visited (I , ))) I I It is to observe that we no longer need the passive voice of the verb, whose syntactic function is to enable us to alter the relative linear order of the quantifying phrases, whereas in the semantic analysis we are only concerned with their scope order. The right-hand ends of the links show that the patient is the visited and the doctors the visitors, given only the convention regarding the meaning of 'visited' established for (8A). The right-hand ends of the links can also be switched; thus there are four possible combinations of the two quantifying phrases and the verb, each of which has a different meaning. To draw out the links in the manner shown above would yield a notation clumsy to work with and even, on occasion, difficult to read, for instance if there were several quantifying phrases and perhaps crossing links. So it is neater to label each link - say as the x-link, the y-link and the z-link, with further letters as required - and then just to write the label at each end of the link, leaving it to the reader to connect the two ends in his imagination. We should then get, instead of the two representations above, a dog:x (barked (x)) every doctonx (at least one patient:y (visited (y,x))). This is, in essentials, Frege's notation for generality. Moreover, since each quantifying expression requires a link, the latter is an integral part of the notation for such an expression. The quantifying expressions in the examples above, for example, would have to be written as 'a dog:x (0(x))\ 'every doctor:* ((*))' a n d (*))', and the 8

The notation for links used in this paragraph derives from the lines of identity' in Peirce's existential graphs (1960, vol. 4, sections 385 and 407-8), as subsequently used by Bourbaki (1954) and, in a numerical sub-script variant, Quine (1965, pp. 73-4). It is not found in Frege.

80

Logic: trees

two quantifiers as 'everything:* ($(x))' a n d 'something:* (4>(x)y respectively. We can now see how Ajdukiewicz came to assign quantifying phrases to category S(S). When a schema of category S(N) is supplied as operand to a quantifying phrase, the second link letter in the latter takes the place of a proper name which, in the simplest case, would be required in order to obtain a proposition from the schema. Frege himself was responsible for borrowing the term 'variable' from mathematics (a loan which he later regretted) and the second occurrences of link letters were subsequently called 'bound variables'. This has encouraged people to think of them as constituents of logical formulas, rather than merely as structural signs showing how the constituents are combined. It then seems reasonable to ask to what category the bound variables belong and to assign them to that of names. So we get, for example: A dog:x (barked (x)) S(N) N S(S) S S

in which the categories combine to yield a proposition. But, according to this, 'barked (x)' is a proposition. Yet it is neither true nor false, so how can it be one? The issue is commonly fudged by calling it an 'open sentence', but what that really means is a schema, for the V in 'barked (x)' marks a place where a proper name may be placed to yield a proposition.9 If it is open, it is not a proposition; and, if it is a proposition, it is not open. Moreover, as Geach has pointed out in his translation of Ajdukiewicz's paper (1967, p. 635; see Ajdukiewicz, 1935), if'barked (x)' belongs to category S and 'A dog:*' to category S(S), then 'A dog:z (barked(x))' should also represent a proposition. Ajdukiewicz's category assignments offer no explanation why the two link letters must be the same.10

The assignment of quantifying phrases to category S(S(N)) introduces a new distinction between schemas (and also, hence, between 9 10

See Dummett (1973, pp. 16-17). To be fair to Ajdukiewicz, he does impose as a condition upon quantifying phrases (here called 'operators') 'dass in dem Argument eines jeden Operators, d.h. in dem Ausdruck, auf welchen sich der Operator bezieht, jeder durch die Gestalt des Operators angezeigten Veranderlichen eine innerhalb dieses Arguments nicht gebundene gleichgestaltete Variable entspricht' (1935, pp. 22-3) ('that in the argument of each operator, that is, in the expression to which the operator relates, to every variable indicated by the form of the operator, there corresponds an equiform variable that is unbound within this argument'). And, to signal this, he wrote

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operators). So far they have been differentiated by degree, but now we can also classify them by level. The level of a schema is always one greater than the level of its operands, and a schema whose operands are expressions of a basic category (for Frege, proper names) is of first level. Schemas whose operands are of different levels are of mixed level. So 'barked' and 'visited' are the operators of first-level schemas, the first of degree 1 and the second of degree 2, while quantifying phrases are the operators of second-level schemas of degree 1. Another, simple way of determining the level of a schema is to count the number of nested pairs of parentheses in its category name. 'SfN)9 and 'S(N,Ny both have only one pair, and so are first level, 'S(S(N)y has two and so is second level, while 'S(S(S(N))y would be third level. Schemas of second or greater level are customarily called higher-level schemas. The classification of schemas by levels is a way of showing that the notion of a part or constituent in a semantic structure is relative. Operands are parts relative to the schema of which they are operands. Expressions of basic categories are the only ones which are always parts, because they can only occur as operands. To some extent, this is reflected in Frege's notation, in which there is a sharp distinction between expressions of basic categories and schemas, so much so that, in his mature work, he never writes operators apart from their schemas. Thus by his precept we should write, for example, '( barked' instead of just 'barked', '( visited £' instead of just 'visited' and 'every doctor:* ((*))' instead of 'every doctor'. It is to observe that the schema always shows what kinds of expression are required as operands in order to form a proposition (or other expression of a basic category). Thus the link letters in 'every doctor.x (<^>(x))' show that a first-level schema must be substituted for '(. . .)', while in the third-level schema 'every property:f (Q:x (f(x))', the link letters again show that a second-level schema must be substituted for '£l:x (. . .(x))'. Once we introduce higher-level schemas into categorial grammar, Ajdukiewicz's rule becomes inadequate. We can see this by considering a proposition in which 'not' occurs within the scope of a quantifying phrase, that is, one whose semantic structure could be represented by the schema 'everything:x (not ($(x)))' or 'something:x (not (<£(x)))'. Suppose \— Is

instead of

—

s

for the category of quantifiers. But there is nothing in this notation which shows that the operator is linked to its operand, and in the examples he gives the category name preceded by the vertical stroke combines with other category names in exactly the same way as that without the vertical. Moreover, the second and subsequent occurrences of the link letter are still assigned to category N.

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that the schema be completed with a first-level schema 'F(', of category S(N). But the operand of the quantifier, 'not (F()\ must also be a schema of that same category, while the category of 'not' is S(S). So negation should be able to operate upon a schema of category S(N) (and, in general, upon any propositional schema) to yield a new schema of the same category as its operand. Yet Ajdukiewicz's rule does not license S(S) S(N) » S(N). In order to provide for this combination, a xvXt-thema is required, that is, a rule which says that, given a certain combination, a new combination is permitted. 11 Geach (1972, p. 5) proposed to remedy this defect with the rule: (D)

If Co C, » C2 then Co C,(C3) » C2(C3).

This will, indeed, produce the combination cited above, but it does not cater for operators of degree > 1. Suppose, for example, we want to combine QS(S,S) S(N) S(N)\ It will give us S(S,S) S(N) » S(S,N), but then leaves us high and dry with no means of combining 'S(S,N) S(N)\ In order to deal with this problem, Geach provided a further rule, (E)

If Co Ci C, » C2 then Co Q(C 3 ) C,(C3) » C2(C3).

But this produces the wrong result. It gives us, for instance, S(S,S) S(N) S(N) » S(N), whereas the correct combination is S(S,S) S(N) S(N) » S(N,N).12 But by what right do I speak here of a 'wrong result' and the 'correct combination'? By reference, of course, to Frege's ideography, which, 11

12

The motivation which Geach gives for adding a rule-thema is that we should want to account 'does not fly' a coherent sub-string of 'Socrates does not fly' and 'not every man' a coherent sub-string of 'not every man flies' (1972, p. 484). But although Ajdukiewicz cannot do this, he can show that the two sentences are of category S, so the extra rule is not absolutely necessary on these grounds alone. It is only when the sub-string is used as the operand of a higher-level operator, as in 'some man does not fly' or 'some people can drive every make of car', that the rule-thema becomes essential. Some other attempts at a comparable rule are even more disastrous. Thus Ades and Steedman (1982, pp. 526-7) have a rule of Forward Partial Combination, for which Geach is cited as a precedent, which will not combine lS(S.S) S(N) S(N)' at all. Allowing the Ajdukiewicz category 'S(S)(S)' for 'S(S,S)\ the rule permits combination with one *S(Ny to yield *S(S)(N) S(N)\ but this is irreducible. Nor would it help to reverse the order of these two category names, for although Ades and Steedman include

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providing we stick to Fregean categorial grammar, is a touchstone by which our rules can be judged.13 So, now, consider a schema like 'if n thenCT\Instead of substituting a proposition for each of the schematic letters, let us substitute a propositional schema of category S(N), say '( snores' and '£ yawns'. Then we shall obtain 'if ( snores, then £ yawns'. This contains two schematic letters for each of which an expression of category N may be substituted and, clearly, we must be allowed to substitute different names for each, in order to obtain, for example, a proposition like 'If Philip snores, then Andrew yawns'. It seems that Geach was misled on this point by cases in which a complex expression of this kind appears to serve as the operand of a quantifying phrase, as in 'Everyone who snores, yawns', represented in Fregean notation by: everyonerx (if x snores, then x yawns). For if it is the operand of a quantifier, must it not be of category S(N), that is, of degree 1? The answer is that there is a difference between the schema 'if ( snores, then £ yawns', which is of degree 2, and the schema 'if C snores, then ( yawns', which is of degree 1. In the latter, the same schematic symbol occurs twice, which signifies that the same name must be substituted for both of its occurrences; in the former, we may substitute the same name for both schematic symbols, but need not do so. The operand of the quantifier in the above example is the schema of degree 1, not that of degree 2. In categorial grammar, a special operator will be required to convert the schema of degree 2 into the corresponding schema of degree I.14 The opportunity of re-formulating Geach's rules may be taken to revert to production rules for a tree grammar instead of combination rules. A minor variation on the expansive form rules of section 2.1 will, however, be that category symbols will be used throughout, so that a tree is generated whose nodes are labelled entirely by category symbols. Propositions may be thought of as being obtained by substitution for the category symbols in the tree; this leaves the assignment of terminal

13

14

rules of backward combination, they say, astonishingly, that 'English does not appear to include a rule of 'Backward Partial Combination" (p. 528) and hence do not provide one. The source of the inadequacy in their treatment seems to be that they have only thought about operators of degree 1. Levin (1982) complains that Geach does not justify his combination rule except on the ground of utility, and offers an alternative justification based on a functional interpretation of the notation. That, however, stands or falls with the interpretation (with which Geach has little sympathy), and Levin has missed the implicit reliance upon Fregean principles of analysis. How this would work is spelled out in detail in Potts (1979).

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symbols (linguistic expressions) to categories for separate discussion in a later chapter. A notational convention will allow considerable simplification of the rules: if a rule yields a category name of the form C0(C],. . .,Cn)(Cm), this is to be taken as equivalent to C0(CIt. . .,Cn,Cm). The production rule corresponding to Ajdukiewicz's original combination rule then becomes the rule-schema: (Rl)

Co =» Co(C,)Ci.

So, starting with *S\ we could obtain 4S(N) N' and then, taking Cj = N again and C0 = S(N), obtain *S(N)(N) N N \ which reduces to 4S(N,N) N N' This, in general, will be the technique for obtaining trees containing operators of degree > 1. The production rule-thema corresponding to Geach's intentions but, now, yielding results which are correct on Fregean principles, is: (R2)

If Co =» C, C2, then C0(C3) => Q C2(C3).

Since we have

S => S(S,S) S S S(N) => S(S,S) S(N) S

by 2 applications of (Rl), by (R2).

In order to allow a second application of (R2), one further modification is necessary: it must be legitimate, in the course of a derivation, to switch the order of operands. This cannot have any significance, because the category symbols are not interpreted. So the last line above may be rewritten as: S(N) => S(S,S) S S(N) and then S(N,N) => S(S,S) S(N) S(N)

follows by (R2).

This rule also solves another problem, that a quantifying phrase should be able to combine with a first-level schema of category S(N,N) to form a first level schema of degree 1: Since we have

15

S =* S(S(N)) S(N) S(N) =» S(S(N)) S(N,N)

by (Rl), by (R2).15

Introducing (R2) does not solve a corresponding problem about the meanings of 'not' and the propositional connectives which arises as soon as we assign them to categories S(S) and S(S,S) respectively. Wittgenstein stated the problem as follows: If logic has primitive ideas these must be independent of one another. If a primitive idea is introduced it must be introduced in all contexts in which it occurs at all. One cannot therefore introduce it for one context and then again for another. For example, if denial is introduced, we must understand it in propositions of the form 'not (p)\ just as in propositions like 'not (or (q, p))\

Quantification

85

Unfortunately, the addition of this production rule-thema is still not enough. In classical logic, it is possible to define either one of the quantifiers in terms of the other plus negation, for example something:x 0(x) | not (everything:x (not ((x)))). Now we know that negation does not change the category of its operand, so, since the category of the definiendum here is S(S(N)), that must also be the category of 'everything:x (not (0(x)))\ as may be seen from inspection of the Fregean notation. But rules (Rl) and (R2) are not enough to prove that S(S(N)) =* S(S(N)) S(S). For that, we need a third rule, again a thema 16 : (R3)

If Co(Ci) =* C2(C3) C4, then Co(C,(C5)) =* C2(C3(C5)) C4.

The proof is then: [1] [2] [3]

S =» S(S) S S(S) => S(S) S(S) S(S(N)) =» S(S(N)) S(S)

by (Rl) by (R2) by(R3)

Of the two rule-themas which have now been introduced, (R2) cannot apply to expressions of basic categories and (R3) cannot apply either to expressions of basic categories or to first-level schemas. This prompts the 'something:x (not (Fx))' and others. We may not first introduce it for one class of cases and then for another, for it would then remain doubtful whether its meaning in the two cases was the same. (1922, 5.451, with notation amended to that used in this book)

16

So if we explain the meaning of 'not' by saying that it turns a true proposition into a false one and vice versa, we have no legitimate way of explaining its meaning in the schema 'not (barked ())', for example. This is, of course, an implicit criticism of Frege. Martin (1984) tries to rescue Frege from it by giving a substitutional interpretation of his ideography. That may be a promising approach, but Martin's attempt is unfortunately vitiated by definitions of negation, implication and the universal quantifier which are impossible on Frege's principles, since they have schemas as deflnienda but complete expressions as definientes (p. 148). This is a glaring example of the confusion between concept and object which Frege so often denounced. It is to observe that the motivation for introducing this rule is different from the motivation for (R2). That 'everything:x not $(x)' does not correspond to any sub-string is a merely superficial difference, to do with English word-order conventions, resulting in a gap between 'everything' and 'not', to be filled by an auxiliary which is part of the operand; but the schema shows that it belongs to category S(N). However, the grammar with only rules (Rl) and (R2) will be enough to show that propositions to which this schema contributes are such. The reason for introducing a further rule is, rather, that 'nothingix $(x)' can be defined as 'everything:x not 4>(\)\ so the latter must constitute a semantic unit. In general, our grammar must allow us to explain the meanings of expressions, where appropriate, by breaking them down into components which are combined in a certain way (componential analysis).

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Logic: trees

question whether (R2) and (R3) are just the first two members of a series of rule-themas which will ascend through the hierarchy of levels, a rule (R4) being required for the correct expansion of third-level schemas, and so on. I do not know the answer to this question, and it is difficult to see how it could be decided in advance of encountering structures in everyday language for which (Rl) to (R3) do not provide. 17 An alternative method of allowing a wider range of combinations than Ajdukiewicz's original rule provides for is by type-raising (see Lambek, 1958; Van Benthem, 1988; Dowty, 1988; Steedman, 1988). This has its origin in Frege, though its recent exponents appear to be unaware of the prototype. Frege held that a proper name can sometimes occur as an operator: thus he cites '0(2)' explicitly as a second-level function name, representing 'property of the number 2', and 'not (if (everything:x (if (0(x), x = 2)), not (0(2))))' as another, representing 'property of the number 2 that belongs to it exclusively' (1893, section 22). There is a further, implicit example in his representation of the proposition 'What holds for every object, holds for A' by 'Everything:f (if (everything:x (f(x)), f(a)))\ since the first occurrence of 'everything' is there the operator of a third-level schema (1893, section 20). So in these examples a proper name has been 'raised' from its normal (basic) category N to the (second-level) category S(S(N)). This has been generalized to the effect that a category Co may be raised to category CjfCjfCo)), for any category C/. The effect of Geach's rule (D) can also be obtained by allowing the type of any schema of category C0(Cj) to be raised to category Co(C2)(Cj(C2)), for any category C2- Thus, in order to obtain S(S) S(N) » S(N), we raise S(S) to S(N)(S(N)).

Similarly, in order to obtain

S(S) S(S(N)) » S(S(N)), we raise S(S) to S(S(N))(S(S(N))). the degree 2 case

However, this will not cater for

S(S,S) S(N) S(N) » S(N,N). for which it would be necessary to raise S(S,S) to

17

S(N,N)(S(N),S(N)).

In the above, I have expounded the three rules in a somewhat simplified form in order to bring out their essential features more clearly. A more exact version, which takes account of the context in which an expression to be expanded may occur, and indicates scope relations by a system of super-scripts and sub-scripts corresponding to the edges in trees, will be found in Potts (1978b).

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87

That would call for a further type-raising rule, and so on for each successive degree or, alternatively, a complicated generalization of the original rule. Even then, we could not be confident that type raising gave us the full power of the converse of (R2). Some authors, for instance, have assigned certain complement verbs to category S(N,S) (for example Prior, 1971, pp. 18-19), so we should want to be able to show that S(N,S) S(N) » S(N,N), for example, when such a verb fell within the scope of a quantifying phrase. It is also difficult to see how any type raising could achieve the effect of the converse of (R3). One could, indeed, raise the category of C4 so that it became the operator, but we do not need (R3) to justify, for instance, the combination S(S) S(S(N)), while in the complex schema 'something:x (not (0(x)))\ 'not' is the operand. Since I first introduced (R3) in (1978), it has received neither mention nor comment in the literature on categorial grammar. Perhaps the reason is that it may be unnecessary in a grammar which does not allow for alternative analyses, which is fairly typical of syntactically based categorial grammars which have appeared since. I have already argued, however, that alternative analyses are necessary to a structural system which is to support an account of meaning, and in the present case it seems evident that we do not want to deprive ourselves of the possibility of defining one of the quantifiers in terms of the other, which in turn demands that the schema 'not (something:x (not (0(x))))' can be shown to belong to category S(S(N)). However, these are largely practical objections to type raising as an alternative to rule-themas and might, perhaps, be overcome by typeraising arrangements of sufficient complexity. Far more serious is that type raising embodies several confusions. First, what exactly is meant by 'raising'? In the case of changing a category Co to category CJ(CJ(C0)), it is a question of raising the level of a category; indeed, it is raised by two levels. In the case of using type raising instead of (R2) and (R3), it is more difficult to say what is going on because we do not know quite how far type raising must go and also because the Fregean distinctions of degree and level become blurred in non-Fregean categorial grammars. Is category S(N)(S(N))9 for example, of degree 1 or 2, and of level 2 or mixed level? The corresponding Fregean category, S(N,S(N))9 is of degree 2 and mixed first/second level. What we are to say about the degree of S(N)(S(N)) is unclear; if it is of degree 1, then it is straightforwardly of level 2, but if of degree 2, then mixed level. So the most that can be said about type raising in this context is that it raises the

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level of a category by 1, at least in part, and may also increase its degree by I.18 Bearing in mind that our categories relate to meaning, difference of category implies difference of meaning. One proponent of type raising, at any rate, is clear on this point, paraphrasing Geach's rule (D): 'if an expression occurs in category A(B), then it can also occur in A(C)(B(C)) (with an evident transfer of meaning)' (Van Benthem, 1988, p. 36). But, of course, this is precisely not the intent of Geach's rule or of other rule-themas in categorial grammar. We do not want to say, for example, that the meaning of 'not' is different in 'Some dogs did not bark' from its meaning in 'Fido did not bark'. This is not like the extension of our notion of number when we move, for example, from natural to rational numbers or from rational to real numbers.19 There is absolutely no evidence that someone who understands the meaning of 'not' in 'Fido did not bark' and also understands the meaning of 'Some dogs did bark' must learn something further in order to understand the meaning of 'Some dogs did not bark'. And, lest anyone should consider looking for such evidence, if 'not' and other terms are multiply ambiguous in the way that type raising would demand, we can say goodbye to most logical inference: it will be invalid for ambiguity. The whole point of rule-themas is to allow a wider range of generation or combination than the rule-schema alone would provide, without recourse to re-categorization; that the same results regarding semantic coherence can be achieved by type raising instead, is irrelevant. Type raising is also a misleading way of explaining differences of two levels in category assignments. An example commonly cited is Montague's assignment of proper names to category S(S(N)) instead of to category N, as though he first of all assigned them to the latter 18

19

Some versions of categorial grammar resolve this ambiguity by only allowing categories of degree 1. The level of a category C0(C]) is then defined as whichever is the greater, the level of C/ plus 1, or the level of Co. So, on this account, type raising is always of level and never of degree, there are no mixed-level categories and category S(N)(S(N)) is of level 2. A double price must be paid for this solution. First, the flexibility of analysis which, as argued at the end of section 2.4, logic requires, is lost. Second, the linguistic generalization expressed by distinctions of degree is also lost, even if we allow that Frege's assertion that 'Functions of two arguments are just as fundamentally different from functions of one argument as the latter are from objects' (1893, section 21; cf. 1892, p. 193) is difficult to sustain if one admits functionals (functions which have functions as their values). Though some categorial grammarians suppose that it is, seduced, perhaps, by analogies which they have found between their systems and parts of the lambda calculus (see Van Benthem, 1986, chapter 7). But mathematical ingenuity is no substitute for philosophical argument and the lambda calculus is, in any case, worthless as a tool in the theory of meaning (a justification of this judgment may be found in Potts, 1979).

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category, then allowed them to be 'raised' to the former when testing for semantic coherence (see Dowty, 1988, p. 159; Van Benthem, 1988, p. 37). But this is simply incorrect. He never assigned proper names to category N (his category e) nor, indeed, anything else to that category: it was an empty category, while proper names were assigned with quantifying phrases to category S(S(N)) (his category t(t(e))) from the start, with a corresponding account of their meanings (Montague, 1974, pp. 249-50).20 The examples from Frege cited above are more serious contenders for type raising but, even there, we must not lose sight of an important distinction. Frege always accounts numerals proper names (except when used adjectivally), and what he assigns as second level is not the proper name '2' but the schema '(2)\ The latter, like the schema 'everything:x (0(x))\ contains a schematic symbol for which a first-level expression must be substituted and, hence, is by Frege's normal criterion for levels a schema of level 2. But it is a degenerate second-level schema, because it contains no link letters. So, here again, there is no question of raising or, indeed, in any way changing the category of an expression. Nevertheless, the Fregean examples present a difficulty and, lest anyone should baulk at their mathematical nature, Dummett points out that 'higher-level quantification is extremely common in natural language, and would not be regarded as puzzling or odd by its unsophisticated speakers' (1973, p. 218). From a variety of examples which he gives, I cite (10)

Paul is what I am not,

namely (perhaps), thrifty. As a Fregean representation of (10) he gives (p. 215): (10F) Something:f (and (not (f (I)), f (Paul))), which is strictly comparable to Frege's '$(2)' example. Now can we insist that the proper name 'Paul' does not occur in (10F), but only the secondlevel schema '^(Paul)'? It hardly seems plausible to claim that the proper name 'Paul' does not occur in (10) and it will not help much to say that it only occurs there as the operator of a second-level schema. For surely we can form the first-level schema '( is what I am not' from (10) by removing 'Paul', which is then being treated as a proper name? Moreover, if it is the same proper name which can be considered in (10) either as occurring as a

20

A rule allowing expressions of category N to be raised to category S(S(N)) is more properly credited to Cresswell (1973, pp. 130-3), where the motivation is to allow expressions like 'some economists and Mrs Thatcher' to be assigned to category S(S(N)) while maintaining S(S,S) as the category of'and'. I shall discuss such cases in section 7.3.

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proper name or as the operator of a second-level schema, how can it belong to two different categories without change of meaning? Ultimately Frege does seem to be committed to re-categorization in this case, with its attendant problem of a change in meaning where, intuitively, there appears to be none. Now re-categorization could be avoided by adding a further rule which will allow a schema to take an operand which is three levels, instead of one, below it (our example is of a third-level schema taking an expression of a basic category - level 0 - as operand). In the case of (10), we want to be able to derive S =* S(S(S(N))) N instead of S =» S(S(S(N))) S(S(N)) The latter is, however, in a sense the condition of the former, and the operand of the latter must be at least of second level. So, generalizing, we obtain the rule-thema: (R4)

if Co =* C, C2(C2(C3)), then Co =» C, C3

Examples involving operators of degree > 1 will be taken care of by successive applications of the rule, as with the previous rules (R1)-(R3). This rule marks a slight departure from Frege, but licenses structures which he found it necessary to admit without positing the presence of schemas raising problems about meaning. As to type raising, we may dispense with it for the purposes of semantic analysis. One concession, however, is in order. It may be that a system of categorial grammar with type raising is formally equivalent to one with recursive rules. In such a case, it may be technically convenient to prove some property of the latter indirectly, by proving it first directly of the former. Clearly there could be no objection to invoking a system with type raising for this purely technical purpose. A similar remark applies to the Lambek calculus (Lambek, 1958), which exploits an analogy between rules (Rl) and (R2) and part of intuitionist propositional logic. In this system we can, indeed, prove the two derivations which I cited earlier as obstacles to type raising alone, but it seems that neither (R3) nor (R4) would be obtainable, though the converse of (R4) is. Again, there can be no objection to such a calculus if it is used for purely mathematical purposes, but there must be every objection to it if used as a straightjacket to confine everyday language. Little interest was shown in categorial grammars during the 1960s and early 1970s because of a result published by bar-Hillel, Gaifman and Shamir (1960), now known as Gaifman's theorem, to the effect that context-free string grammars are equivalent in power to Ajdukiewicz's categorial grammar. This was unfortunate: in the first place, because the

Quantification

91

possibility of generating the same expressions by a string grammar as by a tree grammar does not show that it is indifferent so far as the meanings of those expressions are concerned whether we regard them as having tree structures or as merely having string structures; and, in the second place, because the result does not hold for Fregean categorial grammar - as soon as the recursive rule-thema (R2) is introduced, the categorial grammar becomes more powerful than any context-free string grammar. Some, indeed, have complained that Fregean categorial grammar is too powerful, generating structures of such complexity that no linguistic examples of them can be found. I cannot see that this matters greatly: with each minimal linguistic expression assigned a category in the lexicon, the grammar could be used both to generate and to parse complex expressions even though infinitely many possible categories were never called into play. Of course, it would be very interesting to know that language only uses certain categories of the grammar and, even more, to know why; but it is not possible at this stage in the development of categorial grammars even to put a tentative limit to category usage, and it is surely better to have a tool box so well equipped that some tools are seldom, if ever, used, than to have so few tools in it that jobs are constantly turning up which cannot be undertaken. Quantification, however, poses a difficulty for tree grammars which seems to exclude them altogether as a means of representing semantic structures. Trees are, indeed, an improvement upon strings because they embody the notion of one expression working upon the meanings of others with which it combines, but they are still inadequate to represent quantification as conceived by Frege. This has been concealed up to now because we have been using a linear notation for tree structures, but now let us revert to the planar notation with which this chapter began. Remembering that we decided that a schema should be represented by an operator plus the edges associated with it in virtue of its degree, we obtain the following diagrams for propositions (5) (7) and (9) respectively: (5A)

A dog

(7A)

at least one patient

barked

I

(9A)

Every doctor

every doctor

at least one patient

visited

visited

I

I

I

L

1

None of these is a tree, because the leaves are missing. Moreover, it is evident that we cannot tell from (7A) whether at least one patient was visited by every doctor or at least one patient visited every doctor.

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Logic: trees

Similarly, we cannot tell from (9A) whether every doctor visited at least one patient or was visited by at least one patient. Some authors (for example McCawley, 1971, section 3; Montague, 1974, chapter 8) have dealt with this by dispensing with the edges associated with the lowest operators in the tree, but including the link letters of Frege's notation, for example (5B)

A dog:x

(7B)

At least one patient:y

x barked

(9B)

every doctonx

Every doctonx at least one patient:y

i

i

x visited y

x visited y

This does, indeed, resolve the ambiguity of (7A) and (9A) and it could be claimed that inclusion of the link letters with the operators now at the leaves of the tree shows that they are the operators of first-level schemas. However, those link letters belong, not to the operators at the leaves of the tree, but to the schemas of the second-level operators (the schema of which 'every doctor' is the operator is 'every doctor.x (c/)(x))\ which includes both link letters). So this convention is, at best, misleading and, to see the structures with which we are dealing more clearly, we need to replace the link letters with actual links. Assuming that the edge from the left of 'visited' relates to the visitor and the edge from the right to the visited, this yields the following results: (5C)

p A dog

[i

- barked L-haricen

(7C)

At least one patient

(9C)

r

at least one patient-

r- every doctor p every doctor I

vi«iitpH ———

Every doctor

1

visited -

This also restores the edges showing the degree of each of the first-level operators. But now, instead of being descending edges like those in a tree, these edges ascend from the first-level operator back to a second-level operator - that is, if we think of the edges as having a direction. These representations, however, are no longer trees, though they are graphs. A grammar powerful enough to generate such structures will, accordingly, have to be a graph grammar and Fregean categorial grammar, in spite of initial appearances, is indeed a graph and not just a tree grammar.

Quantification

93

We could always sacrifice quantification in order to remain within the simpler world of tree grammars; but it was precisely this structural analysis which enabled Frege to give a correct account of the logical relationships between propositions containing more than one quantifying phrase. Frege's originality with respect to structural analysis did not lie in the replacement of subject and predicate by operator and operand, but in his quantifier notation, although introducing the operator/operand distinction was a necessary preliminary. To logicians, to computer scientists and to most philosophers it would be unthinkable to abandon such hard-won gains now.

Computer science: graphs

3.1 GRAPHS AND GRAPH GRAMMARS A graph is a structure consisting of nodes (represented as small circles) connected by edges. The limiting case of a graph is a single node. The edges may be given a direction, in which case the structure is called a digraph. The nodes may be labelled, for example with words, and so may the edges. So both strings and trees will be special cases of graphs. Graphs allow for any connexions between their nodes and, in particular, for circuits, that is, a sequence of distinct edges and distinct nodes in which, however, the first and last nodes are the same (if they are different, it is just a path). The simplest case of a circuit is provided by a triangular graph; thus (G1)-(G3), all of which are graphs, each contain several circuits. (Gl)

(G2)

(G3)

Graph (G2) differs from the cube graph (Gl), however, in that none of the edges crosses in the former, whereas some of the edges do cross in the latter. On this account, (G2) is a plane graph, whereas (Gl) is not. However, if we consider a graph to be defined simply by the number of its nodes and the connexions between them, rather than by the way in which it is drawn, we can see that (Gl), though not plane, is another way of drawing the same graph as (G2). Thus the notion of a plane graph is not very useful and we need, instead, that of a planar graph, viz. any graph which can be drawn in such a way that none of its edges crosses. Both (Gl) and (G2) are then planar, but (G3) is not: there is no alternative way 94

Graphs and graph grammars

95

of drawing it on a plane surface so as to eliminate all of the edge crossings. This notion has been generalized in order to yield a classification of graphs. Planar graphs can also be embedded on the surface of a sphere; it is evident that the cube graph (Gl) fulfils this condition. From a sphere we proceed to a torus, that is, a ring. Graphs which can be embedded on the surface of such a body but not on the surface of a sphere are termed graphs of genus 7, or toroidal graphs, of which (G3) is an instance. If we think of the torus as a sphere with a hole in it, then we can think of more complex surfaces obtained by making further holes; for example, one with two holes would be like a closed pot with two handles. So a graph of genus n will be one which can be drawn without edge crossings on the surface of a sphere with n holes or handles, but not on the surface of one with n—\ holes or handles. We can build onto this a definition of dimensionality for the structures which things have which has an intuitive basis. If the structure can be represented by a planar graph but not by a linear graph, we can regard it as two-dimensional; if by a graph of genus 1 but not by a planar graph, as three-dimensional; if by a graph of genus 2 but not by a graph of genus 1, as four-dimensional; and so on. The graphs to be employed in this book will mostly be planar, but there will be a few cases in which we shall need a toroidal graph. I do not know what is the highest genus of graph needed to represent semantic structures for everyday language as a whole, since the scope of the present work is severely restricted. In particular, it would be unwise even to make a conjecture about what may be required until a satisfactory representation of semantic structures for propositions containing psychological verbs has been found. Graphs are generated by graph grammars. Naturally, the rules of graph grammars allow us to replace graphs by graphs, a single node being recognized as the limiting case of a graph. Thus a typical graph will contain sub-graphs: not only individual nodes, but also configurations of nodes and edges. For example, several square sub-graphs can be extracted from the cube graph (Gl). At their most general, then, the rules of a graph grammar will allow us to replace either an entire graph or a sub-graph by a new graph. This may complicate, or it may simplify, the original graph (replacement of a complex sub-graph by a single node would simplify). Such rules are clearly likely to be much more complicated than the rules of a string or of a tree grammar, because they must ensure that the new structure is a correctly constructed graph. Obviously we want to restrict these rules to what is absolutely necessary in order to generate semantic structures, so that they be kept as simple as possible. It would therefore be very premature to essay a graph grammar

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Computer science: graphs

for the semantics of everyday language until we have a fairly clear idea of the range of structures involved. Graph grammars, however, are divided into sequential and parallel. Sequential grammars only allow replacement of one sub-graph at a time, whereas parallel grammars allow for several to be replaced simultaneously. The primary application of the latter is to biology, where it may be important to represent simultaneous changes in different cells of an organism. But a linguistic grammar is not intended to represent any actual process, only to determine what configurations of symbols are correctly formed from a semantic, syntactic or phonetic point of view. So there is no theoretical reason why we should need a parallel grammar for linguistic purposes and, indeed, the syntactic grammars so far proposed are all sequential, as is the application of the recursive definitions of a formula specified in logic. From a practical point of view, it might, of course, be more convenient, if one wanted to use parallel processing, to have a parallel grammar. But that need not concern us here and so we may safely conclude that a sequential graph grammar will take care of our needs. That is good news, since sequential grammars are less complicated than parallel ones. 3.2 S E M A N T I C N E T W O R K S Computer scientists began to use graphs for the representation of meaning from 1966 onwards, for the most part quite independently of formal methods in linguistics and logic, under the title of semantic networks (Quillian, 1966). The edges and even the nodes, however, have been very variously labelled. The earliest semantic networks represented definitions of concepts, drawn from dictionaries (Quillian, 1966, 1967). Later, these were combined with proper names and, sometimes, logical operators in order to represent propositions. A persistent theme is that of breaking down concepts into structured primitive elements, so that one could regard semantic networks as making a contribution to componential analysis (see section 3.3, note 7). Commonly, the nodes of the graphs are taken to represent concepts and the edges relationships between them. The relationships include semantic roles (as in case grammar), but much more besides, such as propositional connectives, modalities, types of attribute (for example: size, shape, having as a part) and set relations, like being a sub-set of (see Simmons, 1973). In particular, they include a relationship being a token of. The terminology is borrowed from Peirce (1906), but is not used in the sense of his type/token distinction; rather, being a token of some concept is being an instance of it. Sometimes this is used in contexts such as 'token

Semantic networks

97

(Rex, dog)', to say that Rex is a dog, but also 'token (dog, animal)', to say that the dog is an animal (with 'is a' as an alternative to 'token'; see Scragg, 1976, p. 104). At other times, however, it is used with numbered nodes, as in 'token (C2, John)', where the intention seems to be to pick out an instance of people named 'John', that is, *C2 is a John'; of course, the individual, thus conceived, cannot be given an ordinary proper name, since proper names are here being deemed common count nouns (on the ground that many people are called by the same name: more of this in section 6.2). All this is confused enough: in the first case, an object falling under a concept being muddled with one concept falling within another (to use Frege's way of stating the difference); in the second case, a categorial confusion between proper names and count nouns. But to make matters worse, the token relationship is also used for instances of actions, such as 'token (Cl, break)', to say that Cl is a breaking, that is, an instance of breaking. An example shows how this is applied (Simmons, 1973; Scragg, 1976, p. 122; the semantic roles have been altered to accord with section 1.2): (1)

John broke the window with a hammer

(G4)

O hammer John O O with

window This ignores the tense of the proposition, NBR (number, values siNGular or PLURal) and DET (determiner, values DEFinite and iNDEFinite) are borrowed from syntax, PREP (presumably preposition) appears in the example but is not otherwise explained. Finally, we have a token of a breaking (Cl), of a window (C3) and of a hammer (C2), and of a John (C4). Such semantic networks can also be represented in the notation of the programming language LISP (LISt Processing), in which lists are shown

98

Computer science: graphs

within parentheses (which may be nested) and the first item in any list may be defined as a function with the remaining items naming its arguments. Following Allen (1987, ch. 7), (1) could be represented by (1L)

(PAST C l BREAK (AGENT C 4 (iNDEF/SING C 7 JOHN)) (OBJECT C3 (DEF/SING C6 WINDOW)) (INSTR C 2 (INDEF/SING C 5 HAMMER))).

This is not a straight 'translation' of (G4).1 First, the TOKEN relationship is omitted in favour of a convention that the second item in each string is always a token. Second, DETerminer and NUMBer are also omitted and their values combined into a new operator, which also demands a new token-name as its first operand. It is then quite unclear what the extra token-names name; for example either 'C4' or 'C7' is the name of the Agent, but what, then, does the other name? Third, the preposition 'with' is omitted from the representation; this is an improvement, since iNSTRument tells us the semantic role of the hammer and provision of 'with' can be considered a matter for syntax. It will be evident that the nodes in graph (G4) represent a remarkable hotch-potch of disparate elements and that there is no attempt to distinguish between categories of expression. Similarly, the relationships between the nodes strike one as an arbitrary collection many of which would be difficult to justify as relationships between concepts. Indeed, most of the nodes themselves seem no longer to represent concepts: in particular the nodes C1-C4, but also nodes like 'with' and 'INDEF'. NO justification is offered for introducing a token of breaking, rather than simply replacing the label 'Cl' with 'break', although this may be inspired by Davidson's analysis of action-sentences (see Allen, 1987, p. 213). Finally, there is no indication of how the meaning of the proposition represented is organized by scope. But I do not need to labour these criticisms, for the ambiguities in semantic network notation and its indeterminacy of meaning have already been more thoroughly exposed and criticized by Woods (1975). Woods also observed that most systems of semantic networks are unable to handle quantification properly. At that time, he found only two exceptions. The first (Kay, 1973) represents universal quantifiers by nodes labelled with the second and subsequent occurrences of their link Allen derives representations such as (1L) from a syntactic description of the sentence or expression by means of semantic interpretation rules which are applied in turn to each of the elements in the syntactic description to yield partial semantic interpretations. The latter are then merged into a single semantic representation for the whole expression. Merging resolves any ambiguities of single words in the expression and is clearly modelled upon the projection rules of interpretative semantics described in section 1.5.

Semantic networks

99

letters. Existential quantifiers are eliminated in favour of edges labelled by Skolem functions. These edges are directed from a node labelled with the link letter of the existential quantifier to that labelled by the link letter of a universal quantifier within whose scope it lies. Skolem functions may be explained as follows. Since an existentially quantified proposition will be true just in case something falls under the concept expressed by its operand, we can replace the quantifier by a function whose value is the something in question. That is, instead of 'something:x (f(x))\ we may write 'f(g)\ where g is a function (in this case, without arguments) of whose value / is true. When existential quantification occurs within the scope of universal quantification, however, the value of which the predicate is true will depend upon which case covered by the universal quantifier we happen to be considering. Thus, if every girl loves a sailor, we cannot determine the sailor until we know which girl is in question, for each girl may love a different sailor. So the Skolem function which replaces the existential quantifier in this case must be differentiated for each girl. To this end we write: 'every girl:x (loves (x, g(x)))', where g is a function whose arguments are girls and whose values are sailors. Of course, we may not be able to specify the Skolem function, any more than we can specify what object satisfies an existential proposition. The procedure for replacing existential quantifiers by Skolem functions presupposes that our propositions are in prenex normal form, that is, with all the quantifiers at the beginning. The universal quantifiers are then removed, leaving the second and subsequent occurrences of their link letters to represent them. The example thus becomes: 'loves (x, g(x))\ According to Woods, we can then 'obtain a semantic network notation based on this Skolem function analogy by simply including with every existentially quantified object a link which points to all of the universally quantified objects on which this one depends' (1975, p. 77). Reduced to its barest form,2 though with the count nouns of the quantifier phrases restored, the semantic network which this yields is: (G5)

2

girl Q

Q sailor

Kay's network is more complicated (1973, p. 183). He introduces extra nodes in order to show this case of loving as a member of the class of lovings, and then duplicates the

100

Computer science: graphs

In order to represent the proposition obtained by switching the order of the quantifiers, There is a girl whom every sailor loves', Kay simply changes the direction of the edge labelled with the Skolem function (cf. the example in 1973, p. 184). But this fails to distinguish the above from 'Every sailor loves a girl'. On closer inspection, indeed, one can see that his notation lacks the necessary multiplicity, because a single sign (the edge labelled with the Skolem function) is simultaneously trying to show the scope order of the quantifiers and whether each is universal or existential. A further difficulty (noted by Woods) is that one cannot obtain a network for the negation of a proposition so represented by attaching a negation sign to it; instead, one must first work out the prenex normal form of the new proposition and then construct a new network. More generally, then, such a network cannot be incorporated as it stands into another as a sub-graph. Since Woods's article appeared, several attempts have been made to make the notation more rigorous and to provide for quantification. Thus Hendrix (1975a, 1975b, 1979) severely restricts the labels available for edges and introduces partitions, boxes enclosing groups of nodes (not necessarily forming sub-graphs), as a means of representing logical operators. A link from a node within a given box to one outside it is regarded as falling within the box if its label is written within the latter. Conjunction is then represented by enclosing nodes representing individual states or events within the same box. Disjunction, however, requires a much more complex apparatus: first, separate boxes each containing a node representing an individual state or event are connected to a node D which in turn is linked to a node Disjunctions by a setmembership edge. This is supposed to mean that D is a member of the set of disjunctions (by analogy with this man being a member of the set of men)! The edges from the boxes to the node D are also labelled by setmembership, but qualified in this case to make the boxes disjoint members of D (p. 67, fig. 10). Negation is represented similarly, though with only one box and no node corresponding to D (p. 68, fig. 11). These

Skolem function edge from the individual loving node to the universal quantifier node. But this does not affect my criticism. (G5) contains two innovations. First, duplication of nodes with the same label is avoided by having an edge both from the 'girl' and the Moves' nodes to the 'x' node, and similarly both from the 'sailor' and the 'loves' nodes to the vy' node. Second, the network has directed edges, unlike (G4). These are now usual features of semantic networks. I shall develop them in the next chapter. Meanwhile, it is to observe that no general explanation is given of what a directed edge signifies. Often it appears to relate to the way in which the network is stored on a computer rather than to any logical features of the representation itself.

Semantic networks

101

representations are barely intelligible, and it is quite unclear how transformations such as those in de Morgan's laws might be effected. Hendrix requires no special notation for existential quantification, since 'the occurrence of a structure (that is, a node or arc) in a space [ = box] is taken to be an assertion of the existence with respect to that space of the entity represented by the structure' (p. 69). Since his nodes can represent virtually anything, this introduces implicit existential quantification on a vast scale, for example over the node D described above, but he does not pause to consider any problems which this might raise. Universal quantification is represented in connexion with implication. The latter is shown in a similar way to disjunction, with a node / and another node Implications. But now the edges are from / to the boxes and are labelled <3«te[cedent] and coAis^quent] respectively. For universal quantification, a node representing an individual (whose label is not a proper name) is included within two overlapping boxes one of which is the antecedent and the other the consequent of an implication. This notation is inspired by Frege's use of unrestricted quantification, so that, for example, 'Every man owns a car' is analysed as 'everything:M (if (M is a man, M owns a car))'. The node labelled 'M' would then be placed in the two overlapping boxes (cf. p. 83, fig. 18). The treatment of quantification again raises many unanswered questions. How may 'everything:x (not (F (x)))' be transformed into 'not (something:x (F (x)))'? How can universal quantification be combined with operators other than implication? Moreover, if universal quantification is to be analysed in the Fregean manner, then why is, for instance, 'Some man owns a car' not represented so as to show a hidden conjunction ('something:M (and (M is a man, M owns a car))')? Again, it is totally unclear how other quantifiers, whether numerical adjectives or inexact quantifiers like 'few', 'several', etc., should be represented. These difficulties spring from a single source. Hendrix has not thought about the categories of the expressions which he is seeking to represent, and in consequence has quite disparate and, indeed, incommensurable representations for expressions of the same category: for conjunction and disjunction, both of category S(S,S)\ and for universal and existential quantification, both of category S(S(N)). Moreover, the interpretation of his networks is uncertain, and would become increasingly so the more complex their content, because they contain no indication of scope. Numerous references to Woods's article in the work of Cercone and Schubert show that they have taken note of his criticisms (Cercone, 1975; Schubert, 1976; Schubert, Goebel and Cercone, 1979). In their system, each proposition is represented by a node which is connected to a node for its main verb by an edge labelled 'PRED', and also connected to nodes

102

Computer science: graphs

for each of the verb's operands. Propositional operators are represented by nodes with an edge from a new propositional node which, in turn, is connected by edges to the operands of the propositional operator, for example (2)

Either Jim sat or Sue stood. or

(G6)

Jim

-OSue PRED

PRED sat

stood

Quantification is expressed using Skolem functions, but with one important modification to Kay's method of representing them in semantic networks. Schubert (to whom the notation is primarily due; see 1976) distinguishes universal from existential quantification by using dotted lines for the nodes of the former, leaving nodes representing quantifiers unlabelled. Thus, taking (3)

every girl loves a sailor

again as an example, the Skolemization is, in linear notation, (3F)

if (girl (x), and (sailor (f(x)), loves (f(x),x)))

(G7)

sailor

scope

Semantic networks

103

and the corresponding semantic network, (G7). (Instead of Schubert's dotted node for universal quantification I have used a square node and instead of his dotted edge between the quantifier nodes, simply labelled the edge 'scope'. His dotted edges are unlabelled, the Skolem function which is explicit in Kay's notation becoming implicit. Schubert gives this edge the opposite direction to Kay, so that it is now also the direction of the scope.) Schubert introduces an abbreviated notation which absorbs the propositional nodes, turning predicates and propositional operators at the same time into labelled edges. This greatly simplifies the appearance of his networks. It comes close enough to the notation used in (G7) for us to ignore the differences for present purposes. The abbreviated network for (3) is then (G8). In Schubert's notation the corresponding network for (4)

There is a girl whom every sailor loves

would be obtained simply by deleting the edge from the universally quantified node to the existentially quantified one. But it would surely be clearer to leave the scope edge, though switching its direction, as in (G9). The notation then has the necessary multiplicity: the different types of node distinguish between universal and existential quantification, while (G9)

(G8) Q girl

sailor Q

sailor Q

the scope edge shows their mutual scopes. The presupposition that each network corresponds to a proposition in prenex normal form still prevents us from incorporating networks as subgraphs of larger ones. It can also exclude representation of some propositions containing modal operators altogether. For example, if it is possible that there will (someday) be a woman Pope, it does not follow that there is (already) a woman of whom it is possible that she will (someday) be Pope. Thus there is no way of expressing the first of these propositions in prenex normal form. Schubert proposed to overcome this limitation by using scope edges (subsequently called scope inclusion links)

104

Computer science: graphs

to show when a quantifier falls within the scope of a propositional or modal operator. One of Schubert's examples (1976, p. 180) is: (5)

Mary will receive a scholarship provided that she passes all of the exams.

He represents this in linear notation as: (5F)

iff (something:x (and (scholarship (x), receives (x, Mary))), everything:y (if (exam (y), pass (y, Mary))))

(using 'iff for 'if and only if and slightly simplifying his treatment of 'all of the exams'). This example could, of course, be put into prenex normal form, though the result 'is twice as complicated as the original version and quite incomprehensible when stated in English' (1976, p. 180). The semantic network corresponding to (5F) is (G10). (G10)

iff

scope

receive

/

Kpass

scope

Mary 1 I

O scholarship

exam O

Similarly, a scope inclusion link would be drawn from a modal operator to the first quantifier node which fell within its scope, for instance from the 'possibly' node in the example given above to an existentially quantified node. What Schubert has done is to introduce into semantic networks explicit indications of scope relationships. These are essential to any adequate representation of quantification and of propositional and modal operators, so his notation works where the attempts of his predecessors failed. But it is still deficient in one respect. Because he relies upon different kinds of node in order to distinguish universal from existential quantification, he would have to use a different type of node for each possible quantifier. That might seem feasible, if complicated, if we

Semantic networks

105

restricted ourselves to quantifiers like 'several', 'most', 'many' and 'few'. Once numerical adjectives are included, however, infinitely many different kinds of node would be required. Another version of semantic networks, under the title of conceptual graphs, has been developed by Sowa (1984). Sowa's graphs contain two distinct types of node, one representing concepts and the other relationships between them. But the edges of the graphs are unlabelled, and a relationship node must occur between any two concept nodes, so this appears to be no more than a notational variant upon labelling the edges. Concepts are, however, classified into types, and rules are given for thirty-seven relationships stating which types of concept they may inter-relate. Thus the agent relationship links an action concept to a concept of an animate being, and the cause relationship links two concepts of states (Sowa, appendix B). Many of the concept classifications, however, as well as the relationships posited between them, are controversial, yet are simply presented without supporting argument. Indeed, the original idea that semantic roles are relationships between an action and its participants is mistaken, for we could not imagine an instance of the action in which it was differently related to the participants; the semantic roles are, rather, part of the meaning of the verb (see Potts, 1978). Sowa also provides rules for constructing more complex graphs from those obtainable by applying the rules for relationships noted above (Sowa, section 3.4). They do not, however, allow for introducing negation and propositional connectives, or quantifiers, so he provides for these by adopting the existential graphs of Peirce (1960, vol. 4, pp. 320-410). These contain representations for negation, conjunction and existential quantification only, and are thus limited to classical logic. Perhaps another source of representational inadequacy is the attempt to by-pass language in order to represent meaning directly. From a true premiss that semantic representations should not be tied to a particular language, the false conclusion has been drawn that semantic representations are independent of any means of expression, that they somehow transcend language. Yet the notation in which they are expressed is, of course, itself a language, albeit a technical and not an everyday one. It may, indeed, be justified for the purpose in hand but, since that purpose is to represent the meanings of linguistic expressions, it does not excuse us from carefully relating the notation to the language which it purports to represent. Once we begin to imagine ourselves free of this duty because we are dealing directly with concepts, unjustified assumptions lead to nonsense, while the relationship of the proposed formalisms to everyday language is no longer clear.

106

Computer science: graphs

It appears that the lesson has still not been fully learned that semantic networks must exhibit scope relationships if they are to handle quantification satisfactorily. Thus, to recur to (1L), from a logical point of view, the indefinite article in (1) is a quantifier: not, perhaps, an existential quantifier ('at least one'), but a numerical quantifier Cjust one'). That difference, however, is not to the point here; the trouble is that in (1L), 'a hammer' is represented as falling within the scope of 'break', which is impossible, since it is not a proper name. The standard Fregean representation of (1) would be: (IF)

a hammenx (broke with (x, the:y (window(y), John))),

assuming the definite article to be a name-forming operator. It is, therefore, highly misleading for Allen to describe representations such as (1L) as presenting the iogical form' of propositions, etc.; at the very least we must say, as with the transformational grammarians' notion of logical form, that it is not to be confused with the term as used by logicians. Allen does, indeed, admit that 'the current LF [logical form] cannot represent scoping distinctions' (1987, p. 217). In particular, he has in mind that when a proposition contains two or more quantifying phrases, the notation cannot show which of them lies within the scope of the others. 3 Thus my example from previous chapters, (6)

Every doctor visited at least one patient

would be represented by (6L)

(PAST el VISIT-EVENT (AGENT al (EVERY dl DOCTOR)) (PATIENT p i (iNDEF/SING p2 PATIENT))),

which does not distinguish between every doctor visiting the same patient and every doctor visiting some patient or other, perhaps a different one for each doctor. Allen claims that this is actually an advantage of his notation, in that the ambiguity is resolved by context and should therefore be dealt with separately. But he is not consistent on this score, for when it comes to ambiguous words, such as the adjective 'green' (colour, unripe (of fruit), simple (of persons)), he is careful to distinguish them. Yet here, too, it might be necessary to resort to context in order to resolve the ambiguity, for an apple could be called 'green' to indicate either its colour or that it is 3

This is a consequence of the way in which semantic representations are related to the syntactic ones by the semantic interpretation rules, for the syntactic system uses noun phrase as a category and includes quantifiers as parts of it. Thus the syntactic analysis imposes a framework within which the semantic structure must be developed, and so constrains it.

Semantic networks

107

unripe. This leaves Allen's notion of logical form' wholly unclear; it clearly does not represent the structure of the proposition relevant to its meaning, for then we should require two (or more) distinct structures for a proposition which was structurally ambiguous, even though it might be left to a contextual component of the formal system to choose between them. He proposes, nevertheless, a notation to indicate the scope of quantifiers. It consists in attaching the token-name which follows one of the quantifiers as an index to another quantifier; this is to be understood to show that the indexed quantifier lies within the scope of the other. Thus, in order to represent the sense of (7)

At least one patient was visited by every doctor

in which 'every doctor' is taken to be within the scope of 'at least one patient', we should write: (7L)

(PAST el VISIT-EVENT (AGENT a l (EVERYp2 d l DOCTOR)) (PATIENT p i (INDEF/SING p2 PATIENT))),

whereas to represent (6) or the sense of (7) in which 'at least one patient' is taken to lie within the scope of 'every doctor', we have: (6L')

(PAST el VISIT-EVENT (AGENT al (EVERY dl DOCTOR)) (PATIENT p i (iNDEF/SINGdi p2 PATIENT))).

This device has the appearance of an ad hoc resort to avoid a difficulty. In a proposition containing many quantifiers, it would be extremely unperspicuous. Moreover, since it does not tell us when one quantifier does not lie within the scope of another, there will be cases in which it fails to determine scope unambiguously. Take, for instance, the following type of structure: A:x (B:y (C:z (. . .(x,y,z)) . . . (D:v (E:w (. . .(x,v,w))). We can use the indexing device to show that B and D both fall within the scope of A, and that C falls within the scope of B and likewise E within the scope of D. This will be enough to show that C and E fall within the scope of A, but does not show that E does not fall within the scope of B nor C within the scope of D. Moreover, indexing some of the quantifiers in a representation is not enough to show the scope relationships in the structure as a whole. Allen implicitly acknowledges this when he extends indexing to negation (1987, p. 288). He cites a very common type of example in which 'not' is attached syntactically to the verb in a proposition when its scope is intended to be the whole sentence, for example

108 (8)

Computer science: graphs Everyone is not going to York

for 'Not everyone is going to York' rather than 'No one is going to York'. Without resort to indices, Allen has only one way of representing (8), viz. (8L)

(NOT n l (PRES g l GO-EVENT (AGENT a l (EVERY p i PERSON))

(TO-LOC

11 (NAME yl

CITY

"York")))).4

In order to resolve the ambiguity, 'EVERY' must be indexed as 'EVERYni', to show that it lies within the scope of 'NOT', or 'NOT' indexed as 'NOT P I', to show that it lies within the scope of 'EVERY'. In principle, then, the full scope structure of the proposition would be shown by indexing every expression in the representation written in capitals except the main operator. Moreover, if the meaning of a proposition is in part determined by its scope structure, this would eventually be necessary in order to represent that meaning. Even ignoring uncertainty in the notation, the result would be excessively difficult to read. And why use a basic notation which is designed to express the operator/operand distinction in such a way as positively to obscure it? What is happening here is, of course, that an account of syntactic structure is being arbitrarily imposed upon semantic representations essentially the same mistake as transformational grammar makes. Not all computer scientists, however, treat quantifying phrases as Allen does. One of the most famous question-answering programs, Woods's LUNAR, for use in conjunction with a database of moon rock samples, extracts quantifying phrases from syntactic descriptions based on transformational grammar and preposes them to verb schemas (Woods, Kaplan and Nash-Webber, 1972; Woods, 1977). The schema for quantifiers in this system is (FOR (QUANT) X / (CLASS): (p X); (q X)),

from which a schema for a quantifying phrase may be obtained by substituting a quantifier (including the definite article and numerical adjectives) for '(QUANT)' and a count noun for '(CLASS)'. The Xs are link letters, while for p and for q an expression of category S(N) may be substituted, the former being a restrictive relative clause. Thus Woods assigns quantifying phrases to category S(S(N),S(N)). This works providing that every class name (count noun) is qualified by a restrictive 4

One may well ask what the token-name lnV means in this representation. The idea that an event could have a proper name, as 'el' in (6L), is at least plausible, but of what could 'nl' possibly be a proper name. A not-ing?

Semantic networks

109

relative clause, as in 'Every rock that contains silicon contains sodium', but is unable to handle simpler examples like 4A dog barked'. Quantifying phrases are then extracted from phrase-markers of transformational grammar by the following stages: 1 The main verb is turned into a PREDicate schema by a semantic translation rule which specifies the number and kinds of its operands. 2 Expressions in the phrase-markers corresponding to operands of the PRED which contain determiners are turned into quantifier schemas on the pattern above. 3 If a syntactic structure contains more than one quantifying expression, a decision is made about their relative scopes. This is effected by a procedure based upon the commonest orderings implicit in everyday English (see Hobbs and Scheiber, 1987). The quantifier schemas are then combined by substitution for q in the outermost. 4 Finally, the PRED is substituted for the q of the combined quantifier expression. The problem addressed in the third of these stages engages Allen, too, but at the stage of appealing to context in order to resolve ambiguous semantic representations rather than as a problem in deriving semantic from syntactic representations. Either way, it is a genuine and an important problem, but not one which need detain us here; each meaning of an ambiguous proposition should be represented by a distinct semantic structure, but which meaning we select in a given case may be a matter of the relationship between syntactic and semantic structures or be a matter of context. It is not, however, a question of the kinds of structure used to represent meaning. Shapiro (1971) developed a semantic network analogue of Woods's treatment of quantifiers. In Shapiro's system, quantifiers were represented by special nodes with some new types of edge to the verbs or other quantifiers which lie within their scope.5 Woods allows that it can handle quantification satisfactorily, but objects to it on the ground that it 'breaks up the chains of connections from node to node that one finds attractive in the more customary semantic network notations' (1975, p. 75). He gives the following example: (9)

5

Three look-outs saw two boats.

I have not cited Shapiro's own example, because it is confusing in many respects. He represents 'Every man is human', but describes it as a deduction rule, not as a proposition. Moreover, his network contains some unlabelled nodes whose significance is unclear. Shapiro later abandoned this notation, apparently under the influence of Schubert, and for unexplained reasons (see Shapiro, 1979, p. 189).

110

Computer science: graphs

The most simple-minded graph representation of this, he says, would be: saw

o lookouts

boats

From this, we move to a representation showing semantic roles of the form: lookouts O

agent ««

P Q

patient »» O boats

saw where 'p' labels a special proposition node. Finally, with extra nodes for the quantifiers, we have:6 lookouts QL

class 3 ^ O

prop 2 class ^ O »» O boats patient

Now this final network is extremely close to the graphs proposed at the end of section 2.5, and the directions of the edges can all be interpreted as scope indicators, with the exceptions of the two semantic-role