Contexts, hierarchies, and filters: A study of transformational systems as disambiguated languages 3484301287, 9783484301283

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Linguistische Arbeiten

128

Herausgegeben von Hans Altmann, Herbert E. Brekle, Hans Jürgen Heringer, Christian Rohrer, Heinz Vater und Otmar Werner

Uwe Κ. Η. Reichenbach

Contexts, Hierarchies, and Filters A Study of Transformational Systems as Disambiguated Languages

Max Niemeyer Verlag Tübingen 1983

CIP-Kurztitelaufnahme der Deutschen Bibliothek Reichenbach, Uwe Κ. H.: Contexts, hierarchies, and filters : a study of transformational systems as disambiguated languages / Uwe Κ. Η. Reichenbach. - Tübingen : Niemeyer, 1983. (Linguistische Arbeiten ; 128) NE: GT ISBN 3-484-30128-7

ISSN 0344-6727

© Max Niemeyer Verlag Tübingen 1983 Alle Rechte vorbehalten. Ohne ausdrückliche Genehmigung des Verlages ist es nicht gestattet, dieses Buch oder Teile daraus auf photomechanischem Wege zu vervielfältigen. Printed in Germany. Druck: Weihert-Druck GmbH, Darmstadt.

TABLE OF CONTENTS

0. Introduction

1

1. Disambiguated Languages

3

2. Montague's Definition of a Disambiguated Language

18

3. Phrase Structure and Context 4. Syntactic Operations and Transformations 5. Basic Expressions and Hierarchies 6. References

33 ....

64 87 108

0. Introduction

The present work grew out of a comparison of Montague Grammar with recent models of the transformational school and was strongly influenced by Bowers' Theory of Grammatical Relations. Its central part is a proof, given in chapter 4, that transformational systems consisting of a Phrase Structure Grammar, a lexicon, and a set of transformations, constitute disambiguated languages in Montague's sense. The proof is based on a reinterpretation of Phrase Structure Trees as representations of Fregean properties of expressions called 'hierarchies over expressions'. Hierarchies comprehend hierarchies over combinations of morphological operations as a special class. The latter allow for a redefinition of basic expressions as morphologically fully developed expressional constants of a language and at the same time provide the mechanism to derive basic expressions from unanalyzed root forms. Closely related to the concept of a hierarchy is the concept of a context which, in the form introduced here, allows to analyze expressions into equivalence classes with the help of filters. Filters, however, are not Chomskyan filters, but logical structures defined for Boolean Algebras, and have been known in the

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mathematical literature for quite some time. The three concepts of a filter, a context, and a hierarchy, in the form introduced here, allow finally for an extention of the transformational theory in a way briefly

illustrated

at the end of chapter 5 w i t h some fragmentary rules for German. There again, Bowers' Theory of Grammatical Relations served as a model. Eventual shortcomings, however, are my own responsibility and must be blamed on the theory presented here. A description of the difference between Bowers' and the Standard Theory in formal terms is included in chapter 4.

1. Disambiguated

languages

One of the most outstanding characteristics of a natural language as compared to a formal language is the asymmetric relationship between form, function, and meaning of its expressions, w h i c h is generally perceived as ambiguity. In natural languages ambiguities can be produced and eliminated systematically; they are therefore an inevitable problem for every linguistic analysis. Naturally a linguist will be more concerned w i t h the elimination of ambiguities than w i t h their production. But while in communication they can be eliminated by direct and immediate reference to some real or suggested context of use, analysis can take advantage of such aids only indirectly by description. To describe natural languages then means to a large extent to eliminate ambiguities.

In theory, one may try to eliminate ambiguities by introducing for every ambiguous expression of a language as many artificially designed and unambiguous substitutes or replacements as there are readings it allows for. If we knew for any such replacement exactly which of the possible readings of an expression it is to represent, then a list consisting of an expression followed by all of its newly

introduced

substi tutes could be regarded as a partial description of

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that expression. If we demanded furthermore that every unambiguous expression - provided a language has unambiguous expressions - be a substitute for itself, then a list consisting of all the substitutes of all the expressions of a language together with a relation specifying for every substitute which expression it can possibly replace could be regarded as a partial, although highly redundant, synchronic description of that language. In order to qualify for a description of a language, however, such a system must satisfy a number of additional requirements. Let me use the attribute 'grammatical* to refer to both syntactic and semantic phenomena. Then above all substitute expressions must be constructed in such a way that all the grammatical relations holding between expressions are being preserved, including of course the relation of being composed of. This means that if expression χ in one of its readings bears the grammatical relation R to expression

in one of

the readings of the latter, then there must be exactly one substitute for χ bearing R to exactly one substitute for Of course we know that the number of expressions of a language is potentially infinite, hence a list of the proposed kind could hardly be completed. We know, however, also that grammatical relations holding between expressions can be described in terms of grammatical rules by which we construct complex expressions from less complex ones. Thus if we were

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to design a set of rules which would in principle enable us to construct any of the above substitute expressions from a stock of basic expressions of a language L, then these rules together with all the expressions they operate on would constitute an artificial but autonomous grammatical system, a disambiguated counterpart of L or, as we might simply say, a disambiguated language. The idea underlying an attempt to design disambiguated languages is that any language can be completely described by relating it to a disambiguated counterpart. Other than formal languages in the traditional sense of the word, however, disambiguated languages are usually constructed in close analogy to human languages, hence are mostly the product of an empirical investigation.

Naturally the question as to how one should go about constructing a disambiguated language allows for a variety of answers. Since we know, however, that in some wider sense of language every language has means to eliminate ambiguities all by itself, any answer seems to depend crucially on a prior determination of the various factors by which ambiguities are caused. One may be tempted to postulate two major sources of ambiguities, one of them - and probably the most frequent one - being insufficient information about the context in which an expression occurs. Among the ambi-

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guities arising from this source one would very likely include those that have been called 'lexical ambiguities' as special cases; obviously lexical ambiguities exist as long as an expression is mentioned completely out of context or, to put it differently, w h e n no information about a context is given. The other source would then be uncertainty about the function of an expression within a certain context. Ambiguities arising from the latter source have been called syntactic ambiguities at times. Of course if context does influence meaning, as suggested in the characterization of what I tentatively called 'first source of ambiguities', then both sources can no longer be distinguished clearly. It can be verified that, like ambiguities of the former kind, ambiguities of the latter kind can be resolved by providing more information about the context in question. If this is so, then it should be generally possible to eliminate ambiguities by relating an η-way ambiguous expression to η unambiguous counterparts consisting of an arbitrary but fixed representation of the expression itself followed by an exact description of a particular

context

in w h i c h it may occur. This presupposes, of course, that we first agree on a suitable interpretation of the w o r d

'context'.

In general contexts of expressions are determined by a variety of factors, among them facial expressions, hand and

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body movement, all sorts of psychological and sociological conditions, presuppositions of speakers and hearers and so on. To try to account for all these factors in the way described would not only appear to be extremely uneconomical, but it would also make it necessary that we have already a descriptive language at our disposal which is completely free of ambiguities. We have none of the kind, let alone one that could account for all the psychological factors involved. This may be one reason why linguists have generally restricted their attention to what I shall call 'grammatical contexts'. These are contexts consisting exclusively of strings of expressions and some of a limited number of well defined symbols together with a specification of the syntactic or semantic relations holding between them. Such contexts, or at least a subclass of them, were usually represented by structural descriptions of expressions of a kind similar or identical to the tree structures introduced in the transformational literature. Where the non-pragmatic information provided by these structural descriptions was considered insufficient, additional pragmatic information sometimes entered the description schematically in the form of reference points. These are sequences of symbols representing respectively those pragmatic parameters that were considered relevant for a unique determination of an extralinguistic context of the kind mentioned. Reference points could then

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be attached to structural descriptions as indices.

It might seem odd to think of structural descriptions in terms of grammatical contexts. We are used to think of them as analyses of expressions, i.e. systematic representations of the parallel processes of an expression's being decomposed into or constructed from its elementary components. Sometimes rule markers, attached as labels to the nodes of a tree, would provide additional information about the derivational history of the expression analyzed which, depending on the approach we are working in, would occur in some suitable representation either on top or at the bottom of a tree. From this point of view information about a grammatical context seems to be missing completely. That such a view is rather narrow becomes obvious if we shift our attention away from the expression analyzed and to those expressions that occur as components in its structural description. Then the whole configuration surrounding each individual component expression can be regarded as its respective grammatical context. In transformational grammar configurations of this kind or grammatical contexts, as I called them, have been used systematically to trigger both lexical insertion and transformations.

Now if we think of contexts in a more general way as sets

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of configurations of some kind, then we can certainly speak of empty contexts where in the usual sense of the word no context at all is being given. Technically speaking empty contexts are also contexts. Thus we might be tempted to speak of empty contexts when we actually refer to contexts surrounding those expressions that we have called 'expressions analyzed' with respect to certain structural descriptions. But notice that if in fact we do so, then we exclude all those configurations from the context of an expression that contribute to its own analysis. Translated into a transformational approach this would mean that nothing dominated by a particular node in a tree would belong to the grammatical context of the complete expression generated under that node. Thus a determiner or a noun would not be part of the context of the nounphrase of which they are constituents. Rather the grammatical context of a nounphrase would be the complete tree configuration that was left, after the NP, under which the nounphrase in question had been generated, had been cut out. Although such an interpretation of 'context' seems to confirm our intuitive understanding of the word, it may nevertheless be inadequate for our purposes. Information such as from which constituents an expression has been constructed, by what rules, and in which order the rules were applied, may be vital if we want to eliminate ambiguities, and in the description of a grammatical context must complement the infor-

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mation about the expressions own role as a constituent expression. Whether all of this information can always be supplied by structural descriptions of the transformational kind is open. Still it is essential for a complete appreciation of every grammatical context.

Here, by the way, a well known defect of some traditional descriptive theories, including those that adhere to the popular semantic principle that the meaning of an expression is a function of the meanings of its parts, becomes apparent. Certainly the validity of the latter principle is being restricted by the other principle according to which the meaning of an expression is to some extent also being determined by the function of that expression within a certain context. But while theories in which the analysis ends at the level of a sentence might be able to adequately account for both principles as far as sentence constituents are concerned, they will never be able to do so for sentences themselves. Thus sentences, as the output of structural descriptions, will never be analyzed in the context of other sentences. Therefore, unless we are dealing with dependent clauses, the semantic analysis of sentences will necessarily remain incomplete. I shall assume here, however, that reference points, even though they were introduced as purely pragmatic components, can compensate this defect up to a certain point.

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Notice that a text analysis, for example, is faced with the same dilemma at a higher level, which is one of the reasons why so called 'work immanent' text interpretations have been severely criticized in recent decades.

An interpretation of structural descriptions as specifications of grammatical contexts makes it easy to see that the grammatical relations an expression bears to other expressions of a language are completely determined by the set of all structural descriptions in which it can occur, provided our structural descriptions are complete. Here 'complete' means that they unambiguously represent the respective grammatical contexts in which possible ambiguities of an expression can be resolved. Thus, in a manner of speaking, we disambiguate grammatical contexts rather than expressions by designing adequate structural descriptions. The expressions themselves retain their principal ambiguity and they gradually regain it as bits of the configuration surrounding them in a structural description are step by step removed. A list consisting of an expression followed by all the structural descriptions in which it can occur could therefore be regarded not only as a partial description of the kind mentioned earlier, but as a complete grammatical description of the expression in question. In practice one would of course try to simplify such descriptions by ana-

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lyzing them into equivalence classes or types. Such a type could possibly be defined recursively and could be thought of as representing schematically all those contexts in which a particular expression or a class of expressions retain one and the same function or one and the same interpretation. Subcategorization conditions in Transformational Grammar are examples of context types. Their construction has reached a preliminary peak in sophistication in Bowers 1975, where in addition to specifying a static contextual configuration of some kind, a subcategorization condition also includes encoded information about the rules that can create that particular configuration and the order in which they would apply. The order of the rules in question can then be imposed as a condition on the acceptability of a derivation. For example if a certain contextual configuration can not be constructed from a given configuration by applying certain rules in the order predicted by the subcategorization condition of a lexical item, say a verb, then the derivation blocks, meaning that the item in question can not be inserted or does not fit into the given context. Of course there are other ways to design context types; the concept of a structural description as introduced here is as of yet not committed to a transformationalist interpretation. Since moreover the attribute 'grammatical' in 'gram-

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matical context' was supposed to refer to syntactic and semantic phenomena alike, it may well turn out that the structural descriptions we need here cannot be found in at least that part of the transformational literature that is still committed to the Chomskyan concept of deep structure, the more so since, to my knowledge, an adequate theory of semantic representation within the transformational framework does not exist.

If an expression can indeed be disambiguated by assigning it an unambiguous grammatical context in form of a structural description, and if moreover every expression is indeed completely determined once all the grammatical contexts in which it can occur are known, then every attempt to design adequate structural descriptions for the expressions of a language, however incomplete they may turn out to be initially, must ultimately result - or is at least intended to result in the construction of a disambiguated language. What else should encourage such an attempt if not the belief that grammatical differences between expressions can be explained and ambiguities resolved by means of description? As linguists we may be aware of the fact that the actual construction of a disambiguated counterpart for any language could never be completed. Yet trying to approximate such a hypothetical construct will almost certainly help us gain some insights

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into the mechanisms underlying every communicative process. From this point of view it is truely hard to see why many linguists still regard Montague's claim that there is no important theoretical difference between natural languages and formal languages as sheer blasphemy, while at the same time they have been discussing purely formal languages all along. Certainly languages whose expressions are structural descriptions of natural language expressions are formal languages in the sense that the relationship between form, function, and meaning of structural descriptions is at least intended symmetric. And have such languages not been constructed in order to be able to interprete natural language expressions indirectly via an interpretation of their structural descriptions?

To say that there are no important theoretical differences between natural languages and formal languages, however, is not the same as saying that there are no differences at all. It has already been pointed out initially that one difference between formal languages, as used, for example, in mathematics, and those languages, that will serve as disambiguated counterparts of natural languages, lies precisely in the fact that the latter are mainly the product of empirical investigation. From a theoretical point of view it may nonetheless be of advantage to start a linguistic investigation by considering

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and formulating in exact terms certain minimal requirements a language as a purely theoretical construct must meet in order to qualify for a disambiguated language. In a second step we can then start to construct a disambiguated language not only on the basis of the evidence provided by a thorough inspection of the natural language under investigation, but also in accordance with the theoretical requirements formulated earlier. This is the way Montague proceeded, and in this respect his approach is clearly different from those of the structuralist and transformational schools, for which disambiguated languages, although - or may be because - the concept was not yet known, seemed to be exclusively the theoretical fall out of an empirical investigation of human languages. The definition of a disambiguated language plays therefore a central role in Montague's theory. That it is moreover the central part of his theory of syntax should not surprise us, for if a language is free of ambiguities, then semantic phenomena should be completely reflected in the syntactic structure. Thus the mere assumption of the existence of a disambiguated language for every language seems to lead automatically to postulating a one to one correspondence between syntax and semantics, i.e. in particular to the requirement that the syntactic and the semantic derivation of any particular expression run entirely parallel.

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That the definition of a disambiguated language must not be mistaken for a disambiguated language itself goes without saying. Rather it is part of a linguistic theory of a kind which, in Chomsky's terms, should meet the level of explanatory adequacy. Of course, explanatory adequacy is in itself a rather vague concept, largely because its definition rests, like the definition of other concepts introduced by Chomsky, on the prior definition of the rather individualistic concept: competence of a speaker hearer. Assuming that Montague's definition of a disambiguated language is explanatory adequate would entail that every grammar meeting the requirements set up in that definition is also descriptively adequate, simply because we find nothing among these requirements that could help Us discriminate between good and bad disambiguated languages. This would be a surprising result and, unless we assume that there is one and only one disambiguated language for every language, highly unlikely. Is this then an argument against Montague? I don't think so as long as there is no satisfactory answer to one crucial question: How can a general agreement on descriptive adequacy be reached when our understanding of it too seems to rest to a large extent on our understanding of such concepts as: psychologically correct or: correctly describing the intrinsic competence of an idealized speaker hearer? Since current psychological theories seem unable to offer anything less subjective than our privat*

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understanding of the phrase 'psychologically correct', any attempt to answer that question leads inevitably into the realm of speculation.

A word must be said about the usage of the term 'language' in this context. It may have become obvious already that the concept of a disambiguated language comprehends the traditional concept of a grammar. Thus descriptive adequacy was by no means attributed to languages in the conventional sense, which would of course be utterly meaningless. Montague himself has repeatedly tried to construct a grammar (a disambiguated language that is) for a fragment of English. From a linguist' s point of view these attempts must be called failures. That this is so can, however, not be blamed on the quality of his theory, but must rather be blamed on the fact that he chose to ignore the empirical work that had already been carried out by linguists of various schools and the insights this work can offer to a theoretician. For the following discussion Montague's practical work will be of marginal interest only; instead of losing any more words on it, let us therefore go right back to the core of his syntactic theory and take a closer look at his definition of a disambiguated language as introduced in Montague 1970,1.

2. Montague's definition of a disambiguated language

As can be expected, the basic stock of a disambiguated language in Montague's definition is a set of basic expressions of a language and a set of syntactic operations to construct complex expressions from less complex ones. Right from the start Montague appeals to a more intuitive understanding of the phrase 'basic expressions' as referring to a minimal syntactic unit which is in some conventional sense meaningful. Although the obvious lack of concern with the exemplification of one of his basic concepts may be intolerable from a linguist's point of view, it is still justifiable linguistically. As the input of the syntactic component of a grammar, so one could argue, basic expressions are of course not the result of the application of any of the syntactic operations (see p. 30, Def 1: 4). Rather they are the output of a theory of morphology or even a theory of phonology, if it should indeed turn out that morphological processes can be adequately accounted for within a phonological framework. Thus, one would conclude, the definition of the concept of a basic expression cannot be part of a theory of syntax. In Transformational Grammar the equivalent of Montague's set of basic expressions is the lexicon.

Following a logical tradition, Montague generally rep-

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resents expressions by strings of letters of the English alphabet, enclosed in singular quotation marks. If such a string includes unidentified parts, i.e. names of variables of the m e t a language for expressions of the object language, then the singular quotes are being replaced by corners or quasi quotes. For example if 'walks' is an expression of the object language and χ is a variable for any such expression, then the result of writing

'walks' followed by χ will

be represented by 'walks x 1 rather than 'walks x'. Montague's theoretical work, however, is not at all committed to representations of this kind and in practice we are free to represent both expressions and basic expressions any way we like, provided we do so in a consistent and unambiguous way. The only thing relevant at this point of the discussion is that an analysis of the set of basic expressions

into

mutually exclusive equivalence classes, called 'basic categories', is assumed to have been carried out somehow. In Transformational Grammar such an analysis has been accomplished w i t h the help of categorial features

[+N] , [+V] etc. , which

have been attached to lexical items as labels, thus marking them for class membership. Montague, in order to avoid a committment to the needs of any particular language on the fairly abstract level of the definition of a disambiguated language, refers to basic categories in a rather vage and unspecific way by using symbols

'X ' , where i

represents an

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index chosen from a set

of natural numbers. The use of

natural numbers to distinguish different basic categories is of course by no means necessary. As soon as we turn to the description of particular languages, we may well choose categorial symbols similar to the non terminal or pre terminal symbols of a Phrase Structure Grammar to represent the elements of

In this case it might even turn out that

basic categories can be unambiguously referred to by using indices alene. Here, on the other hand, they can not, because the same set

will be used to index both basic categories

X^ and derived categories, called 'C^1, of those expressions, that are to be constructed from basic expressions by a systematic application of the syntactic operations. In this way we will be able to express that, whatever value i might take, X^ will always be a subset of Cj. Translated into a transformational terminology this means that we will be in a position to say that while all nouns are nounphrases and all verbs are verbphrases, the opposite is not necessarily true. Here sentences will get a special treatment of course. Since all sentences are output of at least one syntactic operation, there are no basic ones. Hence there is no basic category of sentences or, to put it differently, the basic category of sentences is the empty set. The category of sentences will therefore get the special index i®. Naturally X^O is also a subset of C^q·

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Before we turn to the role of syntactic operations within the context of a disambiguated language, some explanatoryremarks on syntactic operations in general might be helpful. An η-place syntactic operation assigns expressions to sequences of η constituent expressions. If f is any operation and is a sequence of expressions, then by 'f(x , ...,x ) = y 1 we express that y_ is the result of applying f to . We call

'the output expression'

or 'the output of f for the arguments

1

. The com-

plete string is called 'input string' or 'input of f 1 . Input strings are ordered sequences, which means that a different arangement of the arguments leads to different results or sometimes to no result at all. An operation can assign only one result to every input string. It can, however, assign the same result to different input strings. If, on the other hand, we were to get two or more different results by applying f to at different times, then f could no longer be called an operation. Rather it would be called a relation in this case. The set of all input strings of an operation f or the set of all strings for which f yields a result is called 'the domain of f', abbreviated *D[f]'. R[f] or the range of f is the set of all of its output expressions. In general the domain of an η-place operation is being given as an η-place cartesian product n, C. = — k » where jeJ, η is the number of places of the operation fj, i^el for all k be a — u' k k is a subtree of tu u' k of degree η relative to t , then t„ is of u v " k degree n+1 relative to t u ·

Let me use the symbol 't^1 to refer to any subtree of t

of

degree n. Notice that tJJ has actually the status of a variable whose range is being determined by the subscript 'u' and the superscript 'n' together. Now if t" u = M V > ) . Let

generated from C < N

N>.

M

is < V P ) V P >

We call

'a filter in C < y p γρ>^_· Then all the trees in

y p > are also contexts of

In general, if IF and IF'

are filters generated from A and Β respectively, and have common elements, then their intersection is a filter generated from A and B. Thus M generated from C < N M

< v p

vp>

N>

< v p

vp>

is a filter in C < y p y p >

and C < v > v > . Moreover, C < V P ) V P > -

= {} which, according to the first

reduction principle, is the same as {}. But notice that there are different filters M < g

g > t generated from

in C < g g > , and all of them are also filters generated from C

< V V>"

orc

*er

to

identify each of them correctly, it is

apparently necessary to know which

they are generated

from. Her the advantage of subcategorization conditions shows drastically. For G the filters in question can be unambiguously identified by the two indices , and . By the first constituent S of every trace they are marked for being subsets of C < g g > . The second constituent informs us that all contexys in question are contexts of a segment with upper rim NP. Since a trace represents always the shortest path from one node

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of a tree to another one, the middle constituent VP in the second trace indicates that the segment in question must be a segment of every subtree typ of the trees generated by G.

Here the question arises as to whether subcategorization conditions need a second constituent at all. It seems that the two filters C < < g > N p > ^ < N p > > and C < < s > v p > N p > ) < N p > > of G are uniquely determined by the first constituents and of their indices alone. While this may be true for G, however, it is no longer true for a Phrase Structure Grammar G 1 including the scheme VP

»V

(NP) (NP). Here the indices of two different filters in C

s

^are

same

first constituent , and

hence can only be distinguished by their second constituents and which represent different segments. The inner brackets in the latter sequence indicate that the trace is to intersect with the second occurrence of NP in . A bracketing of this kind is unproblematic on the understanding that a sequence of segments is also a segment. Such a requirement was avoided in the definition of a segment, basically because a Phrase Structure Grammar does not have to include schemes of the form VP — } V (NP) (NP) in order to gain the gener-

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ative power of those that do. Notice incidentally that by distinguishing different functions of an NP under a VP in the way described we have not made any claims as to which NP is to represent the direct- and which the indirect object. In fact a claim to this effect would make a grammar overly restrictive; as long as we secure that, if either NP dominates a direct object, the other one can not, we have done enough.

Let me at this point introduce a class of particular segments, whose constituents are subtrees of degree η relative to a given tree:

1. A segment is of degree η relative to a context c < s

r>,

if and only if all its

constituents are subtrees of c < s

r>

of

degree n. 2. A segment of degree η is complete, if and only if it includes all the subtrees of c < s

r>

of degree n.

Apparently the upper rim of a complete segment of degree η consists of the end points of all traces of length n+1 of a context c

' 3 »I *

. If C

* S » 1 '

is a rigid filter generated

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from C < u

u>,

and ceC < u

u>,

then there is for every complete

segment sm (of degree n) of c exactly one set of n+1-place traces which all the contexts of c in C, have in common.

' and whose end points define the upper rim r of sm. If sm is of degree m in C < s

r>,

then the upper rims of all com-

plete segments of degree m of contexts of c in C < s

r>

form

a filter of substrings generated from r. This leads to the conclusion that, if c is a context of c', then the function of c' in any of its contexts in C < s Moreover, all the traces in C < g

r>

r>

remains the same.

leading to any of the c'

in c have the same initial segment ending in u. Thus the subcategorization condition for c' in C < u characterizes the function of c' in C < g

u>

r>.

unambiguously If moreover c 1

is dominated by u 1 , and all the traces leading to c' in C

are identical, then the first constituent of the

subcategorization condition for c' can be unambiguously represented by u'. This explains, why in one of our previous examples could be reduced to . There we were discussing trees of a filter in C

generated from C < v p γ Ρ > , and presupposing that all

contexts of in C < g g > include as the only trace leading to V.

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That relative to the same class C < u

u>

different oc-

currences of a tree in one of the contexts from C < u

u>

be

subcategorized differently is implied in the preceding discussion. In general it is sufficient, however, to choose the smallest segment in which a tree differs from all the others to be the first constituent of a subcategorization condition.

It follows from condition 4. of the definition of a Phrase Structure Grammar that Phrase Structure Grammars generate only contexts of type . That the latter play indeed a special role among the trees discussed so far shall be reflected here by what I shall call 'the second reduction principle1 which involves mainly trees dominated by S. Let t , t u , be trees dominated by u and irrespectively. Then, in accordance with the earlier charaterization of contexts, t u -t u , = t u +(-t u ,) = l.u.b.(tw,-tu,). Thus if t u is a context of t ,, then t u "t u i is the complement of t u , in t , i.e. what is left of t after t . has been deleted, u u u' Furthermore, if a, b, c are objects of any kind and b occurs in a, then a^ is the result of replacing b in a by £. The second reduction principle can now be formulated as follows:

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1. t c is in t ; Ο u 2. t , occurs in both t c and t -t P .

Here both 'is in t ' and 'occurs in t 1 stand for 't is u u u a context of'; henceforth all three formulations shall be used interchangeably. The word 'occurs' in 2. expresses furthermore that the subcategorization condition of t , in tg is different from the subcategorization condition of tu , in tU -to, Ο in other words that we are dealing with two different occurrences of t . in t . Again the second u u reduction principle can be supplemented by the second strong reduction principle, according to which

tu W

t*S

, iff

S u' 1. and 2. as above.

The assumption that the second reduction principle is a universal principle is of course as unfounded as the assumption that the concept of a Phrase Structure Grammar is usefull for the description of any language. Together with the first reduction principle, it will, however, take care of such operations as Equi and Raising which can be found typically in transformational descriptions of English

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and German. But it will also help explain 1. the equivalence of constructions such as 'Mary can read and Marycan write' with 'Mary can read and write' and, possibly, 2. the blank representing the head in a relative clause such as 'the man, who 0 shot Billy the Kid' or 'the man I met 0 yesterday', provided we accept the somewhat problematic view that relative pronouns combine with complete sentences before deletion takes place.

That processes such as Equi and Raising are in fact covered by a single principle has also been argued in Bowers' Theory of Grammatical Relations. There, as well as in the transformational literature in general, operations manipulating trees were introduced as transformations. This raises the question as to whether transformations in general, and the two reduction principles introduced above in particular, are syntactic operations in the Montaguean sense. Recall that Montague's operations of a disambiguated language manipulate expressions. Trees, however, qualify for expressions under one condition only, namely the condition that all the symbols Δ are replaced by complex symbols or any suitable equivalent. For a standard transformational model, then, in which transformations apply only after lexical insertion has been completed, the question can

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answered in the affirmative. But then lexical insertion assumes a rather peculiar status. As an operation manipulating trees, it is clearly a transformation. As an operation manipulating trees dominating unreplaced terminal symbols Δ, however, it is not a syntactic operation in the same sense a transformation is. Should we then conclude that there are two different kinds of transformations? Surely not, as Bowers' Theory of Grammatical Relations proves. Bowers' argumentation against the Equi-Raising distinction leads eventually to a model in which lexical insertion and transformations apply in random order. Incidentally it is just this property of the theory he proposes which, in his opinion, renders the concept of Deep Structure, in its common interpretation as a fixed level of grammatical representation, meaningles. In Bowers' theory, then, lexical insertion and transformations are no longer distinct; but neither are they syntactic operations in the Montaguean sense. Notice that the two reduction principles introduced here are transformations in Bowers' sense. For this reason, and because I consider Bowers' approach more organic - or less ad hoc - than the standard theory, I shall assume that his interpretation of transformations is the correct one. But what are syntactic operations then, and what is their role within the system

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proposed here? The answer to this question will finally enable us to prove the initial assumption that transformational systems are indeed disambiguated languages.

It has been pointed ou that terminal symbols Δ are variables marking the place in a tree where lexical items of a certain category have to be inserted. This suggests that we take trees to be expressions including free variables, and hence representations of the syntactic functions we are looking for. Clearly their domain is a class of basic expressions of various categories, and - if the interpretation of expressions as trees with no free variables Δ is correct - their range is in every case a class of expressions. Before going into detail, let me repeat both the definition of a Phrase Structure Grammar with all the changes made and the characterization of a tree.

Def 3': A Phrase Structure Grammar is a system Such that 1. V^ is a finite set and ueV^; 2. Σ c V^ is a set of basic categories; 3. Ρ is a set of productions such that a. ueV^ and veV^*; b. all productions of the same type are neighbors;

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c. ueE

> ν = Δ;

4. SeV^ is the category of sentences.

The definition suggests that the elements of V^ are actually categories of expressions. If so, their members can not be expressions of a disambiguated language, however, but must rather be expressions - in some suitable

represen-

tation - of the language under investigation. Thus 3.c. could be rewritten as follows:

3. c. uel

> veu;

If G is a Phrase Structure Grammar, then t , or a tree of type u (i.e. dominated by u) , is defined as follows:

1. ueV T — » L· 2. u e V L - Z , all k p a s t ,

gegangen p e r £ ;

woman N

, women l u r a l ; r common sleep v , s l e p t 3 > s i n g > p a s t , slept p e r f ;

Now expressions have been described in the model theoretic literature as referring to functions whose values, for some reference point, are again functions - mostly properties in the Fregean sense - or relations. If this is so then it should be possible to combine basic expressions to form expressions in the same way syntactic properties were combined to form basic expressions. Hierarchies would then occur as constituents of other hierarchies which would allow for an immediate generalization of the relation of being over. By demanding that a. every hierarchy is a

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hierarchy over itself and b. whenever F is a hierarchy over G, and G a hierarchy over H , then F is a hierarchy over H, being over turns into a partial ordering: It is reflexive, transitive, and antisymmetric. Moreover if Η is a hierarchy over I, then Η is a context of I and vice versa. For a proof, consider that every hierarchy can be represented as an ordered pair , where u is a name such as Det, N, etc. and ν is its sequence

representa-

tion. Then the equivalence follows from the definition of being over and the transitivity. The trees of a Phrase Structure Grammar are thus hierarchies which finally closes the gap between expressions and basic expressions. Recall that expressions were being defined as trees in which all free variables had been replaced by basic expressions. Both expressions and basic expressions occur thus as the values of a successive application of hierarhies.

Recall now that the properties

(in the Fregean sense)

whose extensions are classes of expressions did not occur as referents of tree configurations, but as referents of predicates constructed from trees with the help of a λ-operator. As in the case of basic expressions, the λ-notation should be avoided, however, as it imposes an

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unnecessary order on the replacement of variables. As the discussion of subcategorization conditions has shown, trees are free of ambiguities and we should be free to decide from case to case if the necessity of ordering indeed arises. From now on trees shall be predicates all by themselves, referring to Fregean properties of expressions or hierarchies as we have seen.

During the discussion of predicates of this kind we focused in on those those we had called FQ

u>

which in-

cluded only free variables and were the first in a > of predicates, and expressions. u j J The extension of H T . is the class of all ·*· > J hierarchies H' for which there is a filter IF such that l.u.b.(H, .,H')eF; ·*·»J Just as the < F j > u > j < m > u e V L did, every family ; j > j < m defines an η-place syntactic operations from expressions into expressions.

Let me now get back to the hierarchies over basic expressions and introduce a number of subordinated hierarchies that play independent roles in a grammar of German and therefore deserve to be given names.

SC

MOD;

=

AUX;

trans V-intr

=

TRANS;

=

INTR;

Det

indef

V

=

PLACE 0

Det

def

=

INF;

^mod V

aux

V

place°

V

inf

N

pro

^common ^proper

s

PRO;

=

CN;

s

PN;

=

EIN;

s

DEF;

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Conj s u b Con

Con

- SUB;

3coord "

jrel "

REL;

C0;

Let furthermore CASE be a variable taking values in {NOM,GEN,DAT,ACC}. Then the following hierarchies are also hierarchies of German.

S

=