Ordering in syntactic representation

Each element in a phrase-marker stands in a ‘privileged’ relation to no more than one other element. This privileged relationship is represented as co-membership in a binary set, and may be asymmetric or symmetric.⁵ An asymmetric privileged relationship is represented by an ordered set containing two members; a sym- metric privileged relationship is represented by an unordered set containing two members.

A phrase-marker is represented as a set of sets. Some of the sets are ordered and some are not. I will say that the ordered set <α, β> has a first member and a second member, while the unordered set {γ, δ} has no first or second member. The terms ‘first’ and ‘second’ will apply only to members of ordered sets. To give examples of both types, the relation between a function and its argument is an asymmetric relation, and so is represented as ordered, while the relationship between members of an adjunction cluster is symmetric, and so is represented as unordered. This means there are no first or second members in adjunction clusters, a point that will become central to the discussion of multiple wh-questions below.

In the semantic tradition that originated with Frege (1892), it has been customary to distinguish three things: linguistic expressions, what they mean, and what they refer to. Also since Frege, it has been customary to consider the function/argument distinction as central. There are function-names, function- senses, and functions, and there are argument-names, argument-senses, and arguments.⁶ I follow this tradition in assuming that the only asymmetric privileged relationship is that between function and argument. The relationship between a function and an argument will be represented as <F , a>, where ‘F’ is a function-name, and ‘a’ is an argument-name. This relation is asymmetric; all ordered sets represent the privileged asymmetrical relationship between a function and its argument in a phrase-marker. It should be understood that ‘a’ need not always be the name of an item: ‘a’ may represent the scope of the quantifier ‘F’. In that case, ‘F’ would represent a second-level concept and ‘a’ would represent not an object, but a first-level concept. If ‘F’ represents the universal quantifier, and everything falls under the first-level concept ‘a’, the result is truth. Ordering indicates asymmetry in this case, just as it does in the case of a verb and its object.

⁵ The assumption that structural relations are fundamentally binary has deep roots, even within the study of natural language. See Yngve 1960 and Kayne 1984. I assume binary branching for convenience only. Ternary branching could easily be accommodated here, as it could be in other systems.

Ordering in Representation 121 5 . 2 . 1 ‘ Ho r i z o n t a l’ o rd e r i n g , ‘ve r t i c a l’ o rd e r i n g ,

a n d t h e c o n t a i n i n g n o d e

In traditional representations, there is both left-right, or ‘horizontal’ ordering, and ‘vertical’ ordering, which is usually called ‘containment’. In the representa- tions <α, β> and {γ, δ}, what is the containing node? Of what categories are they? In traditional tree-structure notation, all nodes are labeled; in traditional bracketing notation, all brackets are labeled, overtly expressing the identity of the containing node. Sometimes the representation of containment is redun- dant; in the representation of phrasal structure, for example, it is redundant to represent both the head and the containing phrasal node.⁷ Since there is no reason to believe that there is anything in phrase-markers that this redundancy represents, the distinct representation of containment must, at least in this case, be eliminated. Set theoretic notation does this; in a notation that uses ordered and unordered sets, there is no need to distinguish two directions of ordering, and no need to represent containing nodes. For an ordered set, the first member of the set is the containing node, while for an unordered set, the containing node is the conjunction of the elements of the sets. In <α, β>, α is the con- taining node, while in{γ, δ}, ‘γ and δ’ is the containing node. In {β, {γ, δ}}, the containing node is ‘β and γ and δ’. In <<α, β>, γ>, the containing node is <α, β>, which is of category α. Recalling that unordered sets do not have ‘first’ or ‘second’ members, these two proposals may be united under the following generalization:

The containing node of a set is the conjunction of all of its elements, minus its second member.

This proposal is incomplete; I will refine it further. Here the point is only that, given the suggested notation, there is no need to represent containing nodes separately: their identities are transparent. Set notation simultaneously indicates both the identities of the members of a set, and the identity of the containing node, without redundancy.

In most traditional theories of syntactic structure, phrasal structure is taken as basic, while adjunction structures arise through the application of an adjunction operation. In the account I advocate, phrasal structure and adjunction structure are on a par. Concerning the identities of containing nodes, it would be desirable that, if bothγ and δ can dominate α and β (whether α and β are horizontally ordered with respect to each other or not), then γ = δ. In other words, it would be desirable that containing nodes be determinable as a function of the contained nodes, both in the phrasal and adjunction cases. The present suggestion concerning the identity of containing nodes achieves this.

But in empirical research pursued using a more standard notational framework, there is the risk that this ideal cannot be met. Suppose it is allowed, in such a framework, both that N may adjoin to V, yielding a constituent of category V, and that V may adjoin to N, yielding a constituent of category N. If the containing node is to be a function from the contained nodes (solely) to the containing one, the ideal must be dropped, at least for structures ‘created’ by adjunction. Furthermore, the proposal that containment is a function from contained nodes to containing ones runs afoul of the desideratum that the well-formedness conditions on phrase-markers be neutral as between phrasal-structure representations and adjunction-structure representations. In the notation suggested here, whether N ‘adjoins’ to V or V ‘adjoins’ to N, we have the unordered notation{V, N} and ‘V and N’ would be the containing node. The prediction is that the structure that results when V adjoins to N and the structure that results when N adjoins to V are of the same category: ‘V and N’. If this is correct, phrase-markers are simpler than previous representations of them have been claiming.⁸

I have emphasized that the ordered (or unordered) constituents in phrase- marker representations indicate only syntactic asymmetry (or symmetry), and are not to be thought of in temporal terms. But there is a relationship between the ordering in sentence-types and temporal or spatial ordering in tokens of these types, which can be expressed as pronunciation rules. One such rule is that all ordered expression-types in a language are pronounced in the same direction. A language may be such that asymmetrically related pairs of expression- types, represented as <F, a>, are pronounced ‘F, a’, or it may be such that they are pronounced ‘a, F’. It is well known that in some languages, such as English, the head temporally precedes its argument, whereas in other languages, such as Japanese, the head temporally follows its argument. These rules have exceptions, and there are many complexities that I am ignoring, but there do appear to be generalizations of this sort applying in individual languages.⁹ Crucially, I express these generalizations as pronunciation rules, not rules of syntax.

This view has implications for parametric theories of language acquisition. It might be thought that if, in English, constituent-types of the form <F, a> are pronounced ‘F, a’, and if, in Japanese, constituent-types of the form <F, a> are pronounced ‘a, F’, then English speakers have nothing to learn concerning ordering, while Japanese speakers have to learn that the ordering of pronunciation

⁸ I assume, merely for purposes of illustration, that there is a syntactic distinction between V and N. That is usually assumed, but it might be questioned. Plainly, empirical questions arise here. One empirical question, which this proposal would force into the open, is what the categories of syntax actually are. Categorial status is conferred by syntactic argumentation, of which this issue is a part. It is not conferred by naming constituents for their use. That is appropriate when naming kitchen utensils, say, but not here. That said, can it be maintained in general that the result of adjoiningα toβ and the result of adjoining β to α yield the same category? What categorial theory would that assumption give rise to?

Ordering in Representation 123 is the reverse of the ordering of the expression-type. But that way of viewing this matter would be confused. The notation <F, a> has no temporal interpretation; it serves only to indicate an asymmetrical relationship between a function and its argument. The correct way to state the matter is the following. There are two empirical possibilities. Either (i) Speakers of both English and Japanese must learn the temporal direction in which constituents of the form <F, a> are pronounced, or (ii) One set of speakers must learn that temporal ordering and the other need not. Suppose the first is true. Then, as far as learning is concerned, the asymmetry of function and argument is not partial as to direction of pronunciation. All speakers know the asymmetry, but must learn the direction of pronunciation, which varies from language to language. If the second is true, then one temporal ordering of asymmetry is unmarked. That ordering would be the ‘first guess’ on the part of the learner. Then the question would be why that should be so; why, in particular, the chosen ordering has priority of place in learning. Of course these matters have been studied empirically, but I am not yet certain that we know which of the possibilities mentioned is correct. Similarly, it is not right to say that English has the traditional phrase-structure rule VP→ V NP, while Japanese has the phrase-structure rule VP → NP V. Phrase-structure, whether given by phrase-structure rule, or by the operation of merge in current minimalism, is assigned to atemporal sentence-types, not their tokens.

Turning to the pronunciation of unordered constituents in particular languages, this account would predict two options: (i) If and where there is no learning, the temporal ordering of unordered constituents would be arbitrary; and (ii) If and where there are rules for the pronunciation of unordered constituents, those rules must be learned. One fairly clear and relatively simple example that might be of the former sort involves the distribution of adverbs in English. It is traditional to say that adverbs adjoin, and that they may adjoin to sentences and to verb phrases. If there are unordered structures such as{IP, Adverb} and {VP, Adverb}, the learner’s first guess would be that the temporal ordering of adverbs with respect to the nodes they are adjoined to is free: that in tokens of these constituents an adverb may precede or follow the constituent it is adjoined to. Amazingly, John left, and John left, amazingly would now be tokens of the same syntactic type, as would John suddenly left and John left suddenly. The positioning of adverbs is a tricky business, however; here I can only gesture toward what I believe to be a promising direction to account for their behavior if the first option is correct.¹⁰

If and where the latter option is correct, rules—possibly language-particular rules—must be stated that specify the temporal ordering of tokens of unordered sentence-types. There are suggestions in the literature along these lines. For example, it has been suggested in various domains that there are certain syntactic

output conditions that filter out illicit orderings. Perlmutter (1971) advances a modern theory of clitics in the spirit of traditional Romance language grammars, which state generalizations of clitic-ordering in terms of person, grammatical case, and so on. If multiple clitics are syntactically represented as adjunction clusters, then this might be a case in which unordered constituents are temporally ordered by a filter stating the order in which the syntactically unordered clusters of clitics are to be pronounced. A similar sort of filter might be appealed to in the cases of multiple overt wh-movement in languages such as Hungarian, Polish, and Bulgarian. Syntactic filters stated in terms of crossing and nesting have been suggested in these cases, but semantic considerations might also be in play. It would be crucial, of course, to distinguish wh-expressions comparable to ‘who’ and wh-expressions comparable to ‘which’, so that matters of scope can be controlled for. Liptak (2001) and Bošcovi´c (1999) have made suggestions along these lines.

There are other approaches to the ‘linearization’ problem, one of which is due to Kayne (1994), who holds that, despite appearances to the contrary, universal grammar imposes a subject-head-complement order on phrases in all languages. Among the tasks that must be faced to sustain this view is the task of stating the relationship between the grammatical structures of sentences and time. To this end, Kayne proposes to associate the string of terminals in a phrase-marker with a string of time slots in a particular way: associated with each time slot will be not simply the corresponding terminal, but the ‘‘substring produced up to that time’’ (1994: 37). The first terminal will appear in all substrings, but the final terminal will appear only in the last substring, yielding an asymmetry which, details aside, allows the conclusion that the interpretation of the set of terminals <x, y>, where x asymmetrically c-commands y, is that x precedes y. Kayne extends this proposal to all phrase-markers in a derivation.

In my view, Kayne’s proposal is not adequate in certain respects. First, in deriving the consequence that asymmetric c-command should be interpreted as temporal precedence, temporal precedence is used to formulate the relationship between terminals and time slots. Secondly, if all phrase-markers are time-slotted, the implication would seem to be that movement rules move constituents in time, presumably from later times to earlier ones, a startling prospect. And thirdly, I can’t help but feel that the mechanism of substrings that Kayne selects could be replaced by a mechanically no less plausible alternative to yield the opposite result. These points would have to be improved upon if the relation between structure and time could truly be said to be derived. More generally, Kayne’s proposal does not make clear what the relationship is between time slots and time. If the time slots are slots in time, and time is the dimension we are familiar with, then the odd consequences just mentioned would seem to follow. But if the time slots are not slots in time, we are left wondering what the relation is between ‘time-slot time’ and time. There is no type-token talk in the sections of Kayne (1994) just discussed, so I think it is probably right to say

Ordering in Representation 125 that there is little relation between Kayne’s goal and my own. I wish to state the temporal ordering of spoken sentence-tokens on the basis of the structures of the sentence-types they are tokens of. Kayne, in contrast, apparently seeks to place sentence-structures, and their derivations, in time. This may mean that Kayne and I also disagree concerning the subject matter of linguistics, but on this topic Kayne is not explicit.

But to return to the topic of horizontal and vertical ordering, I believe, in sum, that the best system of notation for syntactic constituency is one in which phrase-marker representations are notated as sets of sets, which may be ordered or unordered. Ordering represents asymmetry in phrase-markers, which is always the function-argument relation. Lack of ordering represents symmetry in phrase- markers, which is always the relation among the member of clusters. There are pronunciation rules stating the temporal ordering for ordered and unordered sets, but these rules are not syntactic. There may be universal aspects to these rules, such as the generalization that, in all languages, a choice is made as to the direction in which <F, a> is to be pronounced. But the choice of direction, and other matters such as the filters I referred to, might be determined language by language.

5 . 2 . 2 T h e d e fi n i t i o n s o f c o n t a i n m e n t a n d d o m i n a n c e In this section, I will define several relations between elements in phrase- marker representations. Let’s start with two generalizations concerning immediate containment:

i) Given{e1, e2}—or, equivalently, given ‘e1and e2’—e1and e2immedi- ately contain each other.

ii) Given <e1, e2>—or equivalently, given e1—e1 immediately contains e2, and e2does not immediately contain e1.¹¹

In{α, β}, α immediately contains β and β immediately contains α. In <α, β>, α immediately containsβ, but β does not immediately contain α. In a structure such as {{{α, β}, γ}, δ}, α and β immediately contain each other; {α, β} and γ immediately contain each other; and {{α, β}, γ} and δ immediately contain each other. Furthermore, since {α, β} is of category ‘α and β’, both α and β immediately containγ, and γ immediately contains both α and β. The members of an unordered set are both the members of that set and the categories of that set. So, since{{{α, β}, γ}, δ} = ‘α and β and γ and δ’, all of α and β and γ and δ immediately contain each other.

In what environments may a constituent that is of a conjoined category appear? In{α, β}, α and β immediately contain each other, and {α, β} is of the category ‘α and β’. Let the principle be that a constituent that is of a conjoined category

meets a subcategorization requirement if one of the conjoined categories does. To illustrate, suppose{α, β} occurs in the structure <γ, {α, β}>, and γ is subcat- egorized only to allow arguments of categoryα. Then this is allowed. If instead γ is subcategorized to allow arguments of category β, <γ, {α, β}> will also be allowed. Finally, if in <γ, {α, β}>, γ imposes no subcategorization requirement, then <γ, {α, β}> is allowed. However, if γ is subcategorized only to allow arguments of categoryδ, δ = α and δ = β, then <γ, {α, β}> is not allowed.¹²

Let me now define the more general notion ‘containment’. First, immediate containment is a kind of containment:

If e1immediately contains e2, then e1contains e2. The general definition of containment will be:

e1contains en iff they appear in a series of the form ‘ . . . e1, e2, e3, . . . , en−1, en, . . . ’ where eiimmediately contains ei+1.

To see how this works, consider the structure <α, {β, γ}>. {β, γ} is of the category ‘β and γ’. So <α, {β, γ}> = <α, ‘β and γ’> . α immediately contains ‘β and γ’. ‘β and γ’ immediately contains β and ‘β and γ’ immediately contains γ. So we have the series ‘α, ‘β and γ’, β’ and the series ‘α, ‘β and γ’, γ’, in which each member immediately contains the next. So α contains β and α contains γ. The structure <α, <β, γ>> yields the same result. <β, γ> is of the categoryβ. So <α, <β, γ>> = <α, β>. α immediately contains β. And β immediately containsγ. So α contains β and α contains γ. On the other hand, in <<β, γ>, α>, β contains α, but γ does not contain α.