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Pairing and ordering

Note that what is required for adjunct structures to have syntactic properties that they do and to pass the labeling procedure without falling prey of the halti ng problem, which arises in cases of an undeterminacy of structure (whence {XP, YP} structures are supposed to be ‘unstable’ and to pass the labeling test only if one of their immediate constituents is a copy, part of a discontinuous object, hence invisible for the labeling algorithm, or under feature sharing between heads), is to identify the source of the label uniquely—thereby directing the search procedure towards one of constituents in a definite way—and to ‘deactivate’ the other constituent; what is neither necessary nor even wanted is endowing the adjunction with order. No qualms against arbitrariness of the Kuratowski-style analysis of ordered pairs with regard to the ordering relation arise here, as they do when the order of some а, в is at stake, as it is e. g. in analysis of relations and their representations:

The Wiener-Kuratowski procedure may be looked at as appealing to properties that serve to order classes logically... Given the signs ‘(a, b) and ‘(b, a)’, one arbitrarily takes them to stand for one ordered pair rather than the other, i.e. one recognizes the ordering of the signs to correspond to the ordering of the elements. Likewise, one arbitrarily construes the ordered pair (a, b) in terms of a set like [1], or a variety of other alternatives. Finally, if, following Kuratowski, one uses sets like [1] and [3], one chooses which of [1] or [3] is to represent (a, b) and which will represent (b, a). This shows that in the construal of (a, b) as [1] one implicitly takes a and b in an ordering, for one takes the element in the unit set as the first element. (Hochberg 1981: 162)

Determining the order of (XP, YP) is not the task of narrow syntax, though, just as it is not in the case of {XP, YP} structures; the device of ordered pair was introduced to eliminate the symmetry between constituents present in the latter case:

The construction is crucially asymmetric: if a is adjoined to в, the construction behaves as if a isn’t there apart from semantic interpretation, which is not that of standard X- bar-theoretic constructions; island properties differ as well. в retains all its properties, including its role in selection. There is no selectional relation between в and a (...) The adjunct a has no theta role in (a, в), though the structure does—the same one as в. (Chomsky 2004a: 117)

Given these expectations of the account of how adjunction structures are formed and distinguished by narrow syntax, together with current assumptions about the labeling algorithm, it is clear that, as already mentioned above, it would not do to assimilate them to {XP, YP} structures with regard to the structure building operation and locate the difference entirely in the labeling part of the derivational process: although it would be conceivable that the adjunction structure might be determined as such only at the stage when the labeling algorithm operates, there would still be a need to pass to the interfaces the information about (i) exemption of the structure as a whole from the labeling procedure, and (ii) the path to be chosen as the ‘main’ path of labeling. Since labels are no longer explicitly encoded in syntactic objects, and in the case of adjuncts there is no featural configuration or operation, the phase level memory would not be of much help: it may keep the record of the immunization against a labeling failure, yet both XP and YP would be, by themselves, labeled and equally good candidates for being on the main structural spine—a case of illegitimate ambiguity. The ordered pair device provides a required asymmetry: it suffices if it is decided, once and for all, which element is to be treated as the adjunct; no ordering relations follow from that choice.

Definitions of the ordered pair are well known to vary; provided that they preserve the basic property of ordered pairs mentioned above, they are chosen according to specific requirements of the theory they are used in (as e. g. when a version of stratified type theory is employed, and the difference between {a, {a, в}} and [7], negligible for practical purposes, begins to matter; see e. g Holmes (1998), Forster (2007) and Kanamori (2003) for an overview of various proposals), or the notion of ordered pair is introduced as a primitive one, an appropriate axiom taking care of the basic property (a path which Bourbaki (1954) take, whereas Bourbaki (1970) adopt the definitional path and choose the definition of Kuratowski), the concern in general being to fix on the particular functions of the unclear expression that make it worth troubling about, and then devise a substitute, clear and couched in terms to our liking, that fills those functions’, as Quine (1960: 258-259) summarizes the approach. It would seem too far-fetched to enquire into the nature of ordered pairs as generated on the assumptions of Chomsky (2004a) by trying to align them with a definition of an ordered pair—it

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would be like asking whether narrow syntax and components that it sends its output to choose Kuratowski over Wiener or Hausdorff over Quine-Rosser. Still, one may look into ordered pairs to find which definition, if any, would suit best the purpose of securing properties required of adjunction structures when letting the labeling algorithm to work on such set-theoretic construct—(XP, YP) being no more and no less a modeling device than {XP, YP} is: ‘We don’t have sets in our heads. (...) That’s something metaphorical, and the metaphor has to be spelled out someday’, as Chomsky (2012b: 91) remarks—in the hope that it will either make clearer the procedure which applies to ordered pairs or, in the best case, will permit reduction of pair-merge to mechanisms otherwise made available by the theory, in effect making the use of ordered pairs merely a notational convenience. The set-theoretic representation of ordered pairs is thus no more than a heuristic device, a starting point to look for an appropriate representation which might replace ordered pairs as ‘formal objects with their own properties, “primitive objects” for mental computation that are distinct from sets’ (Chomsky 2005: 15-16), thereby reducing the repertory of primitive notions of syntactic theory.

The simplified definition of ordered pairs as {а, {а, в}} is a nonstarter in this regard: a structure {XP, {XP, YP}} would be either generated by external merge of XP to {XP, YP}, hence involving a repetition, and irrelevant for the purpose at hand; or it would be generated by a very local application of internal merge (it should be kept in mind that phase level memory makes the difference visible for the purpose of the labeling algorithm, hence it is unambiguously identified without the need for indexing or other technical devices, dubious on min imalist grounds and discussed in the context of chains in Martin and Uriagereka (2014); the displacement violates antilocality constraints, but seems available under the free merge hypothesis, see also section 2.3.2). In the latter case, the copy of XP would be made invisible for the purposes of labeling, as in (1).

(1)

Invisibility of the copy of XP in a is a welcome result, identifying YP as the source of the label; but the head of the XP-chain would be seen by the labeling algorithm as searchable for the label of the entire structure—contrary to the behaviour required for adjunction structures, in which the adjunct does not participate in labeling. The full Kuratowski definition, on the other hand, would have the head of the XP-chain embedded in a set (its singleton set), hence invisible for the labeling of the whole structure, but the algorithm would encounter the {XP, YP} problem—ambiguity for the search procedure—unless the head of the XP-chain is invisible as much as its copy in a is.

 
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