2. Determinacy as a constraint on rules

Recent work by Epstein et al. (Reference Epstein, Kitahara and Seely2018), Chomsky et al. (Reference Chomsky, Gallego and Ott2019), Goto and Ishii (Reference Goto, Ishii and Cho2019, Reference Goto and Ishii2020), and others develops a principle of Determinacy as a third-factor constraint on syntactic operations. Determinacy may be defined as an avoidance of “ambiguous” (Goto and Ishii Reference Goto and Ishii2020: 5) or “indeterminate” (Chomsky et al. Reference Chomsky, Gallego and Ott2019: 20) rule-application. Put simply, if the input or output of a particular rule does not show a one-to-one correspondence between that input/output and a term of the domain of that rule, then a “Determinacy violation” is the result.Footnote ¹

To illustrate: Goto and Ishii (Reference Goto, Ishii and Cho2019: 93) approach Determinacy as a constraint on the input of the combinatory operation Merge, which takes two objects and combines them into a single set (1a). Merge may apply recursively to its whole output, building hierarchical embedded structures (1b), and it may also select a subpart of its output, yielding syntactic movement (Internal Merge), notated with bracketed “copies” (1c) (see definitions in Chomsky Reference Chomsky and Belletti2004, Reference Chomsky, Sauerland and Gärtner2007, Reference Chomsky, Freidin, Otero and Zubizarreta2008).

1. (1)

   1. a. Merge(X, Y) = {X, Y}

   2. b. Merge(Z, {X, Y}) = {Z, {X, Y}}

   3. c. Merge(X, {Z, {X, Y}}) = {<X₁>, {Z, {<X₂>, Y}}}Footnote ²

The above authors argue that the input to Merge must be unambiguous. In other words, there must be a one-to-one correspondence between a single member of the input and a single member of the domain of Merge. In most cases, determinate inputs to Merge are automatic, since lexical items are selected from the lexicon or the workspace (=the domain of Merge) and indexed uniquely from each other. A problem arises, however, when a situation like the output of (1c) is encountered. In this case, X in the domain of Merge is indeterminate because it exists as two “copies” in that domain, notated as <X₁> and <X₂>. If X is selected for input to Merge again (e.g., in a subsequent movement step), it will be unclear which copy of X actually functions as the input, leading to a Determinacy violation.

Before continuing, it should be noted that this application of Determinacy differs from that presented by Chomsky et al. (Reference Chomsky, Gallego and Ott2019) and Epstein et al. (Reference Epstein, Kitahara and Seely2018). In those works, Determinacy is characterized as a constraint on the output of Merge, preventing the emergence of indeterminate structures. Goto and Ishii (Reference Goto, Ishii and Cho2019) argue against this conception, however, on the following grounds: If Determinacy applies to the output of Merge, this means that no Merge-output may contain an indeterminate item. As a result, Internal Merge is actually blocked by output-Determinacy, since it results in a structure containing multiple copies. Chomsky et al. (Reference Chomsky, Gallego and Ott2019) also identify this problem and attempt to circumvent it by including a version of the “shortest move corollary”, which states that only the structurally “highest” copy in a structure is available for further applications of Internal Merge. This has the character of a stipulation, however, given that Goto and Ishii (Reference Goto, Ishii and Cho2019) are able to develop an account that does not require this additional constraint in order to allow Determinacy and Internal Merge to coexist.

Goto and Ishii use this version of Determinacy to capture “freezing” phenomena in English and other languages. The sentences in (2) and (3) illustrate these effects for subject phrases. Under the assumption that the sentential subject generally moves to Spec-TP in English, any attempt to extract an item from the subject fails due to a Determinacy-violation (=the Subject Condition, Chomsky Reference Chomsky, Anderson and Kiparsky1973, Huang Reference Huang1982). If, however, the subject phrase does not undergo movement, as in expletive constructions with there (3), extraction from the subject is licit (Lasnik and Park Reference Lasnik and Park2003, Stepanov Reference Stepanov2007).Footnote ³

Another straightforward application of input-Determinacy is found in limits on embedded subject extraction in English with and without expletive there. In sentence (4) below, the embedded subject undergoes movement from its base position in VP to the embedded Spec-TP, but cannot move further to the matrix Spec-CP. If, however, as in sentence (5), an expletive is inserted in the embedded Spec-TP, extraction of the embedded subject to the matrix Spec-CP is much improved (examples are adapted from Moro Reference Moro1997a: 126, see also Culicover Reference Culicover1993, Bayer and Salzmann Reference Bayer, Salzmann, Brandt and Fuß2013).

Goto and Ishii (Reference Goto, Ishii and Cho2019) also successfully apply input-Determinacy to capture verb-particle constructions, constraints on vacuous topicalization, that-trace effects, and freezing effects with topics. Going forward, I adopt this conception of Determinacy as a constraint on the input of Merge, and in the next section I will extend it to another syntactic rule: the labelling procedure developed in work by Chomsky (Reference Chomsky2013, Reference Chomsky, Di Domenico, Hamann and Matteini2015).

3. Determinacy and labelling

Chomsky (Reference Chomsky2013, Reference Chomsky, Di Domenico, Hamann and Matteini2015) outlines a framework wherein the combinatory operation Merge is separated from the process of “projection” or labelling, by which the head of a syntactic object is determined. This is a break from prior conceptions of Merge, which typically incorporated projection/headedness into the combinatory operation itself (see (Chomsky, Reference Chomsky, Hale and Keyser1993, Reference Chomsky1995), where label-projection functioned explicitly as a subpart of Merge). Instead, Chomsky (Reference Chomsky2013, Reference Chomsky, Di Domenico, Hamann and Matteini2015) describes a “labelling algorithm” which applies separately to each output of Merge, being triggered at the level of the syntactic phase.

This labelling procedure takes a single syntactic object – the output of Merge – as its input and identifies a unique label for that object according to Minimal Search (MS), another third-factor principle of minimal computation. MS requires that the least-embedded simplex item found inside a syntactic object be selected automatically as the label of that object.

Given the automatic nature of label-selection, Chomsky (Reference Chomsky2013) specifies structural configurations where labelling will be successful and configurations where labelling will fail. Example (6) shows a structure where a label cannot be found by MS, since both X and Y are equally accessible within the input set (see below for further discussion of why this would result in a labelling problem).Footnote ⁴ Example (7), on the other hand, shows a structure where the input set contains X and another subset. In this case, X is found by MS and selected as label immediately, since the contents of the subset would require additional (i.e., non-minimal) search.

1. (6) Input: {X, Y}

   Label({X, Y}) = X?, Y?

2. (7) Input: {X, {…}}

   Label({X, {…}}) = X

Chomsky (Reference Chomsky2013: 43) does not go into great detail about the reason for the disallowance of labelling in (6) beyond stating that “[i]n the best case, the relevant information about [a syntactic object] will be provided by a single designated element within it […]” and that, in contexts like (6), “minimal search is ambiguous”. Importantly, it should be noted that this is by no means the first time that a property of ambiguity or symmetry has been identified as a significant factor in the evaluation of syntactic structures like (6). Chomsky (Reference Chomsky2013) in fact builds on a body of work going back to Moro's (Reference Moro1997a,Reference Morob, Reference Moro2000) Dynamic Antisymmetry, and Kayne's (Reference Kayne1983, Reference Kayne1984) and Pesetsky's (Reference Pesetsky1982) notions of “paths”, among other influences. The point to emphasize here is that, on further consideration, Minimal Search on its own is not necessarily the deciding factor for ruling out structures like (6) in the context of Chomsky's labelling system. In both (6) and (7), MS has, in fact, successfully identified the least embedded item(s) in the input set.

Instead, the implicit reason for failure-failure in (6) is that MS finds multiple options for labelling, and it is assumed that the ideal outcome of MS is to find a single discrete label. As a result, the labelling procedure can actually be defined by two requirements: (i) labelling must proceed according to MS, and (ii) there cannot be multiple options for labelling within a given syntactic object. The contribution of this paper to the discussion is to observe that the latter requirement has the distinct character of a Determinacy-based restriction, since it involves ambiguity: the labelling procedure is governed by a requirement that the item selected to label a syntactic object must be determinate or unambiguous in the domain of the labelling rule. The formalization of this requirement can be worked out as follows:

Application of MS to the structure in (6) finds two simplex terminals, which I have notated so far as the set {X, Y}. In order to draw out the similarity between Determinacy in the case of Goto and Ishii's proposal for copies, and Determinacy in the case of labels, however, I will now notate the output of MS as single unit α. If α is submitted as input to the labelling procedure, a Determinacy-violation will result, since α exists as two separate items (X and Y) in the domain of the labelling rule. This parallels Goto and Ishii's application of Determinacy to syntactic movement in the following way: In a context like (1c) above, if X is submitted as one of the inputs to an operation of Merge, a Determinacy-violation results since X actually exists as two separate but identical items (<X₁> and <X₂>) in the domain of Merge. Example (8) shows the relevant comparison between these two cases.

1. (8)

   1. a. X = {<X₁>, <X₂>}, X is indeterminate for Merge

   2. b. α = {X, Y}, α is indeterminate for Label

Crucially, this understanding of how Determinacy constrains the labelling procedure requires that we view the labelling procedure as “blind”: insensitive to the specific identities of the simplex items found by search, just as Merge is insensitive to the specific identities of the individual “copies” of X in (8a). The procedures of search and labelling are only concerned with finding/selecting simplex items that are determinate.

Having illustrated how a principle of Determinacy can be applied to the labelling procedure to explain why structures like {X, Y} are unlabelable, the next section addresses the question of why copies should be invisible to the labelling procedure and demonstrates that the principle of input-Determinacy can also provide an explanation in this area.

4. Explaining copy-invisibility through Determinacy

Aside from the labelling configurations in (6) and (7) above, Chomsky proposes that a non-labelable situation like (6) can also be rectified if one of the members of the input set undergoes syntactic movement, as in (1c). In this case, the remaining “copy” of the moved term is assumed to be invisible for labelling purposes due to the fact that it constitutes a “discontinuous” syntactic object (Chomsky Reference Chomsky2013:44).

1. (9) Input: {<X>, Y}

   Label({<X>, Y}) = Y

This constitutes a third way that labelling can succeed, and Chomsky exploits the notion of movement-based labelling to elegantly unify a variety of syntactic effects, following ideas from Moro (Reference Moro2000), including movement of the v*P-internal subject into Spec-TP (10a), “raising-to-object” (=Spec-RP) in ECM constructions (10b), and successive-cyclic long-distance movement of phrases through Spec-CP (10c).

Ultimately, Chomsky's use of MS-based labelling as a motivation for certain instances of syntactic movement offers a less stipulative account of effects that were previously attributed to a purely formal EPP- or “edge-” feature. Even so, it remains unclear why precisely a “discontinuous” syntactic object should be inaccessible for label-determination. Given that a principle of Determinacy provides a means of formally distinguishing “discontinuous” (i.e., indeterminate) from “continuous” (i.e., determinate) syntactic objects, an approach that incorporates input-Determinacy is clearly desirable.

In order to build an account of how input-Determinacy explains the invisibility of copies for labelling purposes, it is necessary to consider in more detail the nature of the labelling procedure itself. Thus far, I have focused primarily on the end-result of the procedure where a label is selected, assigned, or determined for an object. It should be emphasized, however, that there is also a search procedure which precedes and directly feeds the application of labelling. This initial stage will be the focus here.

A first pass at formalizing the search procedure is to characterize it as an operation taking an object K as input and outputting the contents of K into the domain of the labelling procedure. However, the procedure employed by Chomsky (Reference Chomsky2013, Reference Chomsky, Di Domenico, Hamann and Matteini2015) requires some additional complexity. Rather than finding the immediate elements of K and terminating, Chomsky's MS also evaluates if further search is necessary. This requires not only finding the elements of K but also identifying them as simplex (i.e., a terminal element) or complex (i.e., a non-terminal element). Furthermore, search of an object K should properly be divided into two parts, given that outputs of Merge are invariably binary. In tree-theoretic terms, this means that searching K requires examination of both branches of K. With these factors in mind, I will characterize the search procedure as an operation “Search” taking two inputs,Footnote ⁵ one for each branch of K (notated as K_α and K_β below), and outputting any terminal nodes found by MS on each branch. The different possible outcomes are laid out in (11) below:

1. (11)

   1. a. If K_α = X and K_β = {…}, then Search(K_α, K_β) = X

   2. b. If K_α = X and K_β = Y, then Search(K_α, K_β) = {X, Y}

In (11a), the object K is of the form {X, {…}}, as in (7) above. This means that the search procedure will only output X, ignoring the contents of K_β, since they require further search. In (11b), however, the object K is of the form {X, Y}, as in (6) above, meaning that search finds a terminal on each branch of K. The output of search in this case is the set of terminals {X, Y}. As discussed in section 2, only the output of (11a) can be successfully submitted to the labelling procedure. The output of (11b), on the other hand, is indeterminate, since it is an object that exists as multiple separate items within the domain of the labelling rule. Recall that this is what is represented by α in (8b) above.

At this point, we can consider why an item that has undergone movement in a syntactic structure would be unavailable for labelling purposes. Example (12) below shows a context where an object K contains a copy-item, K = {<X_n>, Y}, as in the VP of a sentence like What did Susan write?, where K = VP and VP consists of the set {<what>, write}. In this case, the terminal node on K_α exists as multiple copies in the derivation (i.e., <what₁>, <what₂>), making it indeterminate and prohibiting it from being submitted as input to Search. K_β, on the other hand, is fully determinate and able to serve as input, yielding Y as the output.

1. (12) If K_α = <X_n> and K_β = Y, then Search(K_β) = Y

The situation in (12) is substantively different from the situation in (11a) above. In the latter case, both inputs to Search are fully determinate, even though one of them (=K_β) does not yield a terminal element through search. In (12), K_α may not even function as an input to Search, because it is already indeterminate. The outcome in this case is the same as in (11a): the other input to Search successfully yields a terminal element that may be submitted as input to the labelling procedure.