2. Omune (Reference Omune2018) and Nakashima (Reference Nakashima, Bakay, Pratley, Neu and Deal2022)

In this section, we review two instances of ZFC-based reformulations of Pair-Merge/MERGE proposed independently by Omune (Reference Omune2018) and Nakashima (Reference Nakashima, Bakay, Pratley, Neu and Deal2022). Pair-Merge/MERGE has been assumed to be an independent operation that takes two syntactic objects $\alpha$, $\beta$ and forms an ordered pair $ \lt $ $\alpha$, $\beta$ $ \gt $ (Chomsky Reference Chomsky, Martin, Michaels and Uriagereka2000). Omune (Reference Omune2018) points out that Pair-Merge can be eliminated if we adopt the definition of an ordered pair within ZFC as follows:

He argues that an ordered pair $ \lt $ $\alpha$, $\beta$ $ \gt $, which has been postulated to be created by Pair-Merge, reduces to {$\alpha$, {$\alpha$, $\beta$}} created by Set-Merge.

Omune (Reference Omune2018) argues that reformulating $ \lt $ $\alpha$, $\beta$ $ \gt $ as {$\alpha$, {$\alpha$, $\beta$}} allows us to capture the asymmetry between $\alpha$ and $\beta$ in $ \lt $ $\alpha$, $\beta$ $ \gt $ without postulating Pair-Merge. That is, an unordered set {$\alpha$, {$\alpha$, $\beta$}} is sufficient to represent an adjunct structure where $\beta$ adjoins to $\alpha$ and becomes invisible. This is the essence of the ZFC-based reformulation of Pair-Merge as Set-Merge. Given this, Omune (Reference Omune2018) proposes that Pair-Merge of heads (i.e., head-to-head adjunction) is also reformulated by Set-Merge. In his analysis, an unordered set (2a) created by Set-Merge represents a head adjunction structure (2b) created by Pair-Merge, where H₂ adjoins to H₁. Note that Omune (Reference Omune2018) assumes Chomsky’s (Reference Chomsky, Domenico, Hamann and Matteini2015) reconception of head adjunction, in which the host head adjoins to the raised head.Footnote ³

Nakashima (Reference Nakashima, Bakay, Pratley, Neu and Deal2022) independently proposes a type of ZFC-based reformulation of Pair-MERGE/adjunction as Set-MERGE under the latest minimalist model with Workspace (WS) and MERGE, as proposed by Chomsky (Reference Chomsky2019a, Reference Chomsky2019b) and Chomsky et al. (Reference Chomsky, Gallego and Ott2019). Let us briefly review Nakashima’s (Reference Nakashima, Bakay, Pratley, Neu and Deal2022) analysis. Suppose that the WS contains $\alpha$ and $\beta$, as in (3). Goto and Ishii (Reference Goto, Ishii, Asatryan, Song and Whitmal2020) claim that four types of derivations are available, as illustrated in (3). Examples (3a–d) each depict a possible output obtained via a single application of MERGE to the WS.

Nakashima (Reference Nakashima, Bakay, Pratley, Neu and Deal2022) argues that adjuncts are introduced by the outputs of MERGE in (3b–c), in which either $\alpha$ or $\beta$ becomes invisible in the derivation.Footnote ⁴ Although Nakashima (Reference Nakashima, Bakay, Pratley, Neu and Deal2022) does not mention ZFC in his paper, the structures in (3b–c) are syntactically equivalent to the right-hand side of the equation in (1), apart from the fact that (3b–c) are analyzed within the framework of WS and MERGE (Chomsky et al. Reference Chomsky, Gallego and Ott2019, a.o.). For instance, Nakashima (Reference Nakashima, Bakay, Pratley, Neu and Deal2022) analyzes (4), a sentence with a PP adjunct, as in (5), whose structure is reminiscent of that of (1).

Therefore, it is fair to conclude that the proposal of Nakashima (Reference Nakashima, Bakay, Pratley, Neu and Deal2022) in (3b–c) is also rooted in the ZFC-based reformulation of Pair-MERGE as Set-MERGE in (1).

The rest of this article shows that while eliminating Pair-MERGE is theoretically desirable, Omune’s (Reference Omune2018) and Nakashima’s (Reference Nakashima, Bakay, Pratley, Neu and Deal2022) reformulations of Pair-MERGE as Set-MERGE have both conceptual and empirical problems.

3. Determinacy and the ZFC-based analysis of Pair-MERGE

We first point out a conceptual problem with the ZFC-based reanalysis of Pair-MERGE. We point out that head-to-head raising, especially NEG-to-T raising, under Omune’s (Reference Omune2018) ZFC-based analysis inevitably violates the Principle of Determinacy, which prohibits ambiguous rule applications in linguistic computation (Chomsky Reference Chomsky2019a: 275). Under the MERGE-based system, Determinacy is ensured by restricting accessible elements in WS to the minimum (Chomsky et al. Reference Chomsky2019: 245–246). As Reference ChomskyChomsky (Reference Chomsky2019b: 36) suggests, there are already ways to ensure Determinacy within syntactic theory. One is to limit accessibility using the notion of c-command, which reduces to Minimal Search. To illustrate, suppose that X undergoes successive cyclic movement and forms a chain. Given two copies of X in a c-command relation, a subsequent operation deterministically applies to the higher copy by Minimal Search, which renders the lower copy inaccessible.

With this in mind, let us consider (7), which represents a derivation of NEG-to-T raising in English, a typical case of head-to-head raising, under Omune’s (Reference Omune2018) analysis. In English, NEG syntactically head-raises to T and then to C in interrogatives, as illustrated in (6c). In (7a), MERGE takes NEG and T as its input and forms {NEG, T}. In (7b), MERGE takes NEG and {NEG, T} to form {NEG, {NEG, T}} so that the asymmetry between NEG and T is represented in the complex head.Footnote ⁵

Notice, however, that NEG in (7a) does not c-command the lower NEG in the head of NegP; hence a Determinacy violation occurs. The structure in (7), created via Set-MERGE under ZFC in (1), cannot encode a segment-category distinction, which by contrast is a natural consequence of the asymmetric structure created via adjunction or Pair-Merge/MERGE. Therefore, any head contained inside the complex head created via Set-MERGE cannot c-command elements outside of the amalgam.Footnote ⁶

The derivation in (7) is filtered out if we assume the so-called Input Determinacy of Goto and Ishii (Reference Goto, Ishii, Asatryan, Song and Whitmal2020), which states that Determinacy applies at the input of MERGE (Goto and Ishii Reference Goto, Ishii, Asatryan, Song and Whitmal2020: 30). Since Minimal Search is ambiguous between NEG and NEG in (7a), neither occurrence of NEG is made inaccessible by Minimal Search. As a result, MERGE that targets NEG does not apply in a deterministic fashion. Put another way, the structural change obtained by applying MERGE(NEG, {NEG, T}) is not unique; we cannot tell where the higher $\boxed{\textrm{NEG}}$ in (7b) came from.Footnote ⁷

The same derivation violates the Principle of Determinacy even if we assume that Determinacy applies at the output of MERGE (Chomsky et al. Reference Chomsky, Gallego and Ott2019) (so-called Output Determinacy). In (7a), MERGE takes NEG and T, and forms {NEG, T}. At this point, NEG and NEG in (7a) violate the Principle of Determinacy, because neither occurrence is made inaccessible. Therefore, as we have seen, the ZFC-based analysis of head-to-head raising inevitably violates the Principle of Determinacy, regardless of whether we assume Input or Output Determinacy (cf. Chomsky et al. Reference Chomsky, Gallego and Ott2019; Goto and Ishii Reference Goto, Ishii, Asatryan, Song and Whitmal2020).

Although the Phase Impenetrability Condition (PIC) also plays a role in satisfying Determinacy (Chomsky Reference Chomsky2019b), the aforementioned problem cannot be resolved by PIC. If we assume that v and C are phase heads (Chomsky Reference Chomsky, Freidin, Otero and Zubizarreta2008), PIC makes the VP complement of v (shaded in (7)) inaccessible. However, since PIC does not make either copy of NEG inaccessible, a Determinacy violation is not obviated. Since these technical problems arise once the ZFC in (1) is assumed, any ZFC-based reformulation of Pair-MERGE as Set-MERGE violates the Principle of Determinacy (Chomsky Reference Chomsky2019b, a.o.).

We have demonstrated that the reanalysis of Pair-MERGE as Set-MERGE under ZFC inevitably leads to a Determinacy violation. Now, we will discuss how Pair-MERGE in its original definition circumvents the problem regarding Determinacy.Footnote ⁸ Pair-MERGE is a structure-building operation that corresponds to adjunction (Chomsky Reference Chomsky1986), and, unlike Set-MERGE, it guarantees asymmetries between the elements of the ordered pair: $\alpha$ and $\beta$ in $ \lt $ $\alpha$, $\beta$ $ \gt $. Let us take (6) and (7) as an example. If a head adjunction structure is created by Pair-MERGE, the raised head (i.e., $\boxed{\textrm{NEG}}$ in C) takes scope outside of a complex head under the definition of c-command in (9).

It is possible to distinguish between segments and categories within the ordered pair $ \lt $ $\boxed{\textrm{NEG}}$, T$ \gt $ in (8b). Because $\boxed{\textrm{NEG}}$ c-commands the lower NEG in the head of NegP in (8), Minimal Search makes the lower occurrence inaccessible; hence, the Determinacy violation observed in (7) with Set-MERGE does not arise here.Footnote ⁹

An anonymous reviewer brought up an astute point as to why Pair-MERGE makes elements inaccessible or invisible. It is true that without this point being explained, our analysis would end up being a mere stipulation. However, in fact, our analysis does not require that inaccessibility be stipulated for Pair-MERGE. Rather, the assumption of inaccessibility of Pair-MERGE is orthogonal to our analysis. That is, the definition in (9) allows the moved head to c-command its lower copy from inside the complex head created via Pair-MERGE (i.e., head-to-head adjunction); hence, our analysis circumvents the Determinacy violation, unlike the ZFC-based Set-MERGE analysis proposed by Omune (Reference Omune2018) and Nakashima (Reference Nakashima, Bakay, Pratley, Neu and Deal2022).Footnote ¹⁰ Related discussions on this point will be provided in section 4 with references to tree-diagrammatic representations in (18) and (19).

The same reviewer also pointed out that the relationship between Pair-MERGE and the segment-category distinction is not clear; that is, it is not obvious why only a Pair-Merged object, but not its ZFC-based Set-Merged counterpart, results in a segment-category distinction. As the reviewer has suggested, it is also possible to assume that the ZFC-based reformulation of $ \lt $A, B$ \gt $ as {A, {A, B}} results in a segment-category distinction. Specifically, one can presume that A in {A, {A, B}} can c-command from out of the whole complex according to (9). However, this assumption leads to an overgeneration problem. The ZFC-reformulation of an ordered pair {A, {A, B}} is simply created via Set-Merge only. This assumption predicts that a structurally identical syntactic object should also allow such c-commanding relations, say NPs in {v*, {V, NP}} and {XP, {P, NP}} being able to c-command constituents from deep within the syntactic objects containing them. For instance, in (10), it should be possible for boys to bind themselves if boys in {parents, {of, boys}} can c-command constituents outside of the subject NP, based on the above assumption. However, this is not the case. The only possible reading in (10) is that the whole subject NP parents of boys binds the reflexive themselves in the object position. Hence, the data indicate that the above assumption is not tenable empirically.Footnote ¹¹

Some might say that the ZFC-reformulation of $ \lt $A, B$ \gt $ as {A, {A, B}} is fundamentally different from the similar structures above because only the former has two occurrences of the same element, A. However, even under the ZFC-based reanalysis of Pair-Merge, multiple occurrences of the same element are not derived via internal Merger, but via external Merger; otherwise, such internal Mergers would definitely violate the anti-locality condition (cf. Grohmann Reference Grohmann and Boeckx2011). In other words, A₁ in {A₁, {A₂, B}} is not a copy of A₂. Therefore, it is impossible to make a fundamental distinction between the ZFC-reformulation and the similar structures exemplified above.

It has been widely hypothesized that Pair-Merge corresponds to traditional adjunction (Chomsky Reference Chomsky and Belletti2004), which is why we assume that an ordered pair takes over the properties of segment-category distinctions. Although it is difficult to provide a sufficient explanation for why an ordered pair results in a segment-category distinction in a principled way, we have demonstrated that the ZFC-based analysis fails to differentiate segments from categories. As long as one attempts to capture adjunction with only Set-Merge, this problem inevitably occurs. To resolve this, we have argued in this article that one needs to make use of another operation for structure-building: the most likely candidate is Pair-MERGE. As such, our claim that Pair-Merge is preferable over the ZFC-based reformulation of Pair-MERGE as Set-MERGE remains valid.Footnote ¹²

Before closing this section, we touch on other cases of potential Determinacy violations caused by different types of head-to-head movement in Narrow Syntax. To this end, we discuss (i) $\sqrt{R}$-to-v movement in categorization (Chomsky Reference Chomsky, Domenico, Hamann and Matteini2015), and (ii) T-to-C movement in English interrogatives (Roberts Reference Roberts2010). First, in the case of $\sqrt{R}$-to-v movement, $\sqrt{R}$P and v undergo Set-MERGE and create {$\underline{\sqrt{R}}$, v} in (11b). Then, ZFC dictates that $\{\boxed{\sqrt{R}}$, {$\underline{\sqrt{R}}$, v}} is further generated via multiple applications of Set-MERGE in (11c). At this stage, $\underline{\sqrt{R}}$ and $\sqrt{R}$ in (11c) violate the Principle of Determinacy. Specifically, the raised head $\boxed{\sqrt{R}}$ does not c-command the lower occurrence of $\sqrt{R}$ in (11c); hence, the derivation does not proceed in a deterministic fashion.

Once we make use of Pair-MERGE, the abovementioned problem regarding the lack of a c-command configuration is resolved, as in (12). The standard definition of c-command in (9) allows $\sqrt{R}$ inside the complex amalgam $ \lt $ $\sqrt{R}$, v$ \gt $ to c-command the lower occurrence of $ \lt $ $\sqrt{R}$, v$ \gt $, making $\sqrt{R}$ inaccessible for further derivation.

The same issue arises with T-to-C movement observed in English interrogatives, such as (6c). T head-raises and undergoes Set-MERGE with C, creating {T, C} in (13b), which is followed by another application of Set-MERGE in (13c). The ZFC-based reanalysis of Pair-MERGE/adjunction can obtain the complex head {$\boxed{\textrm{T}}$, {T, C}}. Nevertheless, $\boxed{\textrm{T}}$ fails to c-command T outside of the amalgam, resulting in a Determinacy violation.

Again, this problem is solved at once with Pair-MERGE: the definition in (9) allows T inside the complex head $ \lt $T, C$ \gt $ to c-command its lower copy in (14). Consequently, the derivation proceeds in a deterministic fashion, circumventing the problem with the ZFC-based reanalysis of Pair-MERGE as Set-MERGE illustrated in (13).

As we have demonstrated in this section, the ZFC-based reanalysis of Pair-MERGE as multiple applications of Set-MERGE proposed by Omune (Reference Omune2018) and Nakashima (Reference Nakashima, Bakay, Pratley, Neu and Deal2022) inevitably violates the Principle of Determinacy. We then emphasized, by referring to several types of head-to-head raising, that this conceptual problem can be solved at once with Pair-MERGE and the segment-category distinction ensured by the standard definition of c-command in (9).Footnote ¹³ In the next section, we point out that the ZFC-based analysis of Pair-MERGE also faces an empirical problem concerning (head-to-head) adjunction in Narrow Syntax.

4. NEG-raising and the ZFC-based analysis of Pair-MERGE

Let us now turn to an empirical problem. In this section, we take up syntactic NEG-raising in English and its NPI-licensing as a case study. Reference RobertsRoberts (Reference Roberts2010: 10) claims that a head-raised NEG licenses an NPI in its c-commanding domain. Let us assume that this is correct, at least for certain constructions in English.Footnote ¹⁴ Sentences (15a) and (16a), in which NEG does not c-command the NPI, are ungrammatical since the NPIs are not licensed. In contrast, (15b) and (16b) are grammatical because NEG raises to interrogative C and c-commands the NPI.

The standard definition of c-command assumed in this article for the structure of adjuncts is in (9), repeated here as (17), and the tree-diagrammatic representations of (15a) and (15b) are given in (18) and (19), respectively. In (18), NEG does not c-command the NPI anybody, while it does so in (19). In (19), the adjoined does is a segment, not a category. Thus, NEG in (19) c-commands the NPI because every category that dominates NEG – namely, CP – also dominates the NPI, and NEG and the NPI do not dominate each other.

Now, we show that the ZFC-based analysis of Omune (Reference Omune2018) (and the one of Nakashima Reference Nakashima, Bakay, Pratley, Neu and Deal2022) faces a problem in analyzing the data in (15) and (16). To this end, we follow Omune (Reference Omune2018) and assume Chomsky’s (Reference Chomsky, Domenico, Hamann and Matteini2015) system of head adjunction (i.e., internal Pair-MERGE of heads), in which the host head adjoins to the raised head. Thus, NEG-to-T-to-C raising is represented with Pair-MERGE as in (20a–b). In (20a), T (i.e., the host) first adjoins to the raised NEG. In (20b), C (i.e., the host) adjoins to the raised amalgam $ \lt $NEG, T$ \gt $ and becomes invisible. As a result, NEG-to-T-to-C raising yields a complex head in (20b), where NEG is the only visible element.

Based on the assumption in Chomsky (Reference Chomsky, Domenico, Hamann and Matteini2015), NEG-to-T-to-C raising is formulated under Omune’s (Reference Omune2018) reformulation of Pair-MERGE as in (21a–b). Notice that in the complex head formed by Set-MERGE in (21b), NEG only c-commands {NEG, T} or T. Thus, in Omune’s analysis, the raised NEG cannot c-command the NPI in (15b) and (16b). One cannot resort to the definition of c-command in (17) here, since Omune’s (Reference Omune2018) analysis is entirely based on Set-Merge/MERGE, which discards segment-category distinctions. Accordingly, Omune (Reference Omune2018) wrongly predicts that (15b) and (16b) are ungrammatical since the NPIs should not be licensed by NEG raising; hence, an undergeneration problem arises.

In contrast, if we assume that a head-adjunction structure is created purely by Pair-MERGE as in (20), a raised head takes scope outside of a complex head under the definition of c-command in (17), repeated here as (22). Since Pair-Merge/MERGE incorporates the asymmetrical relation between $\alpha$ and $\beta$ in $ \lt $ $\alpha$, $\beta$ $ \gt $, it is possible to distinguish between segments and categories. Accordingly, the data in (15) and (16) are explained since the raised NEG that is visible in the amalgam $ \lt $ $ \lt $NEG, T$ \gt $, C$ \gt $ c-commands the NPI in both cases.Footnote ¹⁵

Again, the definition in (22) does not save Omune (Reference Omune2018) or Nakashima (Reference Nakashima, Bakay, Pratley, Neu and Deal2022) because (22) is incompatible with any ZFC-based reformulation of Pair-MERGE. That is to say, Omune’s and Nakashima’s analyses, which are departures from the classic asymmetric adjunction structure, do not distinguish segments from categories; hence, a head inside the complex head cannot c-command elements outside of the amalgam.Footnote ¹⁶

An anonymous reviewer has raised a question concerning (16b) as to why we do not assume the Vacuous Movement Hypothesis (VMH), which states that local overt wh-movement takes place except for subjects in English (cf. Chomsky Reference Chomsky1986). If VMH is assumed, the wh-subject would remain in the TP-domain, and the auxiliary would not undergo head movement for inversion. This possibility is certainly worth considering. However, we emphasize that our analysis holds true regardless of whether we assume VMH. The crucial point here is whether NEG c-commands the NPI in (15) and (16). Take (16) for instance. Even if wasn’t remains in T after NEG-to-T movement, it still c-commands the NPI anybody – and the phrase by anybody – in the v/VP-domain. Therefore, regardless of whether we assume VMH, the NPI anybody is c-commanded by NEG thanks to the syntactic head movement of NEG; hence, the assumption regarding VMH is orthogonal to our analysis in this article. We thank the reviewer for bringing this issue to our attention.