Numeral differential constructions in Arabic *

In sentences like Mary found three shells more than Joan did , three appears to syntactically modify the object noun shells yet is interpreted as a specification of the difference between the number of shells that Mary found and the number that Joan found. Accordingly, syntactic and semantic analyses of such cases cast the numeral as a modifier of more in the trans-formational base for this construction. In this paper, I present evidence from Arabic indicating that the numeral is a direct modifier of the plural noun in the counterpart construction in that language. I propose that derivational processes sever its thematic relation to the plural noun and reintegrate it as a modifier of the differential argument of the comparative morpheme. What makes this possible is raising of the numeral together with late insertion of the comparative degree phrase (DegP), as previously proposed for superlative DegPs.


Introduction
The English comparative morpheme -er seen in (1a) is conventionally taken to denote a relation between two sets of degrees, one contributed by the thanclause and one contributed by the matrix clause (Cresswell 1976, Heim 1985, and many others). On such an analysis, example (1a) has the LF in (1b), in which -er has moved together with the than-clause to a peripheral position, and whose meaning is paraphrased in (1c).
(1) a. Mary is taller than Joan. b. [er [than Joan is -tall]] [Mary is -tall] c. The degrees to which Mary is tall are a superset of the degrees to which Joan is tall.
As Hellan (1981), von Stechow (1984) and others point out, the comparative itself has a degree argument which represents the 'differential' of the two degree sets. This differential can be specified by a measure phrase like three centimeters in (2a). A natural syntactic account of differentials makes them a modifier of -er, as sketched in (2b).
(2) a. Mary is three centimeters taller than Joan.

b. [[three centimeters] er [than Joan is -tall]] [Mary is -tall]
When -er modifies a plural count noun, a bare numeral may specify the differential argument of -er. I refer to such cases as 'numeral differential constructions'. Bresnan (1973) attributes the base structure in (3b) to the sentence in (3a). The variant in (4a) suggests that extraposition of the than-clause can take the comparative quantifier -er with it (in the form of more, which I take to be an allomorph of -er). It cannot, however, take the differential with it, as (4b) shows.
(4) a. Mary swam five laps more than Joan swam. b. *Mary swam laps five more than Joan swam.
In the structure in (3b), the numeral does not actually modify the plural noun it precedes in the surface structure. It modifies the comparative morpheme -er, and is interpreted as a specification of its differential argument. (3a) only asserts that Mary swam five laps by accident of the fact that it asserts that she swam five laps more than an unknown quantity and that ( + 5) entails (5) by virtue of the downward monotonicity of degree predicates (Heim 1985). The counterpart of (3a) with fewer, in (5), does not 17:2 Numeral differential constructions in Arabic assert that Mary swam five laps, nor does it deny she swam five laps, meaning there is no semantic impetus for the constituency of swim five laps in (5), or by extension (3a).
Syrian Arabic has a construction with the format seen in (4a). However, the construction has properties which suggest that it does not display the constituency in (3b), but rather that in (6), where the numeral modifies the plural noun it precedes directly and the entire degree phrase is adverbial (adjoined to VP). Nonetheless, as in English, the numeral is interpreted as a specification of the differential argument of the comparative. This situation seems paradoxical. I present an analysis of this situation with two main ingredients. One is Kennedy & Stanley's (2009) and Kennedy's (2013and Kennedy's ( , 2015 analysis of numerals, which makes them degree quantifiers that may have scope external to the noun phrase that contains them in the surface structure. The constituency in (6) is transformed by movement into one in which the numeral is local to the comparative morpheme. This alone does not suffice to 'sever' the semantic relationship between the numeral and the plural noun. What accomplishes this is the second ingredient, namely a compositional step found in movement analyses of the superlative, in which a degree quantifier may be inserted between another quantifier and the associated abstraction index (Heim 1985(Heim , 1999. I claim, in effect, that five is integrated with the comparative at LF in (6) in the same manner as subjects of comparison are integrated with the superlative. Section 2 describes the Arabic data supporting the constituency in (6) (for Arabic) and Section 3 develops an analysis along the lines just sketched. Section 4 expands the empirical picture somewhat and Section 5 discusses the prospects for extending the analysis to other languages.

Comparatives and differentials in Syrian Arabic
The comparative morpheme in Arabic is the prosodic template aC 1 C 2 aC 3 , where C 1 -C 3 are the root consonants of the base adjective. By inserting the root consonants into the comparative template we derive, for example, aṫ wal 'taller' from ṫ awīl 'tall', aftaè 'brighter' from fātiè 'bright', aṡ Gar 'smaller' from ṡ aGīr 'small', etc. Regular phonological rules occasionally distort the shape of the template, such as glide deletion in aèla 'prettier' (underlyingly aèlaw) from èilu 'pretty' (underlyingly èilw) or metathesis as in aZadd 'newer' (underlyingly aZdad) from Zdīd 'new'. The comparative combines with a standard phrase headed by the preposition min 'from/of', which I gloss as 'than' here to highlight its standard-setting role in comparatives. Thus, aṫ wal min nabīl expresses 'taller than Nabil'. This is a 'phrasal' comparative; when the preposition min is followed directly by a DP, no additional sentence fragments may follow, suggesting the standard phrase does not contain an elided clause (see Al-Bitar 2019 for detailed arguments to this effect and footnote 3 for an example of the clausal comparative in Arabic, which is morphologically distinguished from the phrasal comparative). The comparative term aktar 'more' is derived from the base adjective ktīr 'much/many', mirroring the composition proposed for English more by Jespersen (1949), Bresnan (1973), Hackl (2000Hackl ( , 2009) and others as much+er. Its antonym is aPall 'less' (underlying aPlal), based on the adjective Palīl 'little'.
In Hallman 2016, I observe that the superlative morpheme in Syrian Arabic may appear at some distance from its scalar associate. This generalization applies to the comparative as well. Comparative aktar 'more' may occur with a standard (min 'than') phrase at the right clause edge, and find its scalar associate potentially at some distance to its left. 1 In (7), for example, the plural noun war @ d 'flowers' may function as the scalar associate for the comparative phrase aktar min nabīl 'more than Nabil' even though they are separated by the oblique indirect object la-sāra 'to Sarah'. The dependency between the comparative phrase and its scalar associate is sensitive to certain syntactic boundaries. Although a plural (or mass) DP may function as scalar associate for the comparative phrase, the dependency may not reach into a DP and find a scalar associate there; the dependency is bounded by DP. For this reason, the adjective èilu 'pretty' is not accessible as a scalar associate for aktar in (7). Presumably, the gradability in quantity of war @ d èilu 'pretty flowers' is a feature of the DP itself, projected from the noun, so that the dependency between the comparative phrase and its associate in (7) does not actually cross over the DP boundary, while the unattested association with the adjective would have to do so. Consequently, the speaker may assert with (7) that Muen gave more flowers to Sarah than Nabil did, 2 but not that the flowers he gave her were prettier than the ones Nabil gave her. 3 Nabil 'Muen gave more pretty flowers to Sarah than Nabil.' 'Muen gave prettier flowers to Sarah than Nabil.' In the course of interpreting (7), nabīl is made the subject of a predicate borrowed from the syntactic context, here gave d-many pretty flowers to Sarah. Example (8) shows that this predicate is copied from the constituent to which aktar min nabīl 'more than Nabil' is adjoined. The only gradable predicate in (8) is èilwe 'pretty (fs)'. The comparative phrase aktar min nabīl 'more than Nabil' may adjoin to it, but then èilwe is the only predicate available to fill in the predicate position in the standard clause, and the resulting interpretation is the pragmatically odd assertion that the shell that Sarah found is prettier than Nabil is. An interpretation in which we assert that Sarah found a shell that is prettier than any shell Nabil found is not available because although in principle the phrase aktar min nabīl may adjoin to the verb phrase, as it apparently does in (7), it may not then take èilwe 'pretty' as an associate, since the dependency between aktar and èilwe would cross over the boundary of the DP ṡ adafe èilwe 'pretty shell', which, as mentioned above (and as (8) shows), is not grammatical. I refer in what follows to the constituent consisting of the comparative morpheme aktar and the standard phrase min+DP 2 Several other interpretations are available for (7) that are not notated here. One has Nabil as the alternative object, that is, (7) can mean that Muen gave more flowers to Sarah than he gave to Nabil. Also, the potential pluractionality of the verb give may function as a scalar associate for the comparative, so that (7) means that Muen did more flower-giving-to-Sarah than Nabil did, or that Muen did more flower-giving-to-Sarah than he did flower-giving-to-Nabil. The quantity comparative reading of (7) entails that Muen gave Sarah more flowers than Nabil did, regardless of how many times they gave her flowers. 3 Syrian Arabic also employs a 'clausal' comparative, but this is morphologically distinguished from the phrasal comparative by the complementizer ma following the preposition. Assimilation with min produces the compound preposition/complementizer mimma. In the clausal comparative, Arabic tolerates ellipsis only of the scalar associate itself, as in (i), which paraphrases the meaning of (7).   (7) and (8) show that the comparative phrase may adjoin to VP in the surface structure in Arabic as long as no DP boundary intervenes between it and its scalar associate (though the scalar associate may itself be a plural DP, as it is in (7)). Nabil 'Sarah found a prettier shell than Nabil is.' (!) 'Sarah found a prettier shell than Nabil found.' As in English, a measure phrase can specify the differential argument of a comparative adjective in Syrian Arabic. But a measure phrase may not precede the comparative phrase on the model of English, as (9a) illustrates. Rather, it follows the comparative phrase in a prepositional phrase introduced by the preposition b-'by ', as (9b) illustrates. It is clear on the basis of (9b) that the comparative adjective aṫ wal 'taller' has a degree argument corresponding to the differential of the two heights it compares. I therefore assume that the ungrammaticality of (9a) is traceable to a syntactic condition, probably related to the fact that Arabic is a relatively strongly head initial language. centimeters 'Sarah is taller than Muen by three centimeters.' And as in English, in the quantity comparative, where the comparative has a plural count noun scalar associate, a numeral may specify the differential argument of the comparative, as (10) illustrates. Nabil 'Sarah found three more shells than Nabil.' But in Arabic, there is evidence for a selectional relationship between the numeral and the following noun that belies an analysis of (10) along the lines of Bresnan's approach to English. Consider a reduced version of (10) without the comparative or the numeral. 17:6 Numeral differential constructions in Arabic (11) sāra Sarah laPi-t found ṡadaf shell COLL

Qa-S-Sat˙t˙.
on-the-beach 'Sarah found shells on the beach.' The noun ṡ adaf 'shells' in (11) is one of a large class of lexically collective nouns in Arabic. It is morphologically singular, but only pluralities fall under its denotation. Singular count nouns can be derived systematically from such collectives by the addition of the feminine singular suffix -e (underlyingly -et). In this manner, ṡ adafe 'shell' is derived from ṡ adaf 'shells', warde 'flower' is derived from war @ d 'flowers', SaZare 'tree' from SaZar 'trees', etc.
Such singulars are sometimes referred to as 'singulatives' (Greenberg 1974, Ojeda 1992, Fassi Fehri 2018. These singulatives can also be pluralized in the regular manner for feminine singulars, which involves lengthening the vowel of the singular suffix (which then lowers slightly) and pronouncing the underlying -t. In this manner, the plurals ṡ adafāt 'shells', wardāt 'flowers' and SaZarāt 'trees' are derived from sadafe, warde and SaZare respectively. The plurals of the singulatives have a different distribution from the underlying collectives. Crucially for the present purposes, the collectives may not be modified by a numeral; the plural of the corresponding singulative is used for this purpose, as (12) illustrates. on-the-beach 'Sarah found three shells on the beach.' That is, the numeral exerts a selectional effect on the morphological form of the following noun. This effect is found even in numeral differential constructions like (10), where the numeral appears to be specifying the differential argument of the comparative rather than the cardinality of the noun. The noun nonetheless appears in the form required by the numeral. A collective noun is impossible in the context in (10), as (13) shows. Nabil ('Sarah found three more shells than Nabil.') Note that the comparative itself is compatible with a collective noun as a scalar associate, as (14) shows. It is the numeral that requires the plural of the singulative form of the noun. Nabil 'Sarah found more shells than Nabil.' The selectional relationship between the numeral and noun has an additional facet: in Arabic, numerals above ten require their nominal restriction to be morphologically singular, or singulative if the base noun is collective. This fact carries over to numeral differential constructions, as (15) shows. Nabil 'Sarah found fifteen more shells than Nabil.' The selectional relationship between the numeral and noun seen in Arabic militates against an analysis along the lines of Bresnan's (1973) analysis of similar constructions in English sketched in (3b). According to such an analysis, tlet 'three' would modify the comparative phrase in the base structure, as schematized in (16a). The surface structure would be derived by post-posing the comparative phrase, stranding the numeral, as in (16b). But in this analysis, tlet 'three' does not modify ṡ adafāt 'shells' at any level of representation, and the selectional relationship between the numeral and noun is not expected. One possible way of reconciling the selectional restrictions the numeral places on the noun with the fact that the numeral seems to specify the differential argument of the comparative is to characterize the whole DP tlet ṡ adafāt 'three shells' as a measure phrase modifying the comparative, so that (10) does not have the constituency in (16a) but rather that in (17).
Here, although three shells is a constituent and therefore compatible with the selectional restrictions holding between the numeral and modified noun, the DP as a whole does not function as the object of the verb found but rather is functioning as a measure phrase specifier of more. But we have seen in (9a) that measure phrases in Arabic are disallowed in the pre-adjectival position. The pair in (9) demonstrate that measure phrases only occur following the 17:8 Numeral differential constructions in Arabic comparative phrase in a prepositional phrase headed by b-'by'. On the analysis of (10) that attributes the constituency in (17) to it, three shells is acting as a measure phrase in the pre-adjectival position, just what (9a) shows is excluded in Arabic. Nor can it follow the comparative phrase in a prepositional phrase, as (18) shows. The fact that tlet ṡ adafāt 'three shells' precedes the adjective aktar in (10) points to the conclusion that it is not functioning as a measure phrase there, but rather as the object of the verb. shells ('Sarah found three more shells than Nabil.') It supports this point that the constituency in (17) does not lend itself to a degree semantic analysis of the meaning of (10), a point that Bhatt & Homer (2019) and Homer & Bhatt (2020) have dealt with in detail. In order to provide the standard phrase with a predicate for Nabil, the entire phrase three shells more than Nabil in (17) must move out of the matrix verb phrase. Then, three shells describes the difference between the denotation of the remnant of movement of the form Sarah found x and the standard phrase Nabil found x. These phrases denote sets of things rather than degrees, whose cardinalities must be compared in some manner, and the sortal specification of the differential carried over to the whole comparison (otherwise (10) would mean that Sarah found as many things as Nabil, and in addition, three shells). Bhatt & Homer (2019) pursue an analysis along these lines for French, where they also find evidence for the constituency of Numeral+NP in numeral differential constructions. They point out that while ordinary quality comparatives with differentials resemble English, as shown in (19a), in numeral differential constructions the comparative phrase, headed by plus 'more', must in turn be introduced by the preposition de 'of', as shown in (19b They conclude that trois livres in (19b) is a DP functioning as direct object of the verb, rather than as a measure phrase modifying plus, and deduce that the comparative phrase de plus que Jean is a modifier of trois livres. They propose that at LF, trois livres de plus que Jean undergoes quantifier raising, deriving the structure in (21), and develop a non-degree analysis of the interpretation of such LFs based on Li's (2015) analysis of similar structures in Mandarin. Specifically, they posit a non-degree counterpart of plus that combines with three individual predicates: que Jean a lu, the set of things Jean read (the VP is recovered from the matrix clause; Bhatt and Homer analyse French comparatives as clausal comparatives with ellipsis), trois livres, the set of pluralities of three books, and finally the matrix clause Marie a lu, the set of things Marie read. Plus says that the second holds of the sum of things in the third that cannot be mapped to a counterpart in the first in a one-to-one manner. That is, if we exhaustively map the things Mary read to things Jean read, three books are left over. An additional condition requires that the mapping preserves the taxonomic level of the second argument trois livres, ensuring that the things Marie and Jean read are books.  Since this analysis does not make reference to degrees, it does not involve the normal comparative degree quantifier, nor the quantity adjectival base the comparative is built on in many languages ('much' underlying 'more'). This is compatible with the morphologically opaque formative plus in French, though it posits an additional meaning for plus only found in numeral differential constructions. For Arabic, this step is particularly suspicious, since the comparative morpheme that occurs in such constructions is the same as that that occurs in ordinary comparative constructions, and this morpheme 17:10 Numeral differential constructions in Arabic is morphologically built on the quantity adjective ktīr 'much/many'. Putative non-degree aktar in Arabic would have the same internal morphological composition as the true degree quantifier aktar, one based on a degree adjective ktīr. This suggests that eschewing degrees in the interpretation of quantity comparatives with differentials might be unwarranted, at least for Arabic.
In fact, Homer and Bhatt themselves reach this conclusion in a follow-up paper (Homer & Bhatt 2020). There, they point out that French comparatives display scope ambiguities that point to a degree quantifier analysis of the comparative, even in numeral differential constructions of the kind seen in (19b). Example (22a), for example, has a reading in which plus 'more' scopes over veut 'want', meaning the number of books Marie wants to read exceeds the number of books Jean wants to read by three, without entailing the existence of any particular books they want to read. The same can be said of the Arabic counterpart in (22b). Nabil 'Sarah wants to read three books more than Nabil.' On a degreeless treatment, moving the DP trois livres de plus que Jean derives the structure in (23), which, on analogy to the discussion of (21) above, holds when three books are left over after we map the books Marie wants to read to the books Jean wants to read. To determine whether this holds, we must map particular books that Marie wants to read to particular books Jean wants to read, but the existence of such books is precisely what (22a) does not entail on the relevant reading. This supports a degree quantifier analysis of plus after all, even in numeral differential constructions. This fits well with Arabic, where the degree-based morphological composition of aktar is transparent. Consequently, Homer & Bhatt (2020) propose that plus is a degree quantifier in these cases as elsewhere, defined in (24). It relates two degree predi-17:11 Peter Hallman cates and a differential (unnumbered in (24), the role that trois livres ends up filling).
Movement of trois livres de plus que Jean leaves a degree denoting trace in an argument position. However, that trace is converted by a 'degree to stuff' operator (Op in (25)) into an individual whose cardinality corresponds to the degree denoted by the trace. This individual serves as argument of the verb. The LF of (19b) has roughly the form in (25)  The standard phrase and the main clause are interpreted as degree predicates ( 1 and 2 respectively in (24)), with an elided VP in the former case, identical to the matrix VP. Plus combines with these two degree sets and a predicate of degree sets (the unnumbered in (24)), corresponding to the differential, trois livres in (19b). For its part, trois livres has a different meaning as a differential than as an ordinary quantifier. The differential trois livres is derived by the operator M defined in (26) which takes the numeral trois and the noun livres as separate arguments. The noun argument serves merely to identify the units of the scale of measurement. The expression ( ) returns the size of and is the point on the scale (of books in (25)) that corresponds to ('three' in (25)). The degree quantifier M (trois)(livres) holds of a degree set when the measure of is three on a scale whose units are books. Here we are using 'book' in an unconventional way, as the unit basis for a scale.
As a result, (25) is true when the measure phrase M (trois)(livres) holds of the set of degrees such that Marie read stuff with cardinality and Jean didn't read stuff with cardinality . The measure phrase trois livres sets the dimension on which the differential is measured to books. Homer and Bhatt claim that this dimension specification is transferred down to the degreeto-stuff operator, ensuring that the stuff that Marie and Jean read is also books. The nature of this transference mechanism is left unspecified but could perhaps be characterized as a kind of binding. More critically, it is unclear whether measuring stuff in book units suffices to ensure that the mea-17:12 Numeral differential constructions in Arabic sured stuff is actually books. Canonical measure phrases like three bottles in three bottles of wine do not have this property; this phrase refers to wine, not bottles. By analogy, we do not expect I read three books of stuff, to the extent this is interpretable, to entail that I read books. But precisely such an entailment is at the heart of Homer and Bhatt's analysis. For these two reasons, it is unclear whether making trois livres a measure phrase in (25) suffices to ensure that we are comparing Marie and Jean in terms of how many books they read, and whether the assumption that we can only use book-units to measure out books and not other things might preclude us from measuring out wine in bottles. Lastly, the analysis gives a different constituency and semantic interpretation to trois livres when it is used as a measure phrase than when it is used as a quantifier, in which trois livres is not actually a constituent, but rather, the numeral and noun are each arguments of a third, covert, element M . Even this term is embedded under the degree-to-stuff operator in the base structure. It is unclear then whether we should expect the term Op(M (trois)(livres)) to pattern like the ordinary object trois livres with respect to, for example, subextraction and other evidence Homer and Bhatt discuss implicating that trois livres acts as a syntactic object of the verb in (19b).
Particularly this last point makes Homer and Bhatt's analysis difficult to carry over to Arabic. If tlet 'three' and ṡ adafāt 'shells' do not form a constituent in (10), but rather function individually as arguments of M , then we have no way of enforcing the selectional restrictions that are found between the numeral and the noun in Arabic. If we were to define M to combine with the whole DP tlet ṡ adafāt we are left with no easy way of excluding tlet from the specification of , so that we would be measuring out shells in units of 'three shells'. A solution to this problem, whatever it may be, would not allay the other concerns expressed above. I conclude firstly that the hypothesis sketched for Arabic in (16a), where tlet 'three' directly modifies the comparative, is untenable because it does not capture the selectional relationship between the numeral and noun. Secondly, the hypothesis in (17), where tlet ṡ adafāt 'three shells' modifies the comparative, is compromised by dissimilarities to bona fide differential modifiers (of comparative adjectives). Further, the semantic composition for this constituency proposed by Homer and Bhatt is both not easily reconcilable with the selectional facts of Arabic and has other potential drawbacks. These considerations warrant an at least cursory look at possible alternatives. I present such an alternative in the following section.

Analysis
I start with the premise motivated above that tlet ṡ adafāt 'three shells' is a constituent in (10) and that the comparative phrase aktar min nabīl 'more than Nabil' is functioning as an adverb, as the examples in (7) and (8) (27). But it is clear that tlet 'three' in this context is interpreted as a specification of the differential argument of the comparative. How is it possible for the numeral in (10), which selects the morphological form of the following noun, to semantically modify not that noun but the comparative aktar 'more'? I claim here that existing tools of semantic analysis provide an answer to this question. Heim (2006) proposes that the core meaning of the comparative morpheme is a containment relation between the denotations of the matrix clause and of the standard phrase (headed by than in English), both interpreted as degree predicates. The English sentence Sarah is taller than Muen is true if and only if the degrees to which Sarah is tall properly include the degrees to which Muen is tall. As mentioned previously, the comparative constructions under investigation here are phrasal comparatives (clausal comparatives are marked differently; see footnote 3). Consequently, I amalgamate Heim's (2006) proposal with an earlier proposal for phrasal comparatives (Heim 1985), where the comparative combines with two individuals and a degree relation (a relation between and degree and an individual). I define the Syrian Arabic comparative morpheme (the template aCCaC) accordingly as in (28), as a relation that combines with two individuals and a degree relation, but where the comparative claim is encoded as the superset relation between degree sets. I will later augment this definition with a differential argument. In all the formulas to follow, the symbols and represent variables of type ('ordinary' individuals) and is a variable of type (for 'degree'). All other variables are subscripted with their semantic type. (7) is repeated in (29)  Nabil (i) 'Muen gave more flowers to Sarah than Nabil gave to her.' (ii) 'Muen gave more flowers to Sarah than he gave to Nabil.' These interpretations are standardly assigned the same base structure, that in (30a), represented with English morphemes for perspicuity. The first interpretation is derived by extraction of Muen from VP, making Muen the subject of the comparison. The second interpretation is derived by extraction of Sarah. In both cases, the aktar-phrase moves to the level of VP, the constituent interpreted as a degree relation. VP is interpreted as a degree relation by virtue firstly of extraction of a DP (Muen or Sarah), which leaves an individual-denoting trace, and secondly extraction of the DegP more than Nabil, which leaves a degree-denoting trace. The interpretation in e.g. (30c) composes as shown in the tree in (31), where movement of the subject of comparison Sarah and the degree quantifier aktar min nabīl 'more than Nabil' is in each case accompanied by insertion of an abstraction index (labeled Op), whose role is to derive a predicate over the remnant of movement; I expand on this below. This semantic analysis for the comparative is premised on the assumption that degree predicates are downward monotone; if a degree predicate holds of a degree it holds of every degree less than . I also assume the preposition min 'of/from' is vacuous. On these assumptions, if Muen gave Sarah five flowers and he gave Nabil three, then the sentence is true, because the set { | Muen gave -flowers to Sarah} is the set {1, 2, 3, 4, 5} and the set { | Muen gave -flowers to Nabil} is the set {1, 2, 3} and the first is a superset of the second. muQīn Qat˙a -war @ d la- While the tree in (31) composes properly to derive the intended interpretation, it displays a derivational step that is unusual for quantifier raising but typical for movement analyses of degree quantifiers. Raising of the subject of comparison Sarah accompanied by insertion of the abstraction index Op is followed by raising of the DegP aktar min nabīl 'more than Nabil' accompanied by insertion of the abstraction index Op . But DegP-raising targets a position between the subject of comparison Sarah and its abstraction index Op . Inserting DegP in this position fails to 'extend' the tree in the way typical of syntactic movement (Chomsky 1993).
This same failure to extend the tree by DegP movement is found in movement analyses of the superlative. Heim (1999) attributes the meaning in (32a) to the English superlative morpheme -est, which combines with a degree relation and an individual. But in order to apply the superlative morpheme to a degree relation in a sentence like (32b) (on the interpretation indicated there), we need to insert the superlative and its abstraction index Op between the subject of comparison Sarah and its abstraction index Op , as (32c) shows. 4 4 Bhatt & Takahashi (2007) suggest that this movement step in phrasal comparatives and superlatives exemplifies a type of movement that Richards (1997)  The possibility of inserting a quantifier between a term and its abstraction index plays a crucial role in the analysis of numeral differential constructions I present below, as does an analysis of numerals as degree quantifiers to which I now proceed. Kennedy & Stanley (2009) and Kennedy (2013Kennedy ( , 2015 a constituent moves to a position directly subjacent to the constituent that moved in the previous movement step, which is either at the left clause edge or itself tucked in under a constituent at the left clause edge. Semantically, the cases of tucking in that Richards discusses (as well as Bruening (2001), who discusses quantifier raising in detail), involve inserting a quantifier beneath the abstraction index of the higher quantifier, not between the higher quantifier and its abstraction index. This means that the kind of movement discussed here and in Bhatt & Takahashi 2007 is different from the classical cases of tucking in. One possibility though, is that tucking in may optionally target a position either above or below the abstraction index of the higher quantifier, on the condition that the result is interpretable. This condition rules out interpolation of ordinary generalized quantifiers. If movement of someone in (i), generating the lambda expression beginning …, is followed by movement of everyone to a position between someone and its abstraction index, generating …, the result is a type mismatch. Therefore, this formulation of tucking in unifies degree operator movement with movement of other quantifiers that undergo tucking in without overgenerating (composable) interpretations. analyze numerals as degree quantifiers that potentially display scope ambiguities with other quantifiers. The numeral three, for example, combines with a set of degrees and asserts that the maximal degree in is 3 (Kennedy 2013: p. 24).
Numerals defined on the model of (33) are scopally flexible. Kennedy (2013) points out that a sentence like (34a) (his example (36), p. 24) can be interpreted to mean that Kim must take exactly three classes, or that three is the minimum number of classes she can take. If three scopes below required the sentence asserts that in all worlds where Kim meets the requirement, the maximum number of classes Kim takes is three. This means she cannot take more than three classes, but neither can she take less than threethe assertion that the maximum number of classes she takes in all the 'required' worlds is three entails that she does in fact take three classes in those worlds. This is the 'exactly three' reading. If three scopes above the modal as sketched in (34c), it means that three is the maximum number such that Kim takes -many classes in all worlds where the requirement is met. This is compatible with Kim taking four or more classes in at least some (but not all) of those worlds. This the 'minimum requirement' reading. The Syrian Arabic counterpart to (34) displays the same ambiguities, which I assume are derived in the same manner. classes 'Sarah must take three classes.' The wide scope interpretation of the numeral is also found in numeral differential constructions. Suppose that Sarah and Nabil are collecting signatures for a petition, and they have set themselves the goal of collecting 100 signatures each. Nabil has already collected 90 signatures, and Sarah 87. In this situation, Sarah needs to collect three more signatures than Nabil needs to collect to reach her goal. The sentence in (36) describes this situation. Nabil 'Sarah must collect three signatures more than Nabil.' In this context, (36) has the same 'upstairs de dicto' interpretation that its English translation has, in which tlet 'three' and the comparative itself are interpreted above the modal, so that the degree predicates being compared include the modal (hence 'upstairs'), but the restriction stays in situ, so that the degree predicates being compared include the restriction tawāqīQ 'signatures' (hence 'de dicto'; this terminology is due to Sharvit & Stateva (2002)). I conclude that as in English, numerals in Syrian Arabic may take scope independently of their restriction, i.e., they may move to a higher scope position, leaving their restriction behind.
Kennedy does not discuss the manner in which plurals are integrated into the argument structure of the verb. I assume that plural objects are integrated by Chung & Ladusaw's (2004) 'Restrict' operation, after movement of the numeral (or type lowering -see below). This operation composes a one-place predicate (the object) with a two-place relation (the verb) by unifying the argument of the one-place predicate with the internal argument of the two-place relation and existentially closing the unified argument. I follow Cresswell (1976) and others in casting plural (and mass) nouns as relations between an individual and a degree representing its 'measure', simply its cardinality in the case of count nouns. The analysis I construct here is compatible with the idea that the degree argument of plurals is contributed by a covert morpheme, e.g., Kayne's (2005) NUMBER or Rett's (2006) COUNT and MEASURE (for count and mass nouns respectively). The base tree in (37a) for the VP collect three signatures (in English; Arabic is parallel) cannot be composed as such. But movement of the numeral three to a higher position (not notated) leaves a degree-denoting trace ′ , which functions as argument of the plural noun, as illustrated in (37b). The result is combined with the verb by the Restrict operation, as (37b) illustrates. Henceforth, I abbreviate the denotation of plural NPs like signatures below as ' .signatures( , )'.

17:19
Peter Hallman As an alternative to movement of three, Kennedy (2015) mentions the possibility for such degree quantifiers to type-lower to a degree-denoting term. Application of Partee's (1987) BE ( ⟨⟨ , ⟩, ⟩ . ( . = )) and iota ( ⟨ , ⟩ . [ ( )]) type changing schemas to the expression in (33) reduces to the degree-denoting expression . = 3 (see Kennedy 2015, pp. 19-20). This term can be interpreted in the base position of three in (37a), allowing the Restrict operation to apply to V and NP without movement, though then three is scopally inert.
To capture the behavior of differentials we of course need an analysis of the comparative that includes a differential argument. The definition in (38a) is based on that in (28) but includes a degree argument corresponding to a measure of the symmetric difference between the two degree sets that are put in the superset relation. The symmetric difference, derived by the operator Δ, is the union of the two sets minus the intersection, as defined in (38b). It is 'symmetric' in the sense that it returns the difference between the two sets regardless of which one is larger, which is crucial for the interpretation of 17:20 Numeral differential constructions in Arabic differentials in the context of fewer, which I discuss in Section 4. The symbol ' ' in (38a) is a function that maps a set of things to the set of degrees representing a tally of the things in . If has three things in it, ( ) is the set {1, 2, 3}. 5 Combining the comparative with the standard-setting individual derives a degree relation modifier: a term that combines with a degree relation and yields a new degree relation. This new degree relation relates an individual to the maximum degree of difference between two degree sets. In treating the comparative phrase as a modifier, this analysis bears a resemblance to Kennedy & McNally's (2005) modifier analysis of the comparative. There, the comparative phrase derives a scale whose minimum value is defined by the standard phrase. In the analysis in (38a), it is that has the effect of recalibrating the scale to the difference between the two degree sets being compared. Also following Kennedy and McNally, I assume that when no overt description of the differential appears, either in the form of a numeral differential or otherwise, then the differential degree argument of the comparative in (38a) is closed by a covert existential quantifier. That is, when no overt differential phrase appears, the comparative requires the symmetric difference to be non-empty.
The last component of the analysis of the numeral differential constructions concerns the base structure position of the comparative phrase headed by aktar 'more'. In the base structure postulated by Bresnan for examples like the ones under consideration here, sketched in (39a) (on the model of (3b) above), the constituent sitting in the degree argument slot of the plural shells is the whole comparative DegP three more than Nabil. According to that analysis, this DegP moves and binds a degree variable left behind in its base position, the degree argument slot of shells. The numeral sits in the degree argument slot of the comparative and if it moves, it binds a variable in that position. In the constituency motivated for Syrian Arabic though, shown 5 I assume that sets of degrees may differ in how coarsely they measure out the stuff they correspond to. If comprises non-discretely measurable material, then a dense set of degrees is put to use and is a correspondence between the linear order of degrees and the subpart structure of . I deal here only with discrete degree sets that model counting. In either case, in (38a) 'measures out' the difference between and as a set of degrees.

17:21
Peter Hallman in (39b), the numeral itself sits in the degree argument slot of the plural in the base structure.  (10), then the DegP corresponding to -er than Nabil is not base generated in the degree argument slot of the plural shells; that position is occupied by three. I show below how the DegP nonetheless manages to bind the degree variable argument of shells at LF. But in the base structure, the DegP -er than Nabil must be base generated in a higher position. The example in (29), where the entire DegP including both the comparative morpheme and the standard phrase occurs to the right of the second argument of the ditransitive verb Qaṫ a 'give', suggests that the DegP occurs in the surface structure at least as high as VP, just where it is interpreted, as illustrated in the tree in (31). I propose it is base generated there in numeral differential constructions. In the numeral differential example (10), repeated in (40) below, the DegP aktar min nabīl 'more than Nabil' compares Sarah and Nabil in terms of how many shells they found. The numeral tlet 'three' is not part of this description; it is interpreted external to the constituent that DegP modifies. This situation is similar to the situation that Kennedy treats in (34): three is not part of the description of what Kim is required to do on one reading of (34). In this reading, three is displaced to a position outside of that description. I claim that similarly, in Arabic, tlet 'three' is removed by movement to a position outside of the VP to which the comparative DegP is adjoined. Recall now that that DegP is semantically a modifier: it maps a degree relation (whose degree argument specifies a number of shells in (40)) to another degree relation (whose degree argument specifies the differential). This sets up the possibility for displaced tlet 'three' to be re-integrated with the VP above the comparative DegP, since the VP derived by DegP adjunction has the same logical type as the underlying VP (that of degree relation ⟨ , ⟨ , ⟩⟩). Nabil 'Sarah found three more shells than Nabil did.' I propose, then, that the numeral tlet in (40) moves out of the VP to a position where it saturates the degree argument of DegP (the specification 17:22 Numeral differential constructions in Arabic of the differential), which is itself adjoined to VP. What makes the semantic reintegration of the numeral as a specification of the differential argument of DegP possible is the fact that quantificational material -here DegP -may be inserted between a moved term and the associated abstraction index, as discussed above in connection with the integration of the phrasal comparative and superlative (see the discussion surrounding (31) and (32c)). There, the comparative or superlative DegP is inserted between a moved subject of comparison and its abstraction index. The constituent derived by adjunction of DegP is itself a predicate, which then takes the moved DP as argument. I propose something similar for numeral differential constructions, except that what moves is the numeral, itself a degree quantifier. The steps involved are spelled out in detail below. We begin the derivation of the sentence in (40) by constructing the VP in (41). In the second step, the numeral tlet 'three' moves and adjoins to VP, interpolating itself between the moved subject Sarah and its abstraction index Op and triggering insertion of its own abstraction index Op , as illustrated in (43), on the model of movement of the comparative and superlative quantifiers in (31)  to type-lower to degree-denoting expressions, as Kennedy (2015) suggests, tlet 'three' could in principle be interpreted in this position, as the degree argument of the degree relation it is predicated on, or it could move further leaving a degree-denoting trace. Crucial for the present purposes is that insertion of the abstraction index over degrees accompanying movement of tlet 'three' sets the stage for insertion of the comparative quantifier in the next step. We can now adjoin the DegP aktar min nabīl 'more than Nabil' to the highest VP in (43), which at this point denotes a degree relation, just what DegP is looking for. This step interpolates DegP between the quantifier tlet, which moved to its derived position in the previous step, and its abstraction index Op . 6 At this point, the denotation of the parent node to DegP can be derived.
6 Insertion of DegP at this point in the derivation bears a resemblance to Bhatt & Pancheva's (2004) 'late merger' of the standard clause in English comparatives. They claim that a sentence like Ralph is taller than Flora starts out as the sentence Ralph is -er tall, tall being the degree relation argument of -er. Then, -er moves (rightward) to a scope position, deriving Ralph is tall [-er] . At this point the standard clause is merged as the second argument of -er, deriving Ralph is taller [-er than Flora] . This differs slightly from the proposal made here, in which the whole comparative phrase, including the comparative morpheme and the standard clause, merges with VP after movement of the numeral. In the analysis of numeral differentials presented here, the comparative morpheme aktar corresponding to -er cannot 17:24 Numeral differential constructions in Arabic aktar min nabil more than Nabil The VP so derived is looking for a degree, but is combined with the degree quantifier tlet 'three'. At this point, we can either lower the quantifier tlet to a degree-denoting expression, or move it, leaving a degree-denoting variable in its place, shown in (45). Now the whole tree can be composed, and this composition results in the desired reading according to which tlet 'three' is be base generated in the degree argument position of the associated scalar predicate (the plural noun in the examples discussed above) because the numeral tlet 'three' is generated there, and movement of tlet derives the degree relation that the comparative is looking for, not movement of aktar '-er' itself. This analysis does, however, have in common with Bhatt and Pancheva's analysis that it orders a covert step (raising of -er in their analysis and of the numeral tlet here) before an overt step (merger of the standard clause in their analysis and of the whole comparative quantifier aktar min nabīl 'more than Nabil' here). Bhatt and Pancheva invoke the copy theory of movement to accommodate this ordering disparity. According to the copy theory of movement, an overtly moved constituent may be pronounced in the position it moved from, mimicking covert movement (Chomsky 1995). On this view, the tree is (44) is derived prior to spell out, but the numeral tlet is pronounced at the 'foot' of the movement chain, adjacent to the noun.

17:25
Peter Hallman interpreted as a differential. In this tree, tlet identifies the differential argument of aktar 'more' and aktar binds the degree variable left by movement of tlet.
This analysis therefore derives the attested interpretation for numeral differential constructions exemplified by (10)/(40). Again, the crucial ingredient in this analysis is one that seems to be necessary for the composition of garden-variety (phrasal) comparatives and superlatives, namely insertion of a quantifier between another term and the associated abstraction index. I claim that this is possible both as a result of movement ('internal merge'), as in the phrasal comparative in (31) and the superlative in (32c), and as a result of base generation ('external merge'), as in my analysis of numeral differential constructions. In the latter case, re-association of a displaced numeral with the comparative seen in sentences like (40) is expected, by virtue of the composition in (45).
The analysis therefore does not invoke novel semantic tools, but draws on those previously proposed to account for phenomena in which quantificational and descriptive material split apart at LF. Kennedy's analysis of numeral raising that I adopt here bears a resemblance to earlier analyses of sentences like (46a) (Kroch's (1998) example (12b), p. 26), which may be understood to be asking for what number the editor decided to publish -many books this year (Cinque 1990, Cresti 1995, Rullmann 1995, Kroch 1998. Here, the interrogative force of how many is external to VP but existential quantification over books is within the scope of decide. Similarly, Sauerland (2004) proposes a theory of the interpretation of traces that treats them as copies of the descriptive but not the quantificational content of the antecedent, so that the VP that is recovered as the content of the gap Δ in (46b) (Sauerland's example (65), p. 87) contains the restriction towns but not the partitive quantifier two of the. The phenomenon I analyse here also bears a resemblance to 'event-related' readings of sentences like (46c). On this reading, (46c) (Krifka's (1990) example (1a), p. 487) holds when four thousand events of a ship passing through the lock took place last year, even if some ships passed through repeatedly, so that less than four thousand ships passed through total. Krifka postulates a null operator that applies to the nominal 17:27 4000 ships and converts it into a quantifier over event descriptions. The net effect is that the numeral 4000 counts events and ships is interpreted existentially within the VP.
(46) a. How many books did the editor decide to publish this year? b. Erik visited two of the towns Polly did Δ. (Δ=visit towns) c. 4000 ships passed through the lock last year.
Like my proposal, these analyses treat discontinuities in the integration of the descriptive content of VP (the verb and its objects) on the one hand, and the integration of information concerning quantification and reference on the other. I have claimed that numeral differential constructions are an instance of this general phenomenon, but that they represent a situation in which the composed descriptive content of VP may be modified by additional descriptive material (the comparative phrase) before the displaced numeral is re-integrated. This is possible by virtue of the fact that the comparative phrase is type-logically a modifier, as Kennedy & McNally (2005) claim, and may intervene between a moved term and its abstraction index, as Heim (1999), Bhatt & Takahashi (2007), and others claim.

A few more observations
In this section, I demonstrate that the analysis fleshed out above extends to the 'negative' comparative aPall 'less' in Arabic and discuss counterparts of many, few and a few in the role of the numeral in numeral differential constructions. Beginning with less than comparatives, example (47) describes the same situation as (40). Sarah 'Nabil found three fewer shells than Sarah.' The term aPall is morphologically composed of the comparative morpheme -the template aCCaC and the negative quantity adjective Palīl 'little/few'. I attribute the definition in (48a) to Palīl, a variation on the analysis in Heim 2006 for English little. It combines with a degree relation and builds a degree relation over the negation of . I define ktīr 'many/much' as its positive counterpart in (48b), based on Solt 2015. I incorporate these terms into the comparative by virtue of the definition for aCCaC in (48c), which is the same as that in (38a) except that it combines first with a marker of 'polarity' -either ktīr or Palīl -in addition to its other arguments. The negative comparative quantifier phrase aPall is defined in (48d) as the combination of aCCaC with Palīl, while aPall min sāra 'less than Sarah', as it appears in (47) and the trees below, has the definition in (48e). At this point we have derived a degree relation to which aPall min sāra 'less than Sarah' can attach, again under the constituent we have just moved, as illustrated in (52).
aPall min sāra less than Sarah As before, at this point we can type lower tlet 'three' to an individual degree or syntactically raise it to a position above the subject leaving an individual-denoting trace, as illustrated in (53), to create a structure that can be fully composed.

17:31
Peter Hallman (53) S 3 = the maximum such that: laPa -ṡadafāt aPall min sāra found -shells less than Sarah 17:32 Numeral differential constructions in Arabic Suppose, as before, Nabil found two shells while Sarah found five. So { | ¬Nabil found -shells} = {3, …, ∞} and { | ¬Sarah found -shells} = {6, …, ∞}. Nabil's set is a superset of Sarah's set so the comparative claim is true (that Nabil found fewer shells than Sarah did, or to be exact, that Nabil is greater than Sarah in terms of how many shells he didn't find). The symmetric difference between the two sets (which again is the union {3, …, ∞} minus the intersection {6, …, ∞}) is {3, 4, 5}, just as in the 'positive' counterpart in (45). The function applied to this set returns {1, 2, 3}, the maximal element in which is 3. So the comparative claim and the differential claim are both true in this situation, as desired.
This analysis attributes a different type of denotation to ktīr 'much/many' and Palīl 'little/few' than to the numerals like tlet 'three'. The former are degree relation modifiers (they map a degree relation to a degree relation) and the latter are degree quantifiers (they map a degree predicate to a truth value). This raises the question of how ktīr and Palīl behave as differentialspecifiers. At issue here are the Arabic equivalents of sentences like (54) in English. To this end, I expand briefly on the distribution of ktīr and Palīl.
(54) Sarah found many/few more shells than Nabil.
The terms ktīr and Palīl may occur as sentence predicates, as (55) shows (here in their plural form ktār and Plāl respectively) in addition to their use as a base for the comparative.  The term ktīr occurs naturally as an adverb but not Palīl. Again, ma . . . ktīr 'not …much' is preferred.  The notion 'a few/a little' is expressed by Sweyyit in Syrian Arabic, etymologically the diminutive form of Si 'thing'. It occurs prenominally, as in (59a) (parallel to ktīr in (56a)), as an adverb, as in (59b) (the -it suffix drops predictably when Sweyyit does not directly precede a noun), but not predicatively, as (59c) shows (parallel to English 'a few' in the translation), nor as a post-nominal adjective, as (59d) shows. The term Sweyyit makes an existential claim. Native speakers report that if I have no friends at all, I cannot say (59a) even facetiously. In this respect, Sweyyit corresponds to a few and not few, a correspondence that is supported by the impossiblity of its occurrence in predicate position (59c), just like English a few. In principle, Arabic has a counterpart to few, namely Palīl, used productively as a predicate and as a base for the comparative (and the superlative, not discussed here), but for reasons that remain unclear, not as an adjective.
To return to differentials, noun-modifying Sweyyit can be used readily to designate a differential, and patterns like the numerals in this respect, though unlike the numerals, it is compatible with a collective noun. (60) describes the difference between the number of shells Sarah and Nabil found as amounting to 'a few'. Nabil. 'Sarah found a few shells more/fewer than Nabil.' I propose in light of this that Sweyyit is semantically the same kind of thing as the numerals are -a degree quantifier that binds a degree variableand displays the same scope possibilities. It combines with a degree property and asserts that the maximum value the degree property holds of is greater than 0 (because it makes an existential claim) but less than a contextually given value . This definition can be plugged into the trees above to generate denotations for (60) with either aktar or aPall.
The judgments concerning ktīr as a differential are somewhat less robust than those for Sweyyit. Examples such as those in (62)  Nabil (i) 'Sarah has many more cards than Nabil.' (ii) 'Sarah has many cards -more than Nabil.' The two possible interpretations of the examples above are teased apart if we replace aktar 'more' with aPall 'less'. In this case the judgments are mixed.
While some speakers accept the sentences in (63) others reject them, while pointing out that they can be salvaged by inserting bass 'but' before aPall, so that e.g., (63a) reads 'Sarah found many shells, but fewer than Nabil'. This last point suggests that those who reject the examples in (63) only accept the examples in (62) on a conjunctive reading of ktīr 'many' and aktar 'more', so that e.g., (62a) corresponds to 'shells that are both many and more than the number of shells Nabil found'. In (63), this reading is awkward unless facilitated by bass 'but' in Arabic as in English (the (ii)-sentences below are awkward in English). In neither case does ktīr function as a differential specifier. Nabil (i) 'Sarah has many fewer cards Nabil.' (ii) 'Sarah has many cards -fewer than Nabil.' 17:36 Numeral differential constructions in Arabic I conclude that for some Syrian Arabic speakers, the term ktīr is ambiguous between the reading in (48b) and another reading analogous to Sweyyit, where ktīr functions for all practical purposes like a numeral and has the same scopal options as a numeral. While Swayyit says of a degree predicate that the maximal value it holds of is less than a certain contextually fixed value, ktīr, on this interpretation, says it is greater than a certain contextually fixed value, as defined in (64). Like Sweyyit, ktīr as defined in (64) may be plugged into the trees in Section 3 to derive its use as a differential specifier in constructions like (62) and (63).
For other Syrian Arabic speakers, the interpretation in (64) is not available for ktīr. For these speakers, ktīr may only be interpreted as defined in (48b), and on this interpretation, ktīr is scopally intransigent. It does not pattern like a numeral and does not participate in the derivational steps described in Section 3 that derive numeral differential constructions. These speakers do not allow ktīr in (62) and (63) to be interpreted as a differential specifier. In summary, ktīr is potentially ambiguous between a degree quantifier that patterns like a numeral, may have wide scope and accordingly may occur in numeral differential constructions (in the role of the numeral, though it does not specify a specific quantity), and a degree relation modifier. This latter use is scopally restricted and so does not participate in numeral differential constructions. Some Syrian Arabic speakers admit only this latter use, and do not allow ktīr to be used in the role of a numeral. All speakers admit Sweyyit 'a few' in this role. English much/many and little/few seem to be systematically ambiguous between these two uses.

Cross-linguistic implications
The discussion of French in Section 2 mentioned some aspects of Homer and Bhatt's analysis of the French pattern that are difficult to reconcile with the Arabic pattern. But Homer and Bhatt show convincingly that the numeral and noun are combined to the exclusion of the comparative phrase in French, and function as an object of the verb, just as I have claimed for Arabic. This raises the question of whether the analysis of Arabic above might be extended to French. Further, a reviewer points out that English displays a selectional dependency between numerals and nouns that is reminiscent of the selectional relations shown to hold in Arabic, which supports treating Numeral+NP as 17:37 Peter Hallman a constituent in numeral differential constructions. Specifically, the number feature of the noun depends on the numeral, being plural except when the numeral is one, as (65) shows. This is the case regardless of whether the comparative quantity adjective more precedes or follows the noun. Recall, too, as mentioned in Section 1, that extraposition of the comparative clause in English, seen in (66a) cannot take the numeral functioning as differential with it, as (66b) shows. The numeral is 'locked' in construct with the following noun. It appears that Bresnan's analysis, which makes the numeral an argument of comparative more in the base structure, is based more on semantic perspicuity than on any empirical facts motivating that constituency.
(66) a. Mary swam five laps more than Joan swam. b. *Mary swam laps five more than Joan swam.
But the analysis presented here has an ingredient that is difficult to reconcile with the syntax of comparative constructions in English and French. It is that in Arabic, the entire DegP aktar min . . . 'more than …' is base generated as a VP adjunct, not as an adjunct of the plural NP that functions as its scalar associate. That VP functions as a degree relation by virtue of movement of the numeral, not movement of the DegP. In Arabic, DegP is never within the plural DP that provides the degree variable it binds in numeral differential constructions. Homer and Bhatt analyse comparative DegPs (e.g., the phrase de plus que Jean lit. 'of more than Jean' in (19b)) as adjuncts of NP in their analysis of French. A facet of the English counterpart points to this same conclusion. Wh-movement of the object in the examples in (65) obligatorily pied pipes the comparative morpheme more and optionally the standard phrase. This seems to call for an analysis in which the comparative phrase is within the DP undergoing wh-movement, though the standard phrase may undergo extraposition.
(67) a. How many more shells did Sarah find than Nabil? b. How many more shells than Nabil did Sarah find?
The analysis defended here predicts that sentences like (67b) should be impossible in Arabic, and I argue below that this prediction is borne out.
It is, as expected, possible for the string kam ṡ adafe 'how many shells' to undergo wh-movement, stranding the degree quantifier aktar min nabīl 'more 17:38 Numeral differential constructions in Arabic than Nabil', as (68a) shows. 7 The judgments for the counterpart of (67b) in Arabic, shown in (68b)  Nabil 'How many more shells did Sarah find than Nabil?' b. %sāra Sarah kam how.many ṡadafe shell aktar more min than nabīl Nabil laPi-t?
found-3fs 'How many more shells than Nabil did Sarah find?' This ambivalence about the status of (68b) is reflective of native speakers' judgments about preposing the degree quantifier in declaratives. Example (69a) shows a degree quantifier preceding a gradable adjective, in (69b) it precedes a VP containing a plural scalar associate. Neither of these are consistently judged acceptable in Syrian Arabic, but nor are they consistently rejected.
shells 'Sarah found more shells than me.' This pattern presents an explanation for the judgments on (68b) in Arabic. The degree quantifier aktar min nabīl 'more than Nabil' is not actually part of the noun phrase kam ṡ adafe 'how many shells' there. It is preposed independently of the fronted wh-phrase, and its marginality in that position reflects the general marginality of fronting a degree quantifier across VP in Syrian Arabic. 7 A reviewer raises the question of how sentences like (68a) are interpreted, since the analysis I have proposed here requires the numeral (here in the form of the wh-word kam 'how many') to undergo quantifier raising independently of its restriction. In general, degree questions can be interpreted with the restriction in situ, even when the restriction is pied piped to the left clause edge in the surface structure; see the discussion surrounding (46a) above. I assume then that kam 'how many' in (68a) first undergoes raising to VP, deriving a degree relation to which the comparative phrase may apply, exactly on analogy to numerals, and from there it undergoes wh-movement to the left clause edge. Both steps pied pipe the restriction ṡ adafe 'shell', but this restriction undergoes reconstruction at LF.

17:39
Peter Hallman On one hand, these observations corroborate the analysis of Arabic proposed here. On the other, they would seem to preempt extending the analysis to the English sentence in (67b), where there can be no doubt that the degree phrase more…than Nabil is part of the wh-DP; the comparative morpheme more is sandwiched between parts of that DP and there is no general possibility of fronting a degree quantifier around VP in English.
In principle, DegP and NumP as defined here could combine directly with a plural NP, as illustrated in (70). Here, the numeral three correctly identifies the size of the differential, but the interpretation does not match the attested interpretation of three more shells than Nabil in another respect. It asserts that Nabil is a quantity of shells! It is crucial for the attested interpretation that DegP be outside the NP at the point where is identified. But if we move DegP to a higher position, it will leave a trace denoting a degree variable, which then will saturate the degree argument of the noun shells. Then there is no degree predicate left for three to apply to. Even if we move three to a higher position, it will leave behind a degree denoting trace, which will not be able to be integrated into the NP structure. This illustrates why it is crucial that the DegP is not base generated within the plural DP, but rather higher in the tree. Since it is merged high, it does not itself leave a trace of movement, rather it 'picks up' the trace left by movement of the numeral. The numeral then is free to bind the differential argument of the DegP. While this works elegantly for Arabic, it appears to bode badly for the possibility of extending this analysis to English. 17:40 Numeral differential constructions in Arabic As mentioned briefly in footnote 6, Bhatt & Pancheva (2004) argue that standard phrases in English are base generated (specifically 'late merged') in the scope position of -er, not in its base position within the noun phrase. They remark that examples like (71a) (their (72a), p. 35) pose a problem for that analysis, one that is similar to the problem described above. The apparent occurrence of than Bill between the two objects of the verb (more presents and to Mary) appears to contradict the claim that the lowest position in the clause that than Bill may occur in is the derived position of -er at the level of VP. They claim therefore that than Bill is in fact at the right VP edge but the phrase to Mary has been postposed further still. But this analysis of (71a) is difficult to reconcile with the constituent question facts in (71b) (parallel to (67b)), where the standard phrase seems to have been pied piped with wh-movement of the plural NP associate.
(71) a. John will give more presents than Bill to Mary. b. How many more presents than Bill will John give to Mary?
There is no evidence that quantity comparative more than… is ever interpreted locally to its scalar associate in English. The only evidence that puts comparatives within the maximal projection of the scalar associate is morphosyntactic: in English, more appears internal to the plural NP that functions as the scalar associate. But this morphosyntactic fact is not reflected in either the interpretation nor in binding configurations, as Bhatt & Pancheva (2004) show in detail. The facts in (67b) and (71b) are not easily reconciled with the analysis of Arabic presented here, but neither are they easily reconciled with the interpretational facts that motivate Bhatt and Pancheva's late merger analysis of English. Arabic appears to reflect the LF scope of the comparative DegP in its surface word order, and the present analysis presents a mechanism through which a numeral adjacent to the plural scalar associate for the comparative may function as a differential specifier of the comparative at LF. In light of this, a mechanism that maps the English surface syntax to the Arabic LF would be desirable, but it is unclear at present how that is possible (see the discussion surrounding (70) above). Since Bresnan's (1973) analysis does not predict the pattern in (65) and (66) and Homer & Bhatt's (2020) analysis raises some unanswered questions of its own, as described in Section 2, I consider the question of whether and how French, English and Arabic might receive a unified treatment to still be open.

Conclusion
I have presented an analysis of sentences of the form Sarah found three shells more than Nabil in Syrian Arabic, in which the numeral three forms a syntactic constituent with the object noun but yet is interpreted as a specification of the differential of the comparative. In Syrian Arabic there is particularly compelling evidence for a tight selectional relationship between the numeral and noun, undermining an analysis in which the numeral forms a constituent with the comparative phrase in the base structure, which is then postposed in the surface structure, giving rise to the illusion that the numeral and noun form a constituent. Syrian Arabic displays a robust morphological selectional relationship between the numeral and noun even in such constructions, suggesting that the numeral functions in the syntax as a modifier of the noun. But I have shown that the interpretation in which the numeral nonetheless binds the differential degree argument of the comparative in the logical form can be derived using tools that are already common currency in semantic analysis. One is the premise that DegP movement need not extend the tree, which is a component of standard movement theories of the phrasal comparative and the superlative. Another is the notion that numerals are degree quantifiers that can have scope independently of the noun they modify in the surface structure. Consequently, the surprising interpretation of sentences of the kind treated here is in fact predicted by current assumptions in semantic theory.