May (1977) discusses Quantifier Raising (QR), a syntactic movement operation that reorders quantifiers at Logical Form (LF).2

QR works by adjoining quantifiers to phrasal nodes (e.g., S, VP).3 The relative scope of the quantifier is its c-command domain at LF. Consider the ambiguous sentence in (10), below.

(10) may either be interpreted on its surface scope (there is only one, unique ceramicist who likes every plate) or its inverse scope (all plates are such that a ceramicist likes them). As QR is a syntactic theory, scope is determined through structural manipulation. Before QR has applied, (10) would be assigned the following (simplified) syntactic structure.

To derive the surface scope interpretation, QR moves the lowest phrase containing the quantifier every, and adjoins it to the VP node. The subject quantifier c-commands the object quantifier adjoined to VP, and therefore takes wider scope. The resulting, simplified, LF is shown in (12).

To interpret this sentence on its inverse scope, QR will instead adjoin the phrase containing the quantifier every to the S highest node such that it c-commands the subject. The resulting, simplified LF is shown in (13), below.

As a syntactic theory which supplies representations to the semantic component of grammar LF, QR is constrained by various locality restrictions both syntactic and semantic. See, e.g., Ross (1967) for a discussion of syntactic “Islands”, or Chomsky (2008) for a discussion of syntactic “Phases”. Fox (2000) suggests that syntactic operations are restricted by a semantic “economy” constraint which prohibits movements which are “semantically vacuous”.

QR coupled with Scope Economy predicts that BNs and MNs are only able to take inverse scope if there is a meaning difference. Additionally, the theory of Scope Economy posits an additional rule: Shortest Move.

This restriction on movement predicts that numeral quantifiers should only be able to take exceptionally wide-scope if there is a new meaning as a result. When an indefinite takes scope outside of its own clause, it is interpreted as a presupposed and unique entity (Diesing 1992). That is, it is imbued with a referential reading.

Scope in QR with Scope Economy is equivalent to the c-command domain of the quantifier at LF. The QR theory applies to all quantifiers equally so it predicts that scope taking for bare and modified numerals should be identical. QR only applies when it will affect the semantic interpretation. Additionally, QR is restricted to the local clause unless necessary for referential interpretation.

Beghelli & Stowell (1997) introduces the second syntactic theory of scope taking: Feature Checking (FC). The theory is distinguished by three basic architectural settings:

The most salient design feature of the Beghelli & Stowell system is the classification of quantifiers into five types. This division can be seen as an extension of Liu (1997), who was among the first to realize that the scope taking ability of a quantifier may depend on its type (Kiss & Pafel 2017). Beghelli & Stowell (1997) frame this difference as a direct response to the QR theory of scope. They state that the QR theory is too free. Namely, it is able to move any quantified phrase in any order, thereby generating unattested scopal readings. They suggest that all quantifiers should not be treated equally, and that the mechanism responsible for assigning scopal interpretations is sensitive to the diversity of the quantifiers.4

Beghelli & Stowell’s solution is to posit a series of functional heads, each of which serves as a landing site for a specific class of quantifier. In this way, their theory is similar in spirit to that of “cartography” in syntax.5

There are five classes of quantifiers identified in Beghelli & Stowell’s system. Each is discussed below.

The final class, the CQP, is posited to account for an asymmetry in the scope of numeral quantifiers. The authors claim, “A CQP in object position should never be able to take inverse scope over a GQP or DQP in subject position” (Beghelli & Stowell 1997: 80, ex. 3(d)).

The claim that a CQP in object position should never be able to take scope over a DQP in subject position is not trivial to prove. As discussed in the introduction, whenever there is a universal quantifier in subject position and an existential quantifier in object position, it is impossible to detect any inverse scope reading for the sentence (“Reinhart’s Riddle”). Experiment 1 provides an empirical test of this claim.

To determine scope, Beghelli & Stowell’s quantifier types must move to specific structural positions at LF. The positions are shown in (19), below.

Let us consider an example sentence interpreted with inverse scope in order to see how the Beghelli & Stowell system works.

On an inverse scope interpretation, (20) means that every pot is such that a (potentially different) ceramicist threw it. There are two quantifier phrases in this sentence, a GQP “a ceramicist”, and a DQP “every pot”. Each QP will move to its respective scope-checking position at LF, thus deriving all possible scopal configurations. For the inverse scope reading, the final (simplified) LF will be as in (21), below.

As shown in (21), each quantifier must move to its respective feature checking position at LF in order to derive the various scopal configurations. Notice how the indefinite expression a ceramicist takes scope from the specifier of ShareP, rather than at the two other possible positions: the specifier of RefP or its case checking position, AgrSP. The reason for this is as follows: if the GQP were to take scope from the Spec of RefP, then the indefinite would be interpreted as taking the widest scope, and it would be endowed with a referential reading making it a specific/unique individual. The meaning would be, roughly: a single, unique ceramicist (known to both speaker and hearer) has the property that they threw every pot. If the indefinite were to take scope from its case checking position, Spec of AgrSP, then it would not be out-scoped by the DQP, and it would be assigned a non-specific interpretation. On this interpretation, the sentence would roughly mean: some ceramicist has the property of throwing every pot. This is the most uninformative reading. If the GQP takes scope from the Spec of ShareP, then the correct interpretation and configuration obtains. The GQP is out-scoped by the DQP, and it is assigned a “presupposed individual” interpretation. The meaning is roughly: Every pot is such that some ceramicist (that the speaker assumes the hearer knows) threw it. The Beghelli & Stowell system has 3 positions for GQPs which encode their relative scope and correlate with an interpretation. The details of how these readings emerge from these positions or how the system knows which indefinites must be assigned certain readings, and why these readings necessitate these scopal configurations is left open. The authors state that:

Unlike the QR theory, the feature checking theory is not free: it does not allow any quantifier to appear in any order. The scope taking potential for each class of quantifier is different; this system predicts that there will be differences between bare and modified numeral quantifiers. Specifically, it is predicted that modified numerals quantifiers (CQPs) should not be able to take inverse scope over a bare numeral quantifier, because bare numeral quantifiers are GQPs, and GQPs always out-scope object CQPs.

Indefinite expressions such as some and a are capable of taking scope outside of a finite clause, while distributive/universal quantifiers such as every may not.

(22) cannot mean that for each student, there was a (potentially different) teacher who said that they glazed their sculpture. However, if the indefinite expression a certain takes wide scope, then the sentence can mean that there is single, unique student such that some teacher said that the student glazed his/her sculpture.

This ability has been widely acknowledged by a number of researchers; the reader is referred to Fodor & Sag (1982) for the seminal discussion of these facts. Within Generative Linguistics, the most popular mechanism for modeling the scope of an indefinite expression is the choice function. Choice functions, as described in Reinhart (1997), are functions which return a member from any non-empty predication set. More formally, Winter (1997) defines choice functions as in (23).

A choice function selects some element from a set, and the element chosen by the function is capable of participating in predication. Consider the ambiguous sentence in (24).

Under the sentence in (24), there are two interpretations translated using choice functions. (24a) means that if any student glazes their own pot, then Chablis will be glad. This is so, because the scope of the choice function is contained within the protasis of the conditional. This is the reading assigned to the sentence when the indefinite takes scope within its local clause. (24b) means that Chablis will be glad if a specific student glazes their own pot; this reading corresponds to the widest scope reading for the indefinite. The formula captures this fact by having the choice function scope over the entire conditional. Namely, the formula requires that a specific student is chosen; the student who would make Chablis glad if s/he glazes their own pot.

Choice functions work by returning a member of some non-empty set of predication, and depending on the scope of the choice function in the semantic formula, derives the scope of the indefinite. This theory does not require any “movement” of indefinites in the syntax, unlike the QR or Feature Checking theory. In this way, choice functions are a purely semantic theory of scope. They have no reflex in the syntax, and thus the curious availability of indefinites to “escape” scope islands (see (24)) can be explained in a principled way. Indefinites are not subject to clause boundary and island restrictions, because indefinites never “move” in the syntax, they always remain in situ; their scope is calculated in the semantics alone using choice functions. The choice function theory predicts that bare numerals and modified numerals should not differ in ther scope taking ability as they are both indefinite quantifiers. Additionally, choice functions predict that exceptionally wide scope will always be available for indefinite expressions like bare numerals and modified numerals, so their scope should not be clause-bound.

To investigate the scope taking behavior of modified and bare numeral quantifiers, a sentence-picture verification task (SPVT) was chosen. In this SPVT, subjects were presented with sentence–picture pairs and were tasked with reporting whether or not the sentence was a good description of the picture. This experimental paradigm was chosen for maximal similarity to previous work on quantifiers e.g., Chemla & Bott (2015); Raffray & Pickering (2010) and for its simplicity. Scope is an abstract concept and elicitation of reliable judgments is notoriously difficult (Szabolcsi 2010), so, it was decided that the simplest methods should be used. Namely, when presented with a sentence–picture pair, the meaning of the sentence can be quickly checked against the picture rather intuitively.

For each subject, the ratings assigned to each condition were submitted for analysis in Python: IPython (Pérez & Granger 2007), Pandas (McKinney 2011), NumPy (Van Der Walt et al. 2011), SciPy (Jones et al. 2001), Seaborn (Waskom et al. 2018), StatsModels (Seabold & Perktold 2010), and Scikit-learn (Pedregosa et al. 2011). Analysis consisted of a linear mixed-effects model with scope (surface vs. inverse) and quantifier type (bare numeral vs. modified numeral) as the factors; additionally, each subject was incorporated as a random effect. Bootstrap confidence intervals at 95% were calculated using 10k samples (Efron 1992).8

The mean ratings with confidence intervals at 95% in each condition are presented below:

Consideration of the means in Experiment 1 reveals that bare numerals in inverse scope condition received the lowest rating; while all other confidence intervals overlap. These results are visualized in bar-plots and violin plots (Figure 3). By examining the distributions of the scores in each condition, we can see that the low scores assigned to bare numerals in inverse scope conditions are concentrated to the middle range of the scale, while all other conditions have ratings distributed across the top half of the scale.

Upon submitting these data in a mixed-effects model with subject as a random factor, we found two main effects and no statistical interactions. This was a surprising finding given the confidence intervals computed above. There was a main effect of quantifier type (p = 0.002) such that bare numerals were assigned lower ratings than modified numerals, and a main effect of scope (p = 0.000) such that inverse scope conditions were rated lower than surface scope conditions. While not significant, there was almost a statistical interaction (p = 0.058) between scope and quantifier. These results are summarized in Figure 4.

These results are surprising given the claims in the literature. We would expect that modified numerals should resist inverse scope and bare numerals should allow for inverse scope interpretation; however, we found that bare numerals were rated lower than modified numerals across both surface and inverse scope conditions. This suggests that the theory of Feature Checking in Beghelli & Stowell (1997) is incorrect: modified numerals should not be barred from taking inverse scope. But, nonetheless, the idea that quantifiers might differ is on the right track.

It is useful to consider subject level performance on this task as there was considerable subject level variation (see Figure 5). Consider the main effect of quantifier type. Only subjects: 15, 25, 29, 31, 33, 37, 39, 43, 45, 47, 57, 61 show an obvious preference for modified numerals, but this is only within the inverse scope condition. Only subjects 35, 45, 47, and 61 show a slight preference for modified numerals in surface scope.

With our model and subject-level plots in hand, the reason for the preference for modified numerals seems to be the result of subject level difference; more data is ultimately needed to fully understand this effect.

These data suggest that the claims in the literature regarding the inability for modified numerals to take inverse scope are unwarranted: it is clear that modified numerals may take inverse scope. This result argues against the Feature Checking theory which posits a separate class of quantifiers to account for this supposed asymmetry. While the specific prediction regarding modified numerals in Beghelli & Stowell (1997) was not borne out in thees data, it appears t quantifiers might differ in their scope taking behavior, but more data is needed to fully understand what may be responsible for such differences.

These data do not clearly distinguish between Choice Functions or QR. Both theories predict that numeral quantifiers should not differ in their scope taking behavior; however, we discovered statistically significant differences between BNs and MNs. We observed that bare numerals were rated lower than modified numerals across both surface and inverse scope. We thus require more data to decide between QR and Choice Functions.

Identical to Experiment 1.

For each subject, the ratings assigned to each condition were submitted for analysis in Python (see 4.2). Analysis consisted of a linear mixed-effects model with quantifier type (BN–MN) and scope (Surface–Inverse) as the only factors and using subject as a random effect. Bootstrap confidence intervals at 95% were calculated using 10k samples. Mean ratings for each condition are reported below.

From these means and their confidence intervals, it is clear that inverse scope conditions were dis-preferred regardless of quantifier. Consideration of the raw values indicates that subjects found the differences between the surface and inverse conditions in experiment 2 to be more prominent than the differences found between bare and modified numerals in experiment 1. Consideration of bar and violin plots (Figure 7) confirms the above. The scores assigned to sentences in surface scope condition are concentrated at the top of the scale while scores in inverse scope conditions are clustered at the lower end of the scale.

Analysis revealed a significant main effect of scope (p = 0.000) such that inverse scope conditions were rated lower that surface scope conditions. No other comparisons emerged as significant. These results are summarized in Figure 8.

The results of experiment 2 demonstrate that quantifier scope is clause-bound. This pattern was robust between subjects and within (Figure 9) (save for subject 62 who assigned low ratings regardless of quantifier type or scope and subjects 48 and 54 who appear to have slightly preferred the inverse scope sentences.)

The results of experiment 2 argue strongly against the theory of Choice Functions which allow for indefinite expressions to take extra-clausal scope. It was clear that the majority of subjects (18/20) did not accept extra-clausal scope for numeral quantifiers. With the results of Experiment 1 in hand, the mounted evidence suggests that the theory of QR with Scope Economy best accounts for the facts.