2.1. Degree predicates

Generally speaking, predicates expressing some form of measurement or comparison, e.g. indicating degrees of difference or similarity, form good NAD constructions. This is true of the following constructions: comparatives, superlatives, equatives, proportionals (see (4)), as well as expressions of excess and sufficiency, such as be too much or be sufficient, which are the ones we focus on in this paper. We have already seen what such NAD constructions look like in (1b) and (2b), characterized syntactically by a seemingly disrupted agreement pattern.

We focus now on the properties and restrictions of the three pieces that make up the predicate of this construction: the copula BE, the degree head and the nominal complement of the degree predicate.

2.1.1. The copula

We begin first by noting that NAD constructions are strictly limited to (i) predicational copular sentences with (ii) degree predicates that express some form of measurement or comparison. Therefore, the examples in (6) are not NAD constructions, since their predicates do not involve any form of measurement or comparison and, furthermore, they only show ordinary agreement patterns.

It is important to remark that the limitations of the non-agreeing variants in (6) to form grammatical NAD constructions are syntactic. For instance, one could imagine that a NAD construction in (6a) could mean that a number of defenders equal to five has the required ability to stop the rival team. Such meanings are perfectly expressible via bona fide NAD constructions, as in (7).

We take it, thus, that NAD constructions require predicates that overtly establish some form of measurement or comparison. In fact, some predicates actually are found to be preferred as NAD constructions in their most common uses. For instance, dimensional nouns that can be predicates directly of units of measurement – such as distance, volume, weight, etc. – yield better results as NAD constructions in spite of their disrupted agreement, in contrast with ordinary agreeing counterparts.Footnote ⁶ Consider the data in (8) and (9).

It seems clear then that expressing some form of measurement/comparison is necessary, but as it turns out, this is by no means sufficient. Consider measure verbs such as pesar ‘to weigh’, etc. These types of predicates may form interesting semantic pairs with NAD constructions, as illustrated by the pair be enough weight vis-à-vis weigh enough; for instance, if 10 kgs is enough weight (for some purpose), then certainly 10 kgs weighs enough, too. Nevertheless, these types of predicates are ruled out in NAD constructions.

In both (10) and (11), we find the same exact contrasting pattern. The two (a) examples show that full plural agreement with the predicate is ungrammatical. These are the baseline NAD constructions. Instead, the (b) cases involve a verbal predicate modified by the degree predicate of excess demasiado ‘too much’ expressing that its subject exceeds a certain threshold along the dimension determined by the verbal predicate itself. These only admit ordinary subject-verb plural agreement patterns, which we take as evidence that they cannot form NAD constructions. In other words, verbal predicates like pesar ‘to weigh’ and durar ‘to last’ cannot partake in NAD constructions even though its nominal counterpart peso ‘weight’ and tiempo ‘time’ can.Footnote ⁷

We must be careful, however, when assessing the semantic status of the two grammatical variants of pairs, such as those presented in (10) and (11). Consider for clarity the following minimal pair, formed by the NAD construction in (12a) and the agreeing variant in (12b).

The meaning difference between (12a) and (12b) is ostensible, and thus the two sentences cannot be taken to be semantically analogous. (12a) states that, on a scale of weight, three books, any three books, exceed some contextually determined threshold of weight. (12b) instead states that there are some three books, say b1, b2 and b3 that weigh too much (either individually or jointly). Conceptually, (12a) is a statement about weight, where three-book objects are claimed to exceed a certain weight threshold, whereas (12b) is a statement about the weight of some three books.

2.1.2. The degree head

There is an interesting limitation to note about the requirement to have a degree expression in NAD constructions. Most strikingly, NAD constructions must always be interpreted as modified by a degree expression, either overtly or covertly. Consider (13).

While the examples in (13) without the degree predicate demasiado or suficiente are not ungrammatical, they only accept an interpretation according to which three books / cars count either as an amount of weight (in (13a)) or an amount of money (in (13b)). This type of interpretation is more natural with nouns such as money, which retains its amount interpretation also under expressions of paucity, such as apenas and negation:Footnote ⁸

What cases such as (13) and (14) cannot mean is that three books have the property of being weight, three cars have the property of being money and three millions have or lack the property of being money. The conclusion is that NAD constructions do not admit a bare counterpart: they must always be interpreted as degree expressions, either because they include an explicit degree expression, or because such degree expressions are covertly understood in the context.

2.1.3. Nominal complement

NAD constructions place virtually no restrictions on the type of nominal that may appear as complement to the degree predicate. Example (2b) above provides already a good array of different nominals (peso ‘weight’, dinero ‘money’, trabajo ‘work’, esfuerzo ‘effort’, lectura ‘reading’). Note that these nominals need not be either abstract or gradable, and thus also include sortal count nouns, such as huerto ‘garden’ or ordenador ‘computer’, as illustrated in (15).

While any nominal may partake in NAD constructions, no non-nominal may. We have already discussed (10b) and (11b) with measure verbs. One could also imagine that adjectives, being the prototypical way to express a gradable property, would be grammatical in such contexts. They are not, however: simply swapping the nominal with an adjective results in ungrammaticality. Consider the minimal contrast in (16).

In fact, any such alternations lead to ungrammaticality. Moreover, with adjectival predicates, subjects must always agree with the copula:

Adjectival predicates like pesado ‘heavy’ and caro ‘expensive’ may form ordinary agreeing predicative structures, such as those in the agreeing variants of (17). Because of this limitation, NAD constructions where the complement to the degree predicate is not explicitly provided (see e.g. (1b) above) are such that only a nominal expression may be recovered, but not an adjective. A reason for this behavior might be that the associated scale of adjectives is conventional by hypothesis, that is, they are lexically fixed to some scale. The associated scale of nominals instead depends on the nominal itself and, in most cases, on the context. For instance, although weight is limited to denote along a single dimension, scales built on top of more abstract dimension names, such as work, effort, trouble, etc., provide different ways to determine what counts as being ‘work’, ‘effort’, etc.

Summing up, the best NAD constructions are formed by predicational copular sentences with some form of degree head, namely, a head that expresses excess or sufficiency, and a nominal complement. NAD constructions may also take a possibly covert nominal as its complement. From a semantic perspective, NAD constructions provide a way to use ordinary nominals as non-conventional units of measurements, that is, units of measurement that are made on the fly without prior agreement as to how their corresponding scale is determined.

2.2. Subjects

When it comes to NADs’ subjects, Quantifier Phrases (QPs) headed by cardinal numerals, either modified or not, make the best and most natural subjects, as illustrated in (18).

In sharp contrast with how naturally such examples are construed, other existential as well as universal quantifiers cannot be subjects of NAD constructions.

As an anonymous reviewer pointed out to us, singular subjects like the indefinite in (20a) and (21a) below give rise to similar interpretations as those conveyed by NAD constructions, but notice that in these examples the number morphology on the copula must be singular. This is independent of the possibility of ascribing (20a) and (21a) a NAD-like interpretation, one whereby a single apple (any one apple) and a single mistake (any one) exceeds the relevant threshold of fruit amount and stir amount. (20b) and (21b) parallel those in (1a) and (2a) with non-canonical agreement with the postverbal Noun Phrase (NP).

If we look at definite determiners, it may appear at first glance that definite descriptions are also ruled out. Determiner Phrases (DPs) headed by either the definite article or other definite demonstratives yield ungrammatical results as subjects of NAD constructions:

Still, there are cases where definite descriptions may form good subjects of NAD constructions. This is especially apparent when the nominal inside the definite description is abstract, typically deverbal, resulting in an eventive denotation of the subject. Example (23a) below shows that conjoined definite DPs must agree with the main predicate of the sentence, as indicated by the ungrammaticality of the singular copula (i.e. (23a) corresponds to a non-predicational identificational copular sentence). The critical example in (23b) shows that with NAD constructions an agreeing copula is not required, conforming thus to the familiar predicational NAD pattern we have seen so far (e.g. (1b) and (2b)).Footnote ⁹

While highly reminiscent of (22), examples like those in (24) (provided to us by an anonymous reviewer) present interesting contrasts with (23):

On the surface, the main differences between (23b) and (24) are two: a lack of optionality in agreement and an ordinary non-eventive head noun in subject position in the latter. Were (24) grammatical with the singular copula, it would seem to suggest that our characterization of subjects in NAD constructions was incomplete or mistaken: as shown in (22)/(23b), only a reduced number of definite descriptions yield grammatical results, and DPs such as las escaleras in (24) is not one of them. The ungrammaticality of (24) in singular, when compared to (23b), suggests that, in fact, it is syntactically distinct to our NAD constructions. The pattern is, moreover, quite generalized: any type of plural definite phrase may appear as a subject in (24), but not in NAD constructions:

There are also important semantic differences between (24)/(25) and NAD constructions: unlike subjects of NAD constructions, the subjects of (24)/(25) refer to (extensional) individual entities, particular set of stairs.Footnote ¹⁰ These differences in agreement and interpretation lead us to believe that NAD constructions, understood as a natural class, exclude cases such as (24), which, in turn, may require a separate treatment from the one we offer in this paper.

In a very similar type of variation on the eventive definite DPs illustrated in (23) above, we find that clauses (both non-finite (26) and finite (27)) may also appear in subject position of NAD constructions. We first show in (26a)/(27a) below that conjoined clauses trigger plural agreement with the copula, whereas in (26b)/(27b), we point out how in the context of NAD constructions, the agreement pattern is reversed. Interestingly, note that conjoined infinitives (26b) and conjoined clauses (27b) do not admit plural copulas, in spite of being subjects of degree predicates. We do not have an explanation for this restriction, but what is critical for us is the possibility of the singular variant, as attested above (see footnote 3, too).

In general, the broader observation is that event-denoting expressions, such as non-finite subjects and propositional-denoting expressions, such as full clauses are good NAD subjects. In addition to these, we can also include deadjectival nominals, as shown below:

There is yet one more type of subject that is allowed with a subset of NAD constructions. These are QPs and DPs introduced by the preposition con ‘with’, which combine with degree predicates whose head is a verb conveying sufficiency, such as bastar, alcanzar, llegar ‘to be enough’, or excess, such as sobrar ‘to exceed’ (see footnote 7). The preposition con may appear modifying numeral QPs, definite DPs and non-finite clauses that otherwise would be ungrammatical. Its most remarkable feature is its ability to rescue two types of statements. First, it may rescue cases where the source of the ungrammaticality lies in the type of subject employed, as in (19) and (21) above, and forces a singular copula.

In addition, the preposition con may also rescue cases where sufficiency is lexically specified in a single verbal head, possibly as the result of spelling out the full degree predicate.Footnote ¹¹

In sum, NAD constructions are but one in a family of constructions characterized by the ‘disruption’ of a canonical agreement pattern, where subject and predicate all share the same phi-feature specifications, a not uncommon property of Spanish copular sentences (see RAE 2009, chapters 33, 37).Footnote ¹²

2.3. Semantic building blocks

So far, we have limited the discussion to the syntactic components and distribution of the different phrases partaking in NAD constructions. It is nevertheless important to note that the semantic effects observed are homogeneous across all these different manifestations of NAD constructions: the expression of a dimension, a degree and a unit of measurement are always required. Let us illustrate this with (1), repeated for convenience.

We are interested in determining what the meaning of (1b) is and how it differs from its mandatorily agreeing counterpart in (1a). The intuition we pursue is that, while statement (1b) is about books, (1b) is about something else: for a writer who signed a contract, it could be work, commitment, effort; for a pre-schooler who has to carry them, it could be too much weight; for a Barnes and Noble executive, it could be too many to give away; for a struggling worker, it could be too expensive to buy, etc. None of these possible interpretations are available in (1a).

This description provides a good hint as to what the semantic ingredients of NAD constructions might be. For good measure, consider first what the role of ordinary gradable adjectives is in sentences like Jane is too tall. The main predicate tall expresses a relation between an (extensional) individual and a degree along some conventionally determined dimension – height in this case. The modifier too expresses that the degree provided by the adjective exceeds a certain contextually provided threshold. With this in mind, consider now the following case, taken from (13b) above:

Clearly, we do not typically use cars to measure amounts of money, and yet that is precisely what the NAD construction in (13b) expresses: that a cost-unit consisting of three cars exceeds a certain contextually determined threshold. The result is similar to the task performed by ordinary gradable adjectives, albeit one achieved by different means. The semantic ingredients required to capture such an interpretation for (13b) above seem to include three components:

• • The source of a dimension. This is the role of the nominal complement that is mandatory in NAD constructions (either explicitly provided or recovered from the context). In (13b), this is provided by the nominal dinero ‘money’.

• • A degree head like demasiado ‘too much’ and suficiente ‘enough’ (in addition to mucho ‘much’, comparative or superlative morphology, etc.). The role of this head is to introduce some form of comparison between a contextually provided degree, namely, a threshold on the dimension provided by the nominal; and

• • A subject that acts as a unit of measurement, that is, a subject that provides the measure that must be interpreted on the scale built upon the dimension provided by the nominal in predicate position.

From a semantic standpoint, we suggest that the copula BE in predicational sentences denotes the identity function for $ \left\langle e,t\right\rangle $ functions: the function of type $ \left\langle \left\langle e,t\right\rangle, \left\langle e,t\right\rangle \right\rangle $ that maps any $ \left\langle e,t\right\rangle $ function to itself.

The semantic task performed by the subject tres coches in (13b) above is less like the type of relation between individuals and degrees that has been posited for adjectival gradable predicates, and more analogous to numeral QPs with nominals that conventionally (i.e. lexically) denote units of measurement, such as kilo. Of course, different degree heads may contribute different semantic roles or even require different analytic pieces. But the general semantic contribution of the three pieces required to capture the interpretation of NAD constructions prevails across configurations. Thus, the difference between NAD constructions and sentences with standard gradable main predicates (such as adjectives) seems to be that while adjectives denote relations between extensional entities and degrees along a conventionally associated scale, nominal complements of the degree head in NAD constructions simply provide the name of the dimension that the degree head employs to express the threshold it denotes.

In what follows, we will assume this proposed characterization and take for granted that subjects of NAD constructions are semantically interpreted as providing a measuring unit, used to locate a degree on the basis of a scale whose dimension is either retrieved from the context or provided by the nominal in the post-copular position.

3.1. NADs as predicational copular sentences

As we have seen in the previous section, one of the most notorious definitory properties of NAD constructions is their limitation to copular environments. Copular sentences have been traditionally divided into predicational (30a) and non-predicational ones, the latter including equative (i.e. identity) (30b), identificational (30c) and specificational sentences (30d).

According to Higgins (Reference Higgins1973), the subject and predicate of these four types of copular sentences are assumed to have different referential properties, which – after Mikkelsen’s (Reference Mikkelsen2005: 130) type-theoretic unification – are reduced to (31).Footnote ¹³

We submit that NAD constructions fit better the type of predicational copular clauses. This is better seen in comparison to otherwise semantically equivalent variants, such as (32), with a definite description in subject position and predicates such as be too much.

A predicational analysis of NAD constructions involves a referential subject of type $ e $ and a predicate of type $ \left\langle e,t\right\rangle $ , raising the issue as to whether numeral DPs, such as three books, can truly perform such a semantic task. As an anonymous reviewer suggests, this could be indicative of a new type of copular construction in Mikkelsen’s (Reference Mikkelsen2005) classification, one where both constituents are of a predicative $ \left\langle e,t\right\rangle $ type. Nevertheless, we believe that subjects of NADs are indeed referential: their referents are abstract in the same way that the referent of (32) above is abstract. Our own analysis, presented in detail in Section 4, involves lowering the subject to a referential type.Footnote ¹⁴ Notice that lowering the predicate to a referential type amounts to saying that expressions such as be too much weight, be the same as two pizzas, etc. are referential, which does not look like a tenable analysis for these cases.

From a syntactic standpoint, assume that in copular sentences, the predication is mediated by the projection of a functional head Pred (Bowers Reference Bowers1993, Svenonius Reference Svenonius1994, Adger & Ramchand Reference Adger and Ramchand2003 and others), as represented in (33).

In the specific case of NAD constructions, the Pred head corresponds to the copular verb BE, which will move to the head of the Functional Phrase responsible for Tense. The predicate XP, which in regular predicational copular sentences may either correspond to nouns, adjectives or prepositions, in the specific case of NAD constructions must be a Degree Phrase containing an NP that introduces a nominal scale (through the mediation of a required Measure Phrase). Finally, the subject of NAD constructions must be a QP with a (modified) cardinal head or (non-extensional) DP (as exemplified in Section 2.2). This means that the predication characteristic of NAD constructions is uniformly represented as in (34).Footnote ¹⁵

Note that the question of flexible agreement of NAD constructions is not entirely resolved by the syntactic properties introduced so far. Plural agreement can be claimed to correspond to the pre-copular canonical agreement, after a (plural) QP/DP is raised to Spec,TP: the [i $ \phi $ ] and [EPP] (Extended Projection Principle) features of QP/DP will check the [u $ \phi $ ] and [uEPP] features of T. However, it is unclear how singular agreement could be derived. Bearing in mind that agreement with the NP in post-copular position is not uncommon cross-linguistically in inflectional languages, one might hypothesize that singular agreement corresponds to post-copular non-canonical (inverse) agreement with the NP in complement position. However, there are two main problems with this kind of approach: for this to happen, the NP1 is expected to be defective for phi-features and NP2 is expected to be referential.Footnote ¹⁶ Given that none of these conditions apply in NAD constructions, we abandon this approach. In Section 4.1, we consider an alternative, according to which, once the NP1 is nominalized, plural agreement is no longer a requirement; in other words, singular number on the copula is a post-syntactic option after a semantic nominalization operation has occurred at Logical Form (LF).

3.2. The basic semantics of sufficiency and excess

In this section, we focus on the contrast between gradable phenomena expressed by means of adjectives and by means of nominal scales.

3.2.1. Adjectival predicates and degree modifiers

A common way to capture gradable phenomena in natural languages is to assume that certain expressions, such as adjectives, degree modifiers, etc., contain degree arguments that are lexically associated with scales.Footnote ¹⁷ Degrees provide a direct way of representing how much of something an object has. For our purposes, we will assume that degrees are primitives, atomic types in the model (of type d).Footnote ¹⁸ With degrees, gradable adjectives are typically expressed as relations between individuals and degrees, (35a). The value of the degree argument can also be additionally utilized by degree morphology (such as the comparative morpheme -er, degree modifiers like very, etc.) to impose further restrictions on its interpretation. Bare gradable adjectives, that is, with no such overt degree morphology, are often proposed to combine first with a null degree morpheme POS, for ‘positive form’ (originally introduced by von Stechow Reference von Stechow1984, (35b)):

POS serves a double function: it allows the whole Adjective Phrase to be predicated of an individual, and it relates the degree argument of the adjective to an appropriate contextually supplied standard of comparison.Footnote ¹⁹ Such standard of comparison is usually understood as an average, a prototypical value or the norm that is expected on the dimension provided by the gradable predicate.Footnote ²⁰ This is typically computed on the basis of the comparison class identified in the context. Consider a sentence like (36):

If Jane happens to be a third grader, then the comparison class to properly compute the truth conditions of (36) will help determine what is the standard or typical height of third graders, disregarding other individuals. We can thus represent the meaning of (36) as follows:

Excess and sufficiency expressions like too and enough express something slightly different, however: They express that the degree of an individual on some scale is not above or below what is standard or prototypical but some other conventionally determined threshold. Both standards and thresholds are most often underspecified and vague and can thus be questioned, disagreed with, etc. But there are some intuitive differences between them as well. Consider the examples in (38).

These examples, while still vague and underspecified, locate the source of their vagueness on a notion different than the standard of comparison. The statements simply assert that Jane exceeds or does not reach a certain threshold that may, but need not coincide with the standard. That standards and thresholds may be different is supported by contrasts such as those illustrated in (39).

Only the first example yields a contradiction, since Jane cannot be both tall and not tall (not at least without further qualifications). This shows that unmodified positive adjectives rely on fixing a comparison class to be interpretable, since otherwise, they would lack the means to find a suitable standard. But the follow-ups in examples in (39b, c) above do not rely on a comparison class. We must only determine whether Jane exceeds or not the relevant threshold.

Let us assume then that interpreting (39b, c) requires comparing Jane’s degree of tallness to some contextually supported threshold $ {TH}^C $ . Given that we are interested in upper (for too) and lower (for enough) bounds, we determine two context dependent boundaries, $ {TH}_{max}^C $ and $ {TH}_{min}^C $ . We propose the following lexical entries for too and enough (which we take to be identical to demasiado and suficiente in Spanish):Footnote ²¹

Essentially, too means ‘more (than)’ and enough just means ‘not less (than).’ The difference is that the comparison is not relative to a standard (as with bare adjectives) or some other linguistically determined degree (as with comparatives). Instead, the comparison is done relative to some conventionally (or possibly conversationally) determined threshold.

We can now see how two expressions such as too tall and tall enough are interpreted relative to different threshold degrees: a maximal threshold for too and a minimal threshold for enough. We illustrate this by providing the interpretations of the two sentences in (38) above.

3.2.2. Degree heads and nominal scales

We turn next to the case of nominals modified by degree expressions. Consider again (2a):

Under the assumption that ordinary nominals typically denote (extensional) properties, we face a problem if we try to combine too and enough directly with an ordinary property like books, since the lexical entries provided in the formulas in (40) are tailored to combine with gradable predicates. A usual reaction is to posit some type of phonologically null syntactic structure that nevertheless acts as a semantic glue between the degree predicate and the NP, in this case, by introducing a degree argument and linking it to the nominal predicate.Footnote ²² Following the syntactic structure introduced in (34) above, we will call this projection a Measure Phrase, whose head M takes a nominal argument of a predicative type and returns a gradable predicate, as the formula in (42) expresses.Footnote ²³

The semantic task of M is thus to take an ordinary property of individuals, such as the denotation of an ordinary nominal like books, and introduce a degree variable d that can later be used up by other degree predicates and modifiers, such as demasiados and suficientes. The gradable predicate mimics the ordinary meaning of adjectives, with the only difference being that the dimension of measurement is set by the nominal itself, rather than being lexically conventionalized. When applied to some individual, M will produce a set of degrees that represents their corresponding amount along some scale. The determination of the scale and its dimension, although context dependent, is not entirely unconstrained, however. In the case of plural nominals, as is the case with (2a) above, the relevant dimension of the scale is invariably a cardinality one, and the dimension is fixed simply to express a scale of a number of individuals, $ \mid x\mid =n $ (see, e.g. Rett Reference Rett2014, Solt Reference Solt2009, Reference Solt2015, Wellwood Reference Wellwood2019 a.o.).Footnote ²⁴ Composing the meaning of (2a) up to the level of Pred’ looks as follows (where ‘*’ corresponds to the pluralizing operator in Link Reference Link, Bauerle, Schwarze and von Stechow1983).

In order to interpret the numeral QP in subject position, we will adopt the view that numerals express nominal modifiers, as defended at length by Ionin & Matushansky (Reference Ionin and Matushansky2006, Reference Ionin and Matushansky2018) (see also Bylinina & Nouwen Reference Bylinina and Nouwen2020), yielding the following denotation for the full QP.Footnote ²⁵

We now have two predicates, PredP and QP of type $ \left\langle e,t\right\rangle $ that cannot, without further assumptions, combine in the usual way via Function Application. Here, we will simply assume the type-shifting operation A from Partee (Reference Partee1987), that shifts the type of an $ \left\langle e,t\right\rangle $ predicate into a generalized quantifier (type $ \left\langle \left\langle e,t\right\rangle, t\right\rangle $ ), introducing existential quantification on the way.

The resulting composition of (2a), after this lifting operation has applied over the subject, is as follows:

The resulting denotation of (2a) in the formula (46b) expresses that there is a cardinality of three books that exceeds or is close to some contextually determined threshold of book quantities. To be sure, the only available interpretation is one where three books count as too many / enough books; no other interpretation is possible.

The meaning composition of (1a) would parallel the derivation in (46), with the only difference that the nominal complement of the degree head appears to be elided.

Note that the difference with adjectival constructions (see Section 3.2.1) is quite apparent. Adjectives require contextually supplied information at two different levels in order to be successfully interpreted: They require a comparison class and a standard of comparison. For example, in order to interpret Jane is tall in (36), we first noted that the standard (average) of tallness varies depending on whether our comparison class is comprised of, for example, NBA players or 10-year-old children. Nominals in constructions like (1a) and (2a) do not show the same dependency on comparison classes. There is no relevant subset of books to be detected nor standards of comparison. There is again no notion of what counts as an ‘average’ number of books. Instead, the interpretations of (1a) and (2a) rely crucially on the notion of threshold, but this is not associated with the nominal itself but with the degree predicates too/enough.

While this difference between nominal and adjectival predicates might seem too obvious to mention, a caveat is in order here. The dimension of the nominal scale is set by the joint action of the context and information provided by the nominal. In the case of plural count nouns, the associated scales invariably range over quantities of things, that is, the corresponding dimension is always set to express cardinalities, and thus fixing such scales does not require much contextual support. This is so because on a scale of book quantities, three books rank higher than two books, which rank higher than one book, and so on, and little to no context manipulation will be able to alter those facts.

4.1. A general theory of NAD constructions

Consider (2b) as a working case.

In NAD constructions, complement nominals, such as weight, work, etc., are used to establish a dimension of measurement, with the help of context. Take for instance the case of money in (2b). What makes three book entities count as too much or enough money? We have well established units of measurement for scales along the dimension of cost/money (say dollars or euros), but clearly (2b) is not deploying those units of measurement. Instead, the role of money is to determine some aspect of cost, of which three books can be meaningfully said to determine one such unit of measurement. While this is relatively apparent in the case of money, things get more underspecified with other types of (typically abstract) nouns, such as work or effort, or even sortal count nouns, such as huerto ‘garden’ or ordenador ‘computer’ (see (15)). In what sense do we mean that three books counts as d-much / d-enough on a scale build upon the dimension of ‘work’? There is no single answer to this question, and different contexts will support different interpretations. For the proofreader, it will mean that three books exceed some threshold of proofreading work (or are sufficient with respect to it), for the author, the dimension will be set according to a dimension of amount of work in terms of book writing, and so on. In a way, any sense that falls within the denotational space of a nominal and can be meaningfully said to be part of its conceptual space might be exploited in order to build an ad hoc dimension.Footnote ²⁶ And this is, we argue, precisely the main raison d’être of NAD constructions: to provide the means of constructing scales built upon ad hoc dimensions. In the next paragraphs, we explain how to capture the semantic properties of NAD constructions with our current assumptions.

In examples such as (2b), the semantic role of the two nominals is always the same: N1 (the subject) is interpreted as some unit of measurement that is located along a scale built upon the dimension provided by N2. At the end of the day, the only limitations to the pair of nominals we find in NAD constructions have to do with notional or conceptual considerations about what makes sense to be predicated of what: while it may be sensible to use physical books to measure weight (or even time, by considering the time it takes to read a book), it may not be so natural to use books to measure viscosity, electric charge and so on; however, whether this is possible or not, it is ultimately only determined by context.

With these considerations in mind, composing the meaning of (2b) for a complement noun such as peso ‘weight’ up to the level of Pred’ looks as follows:

The meanings in (48) are fully parallel to those in (43): the nominal weight, although interpreted as a predicate of individuals in (48a), provides the dimension to build a measure of entities on a contextually determined scale in (48b) and (48c). Now, recall that a sentence such as (49a), to which we assign the structure in (49b) would not be appropriately represented by means of the logical form in (49c).

Interpretations like (49c) miss a general point, since (49a) is not a statement about books: It is a statement about weight, using books as weight units; in other words, books are the entities whose weight can be measured on a scale of weight. This is, in full, parallel to statements such as Three kilos is too much weight: it is not a statement about (any particular) kilos, it is a statement about weight. The formula in (49c) fails to capture this fundamental property of NAD constructions. Note also that the NAD construction in (49a) has the inference that there are sets of three books, not any three particular books whose weight exceeds (or is sufficient for) a certain threshold; in other words, to properly capture the truth-conditions of a NAD statement, we must avoid a requirement that a quantificational statement should have witnesses. The statement in (49a) might be true even if it is not possible to find values to the variable bound by three books, such that these values are books and are too heavy or heavy enough. This observation is crucial to rule out other types of valency shifting operations on the subject. The formula in (49c) shows that this is the case for type lifters such as A, as defined in (45), which do not avoid such unwelcome existential entailments. But so do a number of type-shifting operations that nominalize the subject from a property type to an individual type (i.e. shifts from $ \left\langle e,t\right\rangle $ to $ e $ ). We provide three common possible alternatives below: Choice Functions (Reinhart Reference Reinhart1997, Winter Reference Winter1997), the iota definite description operator (Sharvy Reference Sharvy1980, Link Reference Link, Bauerle, Schwarze and von Stechow1983) and the epsilon indefinite description operator (Hilbert Reference Hilbert1922, Bierwisch Reference Bierwisch, Bierwisch and Lang1989, von Heusinger Reference von Heusinger1997):

As the paraphrases above indicate, none of these alternatives will do: (50a) conveys something too strong, it commits us to a way of choosing a three book individual that equals or exceeds the relevant threshold, which is not the intended meaning. (50b) fails because it requires finding the unique maximal individual consisting of three books, which need not exist. (50c) is weaker than (50b) in that it does not presuppose any uniqueness, but all the same, it commits us to the existence of one three book individual or other that is then said to equal or exceed the relevant threshold. In short, neither of the alternatives in the formulas in (50) account for the fact that three books is not interpreted as referring, predicating or quantifying over ordinary individuals.

Our solution to this problem involves treating the subject of NAD constructions non-extensionally. The idea is simple: Subjects of NAD constructions denote entity correlates of properties, interpreted via a nominalized function (à la Chierchia Reference Chierchia1984, McNally Reference McNally1997). Consider the definition in (51).

What is crucial about this proposal is that subjects of NAD constructions do not denote or quantify over ordinary token entities. Rather, they denote or quantify over something more abstract, some form of higher order entity, which McNally (Reference McNally1997) modeled formally as a nominalized function (or entity correlate of a property) as expressed in (51), drawing from Chierchia & Turner’s (Reference Chierchia and Turner1988) property theoretic semantics. The main motivation for Chierchia (Reference Chierchia1984) and Chierchia & Turner (Reference Chierchia and Turner1988) to introduce entity correlates in the theory was that we can ascribe properties to other properties, as in fun is fun and goodness is good. They capture this by turning properties-as-functions (which correspond to their predicative uses) into their entity correlates (which correspond to their uses as arguments), thereby ‘nominalizing’ them.Footnote ²⁷

An entity correlate is a sort of abstract individual, the reification of the property it is a correlate of. It is thus sortally different from Carlson’s (Reference Carlson1977) object-level entities (sometimes also referred to as tokens). Entity correlates are also similar but ultimately different from the notion of ‘trope’ in that entity correlates are not particularized properties: tropes are particular manifestations of properties in individuals, and thus they serve as natural referents of nominalizations such as Jane’s height, Socrates’ wisdom, etc. (Moltmann Reference Moltmann2004, Reference Moltmann2009). Entity correlates are not specific in the same way, as they do not invoke any one particular manifestation of a property itself but resemble instead an ideal of such property.

In this respect, our conception of an entity correlate resembles somewhat that of a kind (Carlson Reference Carlson1977), an analogy heightened by the similarity between our nominalizing function ‘ $ {}^{\downarrow } $ ’ and Chierchia’s (Reference Chierchia1998) popular rendition of the kind forming operation, ‘ $ {}^{\cap } $ ’. Nevertheless, we believe there are empirical as well as formal reasons to avoid treating subjects of NAD constructions as kinds, derived via the operator ‘ $ {}^{\cap } $ ’. Most notably, when QPs of the form ‘Numeral NP’ are interpreted as denoting kinds in Spanish, only a sub-kind interpretation is available (Borik & Espinal Reference Borik and Espinal2015): in (53), the numeral quantifies over whale-subkinds.

If subjects of NADs were interpreted as kinds, we would expect them to quantify over sub-kinds of the noun they modify, just like in (53) above. This is not so, however: NADs like (1b) and (2b) do not convey that there are three kinds of books exceeding a certain threshold of weight, nor do they mean that there are three kinds of books whose instances exceed such and such threshold. All else being equal, it does not seem right to subsume the referents of NADs’ subjects under kind interpretations.

Our alternative nominalizer ‘ $ {}^{\downarrow } $ ’ does not commit us to kinds, since entity correlates may but need not be kinds. Entity correlates are formal objects, abstractions of properties, bringing in no a priori ontological commitment. As long as we are able to find (or make up on the fly) a referent for them, they shall not fail to refer (they are, in this sense, not very different from situations, degrees, and other abstract objects typically employed in semantic analysis). In this respect, we take kinds to be a proper subclass of entity correlates of properties, and this is the difference that the distinction between our ‘ $ {}^{\downarrow } $ ’ and ‘ $ {}^{\cap } $ ’ (as discussed in Chierchia Reference Chierchia1998) is intended to capture.Footnote ²⁸

With all this in mind, we return to the interpretation of (2b) by means of our nominalizing operator ‘ $ {}^{\downarrow } $ ’, represented in (54) below:

Feeding an entity correlate to the degree predicate, we obtain a statement about weight, not about books and, moreover, we avoid making incorrect existential claims about books. The denotation of the representation in (54b) then amounts to saying that the weight of the entity correlate of the property of being three books is at least as great as some contextually determined weight threshold. In other words, we use books as units of a measure of weight, as desired.

How exactly does an entity correlate then turn into a unit of weight? Ordinary nouns such as books cannot ordinarily be used as units of weight measurements; for instance, one cannot typically say that they bought three books of vegetables, meaning that they bought vegetables weighing as much as three books. We suspect that part of the reason for this behavior (and the reason why conventionalized units of measurement are helpful in general) is because books are not uniform objects with respect to dimensions such as weight, length, etc., as not all books weigh the same. Nevertheless, by obtaining the entity correlate of the property of being three books, we are indeed referencing an abstract object that is nonetheless uniform: uniform with respect to quantity, as the property that leads to the nominalization is comprised of three-book individuals only (Scontras 2017). We suspect, however, that such entity correlates are not just uniform with respect to quantity but with respect to other dimensions of which they may sensibly be predicated. Thus, being an idealization, the entity correlate of three books will, in addition to being uniform with respect to a quantity, involve a uniform weight, length, etc. but also a uniform amount of work, money, effort, etc. In this uniformity resides their ability to serve as units of measurement.

Extending this analysis to NAD constructions with null nominal complements (see (1b)) is straightforward. Assume first a contextually determined phonologically null nominal predicate DIM $ {}^C $ , complement to the head M.

The reasons for positing a null nominal here as well are mainly two: (i) as we have discussed in previous sections, only nominals are allowed in degree predicates of NAD constructions; and (ii) semantically, we need a nominal that provides the name of the relevant dimension and that can be measured. We write DIM ^C to remind us of this dimension-setting role of the nominal. When combined with M, the whole Measure Phrase denotes a relation between degrees and individuals, but one where the dimension is not set conventionally (lexically) but instead compositionally, by combining together M and its nominal complement.

Consider once again (2b) with a singular copula:

The way we account for it is as follows: (i) the postulated nominalization operation applies to Spec,PredP; (ii) this semantic operation gives as output an abstract entity, which makes it possible that (iii) at Spell-Out, a default third-person singular be instantiated on the copula. The relevance of this analysis is that an LF operation appears to be relevant at the time of the mapping from the structural representation of meaning to exponence.Footnote ²⁹

To sum up, in this section, we have argued what the derivation of NAD constructions is, with overt as well as covert nominals in complement position of the degree head. We have shown that the meaning of NAD constructions is heavily underspecified, and without further contextual aids, there is no telling what the relevant scale is. This is so because, referring once more to our (1b) and (2b) examples, any aspect of being a book, or any possible relation to being a book that may be brought by invoking the name book can be retrieved to and utilized as the relevant dimension to fix a measuring scale. It could be that any unit of a number of books is too heavy, or too expensive or too much to keep up with, etc. This interpretation is, in turn, completely devoid of any commitment to the existence of three books, and, in fact, any plurality of three books would be taken to exceed (or be sufficient with respect to) the set threshold. These are fundamentally evaluative scales that rely on the multiple senses that are attributed to the interpretation of lexical nominals.

4.2. General assessment

We now have an analysis of NAD constructions that has been developed on the basis of numeral subjects. NAD constructions, however, come in different shapes, and it remains to be seen how our proposed analysis extends to accommodate the rest of cases explored in Section 2, as well as the observed limitations.

Looking at subjects, we observe that, from an analytical point of view, the only well-forming requirement our analysis requires is that the subject must be nominalized. Our analysis imposes this restriction on the grounds that subjects must be referential, but intensionalized entities, interpreted as entity correlates of properties in a predicative copular construction, where they simply provide the argument the degree predicate is seeking. Consequently, we expect expressions for which nominalizing operations are readily available to form good NAD constructions and, conversely, we expect other types of expressions for which nominalization is not an option to yield ungrammatical results.

We begin discussing the case of subjects that are ruled in by our analysis. We have now seen how unmodified numeral constructions are easily treated as we suggest, but the account extends seamlessly to cases of modified numeral constructions as well, which can also be constructed as properties. From a semantic standpoint, this is unsurprising: While numerals establish an equality relation to the cardinality of an object (e.g. $ \left\{x:\hskip0.35em |x|=4\right\} $ ), modified numerals simply vary in terms of the type of relation they express (e.g. more than four expresses the property $ \left\{x:\hskip0.35em |x|>4\right\} $ , etc.). Moreover, the fact that like unmodified numerals, modified numerals may appear in positions admitting only properties confirms this point:

Interestingly, this view also correctly sanctions the fact that, while más de la mitad ‘more than half’ is a good NAD subject la mayoría de ‘most / the majority of’ is not: While the former can be construed again as a relation between a number and the cardinality of an object, the latter is a definite description and thus imposes more stringent conditions on its denotational space and its ability to be modified by further semantic operations.

So what kind of definite descriptions are then ruled in versus out of NAD constructions by our account? First, definite descriptions headed by ordinary extensional object denoting NPs (including demonstratives and possessives) are correctly ruled out. These are typically extensional objects that refer to actual objects in the world and impose, in addition, certain requirements on the domain (such as presupposing the existence and uniqueness of their referents). We have argued extensively that NAD constructions require, instead, intensional individual objects, ideals of properties that do not have to be realized in the actual world and place no additional constraints in the context. Ordinary definite descriptions have actual referents, and thus do not fit the bill (see discussion around (22) through (25) above).

There is, nevertheless, a natural class of definite descriptions that yield grammatical results as subjects of NAD constructions. These are definite descriptions headed by event-denoting (deverbal; (23)) or dimension-denoting (deadjectival; (28)) noun phrases. That deadjectival subjects perform well is perhaps unsurprising, given that they typically constitute the same type of nominals that may appear as complements to the degree expression, such as height, weight, etc. Event-denoting deverbal nouns instead are closer to non-finite subjects, which have long been argued to have nominalized interpretations (as first discussed by Chierchia Reference Chierchia1984). Our analysis extends seamlessly to such constructions, and, in addition, correctly captures the truth-conditional equivalence between the two types of subjects – when such minimal pairs exist. Notice that speakers allow ‘overtly’ nominalizing non-finite clauses with the definite determiner.

Similar observations can be made of clausal subjects as well (see (26b)/(27b) above). Summing up, our analysis accounts for (i) clausal subjects, both finite and non-finite, either with an overt definite determiner or without; (ii) nominalizations morphologically derived from adjectives and verbs and (iii) numeral NPs of various kinds (by our own hypothesis). Note, additionally, that nothing in our account rules out, in principle, singular subjects as long as they conform to any of the cases in (i) through (iii) – although in these cases, plural copulas are ruled out, as there is no single target for a goal seeking to check a plural feature; this is exemplified in (20a)/(21a) above.

Lastly, we focus on the case of subjects that our analysis rules out. Chief among these are quantified phrases, such as todo ‘all’, cada ‘each’, algunos ‘some’, varios ‘some’, etc. (see examples in (19) above) in subject position. These expressions quantify over instances of objects and do not have sensible interpretations of type $ e $ (Partee Reference Partee1987). In addition, our nominalizing function only applies to predicates (type $ \left\langle e,t\right\rangle $ ), but quantificational expressions also lack sensible interpretations as predicates, as explicitly shown by the examples in (58).

This means that, as a consequence, our nominalizing operation would not be able to apply to such quantificational expressions when they appear in subject position of NAD constructions, even if we were to assume a prior shift from a generalized quantifier type to a predicative type ( $ \left\langle \left\langle e,t\right\rangle, t\right\rangle $ to $ \left\langle e,t\right\rangle $ ).

Notice, too, that nothing in our analysis precludes singular NPs from appearing in subject position, although they do not conform to the narrow criteria we provided for NAD constructions – since they do not allow singular-plural alternations on the copula. Our semantic analysis, however, extends all the same to such constructions: nothing in the semantics of the nominalizing function precludes applying to, e.g. una manzana ‘one apple’ and un fallo ‘one mistake,’ with the numeral one (see (20a) and (21a)).

Our account most straightforwardly sanctions the class of expressions that may appear in subject position of NAD constructions, but as we saw in Section 2.1, there are certain limitations that affect the class of predicates that are allowed in NADs. Concretely, neither adjectives (e.g. heavy) nor verbs expressing some form of measurement (e.g. weigh) may form NAD constructions. Our account is rather agnostic as to why this is the case, and thus compatible with a number of analytical possibilities.

For one, we suspect that the reason why adjectives may not form good NADs is related to the fact that they perform fundamentally different semantic tasks. Adjectives being relations between (extensional) individuals and degrees, they are not easily predicated of other types of expressions. For instance, the most natural way to express that three kilos is an amount of weight exceeding some threshold is through a NAD construction:

Thus, even nominals expressing units of measurement along the scale conventionalized by a given adjective may not be subjects of said adjective.

Verbal non-copular verbs seem to be limited in a different fashion. As an anonymous reviewer suggests, it seems that verbs expressing measurement heavily rely on fixed units of measurements, and they may take as arguments no nominals other than those expressing such conventionalized units.

Overall, we believe that our analysis provides an accurate descriptive and explanatory analysis of a construction that has been unnoticed in the literature on Spanish, one which raises interesting theoretical challenges for the theory of grammar: the possibility that degree predicates introduce a non-conventional nominal scale, the possibility that subjects be interpreted as non-conventional units of measurement and the possibility that a singular number on the copula verb be a post-syntactic option after a semantic operation of nominalization has applied at LF. Our analysis also offers promising venues to fully understand the limitations we find in the distribution of predicates, but we leave for future research aspectual restrictions as those mentioned in footnote 10.