2.1 Forward and reverse proportionality

Proportional readings in the nominal domain have mainly been discussed in the literature on the basis of examples with many and few. It is known since at least Westerståhl (1985a) that next to a cardinal reading, as in, e.g., (4a) (Partee 1989), which requires that the cardinality of the faculty children who attended the party is below a contextually determined standard, few (and many) can give rise to a proportional reading, as in, e.g., (4b) (Partee 1989), according to which the ratio of the cardinality of egg-laying mammals who suckle their young to the cardinality of all egg-laying mammals is below a contextually determined standard. Next to this forward proportional reading, however, proportional uses are also known to give rise to a reverse proportional reading (Westerståhl 1985b), as in, e.g., (4c) (Herburger 1997) according to which the proportion of the cardinality of cooks who applied to the cardinality of all applicants lies below a contextual standard.

The simplest way to capture the various uses of few/many is to assume a three-way lexical ambiguity, as in (5) and (6), where n is a contextually determined cardinality and p a contextually determined proportion. The difference between forward and reverse proportional readings, then, depends on whether it is the restrictor or the scope of few/many that appears in the denominator of the fraction.

Many attempts have been made to reduce the number of entries and avoid lexical ambiguity as much as possible. Such analyses have particularly focused on deriving reverse proportional readings on the basis of many_CARD/few_CARD or many_{F_PROP}/few_{F_PROP}, since the entries in (5c) and (6c) seems to be counterexamples to the Conservativity Hypothesis of Keenan & Stavi (1986). One strategy that has been used to that effect is to drop the assumption that the locus of proportionality is the quantifier itself and assume that (reverse) proportionality is the result of manipulating the contextual standard. As pointed out by Westerståhl (1985b) (cf. Bale & Schwarz 2020) proportional readings can then be derived on the basis of the cardinal entries in (5a)/(6a) as shown in (7) and (8).

Another possibility has recently been proposed in Bale & Schwarz (2020), one that places proportionality in the meaning of measure functions. In degree semantics, the job of relating individuals to their measurement in some dimension is done by measure functions (Cresswell 1976; von Stechow 1984; a.o.), which map entities in the domain of individuals to degrees in some dimension of measurement DIM. The entries for few/many can be rewritten as in (9), where the measure function μ measures the supremum of the intersection of the two sets in some dimension of measurement. Crucially, the value of the measure function is not constant, but is rather determined by context. In the case of cardinal readings, the function measures cardinality, i.e. it counts the number of atoms in a given set by counting the atomic parts of the supremum of the set.

To derive proportional readings, proportionality is built in the measure function itself (cf. Solt 2018). Degrees in the scope of a proportional measure function represent the ratio of two measurements, as in (11) (Bale 2022, based on Bale & Schwarz 2020).¹,² The measured individual in the numerator is always the argument of the measure function.

Forward and reverse readings differ on what is measured in the denominator. The forward proportional reading of (4b) is derived by setting the degree d*to the cardinality of the sum of egg-laying mammals, as in (12a), and the reverse proportional reading of (4c) by setting the degree d*to the cardinality of the sum of applicants, as in (12b).³

As Bale & Schwarz’s (2020) show the account can be straightforwardly extended to proportional readings of other measurement constructions, as in, e.g., proportional readings in comparatives. For example, (13) has a proportional reading according to which what is compared are two proportions; the proportion of the number of cooks that applied to our program to the total number of applicants in our program and the proportion of the number of cooks that applied to your program to the total number of applicants in your program.⁴

Since the comparative obviously does not have a standard-related interpretation, an analysis along the lines of (7)/(8) is unavailable. Whereas (13) can be analyzed using both a lexical analysis along the lines of (5)/(6), or an analysis based on proportional measure functions, Bale & Schwarz (2020) use examples like (14) in order to exclude a lexical analysis. Next to a cardinal reading, (14), also supports a reading in which what is compared is the density of road signs in the two routes. If so, (14) can be true even if the number of road signs on Rte 101 is less than the number of road signs on Rte 104.

Bale & Schwarz (2020) propose to capture this reading by means of the proportional measure functions in (15), where the value in the denominators represent the number of miles that constitute the length of the two routes. The crucial point here is that the degrees in the denominators in (15) are not retrievable from the sentence in (14), excluding a lexical analysis. An analysis in terms of proportional measure functions is thus shown to be independently necessary and can be extended to (13), assuming that the relevant proportional measures are as in (16).

As discussed initially in Solt (2018), an analysis in terms of proportional measure functions can also be extended to capture known differences between the proportional readings generated by different nominal measurement structures. As known since at least Partee (1989), the presence of partitive of affects the available readings. Consider, for example, the examples in (17), which, as Bale & Schwarz (2020) and Bale (2022) show, lack the proportional reading available in (13) and (14). As discussed above, e.g., (13) can give rise to a proportional reading which measures the proportion of cooks that applied to the total number of applicants. With the addition of partitive of in (17a), however, this reading disappears. The only available reading is what we will call a partitive proportional reading, which measures the proportion of cooks that applied to the sum of some plurality of cooks. Similarly, in (17b), the comparative can only be taken to compare the number of the road signs on the two routes, not their density.

One might think that to derive the available readings of (17), it is enough that partitive of introduces the an underspecified measure function and requires that the measured entity is a sub-aggregate of the denotation of its nominal complement, as in (18). This entry is, indeed, capable of deriving readings with simple non-proportional measures, like cardinality. The problem, however, is that non-partitive proportional readings can sneak in if the measure function in (18) is resolved to proportional measure functions like the ones in (15) and (16).

To solve this issue, Bale (2022), building on Solt (2018), proposes that a key ingredient in deriving the distribution of readings is to assume that partitivity is encoded in the measure function itself. This is achieved by restricting partitive measurement to domain-restricted measure functions, defined in (19) (Bale 2022: (12), based on Solt 2018). This ensures that the measure functions in partitives will have a maximum degree in their range, which is equal to the measure of the nominal complement in the relevant dimension.

The key step in eliminating the effects of proportional measure functions is to furthermore assume that partitive of also encodes a type of proportionality, as in (20) (Bale 2022: (14)). Partitives encode the proportion of the degree of the measured individual to the maximum or minimum degree in the range of the relevant measure function. LIMIT_a is function from measure functions to degrees that can be set to either LIMIT_t, in which case it returns the greatest degree in the range of measure function, or to LIMIT_b, in which case it returns the least degree in the range of the measure function.

As Bale (2022) shows this entry nullifies the effects of proportional measure functions. It would take us to far to show this for every possible combination of measure function and LIMIT function. To illustrate, consider the main clause in example (17a), assuming that μ is resolved to the function in (16a) and the LIMIT function is set to LIMIT_t.⁵ If so, the degree predicate denoted by the main clause of the comparative in (17a) is the one in (21), where z the applicants to our program.

The measure in the denominator is only defined if the cooks who applied to our program are a sub-aggregate of the applicants, which is true. By the definition of domain-restricted functions, this measure is equivalent to the non-restricted measure μ_DIM(y). Given the choice of proportional measure, this measure equals $\frac{{\mu }_{\#}(y)}{{\mu }_{\#}(z)}$ . Moving to the denominator, LIMIT_t returns the maximum degree in the range of μ_DIM;z. Since we are dealing with a restricted function, this is equal to μ_DIM;z(z), which, in turn, given the choice of proportional measure function, is equal to $\frac{{\mu }_{\#}(z)}{{\mu }_{\#}(z)}$ . (21) can thus be rewritten as in (22).

By factoring out the denominator values, (22) is equivalent to (23), which is nothing more than a regular partitive proportional meaning based on a non-proportional measure. Bale (2022) shows how this result generalizes to all proportional measure functions.

What we have seen so far is that there exist, in principle, at least three possible analytical options when it comes to the grammar of proportionality and its interaction with the grammar of measurement. One possibility is to locate proportionality in the meaning of a certain functional element, the same one that also introduces measurement. We will refer to this as a lexical analysis. A second option, available for at least examples with standard-related interpretations, is to introduce proportionality by manipulating a contextual standard in a certain way. We will refer to this as a standard-based analysis. The third option, the one argued in Bale & Schwarz (2020) to be able to be extended in all cases discussed here, is to link proportionality not simply with the functional element introducing measures, but with the measures themselves; whereas the meaning of the measure-introducing functional element remains constant, proportional readings are available because the measures themselves can be proportional. We will refer to this as a proportional μ-based analysis.

2.2 Proportionality with percentages

More recently proportional readings have also been discussed on the basis of nominal measurement structures with relative modifiers. Ahn & Sauerland (2015, 2017) identify another case of a reverse proportional reading in cases with explicitly relative/proportional modifiers like percentages. The German example in (24) receives a reverse proportional reading according to which the proportion of students who work here to the total number of workers equals thirty percent. The availability of the reading is morpho-syntactically conditioned. Whereas reverse proportionality is available with the juxtaposed nominal structure in (24), it is not available with the proper partitive in (25). The example in (25) can only give rise to a forward/partitive proportional reading.

Ahn & Sauerland provide an account of these cases that places proportionality solely in the meaning of the relative modifier, as in (26). The difference between (24) and (25) is driven by the different syntax of the relevant measurement constructions and independent properties of the relevant measurement constructions. In the most recent and more thorough investigation of these data, Pasternak & Sauerland (2022), treat the relative modifier n percent as a degree quantifier, as in (27). Whereas MAX(D) returns the maximal degree for which the degree predicate is true, MAX(dom(D)) returns the maximal degree for which the predicate returns a defined value (true or false).⁶

Importantly for current purposes, the modifier is the only place where proportionality comes into place in these analyses. We will refer to this type of analysis as a modifier-based analysis. Let us briefly present the modifier-based analysis of Pasternak & Sauerland (2022). We start with what Pasternak & Sauerland (2022) calls the juxtaposed structure in (24), for which they assume the syntactic structure in (28). As in much relevant literature (starting with Schwarzschild 2006), a functional element, here DEG, is taken to be responsible for combining bare nominals with degree heads, by introducing an underspecified measure function. The determiner that heads the DP is interpreted as an existential quantifier. The assumption throughout is that μ^c is resolved to the cardinality function.

To resolve the type-mismatch between the meanings of NP₁ and NP₂, NP₁ undergoes Quantifier Raising, as in (33). The percentage now composes with the degree predicate in (34). But this raises a different problem, since MAX(dom(D)) is not defined for this predicate, since there is no maximal degree of cardinality.

This is where focus comes in. Assuming that NP₁ is focused, simultaneously achieves two things: (a) MAX(dom(D)) is now defined due to the presuppositions introduced by focus, and (b) MAX(dom(D)) returns exactly the right degree required to get a reverse proportional reading; i.e. the cardinality of the individuals who work here. While focus on NP₁ retains the same ordinary interpretation in (34), it generates the focus semantic value in (36), where P ranges over predicates of individuals. Focus is also taken to introduce the presupposition that at least one of the focus alternatives is true. This presupposition is introduced by the head FPRE, which presupposes the truth of the grand disjunction of the focus value of the propositional constituent it attaches to, as in (37) (Pasternak & Sauerland 2022: (85), based on Abusch 2010). Assuming that among the alternatives to Studierende_F is a very weak predicate which is vacuously true (like human or animate), the presupposition of the degree predicate vP₃ is as in (38) (introduced as a definedness condition). This presupposition renders MAX(dom(D)) defined, since the maximal degree for which the degree predicate is defined will be the cardinality of the individuals who work here. Feeding this predicate to the percentage will generate the required reading, since it will equate the ratio of the cardinality of students who work here to the cardinality of people who work here to thirty percent.

Notice that in this analysis focus is necessary to generate the right reading. Pasternak & Sauerland (2022) claim that this is an empirically desirable result. Not only is focus on NP₁ claimed to be necessary to generate the reverse proportional reading in German, but different focus structures are claimed to have truth-conditional effects. So, whereas the sentence in (39), with broad focus on NP has the same reading as (24) (with the addition of the contribution of westfälische in the ordinary meaning), narrow focus on the adjective, as in (40), affects the value of the denominator.

We move next, briefly, to what Pasternak and Sauerland call the genitive structure in (25), which is assigned the syntactic structure in (41). The structure is essentially treated as a partitive, with the quirk that Pasternak & Sauerland prefer having the relevant partitive functional head shift the interpretation of Prozent from a quantifier over degrees to something that measures individuals. The key component of the meaning of MEAS for current purposes is that it requires (a) the measured individual to be a sub-aggregate of the denotation of the nominal complement of Prozent MEAS and (b) the relevant measure function to be domain-restricted.⁷ This will ensure that MAX(dom(D)) is the cardinality of the plurality of the students as required (assuming again that μ is resolved to the cardinality function).

As we have just seen a key component of existing analyses is to explain differences in the distribution of available proportional readings to the properties of different nominal structures. Particularly, partitives are taken to restrict the range of available proportional readings by imposing some additional restrictions. At the same time, however, seemingly very similar readings (the reverse proportional reading of many, the proportional reading of comparatives like (13) and the reverse proportional reading of juxtaposed structures with percentages) have been analyzed in very different ways, at least regarding the key question of the locus of proportionality in nominal measurement. The rest of this paper discusses to what extent these facts can be given a unified explanation by exploring how prima facie proportional modifiers like percentages interact with other potential sources of proportionality. Notice that a proportional analysis of percentages can, in principle, be combined with any one of the other analyses we have seen so far, multiplying the available analytical options. In this paper we will focus on two empirical domains. First, we look at a case, which, as far we know, has not been investigated before, the use of percentages in differential comparatives. Next, we move to the Greek equivalents of the German facts in (24) and (25). Since the structures we are primarily interested in are not sensitive to contextual standards, standards-based analyses will not be discussed any further. Lexical analyses will also not feature prominently in what follows, since Bale & Schwarz (2020) have already shown that they cannot capture the full range of available proportional readings, as discussed above. Our attention, thus, will focus on how the proportional measure functions of proportional μ-based analyses interact with proportional modifiers like percentages.

For the remainder of this paper, we change the language of investigation from English and German to Greek. By focusing on the Greek facts, we are not only expanding the relevant empirical landscape, but we also motivate a revision of the analysis in Pasternak & Sauerland (2022), particularly in the syntactic structures assumed for juxtaposed and partitive structures and the reliance to focus to generate reverse proportional readings. Before we move to this point, we will first consider the case of comparatives in some more detail. We argue that to account for the full range of available readings of percentages acting as differential measure phrases, a proportional μ-based analysis is necessary, even in the presence of a proportional modifier.

3.1 Differential comparatives

Consider first a regular differential comparative as in (45).⁸ (45) is true in a context in which we hired 10 students yesterday and 7 today.

Following Alexiadou et al. (2021), we adopt a decompositional analysis of Greek comparatives according to which perisoteri ‘more’ is decomposed into the comparative morpheme -ter- ‘-er’ and periso- ‘many’.⁹ The comparative specifies a relation between sets of degrees, as in (46). We choose the variable names T and M to stand as mnemonics for ‘than-clause’ and ‘main-clause’, respectively. periso- ‘many’ introduces a measure function as in (47). In this case, the function measures cardinality, cf. Makri (2018; 2020).

The than-clause and the main clause provide the two relevant sets of degrees, after predicate abstraction in the than-clause and after Quantifier Raising (Heim & Kratzer 1998) the degree phrase formed by the comparative morpheme and the than-clause, as in (48). The numeral tris ‘three’, which we take to name an individual degree in a scale of cardinality, provides the differential argument.

The meaning of (45) comes out as in (49), i.e. it is true if the number of students we hired yesterday exceeds the number of students we hired today by three. The differential measure phrase then simply specifies the difference between two degrees.

3.2 Differential comparatives with percentages

With this background we can now proceed to consider comparatives with percentages as differential measure phrases, repeated in (50). We observe that such examples are three-way ambiguous. In its first and most prominent reading, which we called the relative cardinal reading, the differential measure phrase represents the ratio of the difference between two cardinalities to the cardinality provided by the than-clause. Under this interpretation, (50) says that the positive difference between the number of people we hired yesterday and today equals 50 percent of the people we hired today. (50) is thus true in the context of (51a). In its second reading, which we called the relative proportional reading, (50) receives a proportional interpretation like the one discussed for (13) above. It compares the proportion of students we hired today (relative to today’s hirees) to the proportion of students we hired yesterday (relative to yesterday’s hirees). The differential measure phrase, in this case, represents the ratio of the difference between two proportions to the proportion provided by the than-clause. Under this interpretation, (50) says that the positive difference between the proportion of students we hired yesterday and today equals 50 percent of the proportion of students we hired today. (50) is thus true in the context of (51b). The third reading, which we called the absolute proportional reading, is also based on the same underlying proportional reading. In this case, however, the percentage appears to behave more like a regular, non-proportional differential measure phrase; it does not represent a ratio, but simply the difference between two proportions. Under this interpretation, (50) says that the positive difference between the proportion of students we hired yesterday and today equals 50 percent and is, thus, true in the context of (51c).¹⁰

4.1 Juxtaposed nominal measurement

Juxtaposed measurement structures with absolute measures, as in (74), are comprised of a substance noun, a measure noun, and a numeral. The substance noun and the absolute measure noun, which inflect for case and number, bear the same case. Relative measures like tis ekato ‘percent’ do not inflect for case or any phi-features. In fact, tis ekato ‘percent’ has itself the form of a DP as it is built out of the definite determiner in its plural dative form tis and the numeral ekato ‘hundred’. Morphological dative has been replaced in Modern Greek by the genitive, so we are dealing with a fixed expression.¹⁵ Case is determined by the position of the nominal construction in the sentence. For example, the substance and measure noun bear nominative when in subject position, as in (75) and (76).¹⁶ We conclude that the juxtaposed measurement structure is a Determiner Phrase which is an argument of the verb and case-marked by it, see Alexiadou et al. (2007) for an overview.

Notice that Greek allows a range of word-orders, usually conditioned by information structure. In (75) and (76) above, we used post-verbal subjects. Reverse proportional readings seem to require that the juxtaposed structure appears in post-verbal position. In the presence of multiple arguments, reverse proportional readings require the nominal construction to appear in sentence-final position, as in (77). There is significant controversy in the literature on the distribution and analysis of different word-orders in Greek, see Oikonomou & Alexiadou (2021) for a recent summary. At this point, we cannot offer a more concrete proposal about the interaction of reverse proportionality with the usually subtle information structural effects that condition word-order variation in Greek.

Nominal constructions in adjunct positions also allow reverse proportional readings, as in (78).

The internal constituency of juxtaposed absolute measurement has been an issue of considerable debate in the literature (Alexiadou et al. 2007 for overview).¹⁷ We assume a structure as in (79), where the absolute measure tria kila ‘three kilos’ forms a constituent (labelled MP, Measure Phrase) and is introduced in the specifier of a functional head Meas in the extended projection of the substance NP mila ‘apples’. We extent this analysis to juxtaposed relative measures with percentages in (80). Following Pasternak and Sauerland (2022) we assume that the Determiner Phrase is headed by a covert existential quantifier.¹⁸

The structure we adopt is mono-projectional in the sense that a single nominal is projected (see Stavrou 2003, Alexiadou & Stavrou 2020 for arguments in favor of a mono-projectional analysis of Greek absolute juxtaposed measurement.) We assume that semi-lexical nouns like kilo ‘kilo’ acquire case via agreement with the substance NP. We have assumed here that the numeral and the semi-lexical noun form a constituent. This analysis explains the fact that both the MP and the substance noun can be left-dislocated in Greek. This is so for both absolute and relative measurement, as shown in (81) and (82), respectively.¹⁹

One area where the absolute and relative measures differ in their behavior is verbal agreement. In the case of absolute measurement, verbal agreement in number depends on the number of the measurement construction, which itself depends on the number of the semi-lexical number, as shown in (83) and (84). In the case of relative measurement, however, number on the verb is always plural, as shown in (85). We will provide an analysis of these agreement patterns in section 4.4.

Before moving on, we should mention that reverse proportional readings in Greek can also be generated with an adverbial strategy, as in (86), where the percentage appears as part of an adverbial PP, headed by kata ‘by’. An analysis of adverbial reverse proportional readings lies outside the scope of this paper. The existence of this strategy does raise the possibility, however, that the structures we have considered so far also involve adverbial percentages. In this case it would in principle even be possible to assume that the adverbial percentage attaches to the DP projected by the substance noun, explaining the fact that the whole phrase trianta tis ekato fitites ‘thirty percent students’ has the properties and distribution of a DP.

The main reason to reject such an extension of the adverbial strategy has to do with the fact that PP percentages and bare percentages do not have the same distribution. Recall that juxtaposed relative measurement constructions can appear inside PPs, as we saw in (87) and (88) above, giving rise to a reverse proportional reading. PP-percentages, on the other hand, are ungrammatical in these positions.

Moreover, PP-percentages can appear in more positions within the clause than bare percentages. So, whereas PP-percentages can appear pre-verbally, as in (89), bare percentages cannot, as shown in (90). The same is true of the clause-final position in (91) and (92). We thus reject an adverbial analysis of percentages in juxtaposed measurement structures.

4.2 The distribution of reverse proportionality

Before moving to our analysis, we present further motivation for seeking an alternative to a modifier-based analysis. We first observe a correlation between the availability of juxtaposed measurement structures with percentages and the absolute proportional readings of the corresponding comparatives. We then turn to the issue of focus-sensitivity.

We have seen that whereas readings based on domain-restricted measure functions, as in (93), give rise to absolute proportional readings of differential percentages, readings based on non-restricted measure functions, as in (94), do not.

We observe that this contrast carries over to the availability of proportional readings with percentages in juxtaposed measurement structures. Whereas, as we have seen, (95) is available, it is not possible for (96) to have a reading based on the density of road signs in the highway.²⁰

This pattern generalizes to all non-restricted proportional measures. To give one more example, consider the proportional reading of the comparative in (97) based on the proportional measures in (98). Such measures cannot support an absolute proportional reading, as shown in (99), and neither is it available in the juxtaposed measurement structure in (100). As in the case of absolute proportional readings in comparatives the intuition is that the percentage will only be felicitous in juxtaposed measurement if it can be taken to directly specify a degree in the dimension of the underlying proportional measure function.

Our claim is not that modifier-based analyses cannot account for the unavailability of (96) and (100). For the reading of, e.g., (100), to be generated in such an account, the denominator of the percentage would have to measure the cardinality of the inhabitants of Athens. But recall that since in the account of Pasternak & Sauerland (2022) the denominator is strictly determined by grammatical means, i.e. the landing site of the QR-ed nominal and the focus structure of its sister constituent, there is simply no way to generate the intended meaning. Since, however, such an analysis has nothing to say about the derivation of absolute proportional readings in comparatives, the observed correlation will have to be coincidental. On the other hand, if, as we will propose in the next section, the proportional readings of juxtaposed measurement structures require the presence of a domain-restricted proportional measure function, just like we have argued for the absolute proportional readings of differential comparatives, the correlation follows naturally.

Consider next the issue of focus-sensitivity, which in the account of Pasternak & Sauerland (2022) is necessary to generate the reverse proportional reading of juxtaposed structures. Indeed, the nominals that head juxtaposed measurement structures do receive the main stress of the sentence. Recall, however, that, as we have seen in (76)–(78), there is a strong preference for the nominals to appear in sentence-final position. In this case, it is not clear whether main stress is the result of default stress assignment or the result of F-marking the nominal. More informative are examples where default stress assignment and F-marking produce different stress patterns, like the German example in (40). The moment we move to such examples, however, the empirical picture becomes much murkier. An example like (101) sounds perfectly grammatical to native speakers, but it is almost impossible to figure out what it actually means.

The most natural way to convey the intended meaning is to use a hanging topic, as in (102). Hanging topics in Greek (Anagnostopoulou 1994) are base-generated in their sentence-initial position and receive widest scope. This is confirmed by examples like (103), where the topicalized nominal cannot possibly be linked to any position lower in the structure. Yet (103) has a proportional reading where the percentage specifies the proportional relation of the number of students we hired to the number of waiters we hired.

One cannot simply claim that the structures in (102) and (103) are necessary because F-marked constituents in Greek have to appear in sentence final position. For one thing, there is no such requirement in the language. For example, mono ‘only’ can associate with the adjectival modifier in (104), even if this does not appear in sentence-final position.

Moreover, the analysis in Pasternak & Sauerland (2022) simply cannot generate the observed proportional readings of (102) and (103) even if we do assume that Italus and fitites are F-marked. Take for example (103). For the observed meaning to be generated the denominator should specify the number of waiters we hired. But there is no possible landing site for the QR-ed phrase that would provide the right argument to the percentage. At best, the account generates a reading, where the percentage specifies the relation of the number of students we hired to the total number of people (rather than waiters) we hired. It might be possible to save the account by alluding to the known fact that focus-alternatives simply restrict rather than fully determine the domain of quantification of focus-sensitive quantifiers. But this is not how focus-sensitivity works in Pasternak & Sauerland (2022), where the focus value of the constituent the percentage composes with only enters the truth-conditional meaning via the effects of the focus presupposition on the domain of the degree predicate. There is no space in the account for context sensitivity in its current form.²¹ We will argue in the next section that an analysis that allows context sensitivity to enter the derivation of these readings solely via the contextually supplied proportional measure function faces no similar issues.

4.3 Analysis

We can now proceed to present our analysis of proportional readings in juxtaposed measurement structures. As discussed above, the analysis should (a) explain the correlation between these readings and absolute proportional readings in comparatives, (b) allow enough context sensitivity to derive examples like (102) and (103) where the measures involved cannot be derived solely by grammatical means, (c) but not in a way that sneaks in unwanted readings, like the ones based on non-restricted proportional measure functions in (100). The key ingredient of the analysis that helps us derive objectives (a) and (b) are domain-restricted proportional measure functions. In section 4.3.1 we provide an analysis based on domain-restricted measure functions that also assumes the entry for percentages which we argued is necessary to derive all readings of percentages in differential comparatives. As we will see, however, the context sensitivity introduced in the meaning of percentages leads to an over-generation problem. To solve this issue and still achieve a unified treatment of percentages in both juxtaposed structures and differential comparatives, we will need to revise both the meaning of percentages and the meaning of differential comparatives. We make a concrete proposal to this end in section 4.3.2.

4.4 Absolute juxtaposed measurement constructions

To complete the analysis of juxtaposed measurement, we consider next juxtaposed structures with absolute measures, as in (147). Nothing special needs to be said. The head Meas is the same as in the case of relative MPs. Assuming that the measure function is resolved to a function measuring weight in kilograms, the degree predicate that is the argument of the MP specifies the interval in (152). The MP will specify the value of the length of this interval, as in (153), which is equivalent to saying that the weight of the apples we ate yesterday is 3 kilos.

Before moving to partitive measurement constructions, we show that our analysis can provide an explanation of the observed verbal agreement patterns in number. As we will see, the analysis of these patterns provides some additional evidence in favor of a quantificational analysis of n percent. Recall that this was a major difference between absolute and relative measures in juxtaposed measurement; whereas in the case of absolute measurement, the number on the verb depends on the number of the unit noun, the number of the verb is always plural in the case of relative measurement.

We begin with the assumption that the number of the verb depends on the number of DP via agreement. Moreover, we assume that the number of the measurement construction is determined at the level of a functional projection Num in the extended functional projection of the noun. We assume that Num is located above the unit noun (see Stavrou 2003; Alexiadou & Stavrou 2020 for Greek), as in (157) and (158). For ease of exposition, we treat the absolute MP as denoting an individual degree in a dimension of degrees of weight measured in kilos. Nothing changes when we move to the quantification analysis above.

To provide a concrete implementation, we adopt Scontras’ (2014) analysis of number, according to which the singularity presupposition of the singular feature SG is relativized to a measurement, as in (159). SG carries a one-ness presupposition, so that “every member of a predicate denotation must measure 1” (Scontras 2014: 22).²⁶ The plural feature PL, on the other hand, is semantically vacuous, as in (160). The analysis thus replicates the familiar asymmetry of SG and PL. In contexts where both SG and PL can in principle be inserted, their distribution is regulated by Maximize Presupposition, which favors the derivation with the strictest presupposition, in this case the one with SG.

The singularity presupposition checks whether the degree of every member in the predicate denoted by MeasP in some dimension of measurement equals one. For this analysis to go through, the measure introduced by SG and the measure introduced by Meas are identified. Consider first the case of absolute measurement. In the case of (154), MeasP denotes the predicate in (161). The presupposition of SG is satisfied, so SG is favored over PL. By subject-verb agreement, the number feature of the verb is singular. In the case of (155), on the other hand, MeasP denotes the predicate in (162). The presupposition of SG is not satisfied, so PL is the only option. By subject-verb agreement, the number feature of the verb is plural.

We move next to relative measurement, where number on the verb is always plural. We will see that both analyses with and without proportional measure functions make the right prediction. However, analyses that treat percentages as individual degrees do not. The crucial case is the one with the relative modifier ena tis ekato ‘one percent’ in (156). Why is singular agreement unavailable in this case? Consider first an analysis without proportional measure functions, where proportionality is solely contributed by the percentage. Since the measure function contributed by Meas in this case measures cardinality, the presupposition of SG requires that the cardinality of the students that were hired is one. All we know about this cardinality, however, is that it equals 1% of some other cardinality, the cardinality of the people that were hired. Similar considerations apply in the case of an analysis with proportional measure functions. In that case the presupposition of SG requires that the proportion of the students that were hired to the total number of hirees equals one (or 100%), which, by assertion, is not the case. In an analysis which treats percentages as individual degrees, on the other hand, the presupposition of SG is indeed satisfied, since, in this case, it requires that the proportion of the students that were hired to the total number of hirees equals 1%, which is the case. Such an analysis, then, makes the wrong prediction that singular agreement should be licensed with the relative modifier ena tis ekato ‘one percent’.

5.1 Absolute and relative partitives

Partitives differ from juxtaposed measurement in some obvious ways. First of all, we are clearly dealing with two DP projections, since the inner nominal is definite and projects its own D layer. Moreover, the two nominals, the inner nominal and the measure noun, as seen in the absolute partitive in (164), do not share case. Whereas the case of the measure noun is determined by the syntactic position of the partitive, the inner nominal receives accusative case by the preposition apo ‘from’. An overt PP can also appear in the case of relative measurement. Most frequently, however, the inner nominal appears in genitive case. Notice that genitive did have a partitive function in the diachrony of Greek and can still be found with absolute measurement in some more archaic and formal registers.

Unlike relative juxtaposed measurement, where a definite determiner can only appear under certain conditions and only in the case of absolute measurement, a definite determiner can readily head the partitive, as in (165) and (166), without any apparent difference in meaning. But whereas the number on the definite determiner is always SG with relative measures, in the case of absolute measures, its number value depends on the number of the measure noun.

As in juxtaposed measurement, an overt indefinite determiner is only possible with relative measures and functions as an approximator.

Partitives also differ from juxtaposed measurement with regard to verbal agreement patterns. Recall that in juxtaposed measurement, relative measures only licensed plural number on the verb, whereas absolute measures licensed both singular and plural, depending on the number of the measure noun. Partitives show the same pattern in the case of absolute measurement, as shown in (169) and (170). In the case of relative measurement, on the other hand, the pattern is reversed. Relative measures only license singular number, as shown in (171). Notice, however, that in positions where the partitive controls agreement on the verb the presence of an overt definite determiner (which, as we saw above, is always SG with relative measures) is strongly preferred.

Notice finally that partitives behave like juxtaposed structures in terms of left-dislocation. The PP/genitive DP can be left-dislocated, as shown in (172a) and (173a). Moreover, the numeral and the semi-lexical/measure noun form a constituent that can be left-dislocated, as shown in (173b) and (173b) for relative and absolute measures, respectively.²⁷

It goes beyond the scope of this paper to provide an analysis of partitives that captures all the properties we have observed. In the case of relative partitive structures, a particular challenge that arises is the analysis of structures with overt definite articles.²⁸ Since our current focus is on a different issue, i.e. the locus of proportionality and the interaction of proportionality with partitivity, we put such cases to the side. To be able the to highlight the relevant differences with juxtaposed measurement structures in what follows, we will focus on cases without overt determiners and assume a structure that is minimally different from that assumed for juxtaposed measurement, as in (174) and (175).

As in juxtaposed structures, the covert head D is taken to be an existential quantifier. We assume that semantic partitivity is located in the meaning of the Meas_part head and that the presence of the preposition apo ‘from’ or genitive case is a morpho-syntactic effect with no semantic import. We also assume a mereological account of plurality and adopt a treatment of the definite determiner in terms of the summation operator σ (Link 1983), so that the DP_Gen/PP phrase containing NP will always have the meaning in (176).²⁹

Given the quantificational meaning of D and the quantificational analysis of MPs we have adopted, there are two instances of QR, as shown in (177) and (178).

5.2 Proportional and non-proportional readings of partitive measurement

Following Solt (2018), Bale (2022), and Pasternak & Sauerland (2022) we will introduce partitivity in the meaning of Meas_part by restricting the measure functions introduced by Meas_part to domain-restricted functions. The question we want to consider is whether we can stick to a non-lexical analysis, where Meas_part does not introduce proportionality, as in (179), or we need to move to the entry in (180), as proposed in Bale (2022).

Let us start by considering the entry in (179). What readings does this entry license? Obviously, resolving μ^c to a simple non-proportional measure, like cardinality or weight, the entry will derive the readings of examples with absolute MPs like tris ‘three’ and tria kila ‘three kilos’ in (173). In this case the degree predicate that will be the argument of the measure phrase will be the one in (181) for, e.g., cardinality, where s is the sum of students. Assuming we hired three out of 6 students, (181) specifies the interval in (182). The absolute measure phrase tris ‘three’ correctly specifies the length of this interval.

The same resolution of μ^c can also generate forward/partitive proportional readings, when (181) is the argument of a percentage. Since the function is domain-restricted, the maximal degree in the domain of (181) is the cardinality of the sum of students. Setting the denominator to ENDPOINT_t (the only option, since ENDPOINT_b picks out zero), the percentage specifies the proportional relation of the cardinality of the students we hired to the cardinality of the sum of students.

The entry in (179) also makes successful predictions in case μ^c is resolved to proportional measure functions. Recall that we wish to derive the fact that an example like (163) only allows for forward/partitive proportional readings, not any other proportional readings. Consider first the proportional measure function in (185). Due to partitivity, the function is actually domain-restricted, as in (186). In a context where we hired 3 out of 6 students, the degree predicate in (184) will pick out the interval [0, .5]. The maximal degree in the domain of the predicate is the maximal degree in the range of μ₁;_s, $\frac{\text{|}\bigsqcup \text{STUDENTS|}}{\text{|}\bigsqcup \text{STUDENTS|}}$ , which equals 1. The percentage, thus, specifies correctly the proportional relation of the number of the students we hired to the total number of students.

Assume next that μ^c is resolved to the function in (189), the one that has been responsible for generating reverse proportional readings. Due to partitivity, the function is domain-restricted, as in (190). Crucially, since the domain-restriction is determined grammatically by Meas_part, the domain of the function is restricted to sub-aggregates of the sum of students (not the sum of hirees, as before). In a context where we hired 6 people, 3 of whom are students, the degree predicate in (188) will pick out the interval [0, .5]. The maximal degree in the domain of the predicate is the maximal degree in the range of ${\mu }_{2;s},\frac{\text{|}\bigsqcup \text{STUDENTS|}}{\text{|}\bigsqcup \text{HIREES.YEST|}}$ . What is its value? (191) will only be true in contexts in which the cardinality of the sum of students is equal to the cardinality of the sum of people we hired. Then the value will be 1 and the percentage will provide the right proportional specification of the relation of the two proportions. In this context, however, the forward proportional reading also comes out true since $\frac{\text{|}\bigsqcup \text{STUDENTS|}}{\text{|}\bigsqcup \text{HIREES.YEST|}}=\frac{\text{|}\bigsqcup \text{STUDENTS|}}{\text{|}\bigsqcup \text{STUDENTS|}}$ . More generally, given the grammatically determined domain-restriction on the measure function and the fact that the only available choice for the denominator in the percentage is ENDPOINT_t, any context in which (163) is true under the measure function in (190) will also be a context in which (163) is true under the measure function in (186). The same is true regardless of the choice of proportional measure function, they are all rendered invisible and identical to a measure in which the denominator is identical to the measure of the plural individual denoted by the nominal complement of Meas_part.

The account achieves, in the presence of percentages, the same result that Bale’s (2022) entry in (180) achieves for partitives in all measurement constructions (and in much the same way). The advantage of Bale’s entry, as we discussed in section 2.1., is that it correctly predicts that partitive measurement prohibits non-partitive proportional readings across the board. For example, we saw that (the English version of) (192) only gives to partitive/forward proportional readings.

As we will see, however, there is an issue with simply adopting Bale’s entry instead of the one in (179). The issue does not have to do with the distribution of proportional readings, but with how the entry derives non-proportional readings of partitives.

Let us first reassure ourselves that we can unproblematically combine Bale’s entry in (193) with the proposed semantics of n percent. Consider, again, the example in (163). Under Bale’s analysis, the degree predicate in (194) is the same regardless of the choice of measure function since the denominators in the specification of all proportional measure functions are factored out. Assuming that the LIMIT function is set to LIMIT_t, this predicate is exactly identical to the one in (184), except that the proportional measure is grammatically derived. Feeding (194) to a percentage, then, will lead to exactly the same result, a partitive/forward proportional reading.

Bale claims that the same entry can derive simple cardinal readings when LIMIT is set to LIMIT_b. He assumes that the minimal degree in the range of a domain-restricted measure function μ_DIM;x is 1, the measure of the atoms that are a part of x. If so, the meaning of vP₅ comes out as in (196); it picks out non-proportional degrees of cardinality.

The same setting of LIMIT is said to derive cardinal readings of comparatives like (192). Following Solt (2018), Bale argues that the English version of (192) also has a simple cardinal reading; it can be true in a scenario in which we hired 10 out 20 students yesterday and 8 out of 10 students today. Notice that the proportional partitive reading is false in this scenario.

The first thing to note is that it is not possible to derive cardinal readings in this way under our own assumptions. Since we have been assuming that all the relevant measure functions (domain-restricted ones included) define ratio scales, all the functions discussed here will have a zero degree in their range (even if no individual in the domain of the function is mapped on zero.)³⁰ Setting LIMIT to LIMIT_b, then, will never be an option in the relevant cases above since this will lead to undefinedness. There is at least one reason to think that this is not an unwelcome result. Contra the judgements reported in Bale (2022), and Solt (2018) for English, we do not detect a simple cardinal reading for the comparative in (192). Indeed, we judge (192) false in the scenario above.³¹ The problem is, however, that we still need to be able to derive non-proportional readings to handle partitive measurement with absolute measure phrases like tris ‘three’ and tria kila ‘three kilos’. Under current assumptions, only the entry in (179) can derive non-proportional readings.

To summarize, the entry in (179) generates non-proportional readings, but it over-generates proportional readings in comparatives. Adopting Bale’s entry in (180), on the other hand, derives the right proportional readings across the board, but under-generates non-proportional readings. At this point we have not been able to solve this puzzle by defining a single entry for Meas_part that at the same time (a) only generates the right proportional readings and (b) only generates non-proportional readings in the presence of absolute MPs. Although this issue does not affect the core empirical domain of interest in this paper (i.e. the interpretation of partitive measurement with relative measures), it affects every analysis of partitive measurement the moment proportional measure functions are admitted (as we have argued is necessary in the previous sections). For the time being, we will assume an ambiguity account, according to which both options are available. Bale’s entry is the default entry for Meas_part and the entry in (179) only becomes available as a last resort in the presence of absolute measure phrases. A more principled account awaits to be formulated.