Part-introducing percent in English

Two uses of English percent, called ‘conservative’ and ‘reversed’, have been extensively discussed in the literature. In ‘reversed’ uses, percent introduces a predicate that characterizes a part of a larger whole. This paper points out that there are other constructions in which it does so as well, and illustrates the full range of such ‘part-introducing’ uses, using corpus examples. I then consider how existing theories fare in capturing its distribution, and offer two suggestions for improving the empirical coverage with a uniform treatment of the part-introducing uses. First, I propose a type-shift that converts a non-gradable predicate to a gradable one that tracks mereological parthood. This makes any non-gradable predicate eligible for use with an analysis of percent designed for constructions like 75% full. Second, motivated by cumulativelike readings, I sketch an analysis in a dynamic semantics with plurals in which percent applies to a cross-assignment sum, evaluated after the rest of the constraints in the clause have been applied to the discourse referent in question.


Introduction
As originally observed by Ahn (2012) for Korean, there are at least two distinguishable uses of so-called proportional measure nouns like percent. 1 Sauerland (2014) calls these readings conservative and reversed, respectively (adapted from Sauerland's (2014) attested examples): (1) The most recent class of NASA consists of 50% of the women.
(2) The most recent class of NASA consists of 50% women.
Ahn & Sauerland (2015;2017) show that the alternation is found in German, Korean, French, and English. In the context of the larger cross-linguistic exploration of this special issue, this paper is dedicated to a detailed empirical investigation of the distribution of percent in English using plenty of corpus examples, with an eye toward pushing our theoretical understanding of the phenomenon forward.
The two uses of percent in (1) and (2) both characterize a situation in which there is a part and a whole, where the part makes up some percent of the whole. In (1) (with 50% of the women), the whole is denoted by the women, and there is a part of that whole making up 50% of it that is denoted by the expression 50% of the women. On the other hand, in (2), with 50 women, what comes after percent is a description of the part -there is a larger whole and the part making up 50% of it is characterized as being made up of women. Thus, what immediately follows percent in the 'reversed' case is a predicate that characterizes a part of the whole. In the 'conservative' case, what immediately follows percent characterizes the whole.
Another place where percent can introduce a predicate that characterizes a part of a larger whole is in predicative sentences with adjectives or with nouns: (3) A school system that in 1963 had been 62 percent white became 60 percent black by 1975. (5) The solution consists of 30% acid.
If (2) counts as an example of a reversed use, then so must (5), as it uses the same verb. But because predicative structures are quite different from structures involving transitive verbs, it's a bit odd to refer to examples like (4) as 'reversed' ones. 2 In a predicative structure like (4), we would expect 30 acid to have type ⟨e, t⟩, rather than serving as a quantificational noun phrase, as under the Pasternak & Sauerland (2022) analysis of 30% women. The copula does not normally combine with quantifiers (*The class is few students). I will use the term part-introducing to describe both the reversed uses as in (2) and predicative examples like (4) (the idea being, again, that percent precedes, or introduces, a description of the part).
After a tour through some corpus examples of part-introducing uses, I show that existing theories do not immediately capture all of them, and offer a uniform treatment, treating the predicative uses as basic. First, I propose a type-shift that converts a non-gradable predicate to a gradable one that tracks mereological parthood. This makes any non-gradable predicate eligible for use with Pasternak's (2019) analysis of percent in constructions like 75% full, although it must be modified slightly in order to give the right truth conditions. Second, I sketch a dynamic theory based on Keshet's (2019) PLural Update Semantics, in which plural discourse referents that summarize the accumulated constraints on a given discourse referent can serve as the 'whole' for a percent phrase that both introduces and serves as a predicate. This approach has a number of promising features, including the ability to shed light on certain cumulative-like readings that otherwise elude a compositional treatment.

Part-introducing uses 2.1 Predicative constructions
On the view that I will put forward, the most basic configuration in which percent has partintroducing uses is in predicative constructions. By 'predicative constructions', I mean ones in which a verb selects an expression denoting a predicate of individuals (type ⟨e, t⟩ or, in the case of gradable predicates, type ⟨d, ⟨e, t⟩⟩). There are a number of predicate-selecting verbs; these include not only the copula but also become, remain, and others. There are two different types of predicative constructions involving percent-modified predicates, which I've labeled 'partintroducing' and 'scalar'. These will be exemplified in turn. 2 The verb consists (of) is arguably not an ordinary transitive verb, expressing a relation between two individuals; it's rather more like a copula, insofar as it links an individual with a characterization of that individual. I will return to this in &2.3.

Part-introducing predicative uses
'Part-introducing' uses involve a non-gradable predicate, and carry the implication that this predicate applies to n% of the subject. These are attested with a range of verbs that accept  (6), the non-gradable predicate following percent applies distributively to each individual part in a collection of subparts of a whole making up the indicated percentage of the whole. Furthermore, the whole is rather evenly divided up into these parts. The whole is denoted by the subject of the verb in these predicative contructions.
In other cases where the verb selects a predicate, the predicate is a noun phrase instead of an adjective phrase, and the noun phrase can be 'part-introducing' too. Examples in this category include (4) above as well as the following (again from the COCA corpus): (8) After I did Playgirl, my audience became 95 percent white women.
(9) It must be labeled 100 percent blue agave As with the cases above involving adjectives, these examples have entailments involving partitives making reference to subparts of the subject: 3 Another interesting case that might fit into this category involves the verb divide, which doesn't usually take adjectival complements but seems to in this example: (i) They counted 514 plain M&M's; in a one-pound bag that divided 29 percent brown, 19 percent blue, 17 percent yellow, 15 percent orange, 12 percent red, and 8 percent green.
(10) My audience became such that 95% of the members of it were white women.
(11) It must be labelled such that 100% of the parts of it are blue agave Nominal complements of percent are usually either plural or mass. But it is possible to find singular complements of percent in this category:

Scalar predicative uses
Here are some examples I call 'scalar', taken from the COCA corpus: (15) a. the weapon, dating from the 1570s, was packed 70 percent full with gunpowder, sealed with mud and stored up on the Wall b. Ten minutes of searching your memory to try to pull up what it was you had seen, and after 10 minutes, you said [you were] 40 percent sure. Right? c. The creators extended the work, while remaining 100 percent faithful to the original, and provided an ending that adds a little something extra… These cases are akin to cases like 75% full, discussed by Pasternak (2019) under the analysis to be proposed, these uses are linked. Below, I will propose to account for the difference by using the theoretical assumption that some adjectives are fundamentally gradable (like full), while others (like Democratic) are fundamentally non-gradable and acquire a gradable interpretation through a type-shifting mechanism that tracks part-hood. In fact, I will propose that the same mechanism is at work in the 'reversed' uses.

Attributive part-introducing uses of percent
We turn now to cases involving regular transitive verbs, expressing relations between individuals, instead of selecting a predicate. To set the stage, observe that there is a subtle difference between these two cases: (18) Victoria's Secret… is under pressure to use 50 percent recycled paper in its millions of mailed catalogs and shun paper derived from wilderness regions (19) tobacco farmers won a provision in the recent budget bill requiring manufacturers to use 75 percent domestic tobacco Both contain a string of the form VERB NUM% ADJ NOUN and can be paraphrased using 'NUM% ADJ' as a predicate: (20) The paper VS uses must be 50% recycled.
The tobacco manufacturers use must be 75% domestic.
But the similarities end here. It is more natural to paraphrase (18) by replacing 50 percent recycled paper with paper that is 50 percent recycled than it is to paraphrase ( Among these 'attributive' cases, a distinction can be drawn between 'part-introducing' and 'scalar' uses, just as in the realm of predicative uses of adjectives. Example (18) is a 'partintroducing' use; the adjective recycled is not gradable, and there is a way of dividing this paper up into small, evenly-sized parts such that 50% of those parts have the property of being 'recycled'.
Here are some further examples I would place in this category: (23) a. Hey, firefighters can't afford 100 percent jumbo lump crab meat, OK? b. Runners need at least three to six one-ounce servings of whole grains per day, and eating 100 percent whole-grain bread (as opposed to just whole-grain bread, which may contain some refined grains and flours) is an easy way to meet this requirement since one slice equals one serving.
Both cases just given involve mass nouns, which do not require determiners. Below are cases where the modified noun is a count noun. In these, the noun phrase has an indefinite determiner a preceding the percent phrase: (24) a. In one study, drinkers and smokers "spontaneously abstained" after eating a 67 percent raw diet. b. In 1980-81, 5,000 white students in Henry schools created a 75 percent white system.
Here is a case in which the modifier of the count noun is itself a noun: (25) But I found a 23 percent hydrochloric acid cleanser.
All of these can be paraphrased using a relative clause: 'crab meat that is 100% jumbo lump', 'bread that is 100% whole-grain', 'diet that is 67% raw', 'system that is 75% white', 'cleanser that is 23% hydrochloric acid'. This shows that in these strings of the form VERB (a) NUM% ADJ NOUN, the string NUM% ADJ functions as an intersective, attributive modifier of NOUN.
The same types of examples can be found in subject position: (26) a. Can a 98% white town attract and keep black, Asian and Hispanic families? b. After London, the 100% recyclable sculpture made the rounds on a tour of the U.K. c. Actual rehab of the 95% occupied building took 18 months and $5 million.
These have paraphrases using relative clauses as well: a town that is 98% white; a sculpture that is 100% recyclable; a building that is 95% occupied. Thus, it seems that like part-introducing strings of the form NUM% ADJ, scalar ones can function either predicatively or attributively, and when they function attributively, they can modify a noun in any grammatical position.
To summarize: In scalar attributive cases, (28a) can be paraphrased as (35)  In these cases, the second paraphrase is more precise, but the first paraphrase is also possible, as long as there is some room for contextual narrowing on the subcategorization requirements of the verb. Given an interpretation of the verb get that is restricted in context to the kinds of things The following case involves quantification over the subject position, and each witness for the quantifier corresponds to its own reversed reading.

Distribution of verbs
Even taking the ambiguous cases into account, (potential cases of) reversed readings are restricted to quite a limited set of verbs. (49) a. The committee hired some women, so there were more women. b. # The committee disliked some women, so there were more women.
This is one of the diagnostics that Coppock & Beaver (2015) use for 'entity-introducing verbs'.
Another is the following: (50) a. There are seven women. If the committee hires a woman, that will make eight. b. There are seven women. # If the committee dislikes a woman, that will make eight.
On the other hand, presentational verbs are not good with percent complements: 4 (51) # There appeared 75% women.
Yet they are arguably 'entity-introducing'; although the following sentence is not perfectly natural, it does seem to have an entity-introducing implication.
(52) There are at least seven women. ?If there appears a woman, that will make eight.
So, being 'entity-introducing' may be a necessary, but not sufficient condition.

Subject uses
Before moving on, let us briefly address reversed uses in subject position. Attested uses of percent that could be categorized as 'reversed' are not in abundant supply, but here is a sentence that could be understood that way, offered by a reviewer: (53) Approximately 10% water will remain.
This could certainly be read as a part-introducing use, insofar as what comes after percent is a predicate that characterizes the 10% part. It's clearly not predicative, since 10% water is not functioning as the complement of a predicate-selecting verb. It's also clearly not attributive, because 10% does not combine with an adjectival or nominal modifier to create a complex attributive modifier of a noun. The example can be paraphrased analogously to reversed uses: Approximately 10% of what will remain is water.
So this looks like a 'reversed' use in subject position.
Below, we will review Pasternak & Sauerland's (2022) analysis of the sentence 30% Westphalian students work here-another example with a reversed reading in subject position.
The original example is in German, and it does not feel perfectly natural in English, but I believe it is grammatical and has the relevant reading.
Subject uses can also be obtained by passivizing object uses: (55) 25 percent post-consumer waste was supposed to be used in Coca-Cola's plastic beverage bottles.
The slight degredation of (56) relative to (55) may be due to a subject-verb agreement problem; 75% women does not work well as a plural. (It works even less well as a singular; replacing were with was makes it clearly ungrammatical for me.) But the following sentence strikes me as dramatically worse than (56): (57) * ? 75% women were disliked by the committee.
Thus it seems that the same restrictions on the set of verbs that we saw above applies to subject uses with passive verbs.
Overall, it does seem that reversed uses in subject position are a bit degraded and a bit more difficult to find. This may be due to a requirement for focus on the following predicate (which is predicted under some analyses; see below). As is well-known, English subjects tend to be topics, rather than foci (Erteschik-Shir 2007;Foley 2007;Lambrecht 1994).

Cumulative readings
The final part-introducing use I will discuss involves something like a cumulative reading (i.a. (Scha 1981;Krifka 1986;Champollion 2009: i.a.): (58) In one approach, they mix 10 percent ground-up, dried manure with 90 percent coal (by weight) and then… (58)  What makes (58) like a cumulative reading is that the quantificational elements do not take scope independently of each other. A classic example of a cumulative reading is in the sentence 600 Dutch firms own 5000 American computers (Scha 1981), where 600 Dutch firms and 5000 American computers are involved, but for any given firm, it's not guaranteed that it owns all 5000 computers, nor is it guaranteed that any of the 500 computers are owned by any of the firms. So neither of the quantificational expressions takes scope over the other. As Champollion (2009: 216) puts it, citing Szabolcsi (1997), The following cases, using with, are a bit more involved: (61) Much less biocide is needed per gallon of paint -typically 25 percent along with 5 percent zinc omadine.
(62) biodiesel runs fine in unmodified diesel engines at up to a 20 percent blend with 80 percent petroleum diesel, a combination known as B20.
In considering these examples, a question that arises is whether and to what extent conjunction (or comitative semantics) plays an important role in cases like (58). In that example, it seems that the two percent cases are not actually syntactically coordinated: (63) Q: #What did they mix? A: #10 percent manure with 90 percent coal.
(64) #What they mixed was 10 percent manure with 90 percent coal.
Rather, the with-phrase appears to introduce its own syntactic argument in this case. So while these cumulative-like readings often involve coordination, they need not do so.

Summary
To summarize, we have identified several categories of predicate-selecting percent, each licensing their own paraphrase patterns. A string of the form: There are also cumulative-readings, which don't quite fit into this schema. In the next section, we will apply existing analyses to these cases and see how far we can get.

Conservative uses
At the time of writing, the most recent analysis that has been made publicly available (Pasternak & Sauerland 2022: available on LingBuzz) treats percent as in (74). This analysis is analogous to their treatment of absolute measure nouns like kilo, which they treat as in (73). Both percent and kilo combine with two degree predicates D and D′ (type ⟨d, t⟩) in addition to a number n (which enters into the composition between the two degree predicates).
Note that in (73), I have written n·kg to represent the quantity of n kilograms using the dot (·) for multiplication, assuming that degrees and numbers can be multiplied as described by Coppock (2021); I assume kg denotes a particular degree, the one corresponding to one kilogram.
In partitive, 'conservative' uses, these measure nouns will be selected by a function they call meas (a bit like Solt's (2009) meas, but compositionally different), defined as follows.
Here, μ is to be read as a free variable, set by context to an appropriate measure function. 5 In a partitive example like 30 kilos of the apples, meas combines first with kilos, as in the following derivation (in which some simplifications are used, including rewriting max(μ⟨x⟩) as μ(x) and restating a subset relation between two degree predicates as a less-than-or-equal relation among their maxima): The same happens with meas and percent, giving the following derivation.
Hence n percent of the apples ends up denoting a property that holds of a subpart of the apples making up n percent of it. If this property combines with a silent existential quantifier, this analysis will yield conservative truth conditions for a sentence like 5 of the apples are ripe, in the sense that this sentence will be equivalent to 5% of the apples are apples that are ripe. With the existential quantifier, the meaning can be paraphrased 'there is an x such that x makes up 5% of the apples and x is ripe', which clearly entails that 'there is an x such that x makes up 5% of the apples and x is ripe apples'. 5 Pasternak & Sauerland (2022) write the context-sensitive μ function as μ c , where c denotes the relevant context. I will not use a superscript c for μ in the representation language. Rather, I will assume that it is an indexical constant of the language like i, which picks out the speaker of the given context of utterance; just as ⟦i⟧ M,g,c = the speaker of c, so ⟦μ⟧ M,g,c depends on some feature of context c. This is a purely stylistic matter.

Reversed readings
The 'reversed' reading is derived from the following structure: where dis is short for 'disjunction' and acts semantically as a grand union operator that applies to a set. As shown in the following tree, deg converts a non-gradable predicate to a gradable one by introducing a contextually-determined μ function. The meaning that ∼C attaches to is derived as follows: Hence the value of C will be: 6 Assuming, following Pasternak & Sauerland (2022), that the trivial property λx . ⊤ is among the alternatives to Westphalian, the 30 percent dis C component will have semantics derived as follows: Simplifying a bit, putting these two subtrees together will produce the following: Here is another way of saying the same thing: (83) |we ∩ st ∩ wo| ≥ 30 100 · |st ∩ wo| '(At least) 30% of the students who work here are Westphalian' So this analysis derives truth conditions for a reversed reading in a focus-sensitive manner, using the same lexical entry that is used to derive conservative truth conditions (given the assumption that the trivial property is always among the focus alternatives, and a silent disjunction operator).
Notice that the truth conditions for the reversed case are not conservative, in the sense that 30 Westphalian students work here does not entail 30 Westphalian students are Westphalian students who work here. 6 I'm using an indirect interpretation style here, which involves translating from English to a version of typed lambda calculus. It's not entirely settled how to deal with focus in an indirect interpretation style, as far as I know. Here I'm assuming that the representation language imports a number of set-theoretic devices and that it contains a context-sensitive function alt that takes a natural language string or parse tree as an argument.

Predicative uses
Now let us consider predicative uses like the following: The solution is 30% acid.
(adapted from BNCWeb) These are like reversed cases in that what follows percent is a predicate that holds of a subpart of a whole. Indeed, (84) could be rephrased analogously to (2) above: The solution consists of 30% acid.
But I assume that the copula takes a predicative complement, while consists of does not.
If we assume that 30% acid is type ⟨e, t⟩, and that 30% is raised and interpreted as under Pasternak & Sauerland's (2022) analysis of reversed structures, then we derive the following structure: Expanding the 30% dis C part gives: In words, the truth conditions derived can be expressed as follows: The degree to which x 2 is acid and big (along dimension μ) is greater than or equal to 30% of the degree to which x 2 is big (along dimension μ). Since 'acid' is a unary predicate under this treatment, a given object will be in its extension or not; there is no middle ground. If x 2 is not acid, then the first degree predicate will yield an undefined value, because the maximum of the empty set is undefined. So, assuming that something that is 30% acid is not acid, the sentence is predicted to have a truth value of 'undefined' when it is in fact true.

Pasternak on 75% full
Is there a precedent in the literature that we could look to in order to come to grips with these simple predicative cases like The solution is 30% acid? Pasternak (2019) provides a theory of some predicative cases involving adjectives like the following: (88) The glass is 75% full. (Kennedy & McNally 2005;Pasternak 2019) These are the types of uses labeled 'scalar' above. As Pasternak notes, this construction requires a gradable predicate that is associated with a totally closed scale, i.e., one that has both a minimum and a maximum; cf. #70% tall. where: • A is a gradable predicate associating individuals with degrees • range(A) picks out the set of degrees that A could sensibly associate an individual withand is defined as λd .∃x . defined(A(d)(x)), following Pasternak • defined(ϕ) is true if and only if ϕ is true or false This lexical entry yields a straightforward analysis of predicative cases like (88): The fraction appearing to the left of the equals sign contributed by the lexical entry for percent will have in the numerator the degree to which the glass is full, assuming the min term is the zero element for the 'fullness' dimension. (Here I am embellishing on Pasternak's explanation using the system described by Coppock (2021), following Raposo (2018;, where there are many zeroes, one for each dimension.) The denominator will be the degree corresponding to the size of the full range of possible fullnesses, which will be equal to the maximum possible degree of fullness (assuming again that the minimum is the relevant zero element). Whether or not fullness-degrees are themselves proportions, and in whatever units fullness is measured, this fraction can be represented as a number, since the quotient of two quantities associated with the same dimension (e.g. fullness) is a so-called 'dimensionless quantity', representable by a number without any accompanying unit (JCGM 2012;Coppock 2021). (Here again I am embellishing on Pasternak's explanation.) The truth conditions expressed by that sentence is that the ratio of these two quantities is equal to the number 75 100 . As Pasternak shows, this lexical entry can elegantly be extended to handle conservative cases.
Incorporating ideas from Wellwood (2015), Pasternak proposes the following lexical entry for silent much: is true if ϕ is true, and undefined otherwise.
I have written the presupposition that μ(y) ≥ d using the 'partial operator' ∂ (Beaver & Krahmer 2001). This presupposition is contributed by Pasternak, and it plays a crucial role in the truth conditions, because it controls the range of the derived gradable predicate.
Let us see how it works with a 'conservative' example like (48a) (The committee hired 75% of the women). This silent much combines with of the women to produce a gradable predicate that percent then combines with: The variable A in the lexical entry for percent is replaced by the following gradable predicate: Intuitively, this measures the size of a subpart of the women. The greatest degree for which A is defined for anything can be expressed as μ(σ(women)), because of the presupposition requiring that d to be smaller than or equal to the measure of the women μ(σ(women)). The smallest degree is 0 μ , "the 0-degree of μ" (p. 78). (In Coppock's (2021) terms, this would be the additive identity element for the dimension along which μ measures things.) Referring to this element ensures that the scale is lower-bounded.
Pasternak is not 100% explicit in the text about why zero is the lowest degree in the range, but this result could be assumed to follow in part from the fact that d is comparable to μ(x) using ≥, which would mean that it is a degree of the same dimension as μ(x). (I am assuming that every degree/quantity has a corresponding dimension; cf. Raposo's (2019) dim mapping.) The comparability of those degrees, signalling a common dimension, along with an assumption that there are no degrees below zero along that dimension, could derive the result 0 μ is the lowest d for which A(x)(d) is defined for this A and some x.
If we add a silent existential quantifier to 75% of the women, then for (48a), we derive the proposition that there is a plural individual making up 75% of the women that the company hired. Strictly speaking, this does not rule out the possibility that the company hired more than 75% of the women, but that upper-bounding inference may well be pragmatic. Success! Pasternak (2019) claims that "this analysis can be extended equally well to Ahn & Sauerland's (2017) treatment of [reverse cases]". Under Ahn & Sauerland's (2017) treatment, percent takes an individual as its first argument, representing the 'whole'. In reversed cases, this argument is saturated by a silent definite description whose descriptive content is (a flattened version of) the focus semantic value of the sentence, abstracting over the position of the percent nominal using 'modified trace conversion', which inserts an indexed definite determiner the x in the would-be trace position. Thus (93a) has the Logical Form (LF) in (93b).
Ahn & Sauerland use the following lexical entry for percent: One way to merge their treatment of reverse cases with Pasternak's assumptions might be to combine Pasternak's lexical entry for percent with Ahn & Sauerland's (2017) syntactic assumptions, with the exception that a silent much is inserted above ιc so as to produce an argument of type ⟨d, et⟩ that percent could combine with. I believe this would enable a compositional treatment of relative readings that produces the right truth conditions. This treatment does not immediately account for cases like The solution is 30% acid, though, in part because acid is not a gradable adjective, so it is the wrong type to combine. On the other hand, if acid could be coerced into a gradable predicate, then perhaps a treatment of such cases could be obtained. The question is what sort of coercion operation would yield the right truth conditions.

Some non-solutions
Not every way of converting a non-gradable predicate like acid into a gradable one delivers the right truth conditions. Pasternak's (2019) much is a device we have already used for converting things into gradable predicates, but it expects an argument of type e. Suppose that in order to satisfy that type requirement, we coerced acid into an individual by making it denote the sum of all (contextually-relevant) acid in order to satisfy the type requirements of much. Then we would derive a meaning for The solution is 30% acid that could be rendered back into English as The solution is 30% of the acid. This is not a faithful interpretation.
Alternatively, we might try to convert acid into a gradable property using m-op, from Rett (2018: 105), which shifts an ⟨e,t⟩ meaning to a ⟨d,⟨e,t⟩⟩ meaning: As a minor variant, we could use Solt's (2009) meas, defined as λxλd . μ(x) ≥ d, and combine it with acid as she proposes using 'Variable Identification', which would give the same thing.
Whether we derive the meaning in (96) using m-op or meas, the truth conditions for The solution is 30% acid would then be 'The solution is acid and its measure is 30%.' This is not correct either.

A solution
My proposal for how to link 75% full and 30% acid involves converting a non-gradable predicate into a gradable one tracking parthood of whatever it applies to. For the purposes of discussion, let us imagine that there is a silent lexical item called part that denotes the operation in question: Combined with a non-gradable predicate like acid, part will yield a gradable predicate that holds of an object to a greater degree the more parts of it satisfy the input predicate. That gradable predicate can then serve as an argument to percent. Our part is similar to much, but it combines first with a predicate like acid first rather than the 'whole', and the 'part' element (y) is existentially bound; the gradable predicate that it produces after applying to its first argument applies to a whole rather than a part. The partial operator ∂ is used introduce a presupposition inspired by the one in Pasternak's much. It plays a similar role here: constraining the set of degrees so that they are upper-bounded by the μ-measure of the whole.
To complete the analysis, it will be necessary to modify Pasternak's percent slightly. 8 Applying part to acid yields the following gradable predicate: Let us abbreviate this predicate as A. In Pasternak's lexical entry for percent, the denominator is max(range(A)) -min(range(A)).
The max expression picks out the greatest element in the range of A. should be the weight of the 'whole' relative to which we are comparing the weight of the 'part'.
With the following definition of percent, the ratio in question is between the μ-measure of the acid-part of the whole and the μ-measure of the whole: where: When percent combines with part acid, the greatest value for d such that A(d)(x) is defined is the greatest value for d such that μ(x) ≥ d is true. Hence, since x stands for the 'whole' of which the acid is the 'part', the denominator now correctly picks out the relevant monotonic measure of the 'whole'. Applying the part shift to acid produces the following reasonable-seeming meaning for 30 percent acid: Two key moments in this derivation are the simplification of A ⊤ (x) to μ(x) and the simplification of A ⊥ (x) to 0 μ . In this case A is: The maximum will be μ(x) due to the presupposition, and the minimum will be 0 μ assuming that is the smallest degree on the μ dimension, and that A maps the value of x to either true or false at the smallest degree on that dimension. Another important moment is the simplification of to μ(σ(λy . y ⊑ x ∧ acid(y))). This is licensed because the greatest degree to which an acid subpart of something measures that degree (on some dimension) is the measure of the mereological sum of the subparts that are acid (on the same dimension to imagine that any object in the domain will be associated with the same definedness range.
Hence, we are free to construct the range of A based on any arbitrary member of the domain, rather than existentially quantifying over it, in order to produce the relevant range for cases like 75% full.
That said, some constraints must be put on this part operation. As a reviewer points out, the distribution of the gradable version of acid under consideration here is limited. It does not combine directly with measure phrases, for example: (101) ?The solution is 5mg acid.
(Intended: The acid part of the solution is 5mg.) Furthermore, as the same reviewer points out, The soup is 30% oil does not mean that the temperature of the oil part of the soup is 30% of the temperature of the soup. In other words, this construction only permits measures that grow monotonically with mereological expansion of an object or substance, going from a part to a whole; things like volume and weight. This observation is reminiscent of the constraints on the monotonicity head mon head posited by Schwarzschild (2006). Perhaps mon is involved here. The reviewer observes that part can be derived from more basic parts including mon and a silent partly: ≡mon where μ is a measure that grows monotonically with mereological expansion can be applied to partly to yield a function that composes with mon to produce part: If it is though these more basic operations that the part-tracking gradable predicate is produced, then it should only manifest itself in the syntactic environment of the mon head, and the range of μ-measures that a context might supply should be constrained to those allowed by the mon head (e.g., the montonic ones, if monotonicity is indeed the key property).
This analysis also predicts a mereological/part-introducing reading for The pie is 40% hot, where 40% of the pie is hot. Since the hot scale is not totally closed, we do not get a scalar reading for this example, as Pasternak's entry was carefully designed to ensure that the only predicates that could appear as the complement to percent are totally closed.
We have just seen that this analysis works well for the predicative cases. The same kind of treatment would also straightforwardly produce the right kind of truth conditions for partintroducing attributive cases like 50% recycled paper; the modifier 50% recycled would translate as an expression of type ⟨e,t⟩ in an entirely analogous way to 30% acid, and this could be combined with paper using Predicate Modification.
Our analysis also predicts that expressions of the form n% P, for some predicate P, should be capable of occurring in a pseudopartitive construction or as the complement of a determiner, since such expressions denote predicates. This prediction is borne out (examples from Google, and confirmed as grammatical by the author's native speaker intuitions).
(105) Can I eat a lot of 100 percent dark chocolate without added sugar and still lose weight?
(106) If much material remains on the filter, an ounce of 50 percent alcohol may be used to dissolve it.
(107) How much 50 percent wettable powder formulation should be added per tankful of water?
(108) While most 100 percent fruit juice is 100 percent sugar, you can try tomato juice or a vegetable juice alternative.
Replacing most with all in (108) yields a grammatical example as well. Phrases of the form n% P generally behave predicates, as the analysis predicts.
Pasternak's (2019) much can be adopted in order to gain a satisfactory compositional account of the 'conservative' uses. This option is available under the present proposal as well.
Could we work backwards from this analysis to gain a general account of part-introducing uses, including the reversed cases as in (109)?
Suppose that 75% women, which denotes a predicate on this analysis, could undergo a kind of Quantifier Raising (QR), and that a sum operation took place over the resulting lambda abstract: The following case presents an even more vexing compositional challenge: (111) The manufacturers use 75% domestic tobacco.
As discussed above, this seems to mean that 75% of the tobacco the manufacturers use is domestic. Pasternak & Sauerland (2022)  where P is type ⟨e, t⟩ and ident is a silent lexical item achieving the ident shift: In the next section, I will suggest a way of implementing this basic idea in a dynamic framework using independently-motivated mechanisms developed to account for plural anaphora. I'll also argue that these tools can also be used to shed light on the cumulative-like readings as in mix 10% manure with 90% coal.

Cumulative-like readings
The suggestion in the previous section does not suffice to account for cumulative-like readings, as in the mix example discussed above, repeated here: (117) They mix 10 percent manure with 90 percent coal.
This example could be accounted for fairly easily if the two percent-nominals were coordinated; we could then have a coordinated predicate that holds of x if 10% of x is manure and 90% of x is coal. But as argued above, it seems that the with phrase introduces its own argument of the verb, rather than being syntactically coordinated with the direct object.
But these truth conditions are a bit too weak. All they say is that there is a sum x + y such that 10% of it was manure and 90% was coal (and x was mixed with y). They don't require that 90% of what was mixed in, total, was coal. To enforce that, we can sum over all candidate values for x ⊕ y using Keshet's + operator, which gives the sum over all candidate values for a given discourse referent. In PLUS, the only terms are variables, and variables prefixed by +, whose interpretation is the sum of all current candidate referents for the variable: (121) Interpretation of terms in PLUS • ||t|| g,σ = g(t) • ||+t|| g,σ = sum({g(t) : g ∈ σ}) where g is an assignment, σ is a set of assignments (a state), and for any set S, sum(S) denotes the smallest plural individual that contains every member of S as a subpart. 10 With this operator in hand, we can characterize the semantics of (117) as follows: (122) [x]; [y]; mix(z, x, y); 10%(part(manure))(+(x ⊕ y)); 90%(part(coal))(+(x ⊕ y)) Now, 90% coal applies to the sum of all of the x + y sums such that x was mixed with y. 11 This sum-operator provides an elegant representation of the meaning of the reversed cases as well. For example, it would allow us to represent They hired 75% women as follows: (123) [x]; hired(z, x); 75%(part(women))(+x) 10 Keshet treats the denotation as the set itself, but treating the denotation as a plural individual works better for the purpose of giving a unified account of percent, as 'NUM% ADJ' can clearly function as a property of an individual, as in The glass is 75% full. The + operator is the crucial innovation in PLUS setting it apart from Dynamic Predicate Logic (Groenendijk & Stokhof 1991). 11 A reviewer points out that these truth conditions might be too weak. Suppose Jack has two barrels. In one barrel, there is a mixture weighing 50lbs consisting of 10lbs of manure thoroughly mixed with 40lbs of coal. The other barrel contains 50lbs of coal. Jack combines the contents of the two barrels. Did Jack mix 10% manure with 90% coal? The analysis in (122) would predict it is true in the situation described, but it doesn't seem like an appropriate description. In future research, it would be worth probing whether the sentence is really false in this situation or merely pragmatically infelicitous.
Here +x denotes the sum of all x such that they hired x. For They used 75% domestic tobacco, a natural representation would be: (124) [x]; use(z, x); tobacco(x); 75%(part(domestic))(+x) Here +x denotes the sum of all x such that they used x and x is tobacco. I conjecture, therefore, that part-introducing percent might always apply to a cross-assignment sum, evaluated after the rest of the constraints in the clause have been applied to the discourse referent in question. I suspect, moreover, that there may be some connection between the use of a cross-assignment sum and the restricted set of verbs that give rise to reversed readings, but I must leave that connection unexplored in the present work. I leave it to future research to develop this in a compositional dynamic framework and thoroughly evaluate its predictions.

Conclusion
This paper has explored a wide range of part-introducing uses of percent, including not only the 'reversed' uses (as in The committee hired 75% women), but also predicative (as in The committee is 75% women), and attributive uses (as in They used 50% recycled paper). On the empirical side, this paper also showed that the range of 'reversed' uses is restricted to a limited set of verbs, and that certain cumulative-like examples can be found (as in They mix 10% manure with 90% coal).
In order to give a unified analysis of these part-introducing uses, I made two suggestions.
First, I proposed a type-shift called part that converts a non-gradable predicate to a gradable one that tracks mereological parthood. This makes any non-gradable predicate eligible for use with Pasternak's (2019) analysis of percent in constructions like 75% full. A slide modification of Pasternak's (2019) analysis of percent provides an adequate analysis of the scalar predicative uses that can easily be extended to the scalar attributive uses. With the part operator, we gain an adequate treatment of the non-scalar, part-introducing predicative and attributive uses, as well as the reversed uses. Second, motivated in part by slight compositional challenges in the realm of reversed uses but mainly in order to account for cumulative-like readings, I sketched a dynamic theory based on Keshet's (2019) PLural Update Semantics, in which plural discourse referents that summarize the accumulated constraints on a given discourse referent can serve as the 'whole' for a percent phrase that both introduces and serves as a predicate. While it is only a sketch, it promises to deliver an elegant account not only of cumulative-like readings but also of reversed uses.