2.1 Previous observations on %Qs

Sauerland (2014), Ahn & Sauerland (2015a; b; 2017) and Pasternak & Sauerland (2022) show that the difference between conservative and non-conservative readings of %Qs correlates with different properties cross-linguistically, such as definiteness marking and case marking, as we already saw for English in (6). German behaves like English in this respect (7). While under the conservative construal in (7a), the DP modified by the %Q appears in genitive case and with a definite determiner, under the non-conservative construal in (7b) the now bare nominal agrees in case with the %Q.

S&Co. furthermore observe that some languages pattern with German in allowing for the non-conservative construal with subjects (8), whereas other languages pattern with English, in which the non-conservative construal with a %Q subject is not possible (9). They label this the subject-object asymmetry (SOA).

The proposals that S&Co. ultimately spell out cannot account for this fact and leave the SOA for future research.

Finally, a property that S&Co. view as crucial for the non-conservative construal in all the languages discussed, is its focus sensitivity. For instance, in (10), sentential stress, which they analyze as focus accent, lies on the nominal that appears after the %Q.

Though S&Co. do not explicitly discuss this, in this respect particularly their intransitive examples differ from the conservative construal where (at least under the default accent pattern) sentential stress falls on the right-most constituent (11).

S&Co. take the different accent pattern in the non-conservative cases as marking a particular focus structure that (implicitly) deviates from the default accent pattern. Therefore, the analysis they propose crucially builds on the idea that non-conservative construals are always focus-sensitive. Pasternak & Sauerland (2022) illustrate this with the contrast in (12).

They argue that the difference in what is under focus, i.e., ‘Westphalian students’ in (12a) or just ‘Westphalian’ in (12b), leads to a difference in truth conditions. For example, if the employer in question has 100 employees, out of which 50 are students, and 15 of these are Westphalian, the sentence is false under the conservative reading in (12a), because only 15% but not 30% of the workers are Westphalian students, but it is true under the non-conservative reading in (12b).

The relevant cross-linguistic differences S&Co. arrive at are summarized in Table 1.

While we do not challenge the empirical observation concerning (12) itself, the papers by S&Co. raise several questions.

2.2 Open questions

With respect to S&Co.’s proposal, we have general doubts about the primary role that focus plays in their account, since it is easy to construe non-conservative examples with neutral stress, for which it seems counter-intuitive to talk about focus sensitivity; we will come back to this in §6. Instead, our intuition is that conservative and non-conservative construals have different predicational structures, in some cases leading to different default word order patterns, and that differences in stress are derivative from this difference because of a tendency of neutral stress to fall on the last constituent in a sentence. A more general issue that arises for any formal account is that it has to capture the intuition that the percentage expression first seems to operate on the VP and only then on the NP that it appears with. S&Co. do it by taking focus to be the driving factor. Instead, we propose that it is rather that the %Q needs to stay within the VP to have an effect on it first. Let us see why.

First, according to our intuitions all German non-conservative intransitive examples that S&Co. discuss involve a marked word order, SVPP, whereas the reverse PPVS order sounds more natural. If we only consider examples with unmarked word order and stress pattern, the conservative and non-conservative construals for intransitives do not only differ in definiteness and case marking, but crucially also in word order (13).

These examples with neutral word order also involve default stress on the final constituent. If our intuitions are correct, this could be interpreted in such a way that the differences in focus that S&Co. observe are merely derivative from the difference in word order.³

A related observation is that leaving out the location from the German non-conservative intransitive examples leads to ungrammaticality, irrespective of word order (14).

Returning to English, we observe that changing the structure with intransitives to something that looks like an existential construction leads to a grammatical counterpart; also in this case, leaving out the location results in ungrammaticality (15).

Thus, a first research question concerns the role that word order plays in bringing about one or the other reading; word order has not been addressed in previous research on the topic. A second research question that comes out of these initial observations concerns the source of the SOA in English, as opposed to German, how the SOA can be circumvented, and how non-conservative readings with subject %Qs are connected to existential-like structures.

A third research question is raised by the cross-linguistic generalizations presented in Table 1. If there are languages that lack definiteness marking altogether and also do not show any of the other distinctions addressed in Table 1, how do they express the difference between conservative and non-conservative readings, if at all? In this paper, we will be concerned with Slavic. Except for Bulgarian and Macedonian, Slavic languages lack overt definiteness marking. For example, Polish %Qs uniformly look as in (16), and there is no way to vary the case marking because 5+ numerals are always followed by nominals in the genitive.

We will investigate whether this kind of string gives rise to both conservative and non-conservative readings. Our initial intuitions for Polish are that it does, but that again, word order plays a crucial role in bringing about one or the other interpretation. In §4, we look into the empirical basis for conservative and non-conservative construals in most Slavic languages, both with and without definiteness marking, as well as in German, to explore the role of word order in distinguishing between the two readings. First, however, let us say some words on word order and stress in Slavic and German.

4.1 Empirical research questions and hypotheses

The empirical research questions we explore in this paper are the following.

EQ1 concerns the question whether the same string in Slavic languages without definiteness marking, recall (16) and (22a), can license both readings under examination. We expect that this is the case. If so, EQ2 arises, which concerns the role of word order, as possibly the only means to distinguish between the two construals in these languages. EQ3, then, explores the interaction between definiteness and case marking, on the one hand, and word order, as a means to additionally distinguish the two interpretations, on the other.

Our expectations for both EQ2 and EQ3 are that non-conservative readings involve (syntactically) low/VP-internal %Qs, irrespective of whether they are subjects or objects. A further expectation is that conservative construals involve syntactically high subject %Qs, because all the Slavic languages involve canonical SVO order, and German involves canonical SOV order (+V2). On the other hand, we do not have clear expectations for conservative readings with objects, since there are two options: Either we have low object %Qs, because of canonical SVO/SOV order, or we have high object %Qs, because this would lead to a maximal disambiguation of the readings by word order. Finally, if the expectations for EQ2 and EQ3 turn out to be correct, we can ask EQ4 about whether a particular word order is obligatory or optional for a given reading to arise. We expect that in the absence of other cues, the correlation between word order and reading is rather categorical, and a requirement in Slavic languages without definiteness marking. In contrast, for Slavic languages with definiteness marking and German we expect this correlation to be a tendency, rather than a categorical rule.

4.2 Corpus data

We performed a corpus search for Polish, Czech, and German to get a first impression about the empirical situation. We were interested in whether Polish and Czech %Qs give rise to non-conservative readings, in the absence of definiteness marking (EQ1). Second, we wanted to see whether there is an indication in the corpus data that the non-conservative construal of %Qs in the languages in question prefers or requires them to appear low (EQ2–4).

In the National Corpus of Polish (Przepiórkowski et al. 2012)⁵, we searched for the string procent kobiet ‘percent women.gen’, which resulted in 101 hits.⁶ Six of these were clear cases of the non-conservative construal, in which the %Qs always appeared in postverbal, never in preverbal position. This suggests that the non-conservative reading is available, but that it is not frequent. Furthermore, the findings suggest that it is possible that the non-conservative reading requires %Qs to appear low.

In the Czech National Corpus (CNC) (Křen et al. 2012)⁷, we similarly searched for the string procent žen ‘percent women.gen’, which resulted in 110 hits. Out of these we had 16 clear cases of the non-conservative construal, 15 of which were in postverbal position, and we only had one instance of the preverbal pattern. We will discuss this example in §6.3. Overall, then, these data confirm our initial intuition that the non-conservative construal is available also in Czech and that there seems to be at least a high preference for the postverbal order.

For German, we searched for the string Prozent Frauen ‘percent women’ in the German Reference Corpus (DeReKo) (Kupietz & Keibel 2009)⁸, where Frauen appears as a bare nominal, which should therefore give rise to unambiguously non-conservative construals. This search returned 207 hits, out of which many were double or irrelevant, e.g., because there were punctuation marks between Prozent and Frauen. After filtering, 36 relevant hits remained. We did not find any sentence-initial/higher %Qs. Instead, there were 10 lower subjects of intransitives, which all appeared with sentence-initial PPs. We found only two transitive subjects; none of these appeared in a sentence-initial position, and both were in a nominative-dative constellation (26). The seven transitive objects were all low, but given that objects canonically appear low this is less telling. We found three headlines, all with PPs, and 14 other instances of this string in PPs or adjuncts.

Thus, also this search confirms our intuition that the non-conservative construal correlates with a low position of the %Q, even when it is the syntactic subject (in nominative case). The German data additionally confirm that with intransitives a sentence-initial PP seems to be required or at least the default case. Let us then turn to the native speaker judgments we elicited via a questionnaire.

4.3 Cross-linguistic questionnaire

In order to investigate whether Slavic languages without a formal distinction between possibly conservative and non-conservative construals nevertheless have both readings available (EQ1), and to furthermore compare this with German and with Slavic languages with definiteness marking (EQ2–4), we designed a questionnaire with %Qs as subjects of intransitive sentences and as objects of transitive sentences. In the test items we varied the word order (preverbal/sentence-initial vs. postverbal/sentence-final), and they were judged in scenarios that favor either the conservative or the non-conservative reading. We first built scenarios and schematic test items in English and then translated these into German, Polish, Czech, Slovak, Russian, BCMS, Slovenian, Bulgarian, and Macedonian.⁹

4.4 General discussion

Let us first provide answers to the empirical research questions we started out with in (25):

Both the corpus data and the native speaker judgments confirmed the intuitions we had for German and Polish about the role of word order, and furthermore showed that other Slavic languages behaved similarly. For German, it was clear that the non-conservative reading was best with %Qs in a low position. For transitives, this is also the canonical word order SOV, but for intransitives this results in a potential deviance from the canonical word order, namely in PPSV. For the conservative reading, %Qs were preferred in their canonical word order, i.e., SPPV for intransitives and SOV for transitives.

For Slavic languages without definiteness marking, the judgments were more categorical in that non-conservative readings required postverbal %Qs (PPVS and SVO), whereas conservative readings required (intransitive SVPP) or preferred (transitive OVS) preverbal %Qs. Thus, the differences between these languages and German were twofold. First, a potential reason for the more categorical judgments in Slavic languages without definiteness marking could be that in the absence of a formal distinction, word order is the only means to differentiate between the two readings. The second difference is that in transitives German adheres to the canonical word order SOV for both readings, whereas the Slavic languages without definiteness marking readily accept the OVS order for the conservative reading.

We assume that the reason for this second difference is that even though both types of languages display word order variation to some extent, Slavic languages exhibit even more flexibility (see, e.g., Bailyn 2011; Jasinskaja & Šimík to appear). In particular, whereas the OVS/OSV order in German is rather marked in requiring an explicit contrast and therefore probably involving A’-movement to the left periphery, the OVS order in the Slavic languages under discussion can also be due to A-scrambling that does not require any contrast but could be motivated by other factors (recall §3). However, note that even in the presence of a contrast in both transitive scenarios, which could have facilitated the OVS order (the deviance from the canonical SVO order) in the conservative scenario, our informants still preferred or required the canonical SVO order in the non-conservative scenario, in either German or Slavic.

For Slavic languages with definiteness marking we expected the results to either be similar to German, since both make a formal distinction between the two readings, or to the other Slavic languages, because it is generally assumed that all Slavic languages exploit word order variation to a great extent (for Bulgarian and Macedonian, see, e.g., Rudin 1990; Jaeger & Gerassimova 2002; Tomić 2008). The results suggest that the latter is true for Macedonian and the former for Bulgarian. In particular, the Macedonian speakers had more categorical judgments and a preference for OVS in the conservative transitive scenario (albeit with the additional clitic), whereas the Bulgarian speakers had a strong preference for the canonical SVO order for all transitives and less categorical judgments.

The Macedonian and Bulgarian data raise several questions, to which we do not have an answer at this point. One question concerns the role of the Macedonian clitic gi, whose semantics is not understood (on its syntax, see, e.g., Tomić 2008; Franks 2009). For some of our informants it seemed to require definiteness of the element thus doubled, for others specificity; for some it seemed to be able to pick up both ‘50%’ and ‘the women’, for others only one of these. A more general question concerns the precise semantics/pragmatics of definiteness marking in these languages, in comparison to definite articles in German and other languages. Unfortunately, we are not aware of any formal semantic/pragmatic account of definiteness in these languages, other than a few works on Bulgarian ‘one’ on the grammaticalization path to an indefinite article (Geist 2013; Gorishneva 2016).

Finally, let us briefly comment on ways in which our informants suggested that a marked or unacceptable test item can be rescued in a given scenario. Several German and Bulgarian speakers remarked that sentence-initial objects, which deviate from the canonical SOV/SVO order, were possible in transitive scenarios but that they require explicit contrast, surprise or an element like only. This requirement was stronger for the non-conservative construal than for the conservative one. Similar remarks were made for the other Slavic languages about preverbal non-conservative %Qs, in that a particular information structure or their function as a contrastive topic could improve the acceptability of such items in the non-conservative scenarios. We will come back to this in §6.3.

The empirical generalizations we arrived at lead to five research questions that an adequate analysis of the non-conservative reading has to account for, see (42). We add a sixth one (TQ5), based on a further empirical observation we made in Gehrke & Wągiel (2022).

The account we propose in §5 will provide answers to the first five questions. In particular, we will argue that non-conservative %Qs are always interpreted low, on a par with semantically incorporated nominals. This accounts for their low position (TQ1), the restriction to bare nominals (TQ2) and to have- and existential-like predicates (TQ5). The answer to TQ4 is connected to this, since we propose that %Qs with intransitives are, in fact, existential constructions. Finally, the answer to TQ3 will be that in languages with a rigid word order, like English, subjects have to appear high/sentence-initially, in a position from which incorporation is impossible. This is why there are no ‘true’ subject %Qs in English, and an existential construction has to be used instead (or a passive with transitives). In languages with a more flexible word order, on the other hand, like German and Slavic, subjects (nominals in nominative case) can remain low, in a position from which they can incorporate into the predicate.

Our account will not be able to provide an answer to TQ6, however, but we assume that such examples always involve a marked information structure; we leave this for future research. Instead, as a point of departure for our account we take the unmarked cases with neutral stress and neutral word order. This is a major difference between our analysis and the alternative account by S&Co.’s account, which we will come back to in §6.

5.1 Transitives

Given the morphosyntactic evidence discussed in previous sections, we propose that non-conservative transitive construals involve semantic incorporation of the bare plural noun in object position into the verbal predicate and that the resulting complex predicate is subsequently shifted to a scalar expression on which the %Q operates. The idea is that (44) has a meaning that could (roughly) be paraphrased as ‘The company women-employs to the extent of 50%’.

In this constellation, the grammatical subject is the logical subject of the sentence. We assume that in (44) the noun ‘company’ is interpreted as a group noun of sorts, and thus it denotes a set of singular company members as well as pluralities thereof. Hence, the DP ‘company Ekspol’ provides the argument for the main predicate, the maximal plural individual in the extension of ‘company’, which we will call ce.

We propose that the bare plural is semantically incorporated and combines with the transitive verb via Chung & Ladusaw’s (2003) restrict mode of composition (45a). Under this account, the internal argument is not referential (or quantificational), rather it is a property-denoting expression that does not saturate the verbal predicate but merely modifies (restricts) it. The internal argument variable is then existentially closed (ec) in the subsequent derivation (45b).

For convenience, we will represent the incorporation expression as ‘women-employ’ (46). Notice that we do not apply ec at this stage.

Subsequently, (46) will be shifted to a gradable property associated with a fully closed cardinality-based scale and the %Q will operate on that scale. Building on Bochnak ‘s (2010) treatment of cross-categorial uses of half, we analyse %Qs as scalar modifiers that target an ordered set of degrees provided by the modified expression. This is supported by the fact that %Qs can be used to modify gradable adjectives encoding fully closed scales (47).

Unlike half, %Qs are complex expressions, and thus we decompose them into two components. The numeral simply refers to a natural number (type d) (48), whereas ‘percent’ denotes a function that takes a degree and yields a scalar modifier of type ⟨⟨d,⟨e,t⟩⟩,⟨e,t⟩⟩, see (49), where G is a gradable predicate (an expression of type ⟨d,⟨e,t⟩⟩), S_G is a fully closed scale encoded by that predicate, max returns the maximal degree on that scale, and d is the value provided by the numeral. For instance, (50) specifies that the extent to which a gradable property applies to an individual is 50%.

Given (50), the %Q cannot combine with the incorporation expression in (46) directly, as it combines with the gradable adjective in (47). However, following Bochnak’s analysis of the degree/quantity ambiguity in (51), we assume that in (44) the %Q targets a cardinality-based scale in order to provide a proportion of a plurality.

For this purpose, we adapt Bochnak’s μ operation, which on our account relates the part-whole structure of a plurality denoted by the subject with a cardinality scale, i.e., an ordered set of degrees that can be accessed by the %Q. As defined in (52), μ_r shifts a relation between individuals into a scalar expression of type ⟨d,⟨e,t⟩⟩ by existentially binding the direct object variable and introducing an open degree argument associated with the cardinality of the relevant (plural) individual via the # measure function. After μ_r combines with the incorporation construction, we obtain the gradable predicate in (53).

(53) encodes a cardinality scale, based on the part-whole structure of the λ-bound nominal argument, which will be saturated by the subject. Since that argument represents a bounded individual, the corresponding scale is also bounded, and thus fully closed, which makes it compatible with (50). Consequently, (53) serves as the input for the %Q and the output is the predicate in (54), where S_women-employ is a cardinality-based fully closed scale.

Finally, the predicate in (54) combines with the subject DP, and thus gets saturated by the entity ce. As a result, (55) states that the extent to which the company employs women is 50% of the maximal value on a cardinality scale based on the part-whole structure of ce. These are the desired truth conditions capturing the non-conservative meaning of (44).

Let us now see how this approach allows us to capture non-conservative intransitive construals.

5.2 Intransitives

The overall idea is that non-conservative intransitive construals underlyingly resemble existential constructions. This view is supported by the fact that German and Slavic sentences like (56) can be expressed in English by existentials (57), despite the fact that in English regular declarative non-conservative intransitives are infelicitous, recall (9). In Slavic and German, however, the existential has a structure as in (58), which looks like our intransitive non-conservative %Q examples, except that the verb is ‘to be’.

We assume for German and Slavic that it is the location (or more precisely: a plural individual at that location) that is the subject (see Bassaganyas-Bars 2015 for one such account of existentials). In (56), we take the PP ‘at the Ekspol company’ to denote a set that is pragmatically restricted to include only individuals that work at the company. We take its extension to involve both atomic individuals and pluralities thereof. Consequently, the PP predicate is shifted by the standard iota-shift to the maximal plurality in its extension, i.e., the plurality that contains all the individuals that work at Ekspol (59). Let us call this plural individual atec.

As for the VP, we follow Bassaganyas-Bars (2015) in assuming that in existentials the verb is interpreted as introducing a general pragmatically determined relation π (Barker 1995). Specifically, we propose that ‘work’ in (56) denotes π_work, which is resolved as a working relation between two individuals. Furthermore, we assume that the bare plural ‘women’ in the pivot is incorporated into the verbal predicate via restrict and then the result in (60) feeds μ_r in order to yield the gradable property in (61), associated with a fully closed scale.

The scalar expression in (61) combines with the %Q, which targets the cardinality scale S_women-work (62). That scale is again based on the part-whole structure of the λ-bound variable.

The resulting expression in (62) gets saturated by the maximal plurality of individuals working at Ekspol, i.e., atec from (59), and we arrive at the truth conditions in (63), which state that (56) is true if the extent to which there is a working relationship between women and the individuals working at this company is 50% of the maximal degree on the relevant cardinality scale, i.e., if 50% of the individuals working at this company are women. These are the desired truth conditions for (56) since the proportion of the women is calculated with respect to the individuals working at the company.

To conclude, our analysis captures the empirical facts we arrived at in previous sections and also observed in Gehrke & Wągiel (2022). First of all, it explains the use of bare plurals in non-conservative construals with %Qs, the fact that %Qs in such constructions appear low, as well as the restriction to particular predicates (have and existential-like predicates). The reason is an incorporation structure in transitive constructions and an underlying existential-like structure coupled with incorporation in intransitive configurations. This in turn accounts for the apparent effect of the %Q on the VP. Specifically, the %Q always operates on a gradable expression derived from the main predicate.

A first advantage of our account over S&Co.’s proposals, to which we will turn shortly, is that we have an explanation for the SOA that they observed for languages like English, as opposed to German, but which they could not account for. Under our account, true subject %Qs in sentence-initial position cannot get a non-conservative interpretation because they generally appear too high. Instead they either have to appear VP-internally, which is possible in languages with a more flexible word order, like German or the Slavic languages, or one has to resort to a different construction, such as an existential construction (with intransitives) or a passive construction (with transitives), which is the case in languages with a rigid word order, like English.

A second advantage of our account over S&Co.’s proposals is that we base it on the unmarked cases with neutral stress and word order. Instead, S&Co.’s account starts out with what is in our opinion a marked contrast that leads to a truth-conditional difference, namely the contrast in (12). In this way, they provide an answer to TQ6, which we do not have; at the same time, however, their proposal lacks direct answers to the other questions, which raises additional problems, as we will see in the following section.

6.1 Pasternak & Sauerland’s (2022) analysis

Let us first discuss the syntactic structures argued for by Pasternak & Sauerland. The trees in (64) and (65) represent the conservative and non-conservative NP in German, respectively.

Under both construals in (64)–(65) the measure noun ‘percent’ (N₁) projects an NP (NP₁), which receives case depending on its position in the clause, e.g., nominative or accusative. At the DP-level (not represented here), in both cases NP₁ serves as the complement to a silent indefinite determiner D which introduces existential quantification. Where the two construals differ is in the second NP (NP₂), i.e., ‘the students’ vs. ‘students’. Under the conservative construal in (64), NP₂ further projects to a DP that is the complement of the measure noun (N₁) so that it gets assigned genitive case. Under the non-conservative construal in (64), however, NP₂ does not project to a DP. Rather, it is merely adjoined to the measure NP₁, and thus agrees with it in case. Under this analysis of the non-conservative construal, the measure noun and the numeral, e.g., ‘30%’, form a syntactic constituent, to the exclusion of NP₂. Pasternak & Sauerland show convincingly that the proposed structures make correct predictions concerning constituency.

On Pasternak & Sauerland’s semantic account, %Qs, i.e., measure nouns combined with a numeral, are treated as degree quantifiers. Their denotation of ‘percent’ is given in (66), where n and D are types for numbers and functions from degrees to truth values, respectively. max(D) is the maximal degree of which D is true and max(dom(D)) gives the maximal degree for which D yields a defined truth value. Thus, what (66) does is that for a number n and a set of degrees D, it returns ⊤ (true) if the maximal degree in D is greater than or equal to n percent of the maximal degree in D’s domain.

In order to explain the non-conservative construal in (65), Pasternak & Sauerland’s approach builds on two crucial ingredients, specifically focus sensitivity and QR of the %Q (NP₁). Before we present what we believe to be a flaw of the analysis they propose, let us go through their derivation of (8), repeated here as (67), step by step.

As evident from (65), the measure noun (N₁) combines directly with the numeral which saturates its first argument. This results in an expression of type ⟨⟨d,t⟩,t⟩ (NP₁), i.e., a quantifier over degrees (68).

However, as a degree quantifier NP₁ cannot combine with ‘students’ (NP₂). Thus, Pasternak & Sauerland propose the covert operator deg, as defined in (69), where μ^c is a contextually determined measure function (here, it is cardinality). In our case, deg takes NP₂ and returns a function from a degree to a set of pluralities of students whose cardinality is greater or equal to that degree.

Though the application of deg does not resolve the type mismatch between NP₂ and NP₁, by changing its nature it creates conditions for QR. Consequently, NP₁ moves from the DP to the left edge of the clause (70) leaving a degree-denoting trace, and thus forming a degree predicate (let us call it D_a), true of a degree d iff at least d-many students work here (71).¹²

After D_a saturates the ⟨d,t⟩ argument of the QRed degree quantifier (NP₁), the result is the interpretation in (72). In prose, (67) is predicted to be true if the maximal cardinality degree, i.e., the number, of students working here is at least 30% of the maximal degree for which D_a is defined.

At this point, Pasternak & Sauerland observe that the semantics in (72) faces a problem they call NoMaxDom. In particular, since D_a is a predicate defined for degrees of cardinality, computing max(dom(D_a)) is impossible. The reason is that unlike, e.g., the scale of fullness, the cardinality scale has no maximal degree. Therefore, there is no maximal degree for which D_a yields true or false, and thus max(dom(D_a)) remains undefined.

In order to solve the NoMaxDom problem, Pasternak & Sauerland invoke focus-derived presuppositions, recall the discussion of the apparent focus-sensitivity of %Qs in §2. Adopting a Roothian framework of focus semantics (Rooth 1985; 1992), they assume both a standard interpretation ⟦X⟧ and a focus interpretation ⟦X⟧_f, which is a set of alternatives of the same type as ⟦X⟧. In addition, they assume that a sentence presupposes that at least one of its focus alternatives is true (Abusch 2010). This is ensured by the syntactic head fpre, see (73), where ∨⟦X⟧_f is the grand disjunction of the set of alternatives to ⟦X⟧.

According to the proposal, fpre attaches to the proposition-denoting node (vP). As a result, we obtain (74) as a full syntactic representation of the non-conservative examples in (12) (with the noun ‘students’ modified by the adjective ‘Westphalian’ inside NP₂), which will ultimately allow Pasternak & Sauerland to both capture focus sensitivity and avoid the NoMaxDom problem in cases such as (72).

For the examples in (12), the ordinary interpretation of vP₁ in (74) does not depend on the placement of focus and can be informally described as in (75), where g stands for the variable assignment function.

On the other hand, the focus interpretation differs with respect to where focus is placed. In the discussed examples, the alternatives are ‘at least g(1)-many Ps work here’ and ‘at least g(1)-many P students work here’ for (76) (focus on ‘Westphalian students’, as in (12a)) and (77) (focus on ‘Westphalian’, as in (12b)), respectively.

Now, in order to solve the NoMaxDom problem and to capture the semantic effect on the VP, Pasternak & Sauerland postulate that among the alternatives for ‘Westphalian students’ as well as for ‘Westphalian’ is the vacuously true predicate λx.⊤ (or a similarly unrestrictive expression). Consequently, among the alternatives in the sets in (76) and (77) will be ‘at least g(1)-many individuals work here’ and ‘at least g(1)-many students work here’, respectively. Since they are the weakest alternatives in the corresponding sets they are what the meaning of the grand disjunction introduced by fpre (vP₂) will amount to, recall (73). As a result, once we perform lambda abstraction over g(1) (vP₃), depending on the placement of focus in NP₂ we will obtain the degree predicates in (78) (henceforth D_α) and (79) (henceforth D_β).

Notice that the two degree predicates share the same truth conditions. Specifically, both D_α and D_β would be true for degrees not exceeding the cardinality of Westphalian students working here. Where they differ is their definedness conditions imposed by fpre via the grand disjunction. While the domain of D_α is a set of degrees exceeding the cardinality of individuals working here, for D_β it is a set of degrees not exceeding the cardinality of students working here. In both cases, the domains have the maximal degree. Hence, when D_α and D_β combine with the QRed ⟨⟨d,t⟩,t⟩-type %Q (NP₁), see (80)–(81), the NoMaxDom problem does not arise.

The semantics of ‘percent’, recall (66), allows Pasternak & Sauerland to capture the correct truth conditions in (80)–(81). Since D_α and D_β differ in their domains, (80) is true if Westphalian students constitute at least 30% of the total workforce, whereas (81) is true if Westphalian students constitute at least 30% of the student workforce.

In the discussed cases, Pasternak & Sauerland’s account seems to work neatly. However, closer investigation reveals what we believe are serious problems for their approach.

6.2 Some doubts, or: There is 50% truth to the story

Though the work by S&Co. is an important contribution and an interesting attempt to explain the distinction between the conservative and the non-conservative construal with %Qs as a result of a structural difference, there are a number of doubts we have with respect to their analysis. In general, we agree with S&Co. that in the case of non-conservative construals the truth conditions can vary depending on focus assignment and we acknowledge that the analysis by Pasternak & Sauerland in (80)–(81) does account for the contrast in (12). On the other hand, we feel that their approach does not capture the crucial intuition that what is essential about %Qs is that they have a semantic effect on the VP irrespective of where focus is placed.

As already discussed, the key ingredients of Pasternak & Sauerland’s account are domain sensitivity encoded in the semantics of the %Q, recall (66) and (68), QR, recall (70), and focus sensitivity coupled with presupposition triggering, recall (73). Importantly, all of the non-conservative examples that S&Co. discuss involve stress on (part of) the NP following the %Q (NP₂), e.g., on ‘students’ in (10a) or ‘Westphalian’ in (12b), which is analyzed as marking focus. On their account, it is essential that the focus is within NP₂ for the non-conservative reading to arise. What, then, happens when stress is elsewhere? And what happens with cases involving neutral stress? In this and the following section, we will go through a number of examples that in our opinion pose a problem for Pasternak & Sauerland’s account.

Let us first discuss sentences with neutral stress and word order, which we will illustrate with German, against our background assumptions outlined in §3. As discussed in §4.4, neutral word order for the conservative construal is SPPV (SVPP with V2) with intransitives and SOV (SVO with V2) with transitives. The neutral word order for the non-conservative construal is, in turn, PPSV (PPVS with V2) and also SOV (SVO with V2). This is illustrated in (82) and (83).

What is crucial for our purpose is that under the constellation in (82)–(83) focus can project all the way (e.g., Höhle 1982), which allows for an all-new interpretation, where focus alternatives could be basically anything. Importantly, irrespective of this fact (82b) and (83b) still get non-conservative interpretations. In order to show why Pasternak & Sauerland’s approach fails in such wide-focus configurations, let us go through the derivation of the intransitive sentence in (82b).

Within the subject DP, the deg head, recall (69), transforms ‘women’ (NP₂) into an expression requiring a degree argument as its first input, which, in turn, causes the %Q (NP₁) to QR. For the purpose of this demonstration, we adopt a standard assumption that propositions are functions from possible worlds to truth values. Thus, the ordinary interpretation of (82b) up to the fpre node (vP₁) is as the semi-formal description in (84). In other words, for a world w, (84) yields truth if at least g(1)-many women work at Pirapo in w.

On the other hand, the focus interpretation of vP₁ in (82b) is a set of propositions such that each proposition in that set includes g(1)-many individuals involved in some event or some state. This is illustrated in (85), which contains alternatives of the same form as (84).

So far, everything goes smoothly. Yet, in order to see how a serious problem for Pasternak & Sauerland arises, let us follow their reasoning and include a vacuously true (or similarly unrestrictive) function among the alternatives in (85). For convenience, we will assume a proposition with extremely weak truth conditions (86).

Since (86) is weaker than any other alternative in the set in (85), it will be the output of the grand disjunction applied by the fpre head (vP₂), recall (73). Thus, once the derivation reaches the lambda abstraction over g(1) (vP₃), (86) will serve as the definedness condition of the degree predicate in (87) (let us call it D_p). For convenience, we assume that at this point the world variable w is replaced by the actual world @.

Finally, D_p serves as the argument of the QRed %Q, recall (68). As a result, we obtain (88) as the interpretation of (82b).

Even at first blush, (88) does not feel right. In particular, there are two serious problems with this denotation. First, notice that the NoMaxDom problem still applies here. Since the definedness condition of D_p is extremely weak, recall (87), in effect there is no maximal degree for which D_p would yield a defined truth value (think, e.g., of statements such as Infinitely many prime numbers exist). Hence, the application of max to dom(D_p) would result in an undefined output.

However, for the sake of the argument, let us assume that somehow the NoMaxDom problem could be resolved. This would, nevertheless, by no means help save (88). The reason is that the definedness condition of D_p does not capture the semantic effect on the VP, and thus gives rise to incorrect truth conditions every time the total number of individuals involved in some eventuality is greater than the number of individuals who work at Pirapo. For instance, imagine that in total there are 1000 individuals involved in some eventuality and that 100 people work at Pirapo, out of which 50 are women. In that case, the sentence in (82b) is definitely true. And yet, contrary to fact the semantics in (88) predicts it to be false since the number of female workers at Pirapo, i.e., 50, is less than 500, which corresponds to 50% of the individuals involved in some eventuality. This is a serious flaw and we cannot see how it could be fixed in the proposed system.

In this section, we showed that Pasternak & Sauerland’s account fails to capture non-conservativity understood as the semantic effect of the %Q on the VP in sentences with wide focus. As a result, the proposed derivation either crashes or generates incorrect truth conditions for sentences with neutral stress and word order such as (82b) and (83b). In the next section, we will discuss further data, which we believe show the need to be more explicit about the kind of focus involved in non-conservative readings.

6.3 Another issue: Focus elsewhere

In the previous section, we showed that Pasternak & Sauerland’s (2022) approach fails to account for the non-conservative construal in a wide-focus configuration giving rise to an all-new interpretation, which we consider a serious flaw of the analysis. In this section, we address a further potential issue, namely cases in which focus is not inside the NP following the %Q, as expected under S&Co.’s account, but elsewhere. For example, one of our German informants provided (89) to improve on the sentence-initial order of the non-conservative %Q object. Here, the focus particle nur ‘only’ associates with ‘Pirapo’, which is marked with prosodic prominence.

For examples like these, S&Co. would have to assume a structure with multiple foci. For example, a structure with two foci is illustrated in (90) (from Wold 1996).

In (90), the focus particles also and only lexically associate with focus, which is marked by stress on Sue and on Bill. While this has to be tested empirically, our intuition is that the right intonation for (89) is different, though, since there seems to be just one instance of prosodic prominence; this would speak against S&Co.’s account. Alternatively, it could be that we are dealing with a second occurrence focus on a given constituent, ‘(50%) women’, which would be less prominent or not prominent at all, since given material tends to be defocused (see Baumann 2016).

An example where givenness most likely leads to the NP occurring with a %Q to not be prosodically prominent is one of the Czech examples we have collected within the CNC corpus study. (91b) is the only attested instance of a non-conservative %Q in preverbal position in the entire sample; here it is an object to a transitive verb. As is clear from the English translation of the preceding sentence provided in (91a), the previous discourse discussed the relative ranking of countries with respect to the proportions of women in different parliaments across the world, with higher proportions leading to a higher ranking. Consequently, the whole predicate in (91b) (‘has more than 40% women’) is given (because it is an instance of an apparently higher proportion) and the new information focus is on ‘Sweden’. To comply with the nuclear stress rule, this stressed constituent appears sentence-finally. There is no additional prosodic prominence of ‘women’, but the example nevertheless has a non-conservative reading.

Our cross-linguistic investigation yielded further examples in which focus is on an element other than (part of) NP₂. In particular, a number of informants reported for BCMS, Bulgarian, Czech, German, Macedonian, and Polish, that in non-conservative scenarios, transitive sentences with a preverbal object %Q become more felicitous with heavy stress on the postverbal subject to mark surprisal or contrastive focus, ideally with an adversative continuation. In all of these examples ‘50% women’ can be treated as a contrastive topic, which, according to, e.g., Büring (2006), involves additional prosodic prominence, and this could arguably be treated on a par with second occurrence focus, as discussed in Baumann (2016). For instance, in Czech (92) the NP ‘company Spedex’ is prosodically prominent, which gives rise to a contrastive focus interpretation, further reinforced by the following phrase ‘and not Bustrans’.

Finally, in Polish, the non-conservative interpretation is possible even with fronting of the element that would be required to bear focus under S&Co.’s account. For instance, consider (93) with the %Q in postverbal position. The moved NP ‘women’ surfaces to the left of the VP and is immediately followed by the lexical marker to. In this construction (termed true topicalization), to is analyzed as an overt topic marker and according to Cegłowski & Tajsner (2006) and Tajsner (2015) a fronted element preceding to cannot bear a focal stress.¹³ Crucially, however, (93) definitely gets a non-conservative interpretation.

A complete account of the kind of focus sensitivity involved with %Qs, if any, needs to explain examples like the ones discussed in this section, but it is currently unclear how exactly S&Co.’s approach would capture them. Since our account is not built on focus sensitivity we do not run into issues with the examples discussed here, but would treat them as information-structurally marked. Similarly, we treat the contrast in the ‘Westphalian’ examples in (12) to arise due to a marked information structure in case only the adjective is stressed, and a full account of cases where one deviates from default stress and word order, which we leave for future research, hopefully addresses these issues, without the need to resort to a general focus sensitivity for the unmarked cases.