2.1 Distributive share markers as distributive universal quantifiers

The idea that DistShare markers might be analyzed as a type of universal quantifier dates back to at least Gil’s (1982) typological classification, according to which, DistShare markers, just like DistKey markers, are subtypes of universal quantifiers, as shown by the classification of universal quantifiers provided in Table 1 (Gil 1995: 349). DistShare and DistKey quantifiers differ from “simple universal quantifiers” (e.g. all in English) in that they enforce distributivity (do not permit collective readings).

The unifying assumption underlying this line of analysis is that the distributive interpretation that DistShare markers involve universal quantification. This is a common idea that is present in the work of e.g. Choe (1987) for Korean -ssik, Gil (1990) for Japanese -zut(s)u, Faller (2001) for the Quechua particle -nka, Zimmermann (2002a/b) for German adnominal and adverbial jeweils, po in Czech, Bulgarian and Russian, cîte in Romanian and a dozen other languages, Balusu (2006) for reduplicated numerals in Telugu, or Zimmermann (2008) for reduplicated numerals in Hausa.

Choe (1987), like Gil (1982), was one of the first to stress that the unifying function of distributive markers crosslinguistically is to establish a dependency relation between two arguments of a predicate – serving respectively as the DistKey and DistShare – with distributivity derived by positing a universal quantifier interacting with an existential in its scope. He distinguishes between regular quantificational determiners that form a syntactic constituent with the DistKey argument (e.g. German jeder, or French chaque), from so-called anti-quantifiers which appear to form a syntactic constituent with the DistShare argument (e.g. Korean -ssik, adnominal each in English, and German je(weils)). Zimmermann (2002b) replaces the term anti-quantifier with the notion of distance-distributivity to refer to distributive items that appear (on the surface) to occur at a distance from their DistKey. He contends that distance-distributivity is “a superficial phenomenon. All instances of apparent distance-distributive quantifiers are reducible to regular adnominal quantifiers.” (Zimmermann 2002b: 21). On this proposal, Serbian po (2), Korean -ssik (4), English adnominal each (5) in English, and German adnominal jeweils (6) are all distance-distributive (DD) universal quantifiers.

Zimmermann then develops a uniform compositional analysis of distance-distributivity across languages, taking as a point of departure the analysis of adnominal jeweils in German which, just like Serbian po or Korean -ssik, yields both individual and event distributive readings, as illustrated with (6):

Jeweils in (6) forms a constituent with the subject NP zwei Offiziere “two officers” serving as the DistShare. On the individual distributive reading in (6a), distribution is over the direct object DP die Ballerinen “the ballerinas”, serving as the DistKey. While on this reading, jeweils occurs at a distance from its overt NP-restriction/DistKey, on the temporal/event distributive reading in (6b), distribution is over an implicit DistKey (a plurality of times/events) that is not overtly expressed in the clause, but must be recoverable from the linguistic context.⁵

His proposal is that DD items such as jeweils are QPs embedded inside a PP, itself adjoined to the NP serving as the DistShare. On Zimmermann’s proposal, DD quantifiers, just like regular quantifiers, combine syntactically with their restrictor. The difference is that the restrictor is not a lexical NP, but a pronominal NP which receives its value through coindexation with a DistKey provided by the context. P°, covert in the case of German, provides a relational variable R which specifies the relation that holds between the elements of the DistKey and DistShare. Accordingly, the denotation of the jeweils-DP is given in (7) and the analysis of the individual and event distributive readings of (6) are given in (8a) and (8b), respectively (see Zimmermann 2002b: 250):⁶

On the individual distributive reading in (8a), the denotation of the DistKey die Ballerinen “the ballerinas” provides the value for the set variable Z_i, while the denotation of the transitive verb provides the value for the relation variable R, thus ensuring that the distributive relationship between the elements of the DistKey (here, ballerinas) and elements of the DistShare (here, sets of two officers) is one of accompanying the former home. On the temporal event distributive reading in (8b), the restriction variable Z_i ranges over a set of atomic events and must be anaphorically linked to an appropriate antecedent in (or constructed from) the preceding discourse, and the free relation variable R is assigned a value from the context. For example, if there is a ballet performance every Friday this past year, then on this scenario, the DistKey is a set of Friday ballet performances and the relation between the elements of the DistKey and elements of the DistShare (here, events of two officers accompanying the ballerinas home) is plausibly one of temporal succession. Importantly, R can take on a variety of values, including temporal, causal, subpart, or other contextually determined relations.⁷ As the discussion in section 4.1 will show, the assumption that the values of Z and R are quite free and pragmatically determined will play an important role in capturing the truth conditions for spatial distribution with the DistShare markers po and -ssik.

Another analysis of DistShare markers involving universal quantification is that of Balusu (2006) and Balusu & Jayaseelan (2013) for reduplicated numerals (RedNum) in Telugu. On their proposal, event distributivity is an instance of distributive quantification, involving a Distributivity (D-) operator, a DistKey (the DP that is being distributed over), and a DistShare (the (RedNum)NP that is being distributed). In Balusu & Jayaseelan’s (2013: 10) words, “a distributive operator is essentially a universal quantifier, that has a sorting key, i.e. the quantifier’s restriction, and a distributive share, i.e. the quantifier’s scope”:

Balusu is one of the rare authors, to our knowledge, to address the issue of spatial event distribution and, in particular, the question of how to recover implicit spatial (as well as temporal) distributive keys from the context. Thus, consider the intransitive sentence in (10a), which only yields spatio/temporal event readings, since the only overt argument that can undergo numeral reduplication (and thus serve as the DistShare) is the subject argument. On Balusu’s analysis, event distributive readings involve partitioning of an event into non-overlapping subevents along spatial or temporal dimensions using “a contextually salient method of division”. And it is these spatial/temporal dimensions (event aspects) which serve as the covert DistKey. But how do we partition time and space? They are not inherently partitioned into minimal atomic units, and thus can only be partitioned into units along contextually salient parameters.

Balusu & Jayaseelan (2013: 40) explain the spatiotemporal portioning of the event described by (10) as follows: “What do each time and each location refer to? […] The division of the spatial and temporal regions into units happens according to the context. The units need not be of equal duration in the case of temporal regions or of equal dimensions in the case of spatial regions. […] Suppose that all the monkeys in the enclosure jumped up all at once but they jumped up in pairs holding to each other, then the spatial key reading is easily obtained. It could also be the case that the monkeys were not holding one another in pairs but that they were sitting at different points on the branches of the tree, and that there were two monkeys per branch. In such a situation too the spatial key reading is felicitous.”

2.2 Distributive share markers as event plurality markers

Another prominent line of analysis found in the literature takes DistShare markers to be pluractionals, i.e. markers of event plurality or pluractionality (a term coined by Newman 1980).

Pluractionals are markers on the verb or affecting the verb phrase, indicating event plurality in a wide variety of manners (Cabredo-Hofherr & Laca 2012). For instance, Muller & Negrão (2012) claimed that the distributive numerals in Karitiana should be analyzed as adverbial operators that pluralize the events and restrict the cardinality of the entities they modify. A plural event may involve the same action iterated several times (repetitions), (same) actions distributed in space, time, or affecting multiple participants or objects, either as a group or individually (Cusic 1981; Newman 2012; Lasersohn 1995). An example of temporal pluractionals in English would be adverbial phrases such as again and again or time after time.

Knežević (2015), and Knežević & Demirdache (2017; 2018)⁸ develop a pluractional analysis of Serbian po, building on Lasersohn (1995) and Cable’s (2014) analysis for Tlingit. Pluractional accounts have been defended for a typologically diverse set of DistShare markers, in languages such as Kaqchikel (Henderson 2011; 2014) and Karitiana (Muller & Negrão 2012), or St’at’imcets (Matthewson 2000).

There are two essential advantages for Knežević in using Cable’s analysis of the distributive marker -gaa in Tlingit.⁹ First, Cable provides a single, uniform account of both individual and event distributive readings of sentences with adnominal -gaa, deriving both readings from the same truth conditions. Sentences with -gaa are thus not taken to be ambiguous between individual and event distributive readings. Rather, the semantics for -gaa yields weak truth conditions holding under both readings. What this essentially means is that individual/participant distributivity is subsumed under event distributivity and is not treated as a separate reading. Second, these weak truth conditions predict that the semantics of distributivity with -gaa in Tlingit, just like with po in Serbian or -ssik in Korean, does not involve a DistKey that has to be atomically and exhaustively distributed over (section 2.2.1).

The readings that -gaa yields are illustrated in (11), together with the type of scenarios provided by Cable for individual/participant and event distributivity:

Cable treats distributive numerals (distributive markers) similarly to adverbials such as piece by piece, as analyzed by Beck & von Stechow (2007). This analysis does not posit quantificational D-operators of any sort. (12) gives the semantics assigned to adnominal distributive numerals¹⁰ in Tlingit, and (13) the predicted truth conditions for the sentence in (11) (Cable 2014: 586).

(13b) requires my daughters to have cumulatively bathed pairs of dogs, with each pair being the theme of some sub-event of the larger event. Now, these truth conditions are compatible with either the individual or the event distributive scenarios in (11i-ii), since on a cumulative interpretation, the number of agents (here, one or two) is left unspecified. Importantly, (11) is also correctly predicted to be true on a scenario such as (11iii) which instantiates both participant and event distributivity (e.g. my daughters each bathed two dogs at different times).

Knežević adapts the semantics of -gaa to po distributive numerals in Serbian.¹¹ As shown in (14), just like -gaa, po combines successively with a numeral n and a predicate Q supplied by the modified NP. It then takes as an argument a relation P between individuals and events and returns a predicate of events, holding of an event e iff there is an individual x such that Q holds of x, and the relation P holds between e and x. The conjuncts “e ∈ *e’_nQ & e ∉ e’_nQ” require that e be a sum of events e’ in which exactly nQ dogs participate, but without itself being such an event. The claim is that the events described by a sentence with a po nNP must necessarily involve a number of participating individuals described by the NP that is a multiple of n.

The sentence in (14b) thus comes out to have the truth conditions in (14c). In particular, the events described by (14b) must be constructed out of events in which exactly two dogs participate as themes, without being events of that kind themselves. This means that the events described by (14b) must necessarily involve multiples of two dogs. (14b) will thus not hold in collective scenarios involving exactly two dogs in total. It will, however, be true as long as there are at least two subevents e’ of one big event e, spatially or temporally separated, each containing exactly two dogs. That is, the number of dogs per subevent has to be two, but there may be as many pairs of dogs as there are subevents e’ (the total number of dogs is therefore context-dependent). Furthermore, there is no specification in the semantics of how the agent participants partition across these events: the boys can participate collectively, individually, or in groups in the event e, the only requirement being that there be a bathing of two dogs per subevent e’.

In sum, po, on these proposals, is enforcing weak truth conditions, merely requiring that there be at least two events involving the number of entities denoted by the -ssik phrase. As we shall see in the next section, this crucially means the semantics of distributive numerals should not involve a DistKey that has to be atomically and exhaustively distributed over.

2.2.1 Lack of atomicity and exhaustivity requirements

As previously mentioned, on the semantics developed by Cable and Knežević, sentences with distributive numerals are true under event distributive scenarios such as (11ii), involving a partitioning into two subevents, each with agents acting collectively. This is one of the generalizations that leads Knežević to claim that distributive markers such as po are not distributive universal quantifiers: unlike distributive quantifiers such as each, which enforce a partitioning of their restrictor set (DistKey) into atomic members, DistShare markers like po do not require an atomic partition. So, for example, the Serbian and Korean sentence in (16) should be judged true under all the situations depicted in Figure 1,¹² while the English sentence in (15) is only compatible with the scenario in Figure 1a.

Figure 1

Atomic and non-atomic scenarios (event distributive readings) for the sentences in (15)/(16).

(15)	[Each boy DistKey] is carrying [numNP one piano DistShare].

1. (16)
1. a.

1. Dečac-i
2. boy.PL.NOM
1. nose
2. carry
1. po
2. DISTR
1. jedan
2. one
1. klavir.
2. piano.ACC

1.  
1. b.

1. Sonyen-tul-i
2. boy.PL.NOM
1. phiano-lul
2. piano.ACC
1. han-tay-ssik
2. one.CLF.DISTR
1. nalu-ko iss-ta.
2. carry.PROG.DEC
1. i. ‘(The) boys each are carrying one piano.’
2. ii. ‘(The) boys are carrying one piano at each/different place(s).’

The second generalization that leads Knežević to explicitly argue against a universal quantification analysis concerns another key property of universal quantifiers: the set denoted by the noun phrase serving as the DistKey (the restriction of Q°) must be exhausted. To quote Balusu & Jayaseelan (2013: 10): “the members of the distributive share need not be exhaustively used up when being distributed, whereas the members of the sorting key need to be exhaustively used up in being distributed over.”

Knežević contends that there are no exhaustivity requirements with po. That is, the plurality over which the po numNP is distributed (be it a plurality of individuals, times, spaces, or events) need not be exhausted. Thus, consider again the sentence in (15) and (16), but this time under the scenarios in Figure 2.

Figure 2

Exhausted and non-exhausted scenarios for the sentences in (15)/(16).

English adults judge (15) as true under the scenarios in Figure 2a (since all members of the set denoted by the DistKey (boy) are distributed over) and Figure 2b (the presence of additional, uncarried pianos is irrelevant to the truth conditions of (15) since there is no exhaustivity requirement on the DistShare). But English adults will judge (15) false under the scenario in Figure 2c because the members of the DistKey are not exhaustively used up.¹³

In contrast, the Serbian example in (16) is predicted to be true in all of the situations depicted in Figure 2 on a Knežević/Cable pluractional account of the semantics of distributive numerals. In particular, the scenario under Figure 2c is compatible with the truth conditions provided by Knežević for sentences with distributive numerals: there is an event e constructed out of three (sub)events e’, each involving one piano exactly, and e is an event of boys cumulatively carrying pianos.

These requirements, however, were never a subject of experimental investigation, and have remained until now in the theoretical domain. This paper is thus the first attempt to experimentally test whether there are indeed (no) exhaustivity requirements on an implicit DistKey. We focus here on intransitive sentences in Serbian and Korean. The challenge here is how to establish conditions with a covert (spatial/temporal) DistKey (set of salient spatial/temporal units) that is either exhausted or not. Time and space are not inherently partitioned into minimal atomic units, and thus can only be partitioned into units along contextually salient parameters. The experimental set-up must thus provide some kind of contextual restriction that will impose a partitioning of the event into non-overlapping subevents along spatial or temporal dimensions.

3.1 Experiment 1

Both Experiments 1 and 2 tested two DistShare markers, Serbian po and Korean -ssik. Although Korean and Serbian are typologically very different languages, po and -ssik show similar syntactic and semantic properties and we thus expect them to pattern alike in their interpretations.

Recall that in intransitive sentences, the only overt argument will be marked as the DistShare. This implies that quantification is over events along a spatial or temporal dimension and the DistKey is thus an implicit argument. Participants must therefore contextually infer the DistKey. We sought to control this process with a frame story that limits potential interpretations, and visual materials specifically designed to make spatial units contextually salient. We only used materials depicting spatial distribution (primarily for reasons of simplicity), but we believe our findings would carry over to temporal distribution.

General set-up: We used a children’s visit to the zoo as a frame story for both experiments. This setting allowed us to have cages or caves as identifiable spatial units over which events could be distributed. This method of division into relevant spatial locations is adapted from Balusu & Jayaseelan’s (2013) discussion of (10) above (scenarios involving monkeys on each branch of the tree). We varied the visual contexts (pictures) across subjects to test whether or not the spatial units need to be exhausted.

In our instructions we were careful not to explicitly mention cages/caves in order to ensure that there is no linguistically explicit argument that could potentially serve as a DistKey. Instead, the common context of the experiment was given at the beginning: “Peter and Mary went to the zoo with their class. They brought a camera and took pictures of animals they saw. After the visit, the teacher asked the children from the class to describe their photos. Are the children describing the pictures well?”. Then, in a yes-no picture verification task (a variation of a truth value judgment task), the participants had to evaluate whether the descriptions of the photos were appropriate.

Test sentences: We started from the simplest structure – intransitive sentences, where po/-ssik is syntactically attached to the subject. The subject has to be interpreted as the DistShare, and the only alternative for the DistKey would be the implicit spatial units. In addition, we also used the lowest numeral, so the target phrase was po one/one-ssik. Finally, the test items used a verb-subject word order for Serbian, see e.g. (17a), to avoid po occurring at the beginning of the sentence (since Knežević’s (2015) previous research showed that po-initial sentences sound odd). The word order in Korean remained subject-verb, see e.g. (17b).

Our visual and discourse context aimed to make the cages/caves be considered as relevant spatial units and to serve as the (implicit) DistKey. We designed experimental conditions to try to investigate some basic questions about the different predictions that the theories presented above make. These predictions are spelled out in the section below.

3.1.1 Method and design

Subjects: 53 adult native speakers of Serbian (mean age: 26.35; 37 female, 14 male, 2 unspecified) and 26 native Korean speakers (mean age: 34.00; 17 female, 8 male, 1 unspecified) completed a 5-minute Picture Verification task that was available online. All Serbian participants and 6 Korean participants were recruited through social media and did not receive remuneration. The remaining 20 Korean participants were recruited through Amazon Mechanical Turk and were paid 65 cents for completing the experiment. All participants gave their consent before starting the experiment.

Design and procedure: we used a 1 × 3 design with the same type of test sentence and three different types of pictures as illustrated in Figure 3: A: Exhausted, B: Non-exhausted (empty spatial units) and C: Non-exhausted (not all spatial units contain Ving monkeys). The experiment was preceded by detailed written instructions and a common context for all items, discussed above.

Figure 3

Examples of three test pictures paired with test sentences in Serbian (18a) and Korean (18b), giving three test conditions.

Items: 12 unique sentences (target items) were created using 16 different animals and eight verbs.¹⁵ Each verb was presented no more than twice in a target sentence (see Appendix B for the complete list of items that were used for the target items and fillers). Each condition had four observations and each participant saw only one condition combination per item (see Figure 3 for test pictures where the background has been removed for visual clarity. See Appendix B for an actual picture from the experiment).

Picture A was considered to be the baseline condition, in which spatial units (cages) are considered exhausted since they each contain a dancing monkey. Pictures B and C showed non-exhausted situations. Picture B showed two spatial units with monkeys carrying out the target action, and two empty spatial units. Picture C showed two cages with a monkey carrying out the target action and two cages with monkeys that are not carrying out the target action. We always used two non-exhausted spatial units/cages, rather than one, to avoid singling out the odd situation in the picture. All three conditions showed four salient spaces in a rough 2 × 2 grid formation, and the position of the different types of spaces, e.g. a cage vs. empty area, varied across items.

As previously stated, the test sentences all had imperfective intransitive verbs and the following structure: [Verb [po-1(one)-Subject]] for Serbian (17a/18a) and [[Subject-1(one)-ssik] Verb] for Korean (17b/18b).¹⁶

1. (18)
1. a.

1. Pleš-e
2. dance
1. [po
2. DISTR
1. jedan
2. one
1. majmun].¹⁷
2. monkey.NOM

1.  
1. b.

1. [Wenswungi-ka
2.   monkey.NOM
1. han-mali-ssik]
2. one.CLF.DISTR
1. chwum-ul
2. dance.ACC
1. chwu-ko iss-ta.
2. dance.PROG.DEC

1.  
1. c.

1. ‘DISTR one monkey is dancing (at different locations/each location).’

Control items were 12 numerically quantified sentences with a Verb-three-Subject(plural) pattern (e.g. “Three tigers are sleeping” with one of the three pictures corresponding A, B or C). These items had four yes and eight no responses and were used as true controls – participants who made errors on more than 10% of control items were excluded. Each participant also saw four filler items (see footnote 15). In sum, each participant saw 12 target items, 4 filler items and 12 control items, amounting to 28 items in total.

3.1.3 Results

The results show us that the baseline picture (A) and the picture with non-exhausted spatial units (empty cages/caves) (B) were overwhelmingly accepted (90.1% and 84.4% respectively for Serbian and 96.1% and 97.1% respectively for Korean), unlike the picture with non-exhausted spatial units in which no monkey is V-ing (C) (10.4% for Serbian and 8.6% for Korean) (see Figure 4).

Figure 4

The results show the proportion of YES responses of Serbian (left) and Korean adults (right) with the standard error bars for Picture A (blue), Picture B (red) and Picture C (green) conditions.

We performed a repeated measures mixed-effects logistic regression using glmer() function of the lme4 package (Bates et al 2015) in R (R Core team 2014) for each language. We started all analyses with a full and random intercept model that included Picture Type (with levels A, B and C) as the only predictor (fixed factor) and Subjects and Items as random factors. AIC values¹⁸ were compared to determine which model fit the data best, with a complex model being preferred over a simpler model only if its AIC value is two or more points lower. For both Serbian and Korean, model comparison did not show an effect of the Items, so this random factor was removed. For both languages, the best fitting model retained the fixed factor Picture Type and the random factor Subjects, with a random slope by-Subjects for Picture Type (see the best model formulas for each language and the output in Appendix A). The reference point was Picture A, and we see no significant difference between Picture A and Picture B for Serbian (β = –1.70, z-value = –0.50, p = 0.62) or Korean (β = –0.93, z-value = –0.14, p = 0.89). There was a highly significant difference between Picture A and Picture C for both Serbian (β = –16.60, z-value = –4.79, p < 0.001) and Korean (β = –18.92, z-value = –2.83, p < 0.001).

We also combined data from both Serbian and Korean in one model, including a new fixed factor Language with the levels Serbian and Korean. Models that included Language were not significantly different from models without. This means that there is no significant difference between Serbian and Korean participants in their responses.

3.2 Experiment 2

3.2.1 Method and design

Subjects: 31 adult native speakers of Serbian (mean age: 29.77; 23 female, 6 male, 2 unspecified) and 30 adult native speakers of Korean (mean age: 35.00; 24 female, 6 male) participated in a picture verification task. The experiment was done online and took approximately five minutes.

Design and procedure: we again had a 1 × 3 design but changed two picture types from Experiment 1 (see Figure 5). Picture A was modified to create Picture A1: Exhausted spatial units + monkey V-ing outside cages and Picture C was modified to create Picture C1: Exhausted + no monkey V-ing outside cages showed groups of monkeys not bounded by explicit spatial units.¹⁹ These pictures were designed to assess how necessary spatial delimitation is. Control items and test sentences remained the same.

Figure 5

Pictures used in Experiment 2. Groups of monkeys appear outside clearly delimited spatial units in A1 and C1.

3.2.3 Results

Figure 6 shows that participants accept Conditions A1 (96.8% for Serbian and 77.1% for Korean) and B1 (90.9% for Serbian and 80.5% for Korean), but reject C1 (24.2% for Serbian and 20.8% for Korean). The results match the predictions of DistKey_GROUP, as seen in Table 5.

Figure 6

The results show the proportion of YES responses of Serbian (left) and Korean (right) adults with the standard error bars for Picture A1 (blue), Picture B1 (red) and Picture C1 (green) of the (new) conditions.

Table 5

Predictions and results on all test pictures in Experiment 2.

		Picture A1	Picture B1	Picture C1
Predictions	DistKeyCAGED GROUP	✓	✓	✓
	DistKeyGROUP	✓	✓	✕
Actual results		✓	✓	✕

For analysis, we again performed a repeated measures mixed-effects logistic regression using glmer() function of the lme4 package (Bates et al 2015) in R (R Core team 2014). The analysis and the procedure were identical to those used in Experiment 1. We started all analyses with a full and random intercept model that included Picture Type (with levels A1, B1 and C1) as the only predictor (fixed factor) and Subjects and Items as random factors AIC values were compared to determine which model fit the data best, with a complex model being preferred over a simpler model only if its AIC value is two or more points lower. Model comparison showed that the model with the lowest AIC values for Serbian included the fixed effect of Picture Type and a random effect of Items and Subjects, plus a random slope for Picture Type for Subjects (the reference level (the Intercept) was set to be B1). The model showed that B1 and A1 were not significantly different (β = 0.31, z value = 0.10, p = 0.92), and that B1 had significantly higher rates of acceptance than C1 (β = –11.41, z value = –4.43, p ≤ 0.001). The best model for Korean did not include the Items and had a random slope for Picture Type for Subjects. Again, A1 was not significantly different from B1 (β = 4.31, z value = 1.46, p = 0.15) and C1 was significantly different than B1 (β = –9.99, z value = –3.99, p ≤ 0.001). The full model is given in Appendix A. Finally, when the data was combined, the model with Language as the fixed factor showed no significant difference compared to the model without Language, meaning that Serbian and Korean participants responded similarly to the conditions.

4.1 Distribution over pluralities of non-atomic individuals

Our results provide intriguing evidence for a covert spatial DistKey. The visual context supplies spatially separated groups/triplets of monkeys. This plurality, which is itself made up of plural/non-atomic individuals, serves as the DistKey over which the members of the DistShare (dancing monkeys) are distributed.

These conclusions clearly clash with the assumed predictions of the event plurality analysis. Instead, they appear to provide evidence for the UQ accounts: (i) distributive markers have an essentially relational function, that of establishing a relation between members of the DistKey and members of the DistShare, and (ii) there is an exhaustivity requirement on a DistKey. In our experiments with our materials we found that the members of the DistShare contribute to spatially define the DistKey that has to be exhausted.

To account for these findings, we suggest an analysis of event distributive readings for both Serbian and Korean along the lines of Zimmermann (2002b) as presented in section 2.1.²² Consider again the intransitive sentences in (22b) and (22c), together with their German translation in (22a):

DD quantifiers across languages are assumed to have the uniform underlying structure in (23a). Although all languages with DD quantifiers express the DistShare overtly, languages differ with regards to which parts of the postnominal PP is overtly spelled out. In particular, adapting Zimmermann, po in Serbian would be taken to spell out a complex prepositional head derived via incorporation of a silent universal Q° into the overt P°-head, as shown in (23b). The PP is base-generated right-adjoined to the DistShare NP ([_AP one][_NP monkey]). While in Korean, this PP will stay in its base position,²³ in Serbian it raises to the Specifier of the whole DP (as does jeweils in German), yielding the observed surface order (23c):

We now focus specifically on the analysis of Serbian, but conjecture that a similar analysis could be applied to Korean as well. Zimmermann’s compositional analysis provides a relational variable R specifying the relation that holds between the elements of the DistKey and DistShare. The denotation for the PP headed by po is given in (24a). It will combine with its argument, the value of the numeral NP in (24b), to yield the proposition in (24c):

The numeral NP²⁴ one monkey in (24a) (which forms the DistShare) denotes a property that is predicated of the set variable that is bound by the existential quantifier in the nuclear scope of the universal. This ensures that elements of the DistShare are distributed over elements of the implicit restriction of the universal quantifier.

The indexed variable pro_i in (24) provides the restrictor of the universal quantifier, that is, the DistKey. For event-distributive interpretations, pro_i is simply left free, and understood to denote a salient plurality (of events, on Zimmermann’s analysis of temporal event distributive readings; see discussion of (8b)) that is recoverable from the context. Our experimental results tell us that the DistKey in (24) is understood to be the set of sets/triplets of monkeys made salient by the visual and linguistic input.

Turning now to the value of the relation variable R contributed by po, recall that for participant distributive construals, its value is provided by an overt relation-denoting expression in the clause (e.g. by the denotation of the verb in (8a)). For event distributive interpretations, its value is pragmatically determined by the context (preceding discourse and extra-linguistic context), together with world knowledge (see discussion of (8b)). Zimmermann claims that R can take on a variety of temporal, locative, as well as other values (in particular, causation, temporal succession/overlap/inclusion, homogeneous subpart, etc.).

Focusing now specifically on Serbian, we can extend this analysis and assume that the value of the free relation variable R is constrained/determined in part by the core lexical meaning of po as a locative preposition. Evidence for the status of po as a locative preposition, with plausibly (universal) quantificational content, is provided by the uses of po, illustrated in (25), from Knežević (2015: 134–137) who points out that the earliest diachronic use of po is that of a spatiotemporal preposition. Note, in particular, that the translation provided for po in (25a) is very close to what we want as a value for R in (24), i.e. one dancing monkey per triplet, and that we could paraphrase po in (25a) as per (each). This falls out nicely on the assumption that po spells out incorporation of a universal quantifier into a prepositional head, which we take to have a meaning akin to that of per.²⁵

Following Zimmermann, the value of R is contextually determined and can take on a variety of meanings, that is, temporal, locative, subpart, causal, or other contextual relation. We take the relevant relation that po establishes between the DistKey (the set of triplets of monkeys in the visual input) and the DistShare (a set of one monkey) to be the element-of relation (∈). This constrains each subset/triplet of monkeys Z given in the visual input to contain as one of its members a subset X of one monkey that is dancing.

The truth conditions ultimately derived for (22) are given in (26):

For Zimmermann, event distribution involves distribution of events over an (implicit) plurality of events. Importantly, (26) yields an event-based distributive interpretation, even though distribution here is over non-atomic individuals (groups of monkeys). Recall, in particular, that when quantification is over individuals, the DistShare and the DistKey correspond to arguments of the verb and the value for the relation variable R in the meaning of PP is provided by the denotation of the verb. This ensures that, in the case of (8a) for instance, the distributive relation involves individual ballerinas and groups of two officers, and is one of accompanying the former home. (26), on the other hand, involves quantification over events, since po is establishing a relation between individual events and elements (Z) of the DistKey (Z’_i), and the value of the relation variable R is determined by the context. By imposing few constraints on either Z or R, Zimmermann’s proposal allows us to nicely capture the generalization that the triplets of monkeys provided by the visual input serve as the DistKey, with each triplet having to contain, as one of its elements, a dancing monkey. Since both Z and R are pragmatically determined, the interesting question is then why Z did not end up ranging over cages (with R as a locative relation) so that distribution of dancing monkeys would be over cages and not triplets of monkeys. We offered a speculative answer to this question in section 3.2.4, suggesting that participants use both the visual input (triplets of cageless/caged monkeys) and the linguistic input (mention of monkeys) provided by the experimental materials) to recover the spatial DistKey.

On the proposal defended here, the truth conditions in (26) differ from those for run-of-the-mill cases of quantification over events only in that the members of the spatial DistKey are plural individuals (i.e. groups of monkeys). We take up this issue in the context of Champollion’s (2016b) parameters of variation for distributivity operators.