Studies in Polish Linguistics vol. 17 (2022), issue 1, pp. 1-29 doi: 10.4467/23005920SPL.22.001.15758 www.ejournals.eu/SPL Mojmír Dočekal > https://ordd.org/0000-0002-9993-4756 Hana Strachoňová > https://ordd.org/0000-0002-6897-9409 Masaryk University R-loci and Distributivity: Insights from Czech Sign Language Abstract This article describes a distributivity pattern in Czech Sign Language. The pattern is signed via a reduplication at the R-loci and resembles the distributivity behavior of the binominal each that is known in spoken languages. Nevertheless, there are important differences between the sign language reduplication and the spoken language distributivity that is seen in the binominal each; the most significant concerns the range of readings available for the sign language reduplication. We describe the data we gathered, and then formalize them in the Plural Compositional Discourse Representation Theory. The formal framework allows us to analyze the data and explain certain questions which arise from them. Keywords Czech Sign Language, distributivity, reduplication, R-loci, PCDRT, individual and occasional interpretation Abstrakt Niniejszy artykul omawia konstrukcj^ o interpretacji dystrybutywnej w czeskim j^zyku migowym. Wyraza sie^ ja za pomoca reduplikacji z wykorzystaniem punktow referencjal-nych w przestrzeni migowej i przypomina ona dystrybutywne znaczenie dwurzeczow-nikowego uzycia kwantyfikatora kazdy, znane z jejzyköw mowionych. Niemniej istnieja znaczace röznice pomi^dzy reduplikacja w jejzyku migowym a dystrybutywnoscia w jej-zyku möwionym obserwowana w przypadku tego kwantyfikatora. Najistotniejsza z nich dotyczy zakresu interpretacyjnego reduplikacji w j^zyku migowym. Autorzy omawiaja zebrane dane j^zykowe, a nast^pnie dokonuja ich opisu formalnego w ramach mno-gosciowej kompozycyjnej teorii reprezentacji dyskursu (Plural Compositional Discourse 2 Mojmír Dočekal, Hana Strachoňová Representation Theory, PCDRT). Stanowi to podstaw^ dla analizy materiálu oraz odpo-wiedzi na wynikajace z niego pytania. Slowa kluczowe czeski j^zyk migowy, dystrybutywnosč, reduplikacja, lokalizacje referencyjne, PCDRT, interpretacja jednostkowa i okazjonalna 1. Introduction In this paper, a specific type of sign repetition in the Czech Sign Language (český znakový jazyk - ČZJ) is analyzed as a structure parallel to the so-called binominal each in spoken language.1 Example (1), which is of great significance to our argument, shows that in ČZJ, both numerals and verbs are reduplicated and localized as a morphological marking for distributiv-ity. While reduplicated numerals with various types of distributive readings are not uncommon among spoken and signed languages (Balusu 2006; Cable 2014; Kimmelman 2018; Kuhn 2019; Szabolcsi 2010), our analysis of the ČZJ example (1) demonstrates the connection between the distributive reduplication of a verb and the occasional distributive interpretation of the sentence. (1) DAUGHTER MYTHEY-THREE DOG TWO, TWGv TWOk BATHE; BATHEj B ATHEk 'My three daughters bathed two dogs each/each time.' 1. individual distributivity interpretation 2. occasional distributivity interpretation 3. # cumulative interpretation We suggest that each instance of the numeral or the verbal repetition at the Reference loci (R-loci; Schlenker 2017) is a realization of the distributive operator. This operator is both anaphoric and selective. The framework of the Plural Compositional Discourse Representation Theory (PCDRT; Brasove-anu 2008; Dotlačil 2012, 2013; a.o.) is used to explain the distributive marking in ČZJ, including the multiple marking. Our analysis of ČZJ data supports the non-standard theories of distributivity, such as the PCDRT or Neo-Da-vidsonian event distributivity theories (Champollion 2016a, 2016b). The paper is organized as follows: in Section 2.1 basic observations are considered regarding the types of distributivity markers in natural languages; Section 2.2 discusses the concept of the R-loci in sign languages; Section 2.3 briefly compares two strategies to determine distributive meaning in natural 1 We would like to thank those who attended the Sign Language Syntax and Linguistic Theory workshop at GLOW41 in Budapest, and the two anonymous reviewers of this article for their pertinent comments. We are also grateful to our ČZJ consultants, Mirka Tylová and Ondřej Klofáč, who are Deaf signers exposed to sign language from birth; and to Ivana Kupčíkova for interpreting. R-loci and Distributivity: Insights from Czech Sign Language 3 languages; Section 3 explains the procedure used to collect the Czech Sign Language data and discusses their relevance within the current theories of distributivity; Section 4 shows how the PCDRT framework can be applied to the data and answers the questions posed by the data, with an initial discussion of the cumulative baseline interpretation of Czech Sign Language sentences, including the plurality denoting expressions; Section 4.2 does not include the distributivity markers, whereas in Sections 4.3 and 4.4 a distributive reading is addressed. Section 5 provides a summary of the research undertaken. 2. Morphosyntax of distance distributivity in CZJ 2.1. Markers of distance distributivity The English each in example (2) and the German jeweils in example (3) are instances of so-called distance distributivity markers (or items). They are distant because syntactically, they are detached from their restrictor (each is detached from girls and jeweils is detached from Mädchen) Additionally, they are interpreted as distributivity markers because, from a semantic perspective, the two dogs (distributed share) are distributed over the girls (sorting key) in such a manner that each girl saw two dogs. (2) The girls saw two dogs each. (3) Die Mädchen haben jeweils zwei Hunde gesehen. 'The girls saw two dogs each.' Employing a distributivity marker in this way to express a relation between the sorting key and the distributive share is usually referred to as the binominal each in the literature, and this topic has received considerable attention from those researching spoken languages (Cable 2014; Balusu 2006; Dotlacil 2012; Champollion 2012; Szabolcsi 2010; Schwarzschild 1996; Landman 2000; Link 1983; Safir and Stowell 1988; among many others). With regards to sign language, a particular type of sign repetition and other related phenomena were reported and thought to contribute a similar (distributive) meaning to the sentence (Kimmelman 2017; Quer 2012; Kuhn 2017 for dependent indefinites). In example (4) from the CZJ, the numeral TWO and the verb BATHE are repeated. Each repetition assigns two dogs and a bathing (sub)event to one of the three daughters.2 2 Although there is a terminological difference between reduplication and repetition (see Kimmelman 2018), it is not necessary to discuss this in detail in this paper. Thus, we use them synonymously. 4 Mojmír Dočekal, Hana Strachoňová (4) DAUGHTER MYTHEY-THREE DOG TWO, TWOi TWOk BATHE, BATHEj BATHEk possible translation: 'My three daughters bathed two dogs each.' This data superficially resemble examples from certain spoken languages; see Telugu in (5) and Hungarian in (6). (5) ii pilla-lu renDu renDu kootu-lu-ni cuus-ee-ru these children two two monkeys-Ace saw-3Pl possible translation: 'The children saw two monkeys each.' Telugu (Balusu 2006, ex. 1) (6) A gyerekek ket-ket majmot lattak the children two-two monkey-Ace saw-3Pl 'The children saw two monkeys each.' Hungarian (Szabolcsi 2010, ex. 99) However, there are several differences between the spoken language and the sign language data. The most obvious are the following: i) sign language sign repetition (not necessarily two repetitions) vs. spoken language reduplication (two repetitions); ii) no verbal reduplication in spoken language; iii) overt anaphora in sign language. While the first point is of a more mor-phophonological nature, the remaining two will be discussed in detail. The third point will be considered initially, before addressing the second point in Sections 4.3 and 4.4. 2.2. R-loci in sign language Kimmelman (2017), while exploring quantification in the Russian sign language (RSL), notes that sign language distributive marking is a combination of sign repetition and spatial distributivity. In other words, the distributive meaning effect arises by placing the repeated signs in specific locations within the signing space. These locations are called Reference loci (R-loci) in a framework which is being developed for sign language research by Schlen-ker (e.g., Schlenker 2017, 2018; and several other works). Schlenker, building on the work of Lillo-Martin and Klima (1990), describes the R-loci as positions within the signing space that can realize discourse referents / logical variables. The number and exact positions of the R-loci are in principle arbitrary, but once an R-locus is established; it can be retrieved, e.g., by a pronoun; see example (7) from the American sign language (ASL). R-loci and Distributivity: Insights from Czech Sign Language 5 (7) IX-1 KNOW BUSHa IX-1 KNOW OBAMAb. IX-b SMART BUT IX-a NOT SMART3 'I know Bush and I know Obama. He [=Obama] is smart but he [=Bush] is not smart.' ASL (Schlenker 2017, ex. 5a) Returning to the CZJ example (4), repeated here as example (8), the instances of TWO and BATHE are indexed by i, j, k which means that the first instance of BATHE is articulated in the same location (R-locus) as the first instance of TWO; this is repeated analogically in the second and third instance (see Figure l)4. (8) DAUGHTER MYTHEY-THREE DOG TWO, TWOj TWOk BATHE, BATHEj BATHEk 'My three daughters bathed two dogs each.' Figure 1: Two repetitions The three locations correspond to the three daughters - the sorting key (R-locus-i corresponds to daughter A, R-locus-j to daughter B, R-locus-k to daughter C). The numeral TWO is interpreted as the whole DP (two dogs; the distributed share, as in the spoken language in examples (5) and (6). The distributed share is then distributed over the sorting key as follows: two dogs (dog D and dog E) are assigned to daughter A via index i; two dogs (dog F and dog G) are assigned to daughter B via index j; two dogs (dog H and dog L) are assigned to daughter C via index k. Similarly, the instances of the verb BATHE assign the (sub)event of washing to the respective daughter-two dogs pairs. This mechanism will be discussed in detail in Sections 4.3 and 4.4. The difference between Schlenker's example (7) and our example (8) lies in the details of the mechanism used for establishing and retrieving the R-loci variables. In Schlenker's example (7), R-locus-a is introduced by BUSH and 3 We maintain the notation of the author: IX is a gloss for an index/pointing sign. IX-1 stands for pointing toward the signer (first person), IX-b for pointing toward R-locus b (the location where OBAMA was signed), and IX-a for pointing toward R-locus a (the location where BUSH was signed). We also follow Schlenker (2017) in treating the pointing signs as pronouns. 4 All figures in the text were created by Hana Strachofiova. 6 Mojmír Dočekal, Hana Strachoňová retrieved by a pronoun (pointing to the same location through an index handshape, i.e., IX-a); similarly, for R-locus-b, which is introduced by OBA-MA and retrieved through IX-b. In our CZJ example (8), there is no pointing to an R-loci via pronouns. The R-loci (indexed as i, j, k) are introduced by TWO and interpreted as referring simultaneously to the respective DAUGHTERS. The R-loci are then used by the verb (BATHE). 2.3. Licensing of distance distributivity markers Recently, two closely related phenomena have been discussed in the research focusing on distributive meaning in natural language: pluralizing operators and plurality filters (see overview in Kuhn 2019). The binominal each (as generalized quantifiers with every-type of universal quantifiers) belongs to the first group, and generally is treated as an expression that creates a plurality by summing objects together. The latter group is represented by pluraction-als, distributive numerals, and dependent indefinites. Plurality filters are responsible for the operation of a set restriction, resulting in the output of a set of plural objects (Kuhn 2019). The two types of distributive strategies share certain properties: a) atomicity on the sorting key and b) the required cardinality of a distributed share, see examples (9)-(ll) for a comparison. (9) = example (2) the binominal each: [The girls]key saw [two dogs]share each. (10) = example (5) distributive numerals: [pilla-lu]key renDu-renDu kootu-lu-ni]share cuus-ee-ru these children two-distr monkeys-Ace saw-3Pl 'The children saw two monkeys each.' Telugu (Balusu 2006, ex. 1) (11) dependent indefinites: [BOYS THEY-arc-a]key READ [ONE-arc-a BOOK]share 'The boys read one book each.' ASL (Kuhn 2017, ex. 16a) The first crucial distributional contrast consists in the compatibility with regard to universal quantifiers: pluralizing operators are not compatible, as in example (12), whereas plurality filters are, as in example (13).5 (12) # Every boy had one apple each. 5 Some researchers even claim that distributive numerals and dependent indefinites are licensed by the distributive operator carried by the universal quantifier (e.g., Kimmelman 2017). R-loci and Distributivity: Insights from Czech Sign Language 7 (13) rati pillavaaDu renDu-renDu kootu-lu-ni cuus-ee-Du Every kid two-distr monkey-Pl-Acc see-Past-3PSglit. possible interpretation: 'Every kid saw two monkeys.' Telugu (Balusu 2006, ex. 13) Kuhn (2017) observes that similarly to Telugu or Hungarian, the ASL uses the reduplication of numerals to express a distributive meaning, as in example (14) with a reduplication of the numeral one. In the ASL, the reduplication strategy seems to be semantically equivalent to the use of the arc movement, while articulating the numeral (see example (14) and example (11) above). In Kuhn's seminal approach, examples such as (14) and (15) are instances of dependent indefinites. This is supported, e.g., by the compatibility of the reduplicated or spatially modified (arc) numeral with the distributive universal quantifier as seen in example (15). The agreement in space is glossed by 'a' on EACH and ONE. (14) ALL BOY LIFT {ONE-arc/ONE-redupl} TABLE 'The boys each lifted a table.' ASL (Kuhn 2017, ex. 6) (15) EACH-EACH-a PROFESSOR NOMINATE ONE-redup-a STUDENT 'Each professor nominated one student.' ASL (Kuhn 2017, ex. 16b) In this paper, a slightly different kind of data will be presented. The morphological marking in our CZJ examples (examples (4)/(8)) makes use of several separate R-loci, each having their own referential value.6 On the other hand, in the ASL (example (15)), as far as can be understood from the glosses, the spatial agreement between EACH and ONE operates on a single locus (glossed by 'a'). In such a case, it is not relevant, therefore, whether the hand articulating the numeral moves across the area with or without a reduplication. The arc-movement strategy is also exploited in CZJ, with the same result as in the ASL. Example (16) in CZJ acts in parallel to that example (15) in the ASL: the numerals (and verbs) are articulated with the arc movement and agree in space with the preceding quantifier phrase (glossed by 'a'). Example (17) shows the compatibility of an NP marked by the arc movement with a distributive universal quantifier (EACH), which contrasts with example (18) containing an R-loci reduplication structure. Based on this data, it can be claimed that a distributive R-loci reduplication in the CZJ should not 6 Hence reduplication and localization in Kimmelman (2017) for the RSL. 8 Mojmír Dočekal, Hana Strachoňová be analyzed as an instance of dependent indefinites7 since it shows different morphological and distributional properties.8 (16) GIRL THEY-arc-a DOG TWO-arc-a BATHE-arc-a "The girls bathed two dogs each.' -dink to video (17) GIRL EACH-arc-a DOG TWO-arc-a BATHE-arc-a 'The girls bathed two dogs each.' -dink to video (18) #/?? DAUGHTER MY THEY-THREE EACH-arc-.jk DOG TWO, TWOj TWOk BATHE, BATHEj BATHEk (intended: 'My three daughters bathed two dogs each.') 3. Interpretation of distance distributivity in CZJ Languages across the world vary in the specific type of distributivity expressed by their distributive markers. Our aim is to establish readings for the CZJ data, which are of relevance in the following section. The English binominal each only allows the distributive operator to use the set of individuals as the sorting key, which leads to an interpretation with an individual distribution.9 The German jeweils, on the other hand, is ambiguous. Its distributive operator either takes a set of individuals or a set of subevents as the sorting key, leading to individual distribution (in example (19), to be read as (19b).) or occasional distribution (in (19), to be read as (19c)), respectively.10 The crucial observation is the fact that a cumulative reading is ruled out whenever a distributive marker is present (in (19), to be read as (19a)). 7 Although we present the contrast between the CZJ distributive R-loci reduplication and the CZJ arc-movement strategy in terms of a plurality operator vs. plurality filter dichotomy, this remains primarily at a descriptive level and uses the theoretical distinction more as a guide, because we are aware there are different approaches to the distinct patterns of data, see Cable (2014) or Law (2022). We would like to thank one of the two anonymous reviewers for raising the importance of this issue. 8 In reality, it is not always easy to determine the exact boundary between the localized reduplication and a pluralizing strategy using continuous movement (such as the arc movement). Schlenker and Lamberton (2019) suggest that these two phenomena should be viewed as a continuum. At the same time, their analysis does not exclude different quantificational and referential properties of the mentioned structures. 9 This reading is called a participant key reading, e.g., in Balusu (2006) or a participant-distributive interpretation in Cable (2014). 10 The occasional distribution is also known as an event-distributive reading, e.g., in Cable (2014). R-loci and Distributivity: Insights from Czech Sign Language 9 (19) Die Mädchen haben jeweils zwei Hunde gesehen. a. # cumulative ("The girls saw two dogs.') b. individual distribution ("The girls saw two dogs each.') c. occasional distribution ("The girls saw two dogs each time.') In this section, the availability of the two distributive readings within a variety of CZJ constructions is explored. Sections 4.3 and 4.4 formally analyze the readings. 3.1. Semantic methodology The data reported in this section were obtained through interviews with two Deaf CZJ signers. The survey was designed as a truth judgment task. The two consultants were provided with videos in CZJ, and were asked to evaluate both the 'understanding' (grammaticality) and the 'correctness' (truth) of the utterance in the given scenarios. The scenarios for individual distributive, occasional distributive, and cumulative readings were presented via pictures of 3 girls, 2 or 6 dogs, and bathing events; see Figures 2, 3, and 4." The videos were prepared during earlier sessions with the interpreter, and our target structure was the R-loci reduplication pattern. Some of the predesigned CZJ utterances were later evaluated as 'unnatural' or 'not understandable' by the consultants. In such cases, the signers were asked to initially correct the CZJ utterances and then to judge the truth/false value against the contexts.12 3.2. CZJ data As far as it is known, in spoken languages, the most salient interpretation of a sentence with plurality denoting both subject and object (e.g. The girls saw two dogs) is cumulative. It is a non-scopal reading with very weak truth conditions. Our first step, therefore, is to verify this observation in CZJ. The sentence in example (20) contains a plurality of 'my daughters' (DAUGHTER MY PL) and a plurality of 'dogs' (DOG PL).13 The index 'a' on the plural marker (PL) for 'my daughters' represents the placing of the marker on the right side of the signing space. The index 'b' on the plural marker for 11 Thus, each utterance was assessed once for the grammaticality and three times for its truth value (against the 1. individual distributive, 2. occasional distributive, and 3. cumulative scenarios). 12 The scenarios were inspired by some of the examples found in Cable (2014). 13 It is beyond the scope of this study to state the exact nature of the expression glossed as PL (we would consider a genuine plural marker, a quantifier, a pronoun, a determiner, and a classifier that locates a plurality in space). The results are currently based on the intuition of the CZJ users who interpreted the sequence of a noun and PL as a plurality. The visual representation of the sign is given in Figure 5. Figure 2: Individual distributive reading Figure 3: Occasional distributive reading Figure 4: Cumulative reading R-loci and Distributivity: Insights from Czech Sign Language 11 'dogs' signals the articulation on the left side of the signing space. The verb (BATHE) agrees in space with the NP dogs (see the index 'b' on BATHE). Note that the repetition of the verb is not a case of distributive reduplication (as in example (8) above), since the instances of the verb do not bear different R-loci. This particular repetition is analyzed merely as a very broad type of agreement. It reflects the number of dogs in the given context (see Figure 4 above). Figure 5: Plural (20) DAUGHTER MY PLa DOG PLb BATHEb BATHE^ 'My daughters bathed the dogs.' d. # individual e. # occasional f. cumulative As expected from what is known about spoken languages, the sentence in example (20) can only be accorded a cumulative reading. It avoids both individual and occasional distribution. The sentence was judged as false for the scenarios represented in Figures 2 and 3, and true for the scenario in Figure 4. Now the sentences that express distributive meanings will be addressed. The sentence in example (21) presents a simultaneous solution for distributivity.14 The dominant hand (DH) is given in the first row of the glossed notation and the non-dominant hand (NDH) in the second row.15 The straight line represents the holding of the sign articulated by the non-dominant hand (THREE), while the dominant hand continues articulating. The signs DOG and BATHE are two-handed, and the non-dominant hand joins the dominant hand for the articulation (hence in the glosses, the straight line is 14 By simultaneous we mean an utterance in which each hand articulates an independent sign, as opposed to one two-handed sign. 15 Our CZJ consultant is left-handed; thus, in this particular case, the dominant hand is the left hand. 1 A DAUGHTERS^}] (3u2[\u2\ > 1 A DOGS{u2}](3e[BATHE(e) Aagie.uJ Ath(e,u2)])) b. [uj A lu-J > 1 A DAUGHTERSIu-l) A [u2] A \u2\ > 1 A D0GS{u2] A [e]BATHE(e) A ag(e, ux) A th(e, u2) 4.3. Individual distributivity interpration in PCDRT In this and the following section, the ways distributivity is employed within the PCDRT will be introduced, and applied to the CZJ sentences discussed above. It should be recalled that in the CZJ sentence (example 21) the numeral in the share was reduplicated and so the sentence was judged as true only in the case of individual distributivity, as exemplified in Figure 2. As discussed briefly in Section 3.1, dynamic frameworks (with PCDRT being one of the most influential) do not follow the standard approach to distributivity. Now it is necessary to consider the prerequisites regarding distributivity treatment within the PCDRT. The core ingredient of the dynamic conception of distributivity is the distributivity operator 5U (after Nouwen 2003; Berg 1996; among others), which quantifies over information states, but in its domain are only those assignments (represented as rows in table Table 1 as well as all the following tables) in which the anaphoric dref un (over which 5U quantifies) is atomic are located in its domain. The definition of 5U is found 19 In a full treatment of distributivity within a framework enriched with events, we would have to disassemble formulas such as example (25) and the subsequent examples in a manner so that events such as e in example (25) would contain the pluralities of subevents in which each sub-event would have its atomic #-role bearer. In that way, our proposal would satisfy the usual 1-1 mapping requirement between events and 0-role bearers, which is explicit or implicit in all current event-based approaches to pluralities: see Landman (2000) or Henderson (2012). For the sake of clarity, we do not complicate our formulas, but see Champollion (2016a) for such a formalization. R-loci and Distributivity: Insights from Czech Sign Language 19 in example (26a) and is a simplified version of Dotlacil's (2012) formalization. Unlike the standard theory of distributivity where dist has in its domain the whole predicate (in most cases syntactically the VP), quantifies over information states 5U {XIXJ in example (26a)). Next, the distributive operator 5U updates the atomic assignments (the atomicity condition: #(\JunI) = 1) with nuclear scope. The 8Un operator appears both in the PCDRT formalization of the determiner each (not discussed here), and the formalization of the binominal each, e.g. (26b) (following Dotlačil 2013). The binominal each is anaphoric to its key (notationally as the variable which is A-abstracted over and propagated through the semantic composition until it meets its key argument). Crucially it is distributed over its syntactic argument (share -P(i/m)) and it requires each atom from the key denotation to be paired with the share of the right cardinality. Finally, the distribution is selective and does not scope over the verbal predicate (Q(wm) is not in the scope of SUn). The distributive operator also introduces a new discourse referent (um), so it serves as the existential closure of the property typed share NP. The formalization corresponds well with the ČZJ data: the individual distributivity (e.g. (21)) is signed with the R-loci reduplication inside the share constituent, and the verb remains unpluralized. (26) a. 8Un(D)=mj.unI = ujAVdeunl(#(\)unI) = 1 A £»(/|Un=d)(;|Un=d)) b. IBINOM- eachUm\ = XvrXP(rt)XQ(rt). [um\] A 5v(P(um)) A Q(um) To demonstrate an individual distributivity reading with an information state corresponding to Figure 2, see Table 2. The information state J in Table 2 is an update of the previous information state I where for each atomic entity in the dref uv there is a corresponding share with the cardinality 2, as is the case in Table 2. Table 2. Information state verifying the individual distributive reading of XX-23 Info state J e u2 Ji daughter bathej dogiU dog2 h daughter2 bathe2 dog3U dog4 h daughter3 bathe3 dog5U dog6 The full formalization the meaning of example (21) is in example (27a), with the dynamic version in example (27b): the most important change against the cumulative formalization is, of course, the addition of the distributive operator 5U . As discussed previously, the distributive operator scopes only over the share, with the predicate (syntactically VP) not within 20 Mojmír Dočekal, Hana Strachoňová its scope. Or in other words, it connects the drefs of the key (i/j) with the share (u2), but it does not directly distribute over the denotation of the verb (of type e). The PCDRT framework predicts that the binominal each is in principle compatible with certain types of collective subjects, as the distrib-utivity that begins with the binominal each takes as its two arguments the drefs of the key and the share, while the application of the verb to the bearers of the subject's #-role is not within the scope of 5. Dočekal and Šimík (2020) explore the theoretical consequences of this prediction and show that it correctly captures a number of intriguing patterns of collectivity and that the binominal each grammatically appears in one sentence. Kuruncziová (2020) offers experimental support to this local distributivity of PCDRT prediction in Slovak. Further, and a more general, application of the PCDRT framework to cases of reciprocal distributivity can be found in Dotlačil (2013). (27) a. BuJKI = 3 A DAUGHTERS^}] A 5ul(3u2[|u2| = 2 AD0GS{u2}]) 4.4. Occasional distributivity interpretation in PCDRT This section describes the most intriguing, but also the most revealing, pattern of the CZJ R-loci reduplication. Example (22), which is repeated below schematically as example (28), was reported by both consultants as being ambiguous. An individual distributivity reading (the sentence would be true in the same set of situations as example (21) with this reading) was the primary interpretation. However, an occasional distributivity reading was also reported by both signers as a possible, but not the preferred, interpretation, of example (22) (the interpretation is visualized in Figure 3). As both the glosses and Table 3 demonstrate, the latter interpretation was expressed by two R-loci reduplications, the first reduplicating the numeral modifier in the share, with the second reduplication at the R-loci targeting the verb. (28) DAUGHTER MYTHEY-THREE DOG TWO, TWOi TWOk BATHE, BATHEj BATHEk 'My three daughters bathed two dogs each.' Before formalizing our ideas about occasional distributivity in CZJ, a few general remarks are necessary. First of all, since the reduplication occurs to the verb, examples such as (22)/(28) can be categorized under the pluraction-al agenda. Yet despite the increasing literature on pluractionality (Henderson (2014) for Kaqchikel and Kuhn and Aristodemo (2017) for sign languages, a.o.), pluractional distributivity remains less well-researched and understood b. [u±] A\u±\ = 3 A DAUGHTERS{u±} A Sul([u2] A\u2\ = 2 AD0GS{u2}) A [e]BATHE{e] A ag(e, ux) A th(e, u2) R-loci and Distributivity: Insights from Czech Sign Language 21 than distributivity in the nominal domain. Thus, despite many remaining caveats, our data from CZJ and its analysis can be considered rather as another step in the development of an appropriate approach to the pluractionality phenomena in sign languages.20 The first important property of pluractionality is the plurality of events requirement. We illustrate it here with an example in example (29a) which demonstrates some of the features usually found in pluractional distributivity, which satisfies the requirement in English. Although English has no pluractional verbal morphology, its lack can be remedied with constructions such as one by one or brick by brick (see Brasoveanu and Henderson 2009 for an analysis of such English pluractionality). The first important property of pluractionality, demonstrated in example (29a) is the plurality of events requirement: example (29b) with a singular argument clearly violates the plurality of events requirement and it is therefore ungrammatical. However, intuitively example (29a) requires not only a plurality of events, but also their temporal sequencing since it would be inappropriate in a situation where all the soldiers were killed simultaneously. The plurality of events and temporal sequencing leads to an incompatibility of pluractionality with certain forms of collective predicates, but the incompatibility of pluractionals with collectivity is not a general phenomenon. See example (30) from COCA (Davies 2009). (29) a. The soldiers died one by one. b. *The soldier died one by one. (30) 14 men in secret CIA custody were gathered one by one from locations across the world. COCA Returning to CZJ, in the case of pluractional R-loci reduplication, it is necessary to retain the same type of analysis employed with regard to the numeral R-loci reduplication. Yet it is also clear that the distribution over the atoms in the NP domain would not be possible since this would result in individual distributivity. Consequently, our analysis should be based upon the idea of 20 In this article, we focus on the CZJ distributivity marking in the nominal domain. And even if obviously we touch upon the pluractional R-loci reduplication, an in-depth study requires a further paper. But we note in passing that the CZJ R-loci reduplication allows something similar to a purely pluractional reading in (i): in this case, the R-loci reduplication only targets the verb. The consultants' comment on this example in the following way: it would be appropriate in a situation with a plurality of students, a plurality of dogs, and a plurality of bathing events. But it is not necessary for each student to bathe a plurality of dogs; in other words, we are not dealing with an individual distributivity reading here. We are grateful to an anonymous reviewer for reminding us of the importance of this kind of example. (1) STUDENT PL DOG BATHE BATHE BATHE "The students bathed the dogs.' 22 Mojmír Dočekal, Hana Strachoňová group interpretation. Intuitively, as far as example (22) is concerned, it could be claimed that the occasional reading comes from the plurality of events connected with the group denotation of the key, namely, that example (22) was judged as true in the occasional picture in Figure 3 since the signers interpreted the three daughters as a group atom. This leads to a further issue raised by one of our two anonymous reviewers. As the reviewer correctly remarks, describing pluractional distributivity as involving a collective interpretation (group formation) initially appears to be a somewhat difficult theoretical stance. We agree to some extent: it is clear that some kinds of pluractional distributivity and collectivity are incompatible (but see the discussion regarding examples (29) and (30)). Nevertheless, it is also known that certain sorts of pluractionality and collectivity interact quite well, since the plurality of events and the plurality of participants (the core requirements of pluractionality and collectivity) do not clash. As an example, consider the Kaqchikel example in (31) (cf. Henderson 2014, ex. 100), in this example the pluractional distributivity marker -la' aligns with the collective predicate mol 'group'. Henderson comments on the example as follows: '[it] would be appropriate for describing a situation where I put the beans in a basket one by one' (Henderson 2014: 43). Essentially, the individual events of putting individual beans into a basket do not satisfy the predicate mol 'group,' but this fails to make example (31) ungrammatical. This example exactly mirrors our CZJ example: the group interpretation of the agents in example (22) means that there is a way to link the subevents of bathing to the individual girls constituting the group atoms, and none of these subevents has to be the whole bathing event.21 (31) X-e'-in-mol-o/a' ri kinaq. CP-A3p-Els-group-la' DET beans 'I grouped the beans individually.' Now it is necessary to formalize the ideas introduced above. A sample information state corresponding to Figure 3 can be found in Table 3. This information state verifies the occasional reading; the individual distributivity reading would also be verified by an information state similar to that in Table 2, which differs from the information state in Table 3. The most important difference is that in the case of the occasional distributive reading, illustrated 21 Implementing this would lead us to the partitioning of events and eventual decomposition of the group atoms (most probably via Landman's | operator). This would be very much in the spirit of (Henderson 2014) and his analysis of pluractional distributivity in Kaqchikel. We would like to thank one of our anonymous reviewers for highlighting this possibility. This move would also be needed for the reasons discussed in footnote 16, but we wish to address this in the future since gathering and analyzing the CZJ data related to pluractionality and collectivity would be beyond the scope of the current article. R-loci and Distributivity: Insights from Czech Sign Language 23 in Table 3, the key is not atomic at the level of the entities. It is essential to note a collective atom type shift marked with j (after Landman 2000), a shift which takes as its input the plurality of entities (daughters in our example) and yields a group atom, basically a plurality which from a higher-order perspective resembles an atom. How this is related to the composition and to the individual distributivity reading will now be discussed. Table 3. Information state verifying the occasional distributivity reading of XX-24 Info state J e u2 Ji \ (daugheq U daugher2 U daugher3) bathej dogiU dog2 h \ (daugheq U daugher2 U daugher3) bathe2 dog3U dog4 h \ (daugheq U daugher2 U daugher3) bathe3 dog5U dog6 First, it is necessary to discuss how the individual distributivity reading for example (22) is derived. Our null hypothesis is that each R-loci reduplication signals the presence of the distributivity operator (formalized as the PCDRT 8 operator). A question immediately follows: why are examples (21) and (22) (under one reading) synonymous? Let us start with a hypothesis that the syntactic position where the R-loci reduplication is realized is interpreted sortally: if the reduplication happens in the NP (share), the distributivity is related to the entities, whereas if the reduplication is realized in the verb, the signer distributes over the events, rather than the entities, although the key remains the same, as the reduplication happens at an identical R-loci. Consequently, we hypothesize that the individual distributivity reading is formalized as in examples (32-a) and (32-b) (the final dynamic notation). The individual distributivity reading in example (32) has two distributive operators, 8U, both anaphoric to its key (the dref u^), both requiring each atom to be a corresponding share. The share in the case of the R-loci reduplicated numeral is the NP, and the distributivity requirement can be viewed as the following: for each atom in the (key), there have to be two atoms in the u2 (share). This was discussed in the preceding section. But how can the distributivity of the verb be interpreted? We assume in the same way, but possibly with different consequences: in this case, the distribution operator also requires the presence of (for each atom in the i/j) the right share, but because in this instance the share denotes events, the requirement maps each atom in the u1 to an event in the e, a requirement that is verified in Table 3. Moreover, the verb itself is not pluralized. Consequently, the relationship between entities and events is 1-1. In other words, the two distributive operators collapse into one distributive reading because the distributivity requirement of the share reduplication requires an atomic antecedent for each share (fulfilled), 24 Mojmír Dočekal, Hana Strachoňová whereas the reduplication of the verb requires an atomic antecedent for each event (also fulfilled). (32) a. 3u1[\u1\ = 'iADAUGHTERS{u1]\A8ul{3u2[\u2\=2ADOGS{u2}\) Now we are in a position to explain the occasional reading. The basic insight with which we wish to engage with our formalization is as follows: if the signer also reduplicates the verb at the R-loci, the hearer's attention is drawn to the events and this de-emphasizes the atomicity of the key, or in other words, the key atoms are permitted to be atomic at a higher-order, by which we mean the domain of group atoms. So essentially, we claim that the occasional reading is a result of the key shifting to a group atom so that the 8U quantifies over group atoms (in the sense of Barker 1992; Landman 2000). As in the case of individual distributivity, the meaning of which was discussed in the previous section, the requirement for the atomicity of the key is fulfilled, but in the case of the occasional reading, the atomicity is fulfilled at the level of the group atoms. Therefore, we propose that the occasional reading is formalized as (33a) and (33b) (again the dynamic version). Thus, our solution intuitively builds upon the idea of markedness: if (as in example (21)) the signer reduplicates at the R-loci only in the share, the atomicity requirement is stringent; if the signer reduplicates both the verb and the share at the R-loci (example (22)), the atomicity requirement can be interpreted as, most importantly, requiring a 1-1 mapping between the events and the key atoms, which does not necessarily mean a mapping to additional atoms, but allows the possibility of a mapping to group atoms. For this to proceed, the key has to be shifted to a group atom.22 Such covert shifting is (at least at the processing level) costly, so there is some understanding of the individual 22 One anonymous reviewer raises a question about an alternative and more straightforward formalization of our ideas concerning the occasional reading, and suggests that instead of the distribution over the group atoms, as in example (33), it would be possible for 8 to quantify directly over the events so that the sorting key would be the events instead of the drefs. We agree that this is plausible, but to split the group-atom key idea from the event-key idea, we would have to construct scenarios in which the plurality of the agents either collaborates or not, and depending on the availability of the occasional reading for each of the scenarios, it would become clear whether the shift to group atoms is necessary or not. We leave this for the future, because although our solution may be more complicated than needed, despite working for the examples investigated, only further empirical research can reveal whether the group-atom strategy is really an unnecessary complication. b. [uj A|Ui| = 3 A DAUGHTERS^} Aiul([u2] A\u2\ = 2 ADOGS{u2}) A [e]BATHE{e] A ag(e, ux) A th(e, u2) R-loci and Distributivity: Insights from Czech Sign Language 25 vs. occasional reading preference reported by both consultants. We believe, based upon our analysis, that our work is progressing in the right direction but there are (as always) many questions remaining. One of the more pressing (see also our discussion in footnote 16) concerns the unified treatment of the R-loci reduplication, which (following the CZJ data) we formalize. It is, of course, possible that further in-depth investigations into the CZJ reduplication of verbs would strengthen the link to Henderson (2014) and his treatment of pluractionality in terms of partitioning both the events and their arguments. (33) a. auJKI = 1 A3-D^[/GHr£'i?S{u1}](3u25Uiu1[|u2| = 2 A D0GS{u2}]) (BeSUi [BATHE{e] A ag(e, ut) A th(e, u2)]) b. [ujAlu-J = 1 A 3-DAUGHTERS^} A [u2|SUl([|u2| = 2 A D0GS{u2}])] A [e\SUi([BATHE{e} A ag(e,ut) A th(e,u2)])] 5. Summary In this article, the preliminary data from the exploratory work we conducted with Deaf Czech Sign Language signers is reported. We focused on distributivity in CZJ since the strategies CZJ employs to communicate distributivity are more familiar than the majority of the distributivity markers we can find in spoken languages. One empirical conclusion we draw from our work concerns the different strategies which are used in CZJ to signal distributivity. 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PhD thesis, Amsterdam: University of Amsterdam. 28 Mojmír Dočekal, Hana Strachoňová Appendix One of the two anonymous reviewers raises a question concerning our analysis of the CZJ R-loci reduplication as a sign language parallel to the spoken language binominal each and asks whether an analysis (roughly speaking) of the R-loci reduplication as a dependent indefinites strategy would in fact work. We acknowledge that our research into CZJ is in its initial stages and can be improved in many respects, but we do not believe that the data (at least from what we know) point in the direction of R-loci reduplication as a dependent indefinites strategy. To support the test with the universal quantifier (which is used as a standard test for the classification of an expression as an operator or a filter), as reported in Section 2.3, after determining the contrast between examples (16) and (17), we conducted a further CZJ investigation by running a second standard diagnostic test which distinguishes between distributivity operators (such as the spoken binominal each) and dependent indefinites (or plu-ractional markers). The second standard test concerns the behavior of plain indefinites with respect to distributivity operators and dependent indefinites. Before reporting the CZJ data, we repeated this widely accepted diagnostic test that distinguishes between distributivity operators and plurality filters (dependent indefinites and pluractional markers). First, we considered the contrast between examples (34a) and (34b) from French Sign Language -LSF - (the examples and their interpretation are taken from Kuhn and Aris-todemo 2017, ex. 34)). As the judgments suggest, a distributive quantifier in the LSF can interact with the plain indefinite in the expected manner creating an ambiguity - the relative scope is either V > 3 (many words) or 3 > V (one word). Nevertheless, the situation changes if a plurality filter signed as arc-movement on the index IX is used. Descriptively speaking, this interpretation only permits the wide scope of the indefinite over the plurality filter, 3 > V. This is intriguing, since 3 > V is logically stronger and asymmetrically entails V > 3 (the formula 3x[PxAVy(Qy —► Rxyj] —► Vy[Qy —► BxyPxARxy)] is a valid predicate logic tautology). The lack of the many words reading means that V > 3 would be true only in limited cases where the set Q would be a singleton, in other words in cases where V > 3 is equivalent to the 3 > V scope. Be that as it may, the usual conclusion drawn from data such as this (see Henderson 2014; Kuhn and Aristodemo 2017; Kuhn 2019) is that dependent indefinites cannot induce the covariation of a plain indefinite (unlike distributive operators such as the universal quantifier in example (34a)). (34) a. STUDENT EACH FORGOT ONE WORD. Anany words ^one word 'Each student forgot one word.' R-loci and Distributivity: Insights from Czech Sign Language 29 b. STUDENT IX-arc FORGOT-alt ONE WORD. *many words ^one word 'The students forgot (the same) one word.' With respect to CZJ, we used the second test introduced in example (34). The sentence in example (35) employs the R-loci reduplication strategy on the whole entity classifier representing a girl (glossed as CL). Each locus (i, j, k) refers to one girl. The sentence is interpreted with a wide scope of the plurality in the subject (different bread for each girl): V > 3. This interpretation is more salient than the reverse scope, 3 > V, but certainly example (35) does not only have the 3 > V interpretation reported above as a telltale sign of plurality filters. Based on these judgments, we conclude that the treatment of CZJ R-loci reduplication as a dependent indefinite is highly improbable and that it is a plurality operator with which we are dealing, a plurality operator that we claim is similar to the spoken binominal each. However, we assume the arc-movement strategy reported in example (16) would lead to a data pattern similar to that of the LSF reported in example (34), because CZJ employs the dependent indefinite strategy. And in the future, we would like to focus on the contrasts between R-loci reduplication and CZJ dependent indefinites with respect to their differing scope patterns. (35) GIRL THREE CLt CLk BREAD BUTTER EAT. ^many breads ✓one bread 'Each of the three girls ate one bread with butter.' Mojmír Dočekal Department of Linguistics and Baltic Languages Arna Nováka 1/1 602 00 Brno Czech Republic docekal(at)phil.muni.cz Hana Strachoňová Department of Linguistics and Baltic Languages Arna Nováka 1/1 602 00 Brno Czech Republic email: hana.strachonova(at)gmail.com