This paper investigates the semantics of the reduplicated unitary numeral se:sem in Classical Nahuatl from a formal perspective. In the literature, reduplicated numerals in Classical Nahuatl have been claimed to be distributive numerals. However, in some cases, they behave in an unexpected way for a distributive numeral. For instance, sometimes they seem to mark the key and not the share of the distribution. In distributive relations, the key consists of the individuals, events, or places across all of which the share is distributed, while the share is a property that holds of each individual, event, or place comprising the key (for example, [Every girl]_key [ate four apples]_share). The behavior of se:sem in such cases is unexpected from a distributive numeral, since, across languages, distributive numerals are share markers, not key markers. Despite this seemingly anomalous behavior, we argue that, in Classical Nahuatl, reduplicated unitary numerals are standard distributive numerals that always mark the share of a distributive relation. We base our claim on the fact that, unlike true key markers, reduplicated unitary numerals in Classical Nahuatl do not multiply distributive relations when they occur multiple times in a single clause.

This paper presents an analysis of the distributive unitary numeral in Classical Nahuatl (cn) seːsem, which is formed through the reduplication of the unitary cardinal sem-. Our main claim is that, despite showing a seemingly puzzling behavior, seːsem is a standard distributive numeral that marks the distribution share, as any other reduplicated numeral in cn.

As we will show, the use of seːsem raises several interesting challenges for the semantic theory of distributive numerals for at least two reasons. First, in some cases, it seems to introduce the distributive key, which is not expected of a distributive numeral, since, following Gil (1982, pp. 339–341), distributive numerals always introduce the share. In example (1), note how seːsemilwit͡ɬ ‘each day/daily’ appears to correspond to the distributive key, over which the share is distributed, in kiawi ‘the way in which it rains.’

(1) cēcemilhuitl in quiahui

seː~sem-ilwit͡ɬ in kiawi

distr~one-day.abs det rain

‘Everyday/Daily is the way that it rains’ (Carochi, 2001[1645], p. 390, f. 107)

Second, there are also cases in which the reduplicated unitary numeral (run or runs in plural, hereafter) seems to have scope over an indefinite. In the most plausible reading, the indefinite is interpreted as if its reference covaries with respect to the run (n per each one). This behavior is unexpected from a distributive numeral, since, as claimed by Gil (1982, pp. 118–128) and Kuhn (2019, pp. 6–7), distributive numerals cannot introduce referential covariation of an indefinite under their scope. We illustrate this with example (2), where the indefinites (in this case ‘three’ and ‘four canoes’) under the scope of the distributive seːsemilwit͡ɬ ‘each day’ covary referentially with respect to it: ‘3 or 4 (different) canoes per day’.

(2) in cecemilhuitl popoliuia, aço eacalli anoço naoacalli

in seː~sem-ilwi-t͡ɬ poʔ~poliwia aʔso

det distr~one-day-abs plr~finish.pst.ipfv maybe

eː-aːkal-li anoʔso naːw-aːkal-li

three-canoe-abs or.perhaps four-canoe-abs

‘Perhaps three canoes (of water) or perhaps four canoes (of water) were consumed every day’ (FC09, Sahagún, 1959, p. 48, f. 38r)

It has been noted that in some Mesoamerican languages, unitary distributive numerals behave differently from other distributive numerals analyzed in the literature. For example, according to the analyses by Gómez González (2023) for Matlatzinca and by Chapa and Vázquez Rojas Maldonado (2024) for Ocotepec Zoque, in these languages the unitary distributive can, in some instances, introduce the key of the distribution, thus providing counterexamples to Gil’s proposal.

However, in this paper, we will argue that in NC, seːsem never introduces the key, but rather, it is a standard distributive numeral that always introduces the share. We arrive at this conclusion on the basis of cases with more than one occurrence of seːsem in which, nonetheless, the distributive relations are not multiplied.

To support our analysis, we provide examples in which seːsem does not display the behavior expected of a quantifier marking the key. Furthermore, we present cases in which its behavior aligns with what is expected of a share marker. To account for the instances in which the run seems to be introducing the key (examples 1 and 2), we will propose the presence of a null distributive operator, as has been done in the case of Korean (Oh, 2001, 2006, pp. 22–85).

The remainder of this paper is organized as follows. First, we present some general information about the language, the corpus, and the evidence (§1) and provide some background on distributive numerals in natural languages (§2). We then discuss the morphology of reduplicated numerals in cn (§3), and further, we elaborate on their description and interpretation, focusing on runs (§4). Next, we present our analysis of runs in cn (§5). Finally, we present some conclusions of our study (§6).

Classical Nahuatl is an Uto-Aztecan language from the Nahuan branch. It is a pro-drop language with nominative-accusative alignment and an unmarked VSO order (Dakin, 2021, p. 2177, Table 1; Launey, 1992, pp. 19–23, 33–40, among others). In general, the term “Classical Nahuatl” refers to the written form of the Nahuatl varieties of the Valley of Mexico, mainly from the 16th and 17th centuries (Canger, 1978a, Map 1, 1978b, pp. 4–6, 1980, Map 1, 1988, pp. 45–47; Flores Farfán, 2010, pp. 189–196).

To describe the use of seːsem in cn, we compiled a corpus based mainly on the Florentine Codex (FC), books 1-12, as well as some additional data from the Testaments of Culhuacán (TC, 1984) and the collection of documents in Anderson, Berdan and Lockhart (BC, 1976), all from the 16th century.^[1] To retrieve the occurrences of runs in the FC, we conducted a query on books 1–12 in the electronic corpus of Thouvenot and Fisher ([s.d.]). The corpus size is 251,115 words, in which we have documented 135 cases of runs.

Before proceeding to the description of distributive numerals in cn, it is essential to consider the following two points. First, we must note that this study has exclusively qualitative purposes. We are interested in describing the use of seːsem in the corpus to propose a semantic analysis and discuss how its semantics and distribution contribute to our understanding of the typology of distributive numerals. A quantitative analysis of this form would undoubtedly be interesting; however, it is beyond the scope of this study.

Second, the nature of the language under study (a written language of the 16th and 17th centuries) comes with a series of methodological challenges for semantic analyses, most notably the absence of negative evidence (corpora only offer positive evidence) and the impossibility of accessing acceptability judgments. Thus, following a common practice in historical linguistics, our analysis assumes that the data extracted from a corpus are grammatical, true, and felicitous linguistic expressions in the context in which they are uttered. By contrast, since the absence of evidence does not imply evidence of absence, if a linguistic expression is not documented in a corpus, it is not possible to determine whether it is an acceptable or unacceptable linguistic expression in the language under study. Finally, based on the Uniformitarian Principle, according to which “the rules that govern language structure today are the same that governed language structure yesterday and will be the same that will govern language structure tomorrow”, we assume that typological generalizations can serve as hypotheses in light of which to evaluate the data extracted from our corpus (see Croft, 2003; Walkden, 2019).

Following Cable (2014, p. 563, ex. 1), we adopt the definition in (3) of a distributive numeral:

(3) Distributive numeral: A morphosyntactic construction containing a numeral, whereby (i) the sentence as a whole receives a distributive reading, and (ii) under the allowable readings, the numeral contained within the construction must be interpreted as if it is within the scope of a distributive operator.

We will now briefly explain each component of Cable’s definition. The first part of the definition refers to the distributive interpretation. In a clause with a distributive interpretation, a predicate applies to each and every minimal part of a plurality (or to each and every member of a set). The plurality whose members the predicate applies to is called the distributive key. The predicate that applies to the members of that plurality is referred to as the distributive share (Balusu, 2006, pp. 1–2; Champollion, 2019, p. 290; Choe, 1987, pp. 74–134; Zimmermann, 2002, p. 322). Let us look at an example. The sentence in (4) is compatible with a distributive interpretation, shown in (4a), in which bought a ball is predicated of every child, so that each child bought a (possibly different) ball. Additionally, the sentence has a collective interpretation (4c), in which no distributive relation is established: (4a), in its collective reading, is true when a group of children together bought a single ball.

(4) a. The kids bought a ball.

b. Distributive interpretation: [The kids]_key [bought a ball]_share each.

[∀x.x ∈ ɩX.kids(X)]_key → [∃y. ball(y) & 1-atom(y) & bought(x,y)]_share

c. Collective interpretation: The kids together bought a ball.

There are different types of keys depending on the entities that comprise the plurality. If the key consists of a plurality of temporally distinct events, a temporal key is obtained. In (5a), for example, the key is a plurality of events (‘leaving the classroom’) in which, for each one, three children left the room. In turn, a spatial key is composed of a plurality of locations, as shown in (5b), where the key contains a plurality of spatially distinct corrals, each containing five piglets. Finally, a participant key consists of a plurality of individuals, in which the key is expressed by an argument of the clause. For example, in (5c), the key consists of a plurality of children, each of whom was given three different piglets to feed (Balusu, 2006, pp. 1–2; Cable, 2014, p. 564; Gil, 1982, pp. 11–31; Vázquez Rojas Maldonado, 2013, pp. 90–93).

(5) Types of distribution keys.

a. Temporal key: The children left the room three by three.

Interpretation: ‘In each temporally distinct event, it is true that three children left.’

b. Spatial key: In each corral, there are five piglets.

Interpretation: ‘In each distinct corral, it is true that there are five pigs.’

c. Participant key: Each child fed three piglets.

Interpretation: ‘For each distinct child, it is true that she fed three pigs.’

The second part of Cable’s (2014) definition concerns the scope of the distributive operator: a distributive operator dist universally quantifies the minimal parts that make up a plurality and assigns a property P to each of them (6) (Cable, 2014, p. 563; Champollion, 2019, pp. 293–294; Choe, 1987, pp. 74–134; Lasersohn, 1995, pp. 86–87; Link, 1998, p. 109, ex. 48; Schwarzschild, 1996, p. 61). For example, in the distributive reading of (4a), a distributive operator assigns the property ‘to buy a ball’ to each and every child that composes a plural individual, such that it is true for each child that they bought a ball (4b). The domain over which the distributive operator universally quantifies is called the restrictor, and the assigned property is called the scope. Thus, in (4b), the restriction corresponds to a plurality made up of children, and the scope to the property of buying a ball. From the second part of the definition in (3), it follows that distributive numerals are always part of the expression corresponding to the distributed share, since for a numeral to be distributive, it must always be interpreted as if it were within the scope of a distributive operator (Gil, 1982, pp. 339–341).

(6) ⟦dist⟧ = λP_<et>: λy_e: ∀x. [x ≤ y & atom(x)]_restrictor → [P(x)]_scope

(Cable, 2014, p. 563, ex. 2)

Let us now examine how distributive numerals in natural language align with the definition in (3). Consider the case of Purepecha, where distributive numerals are marked with the suffix -ntani. Thus, for example, in (7) tsimantani is a distributive numeral meaning ‘two for each one’.

(7) Purepecha (Isolate)

Nanaka sapi-echa jikwa-ra-s-ø-ti=ksï

girl little-pl bathe-caus-pfvo-3ind=3pl

tsima-ntani wichu sapirati-echa-ni

two-dist dog little-pl-obj

‘The girls each bathed two puppies.’

(Vázquez Rojas Maldonado, 2013, p. 91, ex. 8)

First, note that, according to Vázquez Rojas Maldonado (2013, p. 91), (7) has a strictly distributive interpretation. In this example, the predicate ‘to bath two puppies’ applies to each and every member of a plurality of girls. The share corresponds to the predicate ‘to bath two puppies.’ The key, in turn, corresponds to the plurality of girls, since each and every girl bathed two different dogs. Second, in Vázquez Rojas analysis, the numeral tsimantani must be interpreted as if it were within the scope of a distributive operator: each girl (nanaka sapiecha ‘little girls’) is assigned the property of having bathed two dogs (jikwarastiksï tsimantani wichu sapiratiechani ‘they bathed two dogs’). In other words, in (7), tsimantani would be located within the scope of a distributive operator, while the restrictor would correspond to a plurality denoted by the girls (nanaka sapiecha). Thus, in Purepecha, a numeral with the suffix -ntani fulfills all the features of distributive numerals, as per the definition in (3).

Before moving on to the cn data, let us stress that distributive numerals are not the only quantifiers that force a distributive reading of the clause in which they appear. Other types of quantifiers also introduce distributive interpretations. However, these quantifiers typically mark the key, not the share. An example of such a quantifier is the universal quantifier every in English (see Gil, 1995 for the different types of universal quantifiers that introduce distributive interpretations).

Before addressing numeral reduplication in cn, we provide a brief description of the generalities of the cn numeral system in terms of Greenberg (1990[1978]). In Table 1, we register the simple atoms of the cn numeral system, that is, the basic numerals that are not bases and that are not the result of an arithmetical operation coded by linguistic means. Note that some simple atoms have free and bound forms, whereas others have only a bound form.

In the numeral system of cn, there are two arithmetical operations whereby it is possible to form complex numerals: addition and multiplication by a vigesimal base (Andrews, 2003, pp. 307–318; Herrera Jiménez, 2023, §4.4; Launey, 1986, pp. 663–684, 1992, pp. 63–70; Molina, 2013[1571], pt. 1, ff. 118v-121v; Olmos, 1875[1547], pp. 190–191). In cn, there are several strategies to perform the addition operation. First, the numerals 6–9 are compounds in which the simple atoms 1–4 are added to the numeral /t͡ʃik^w-/ ‘5’ (8). The numeral /t͡ʃik^w-/ cannot stand alone and only appears as part of the compound numerals 6–9 (Andrews, 2003, pp. 307–318; Launey, 1986, pp. 663–684, 1992, pp. 63–70).

(8) chicunaui

t͡ʃik^w-naːwi

five-four

‘nine’ (Molina, 2013[1571], pt. 1, f. 118v)

Second, in cn, it is also possible to add two numerals by linking them either by affixation of the prefix /om-/ to the second summand (9a) or by using a superessive (9b) or comitative link (9c) (Andrews, 2003, pp. 307–318; Launey, 1986, pp. 663–684, 1992, pp. 63–70; Molina, 2013[1571], pt. 1, ff. 118v-121v; Olmos, 1875[1547], pp. 190–191).

(9) a. matlactlionnaui

maʔt͡ɬak-t͡ɬi on-naːwi

ten-abs plus-four

‘ten plus four’ (10+4=14) (Molina, 2013[1571], pt. 1, f. 118v)

b. Centzuntli ypammatlacpoualli

sen-t͡son-t͡ɬi iː-pan maʔt͡ɬak-poːwal-li

one-four.hundred-abs posr.3sg-on ten-twenty-abs

‘two hundred on top of four hundred’ (1*400+10*20=600)

(Molina, 2013[1571], pt. 1, f. 119v)

c. cempōhualli on-caxtolli īhuān ēyi

sem-poːwal-li on-kaʃtoːl-li iː-waːn eːji

one-twenty-abs plus-fifteen-abs posr.3sg-with three

‘Three with twenty plus fifteen’ (3+(1*20+15)=38)

(Launey, 1986, p. 666, ex. 824)

It is also possible to form complex numerals by multiplying a numeral, the multiplier, by a vigesimal base, which functions as the multiplicand. The multiplier is always a numeral of a lower value than the vigesimal base. Vigesimal bases always appear as the second element of a compound in which the first element is one of the numerals 1–10 or 15 (10). There is also a sporadic product, t͡ɬamik ‘20’, which does not behave as a standard base, that is, it does not appear as the multiplicand in any other product (Andrews, 2003, pp. 307–318; Launey, 1986, pp. 663–684, 1992, pp. 63–70; Molina, 2013 [1571], pt. 1, ff. 118v-121v; Olmos, 1875[1547], pp. 190–191).

(10) ompoualli

oːm-poːwal-li

two-twenty-abs

‘forty’ (2*20=40) (Molina, 2013[1571], pt. 1, f. 119r)

There is a small set of numeral classifiers that always attach to a numeral as the second element of a compound, as shown in (11) for the classifier /te-/. These classifiers are homophonous to other forms that appear in nouns and verbs without a counting classificatory function (Andrews, 2003, pp. 307–318; Launey, 1986, pp. 663–684, 1992, pp. 63–70, 236–238; Molina, 2013[1571], pt. 1, ff. 118v-121v; Olmos, 1875[1547], pp. 190–194; Stolz, 2018).

(11) ōn-te-mê cuācuāuhtin

oːn-te-meʔ k^waː~k^waːw-tin

two-cl-pl pl~eagles-pl

‘two eagles’

(FC08, Sahagún, 1979b, p. 8, f. 6r) apud (Launey, 1986, p. 674, ex. 876)

In cn, reduplicated numerals can be interpreted as distributive numerals. Reduplication is a morphophonological operation that copies and duplicates or reproduces material (Inkelas; Downing, 2015, p. 502; Rubino, 2013). The form from which material is copied is called the base, while the sequence that reproduces the phonological features taken from the base is called the reduplicant (McCarthy; Prince, 1993, p. 66). To illustrate this, consider examples (12) and (13). In (12a), we show the free form of numeral ‘4’, and in (12b), the reduplicated form of this numeral. As can be seen by contrasting (12a) with (12b), the base of reduplication is the numeral naːwi (‘4’), while the reduplicant corresponds to the sequence naː, since it contains the material copied from the base (naːwi). In (13), we show the same process for the numeral ‘2’ (Launey, 1986, pp. 680–681).

(12) a. naui

naːwi

four

‘four’ (Molina, 2013[1571], pt. 1, f. 118v)

b. nanaui

naː~naːwi

distr~four

‘four by four/four each’ (Molina, 2013[1571], pt. 1, f. 119r)

(13) a. ome

oːme

‘two’ (Molina, 2013[1571], pt. 1, f. 118v)

b. oome

oː~oːme

distr-two

‘two by two/two each’ (Molina, 2013[1571], pt. 1, f. 119r)

Note that the base of reduplication can be a root or a morphological constituent larger than a root (see Haugen, 2011, pp. 12–13). Additive numerals formed through compounding provide evidence in favor of this hypothesis. For example, in (14), the reduplication operation can either separately copy phonological material from each of the simple roots of the compound (14a) or from the compound as a whole (14b).

(14) a. chichicunanaui

t͡ʃiː~t͡ʃik^w~naː~naːwi

distr-five-distr-four

‘nine by nine/nine each’ (Molina, 2013[1571], pt. 1, f. 120r)

b. Chichiconaui

t͡ʃiː~t͡ʃik^w-naːwi

distr-five-four

‘nine by nine/nine each’ (Molina, 2013[1571], pt. 2, f. 19v)

The behavior of the numeral maʔt͡ɬakt͡ɬi ‘10’ when subject to reduplication also constitutes an argument for considering that, in cn numerals, the base of reduplication can be a root (Herrera Jiménez, 2023, §8.3). Diachronically, maʔt͡ɬakt͡ɬi appears to stem from the compounding of the roots maʔ- ‘hand’ and t͡ɬak- ‘torso’ (Andrews, 2003, p. 310; Launey, 1986, pp. 560–561). When reduplicated, this numeral copies and reproduces material from the segment t͡ɬak-, one of the roots of the presumed compound from which maʔt͡ɬakt͡ɬi is etymologically derived. To account for this, one could assume that the morphological component does recognize the presence of the roots maʔ- and t͡ɬak-, even though synchronically, the meaning of maʔt͡ɬakt͡ɬi ‘10’ does not seem to be compositionally derived from the combination of these roots. In this case, the reduplication operation would take the root t͡ɬak- as its base. Therefore, we take (15b) as evidence that in cn, numeral reduplication can take a root as its base.

(15) a. matlactli

maʔt͡ɬak-t͡ɬi

ten-abs

‘ten’ (Molina, 2013[1571], pt. 1, f. 118v)

b. matlatlactli

maʔ~t͡ɬaː~t͡ɬak-t͡ɬi

ten~distr~ten-abs

‘ten by ten/ten each’ (Molina, 2013[1571], pt. 1, f. 120r)

In cn numerals, the reduplicant is always a heavy bimoraic syllable (for an Uto-Aztecan perspective, see Haugen, 2008, pp. 35–50). However, it is not clear whether the reduplicant in numerals is always a (C)Vː-type syllable or whether it can alternate between a (C)Vː and a (C)Vʔ syllable. Launey (1986, pp. 680–681) and Andrews (2003, p. 317) argue that the reduplicant can take either the form (C)Vː or (C)Vʔ. For Launey (1986), the reduplicant is of the form (C)Vː when the reduplicated numeral begins with a consonant, as in (12b) above. In contrast, the reduplicant appears as (C)Vʔ if the reduplicated numeral starts with a vowel, which would be the case in (13b). Andrews (2003), on the other hand, proposes that both types of reduplicant can occur with all numerals. However, according to him, there is a semantic difference between reduplicated numerals with the (C)Vː pattern and those with the (C)Vʔ pattern. The (C)Vː pattern expresses distributivity (‘n to each'), while the (C)Vʔ pattern signals repetition or grouping (‘by n’s’ or ‘in groups of n’) (see also Carochi, 2001[1645], pp. 264–281, ff. 70-75v). In contrast, for Launey (1992, pp. 257–266), in reduplicated numerals, the reduplicant always follows the (C)Vː pattern. As shown in examples (12–15), we adopt Launey’s (1992) position for the proposed phonetic transcription.

Finally, it should be noted that the reduplication operation can be iterated (Andrews, 2003, pp. 230–231; Carochi, 2001[1645], pp. 266–270, ff. 70v-72; Launey, 1986, pp. 681, 1031). In (16), for example, the reduplicated numeral expression naːnaːnaːwi ‘four by four/four to each’ undergoes the reduplication operation twice.

(16) mochi tlacatl nananavi ic qujnmjna in mjtl

mot͡ʃi t͡ɬaːkat͡ɬ naː~naː~naːwi ik

all man/person.abs distr~distr~four thus

k-in-miːna in miːt͡ɬ

o.3-o.pl-shot.with.arrow det arrow.abs

‘Every man is shot with four arrows each.’

(FC10, Sahagún, 1961, p. 172, f. 121v apud Launey, 1986, p. 681, ej. 923)

As mentioned earlier, in cn, reduplicated numerals can be interpreted as distributive numerals (see Andrews, 2003, p. 317; Launey, 1986, pp. 680–681, 1992, pp. 257–258), that is, they behave as expected from the definition of distributive numerals in (3). In this section, we first explore the contexts in which we would expect a standard distributive numeral to occur (§4.1). Based on this, we claim that in cn, reduplicated numerals may display the properties of a normal distributive numeral. Then, we will look at some cases where a run should not appear if it were a standard distributive numeral (§4.2). Despite this, we conclude that the best account for the behavior of cn run is one in which they are treated as standard distributive numerals (§4.3).

In this section, we develop an analysis of the meaning of runs in cn. To do so, we first introduce our main theoretical assumptions (§5.1). Then, we demonstrate how these assumptions are implemented to account for the behavior of runs in cn (§5.2) in various contexts. It is worth noting that our proposal is not conclusive and other alternative analysis, such as the one in Champollion (2016a; 2016b), could fit well with the cn data. The purpose of our formal analysis is simply to show how we could account for all cases of runs by assuming that they are share markers, as expected from a standard distributive numeral. However, further research is needed to assess what the best way of formalizing the data is.

In this study, we presented an analysis of the semantics of reduplicated unitary numerals in Classical Nahuatl. Based on a corpus of its usage in various documents from the 16^th and 17^th centuries and considering the vast literature on distributivity, we have argued that in cn, runs are standard distributive numerals and, as such, they always mark the share and not the key of a distribution. Specifically, we propose that: 1) the distributive relation may be introduced by a null distributive operator, which is a universal quantifier over the members of a plurality; 2) runs must be interpreted under the scope of this null distributive operator, and their reference necessarily covaries according to the plurality over which the operator quantifies. Our analysis can account for both prototypical cases in which the run clearly behaves as a standard share marker and non-prototypical cases in which runs seem to mark the distribution key or induce referential covariation of an indefinite in the clause. Lastly, although deeper research is required, we believe that this analysis might be extended to other non-unitary reduplicated numerals in cn.

The authors would like to thank the reviewers for their insights and useful comments on this manuscript.

Conflict of Interest

The author(s) declare no competing interests.

Statement of Data Availability

The data and materials that support the findings of this study are openly available in the electronic corpus compiled by Thouvenot and Fisher ([s.d.]). We queried all instances of the sequence <cece> in the Florentine Codex using Thouvenot and Fisher’s ([s.d.]) corpus. From the results obtained, we selected only those corresponding to unitary distributive numerals. We excluded verbs showing reduplication of sem-, given that in some cases it is unclear whether the reduplication of the base /sem-/ involves a numeral meaning. Note that the form seːsenjakaː ‘every/each’ falls into this excluded group, since, according to Andrews (2003, p. 440), seːsenjakaː purportedly derives from the verbal base senjaʔ- ‘go as one’. Data on distributive numerals other than the unitary type are provided within the article.

AI Usage Statement

We declare that we have utilized AI tools, specifically Grammarly and Paperpal, to proofread the entire manuscript for grammar and spelling, since we are not native English speakers. These tools were not used for content generation, data analysis, or argumentation.

Funding Sources

This research was supported by SECIHTI through project CF2009-11313. In addition, Rafael Herrera Jiménez received support from Universidad Nacional Autónoma de México Postdoctoral Program (POSDOC), under the guidance of Dr. Ana Aguilar Guevara.

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CORPORA

Main Corpus

FC01 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 1: The

gods. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1970.

FC02 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 2: The Ceremonies. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1981.

FC03 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 3: The Origins of the Gods. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1978.

FC04 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 4: The soothsayers and Book 5: The omens. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1979a.

FC05 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 4: The soothsayers and Book 5: The omens. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1979a.

FC06 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 6: Rhetoric and Moral Philosophy. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1969.

FC07 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 7: The Sun, Moon, and Stars, and the Binding of the Years. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1953.

FC08 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 8: Kings and Lords. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1979b.

FC09 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 9: The Merchants. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1959.

FC10 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 10: The People. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1961.

FC11 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 11: Earthly things. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1963.

FC12 = SAHAGÚN, B. de. General History of the Things of New Spain. Book 12: The Conquest of Mexico. Ed. A. J. O. Anderson; C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1975.

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ADDITIONAL CORPUS

ANDERSON, A. J. O.; BERDAN, F.; LOCKHART, J. Beyond the codices: The Nahua view of Colonial Mexico. Berkeley: University of California Press, 1976.

CLINE, S. L.; LEÓN PORTILLA, M. The testaments of Culhuacan. Los Angeles: University of California Press, 1984.

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ANNEX – ABBREVIATIONS

DOI: https://doi.org/10.25189/2675-4916.2026.V7.N1.ID922.A

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ROUND 1

2026-04-13

We would like to thank the reviewers for their useful comments on our manuscript. In the following pages, we will explain how we have solved all the issues raised by the reviewers. We have numbered each reviewer's comments to facilitate the revision. After each comment, we have added our reply.

Reviewer 1

1) One thing that would recommend, and that would help the reader to walk smoothly through the paper, is to provide clear formal definitions and formalised examples for theoretical assumptions and key concepts such as distributive key and share, or restrictor and scope, introduced in section 3; most importantly, I would recommend to explicitly give the LF of each alternative interpretation discussed for run constructions.

Reply: We have added the LF for each alternative interpretation of the relevant examples.

Reviewer 2

1) It seems to me that the structure chosen retraces the process of discovery with hypotheses and refutations and other hypotheses and arguments. This is ok -if tortuous - for sb like me who knows the literature well but extremely confusing for the general reader, who get the impression that hypotheses are proposed that are in fact refuted.

Normally papers are written with hindsight so really starting from the end: the conclusion of the discovery:

This is what is true according to us

This is how else explain the seemingly anomalous examples.

Reply: We have added a paragraph at the beginning of the Introduction that states our main claim to provide the reader with a clear view of our proposal and the challenges in the analyzed data: “This paper presents an analysis of the distributive unitary numeral in Classical Nahuatl (cn) seːsem, which is formed through the reduplication of the unitary cardinal sem-. Our main claim is that, despite showing a seemingly puzzling behavior, seːsem is a standard distributive numeral that marks the distribution share, as any other reduplicated numeral in cn.”

2) As stated in the public review, I think the examples with canoes and baskets are measure-uses of a container noun and matl is clearly a measure and not indefinites that can be multiplied, and the quantificational flavour of the problematic examples seems to be closely linked to the type of noun (at least from the examples that are given in the text). It would be useful to have an appendix with all the examples.

Reply: The reviewer raises a fair point. We acknowledge that it is not possible to rule out an analysis according to which measure phrases are not indefinites in cn, which in turn may explain why a run may induce covariation on them. However, we argue that our analysis is supported by the behavior of measures in other languages, as we show in the following examples in which measure phrases and indefinite behave alike. For instance, in Spanish, a measure phrase such as dos kilotones de dinamita ‘two kilotons of TNT’ multiplies when it is under the scope of a distributive universal quantifier that marks the distribution key (i). Thus, Spanish measure phrases behave in the same way as indefinites when they interact with a distributive universal quantifier (i–ii). Note also that, in Spanish, measure phrases do not multiply when they are under the scope of a pluractional operator such as the frequentative periphrasis andar + gerund (iii) (Laca, 2006). This is another context in which measure phrases and indefinites align (iii–iv). Hence, we argue that cn measure phrases may behave in the same way as the Spanish ones, provided that the behavior of the latter can be extrapolated cross-linguistically.

(i) [Cada mes]KEY explotan [dos kilotones de dinamita]SHARE

‘Every month two kilotons of TNT explode.’

Natural reading: Different portions of 2 kilotons TNT explode every month.

∀e [1–MONTH(e) → ∃y [TNT(y) & 2-KT(y) & EXPLODE(y, e)]]

Non-sensical-reading: The same portion of 2 kilotons of TNT explode every month.

∃y [TNT(y) & 2-KT(y) & ∀e [1–MONTH(e) → EXPLODE(y, e)]]

(ii) [Cada mes]KEY explotan [dos bombas]SHARE

‘Every month two bombs explode.’

Natural reading: Two different bombs every month.

∀e [1–MONTH(e) → ∃y [BOMB(y) & 2-ATOM(y) & EXPLODE(y, e)]]

Non-sensical-reading: The same two bombs explode every month.

∃y [BOMB(y) & 2-ATOM(y) & ∀e [1–MONTH(x) & EXPLODE(y, e)]]

(ii) #Andan explotando dos kilotones de dinamita.

Non-sensical reading: ‘Two kilotons of TNT keep exploding.’

∃E [∀e’ [e’< E → ∃y [TNT(y) & 2-KT(y) & EXPLODE(y, e’)]]]

(iv) #Andan explotando dos bombas.

Non-sensical reading: ‘Two bombs keep exploding.’

∃E [∀e’ [e’< E → ∃y [BOMB(y) & 2-ATOM(y) & EXPLODE(y, e’)]]]

Please refer to footnote 14 for the information added to the paper with respect to this comment.

3) It is not clear to me why 17 is a spatial - as it is cohesion, not the space allocated that defines the group?

Reply. The reviewer is right. We have added a footnote to acknowledge this possibility. See footnote 7: “As one reviewer points out, it may be the case that example (18) is not spatially distributive, but simply cohesion between the singularities that comprise a plurality of five, six or ten women. However, it may also be true that cohesion results from the spatial arrangement between these pluralities, i.e. from the fact that they fill a different space from each other. We do not have enough evidence to argue against or in favour of one of these hypotheses.”

4) There are some typos (y in an author pair; two eagles as translation of sth glossed tree in 10)

Reply: These mistakes have been corrected.

5) Ex 13 is not a reduplication of a complex root but a reduplication of each of its components (that do seem to be roots) a reduplication of both would be five-four -five-four not five five four four.

Reply: The reviewer is correct. We have adapted the wording accordingly: “For example, in (13), the reduplication operation can either separately copy phonological material from each of the simple roots of the compound (13a) or from the compound as a whole (13b).”

6) In the discussion on p. 14 of the phonology it is not explained what is transcribed from the manuscripts nor what conventions the manuscripts follow.

Reply: In most sources, long vowels and the glottal stop are not represented orthographically. Thus, we do not have sufficient evidence to determine whether the reduplication pattern is (C)Vː or (C)Vʔ. Besides, as we mentioned in our paper, in cn there is no agreement on whether reduplicated numerals can exhibit both patterns. Hence, for simplicity, we have decided to transcribe all reduplicated numerals by means of the (C)Vː pattern. Please refer to footnote 6 for the information added to the paper with respect to this comment.to footnote 5 for the information added to the paper with respect to this

7) I don't find the discussion of 19 very clear. I think it is the inherent repetition in the use of day that has the effect that looks like distribution.

Reply: We are uncertain about what the reviewer finds unclear in example (19). In (19), the word for ‘day’ is used only once. Perhaps the reviewer suggests that entities such as days are inherently repeated one after another, which would explain the distributive meaning of (19). However, note that, if semilwit͡ɬ ‘one day’ is not reduplicated, the distributive meaning ‘every day/each day’ does not seem to arise. Instead, we only get the reading ‘one day’ or ‘a whole day’ for semilwit͡ɬ ‘one day’.

Reviewer 3

1) The Null Distributive Operator: The primary concern involves the potential for overgeneration. As the analysis relies on the insertion of a null distributive operator, it is important to specify what constraints govern its distribution. Under what syntactic or semantic conditions is this operator licensed? Does it require the presence of specific plural DPs or VPs? Clarifying these constraints (for instance, in relation to example 1) would strengthen the predictive power of the model.

Reply: The reviewer is right. Note that, for the purposes of this paper, we assume that the null distributive operator requires the distributive key to be a plurality (a sum of individuals or events), as Oh (2001: p. 340; 2006: p. 51) suggests. Please refer to footnote 16 for the information added to the paper with respect to this comment.

2) Encoding the Distributional Requirement: How exactly is the requirement that reduplicated numerals in cn occur within the scope of a distributive operator encoded? Is this envisioned as a formal feature or a specific stipulation within the lexical entry of the numeral?

Reply: This is an important question. To answer it, we adopt the analysis of Rushiti & Dobrovie-Sorin (2024) for distributive numerals in Albanian. According to these authors, the distributional requirement of a distributive numeral is encoded by an uninterpretable syntactic feature [udist] that is checked against the interpretable feature [idist] carried by a distributive operator. For this agreement operation to take place, the distributive operator must c-command the distributive numeral and be local to it (i.e. the upward agree operation is clause-bounded). Please refer to footnote 17 for the information added to the paper with respect to this comment.

3) Lexical Relations: It would be helpful to explicitly state the semantic relationship between reduplicated and non-reduplicated numerals. Are their lexical entries considered identical, with the exception of the requirement regarding the null operator, or is there a deeper semantic distinction?

Reply: This is another helpful and very relevant comment. To address it, we once again adopt the analysis of Rushiti & Dobrovie-Sorin (2024) for distributive numerals in Albanian. According to these authors, distributive numerals carry an uninterpretable syntactic feature [udist] that must be checked against an interpretable syntactic feature [idist]. By contrast, non-distributive numerals do not carry the [udist] feature. Here we assume the same for cn reduplicated and non-reduplicated numerals: reduplicated numerals carry the [udist] syntactic feature, whereas non-reduplicated numerals do not. We also assume that, in cn, reduplicated numerals carry a lexically encoded covariation requirement. Non-reduplicated numerals do not impose such a restriction. Please refer to footnotes 17 and 19 for the information added to the paper with respect to this comment.

4) Formatting of Glosses: The four-line glosses are currently somewhat difficult to navigate. Unless the phonetic transcription is central to the argument, the authors might consider adopting a standard three-line format to improve readability.

Reply: We are aware that four-line glosses may be more difficult to follow than standard three-line glosses. However, since our data are drawn from historical records, we consider the phonetic representation to provide useful information about the language. For instance, in most sources, long vowels and glottal stops are not usually represented in the orthographic transcription. Nonetheless, in cn, reduplication produces a long vowel or a vowel + glottal stop reduplicant. This information would be lost if we omitted the phonetic transcription of the examples. Therefore, even if we acknowledge the downside of four-line glosses, in the benefit of our analysis, we prefer to keep the phonetic representation.

5) Section 4: While informative, Section 4 (“Reduplicated Numerals in cn”') feels disproportionately long relative to the core argument. To maintain the paper’s momentum, the authors might consider compressing this section, moving specific data points to an appendix, or relegating secondary details to footnotes.

Reply: We acknowledge that Section 4 might be long. However, in our view, it contains crucial information for interpreting the data. After a careful review of this section, we consider that all the information in this section is relevant and that relegating parts of it to the Appendices of footnotes would be detrimental to the full comprehension of the data and the analysis.

Review 4

1) Section 6.2 should be expanded to provide a clearer explanation of what the analysis entails and why there seems to be special cases of share and key conflating.

Reply: The reviewer is right. Here we note just some of the consequences of our analysis. Note, however, that a full discussion of them would require much more space, which may exceed the mainly descriptive purposes of this paper.

We have shown that an analysis under which runs are standard distributive numerals that mark the distribution share results in the right truth conditions for different types of sentences containing a run, such as (38–41). Our analysis of (38–41) follows from Gil’s (1982, pp. 123) grammatical relations hierarchy. According to Gil (1982, pp. 123), distributive relations are constrained by the grammatical function of the constituents among which distribution applies: distribution of a constituent A over a constituent B is strongly disfavoured if A is hierarchically higher than B. In the absence of a higher constituent, the highest constituent can only distribute phrasally over its own head, a cover or overt classifier that denotes units of some kind (Gil, 1982, pp. 107–117). This is what happens in cases such as (40–42). In (42), the subject NP is higher than the object NP, which rules out the distribution of in seːsentet͡ɬ kwawt͡ɬi ‘the one-one eagle’ over oːoːntet͡ɬ in kwaːwtet͡ɬ ‘two-two that are eggs’, and forces the distribution of the cardinality predicate seːsentet͡ɬ ‘to measure one atom’ among the denotation of the NP classifier head, individual units that are eagles. The object NP oːoːntet͡ɬ in kwaːwtet͡ɬ also distributes over these individual eagle units, which measure one atom. Thus, the way the meaning of (42) is composed gives the false impression of seːsentet͡ɬ marking the key over which the share oːoːntet͡ɬ in kwaːwtet͡ɬ distributes. Based on this, we suggest that the cases in which key and share seem to conflate are those in which the share can only distribute phrasally over its classifier head, as in (42). Of course, more data is needed to properly test and assess this hypothesis. As for (40–41), note also that our analysis would imply that seːsemilwit͡ɬ in (39–40) is structurally higher than any other constituent and can only distribute over its event-denoting classifier head.

Note also that, in cases such as (40–42), we get the same truth conditions regardless of whether we interpret the run as a key marker or as a share marker that phrasally distributes over its head. However, the semantic composition by which we arrive at those truth conditions differs depending on the way we interpret the runs in (40–42). Here, we have argued for an analysis in which cn runs are always share markers, since distributive relations are not multiplied when several runs occur in the same clause. Nonetheless, we recognize that speakers may have found sentences such as (40–42) ambiguous: in (40–42), runs could be interpreted either as key or share markers and, in both cases, the same truth conditions would be obtained. Note that these types of structures, in which two different semantic computations have the same truth conditions, are especially interesting in diachronic terms, since they could eventually lead to a semantic change, namely the reanalysis of distributive quantifiers to universal quantifiers (see Eckardt (2006) for the conditions under which semantic reanalysis occurs). We leave for further research the fine-grained analysis of the mechanisms behind this semantic change; still, we highlight the relevance of cn runs as an ideal environment for advancing in the understanding of the distributiveQ-to-universalQ grammaticalization path. Please refer to the last paragraph of section §6.2 for the information added to the paper with respect to this comment.

2) Example (10) in page 12 glosses kwa:w as ‘tree’, but the translation is “two eagles”. Is this a mistake?

Reply: Yes, the review is right. This mistake has been corrected.

3) The first paragraph in page 14 is not very clear. It calls example (12b) as a case where a reduplicated numeral begins with a consonant, but that example is of oome which begins with a vowel, an then calls (11b) as a case where the numeral begins with a vowel, but (11b) is an example of naui which begins with a consonant. The paragraph focuses maybe too much on the two different analyses of the syllable in Classical Nahuatl, but it only provides examples of one of them, the one the authors adopt. It is not clear what this discussion brings to the text. Maybe it would be better to only present the analysis that they favour without expanding too much on other possibilities.

Reply: The reviewer is right. The cross-references to the examples were wrong. This has been corrected in the manuscript. As for the relevance of this information, we consider that it is important to acknowledge the different proposals regarding the shape of the reduplicant in cn reduplicated numerales, since some authors have argued that there may be a semantic contrast between (C)Vː and (C)Vʔ reduplicants.

4) In examples (21) and (22), cp and icp are glosses of completive and incompletive aspect, respectively. They are common glosses in Mixe-Zoquean languages.

Reply: The glosses have been added to the Abbreviations List.

Reviewer 5

1) It will be extremely valuable if the authors consider some didactic techniques: to explain in a clear way their relevant categories and terms; describe what they mean, and to recall the readers what they understand by those technical terms. This will help the readers to have a better understanding of the topic and the analysis.

Reply: We are aware that our paper requires readers to have some previous background on formal semantics. However, we believe that all the relevant categories are explained in non-technical terms. Consider, for instance, the explanations of components in Cable’s definition of distributive numerals in Section 3. Nonetheless, if the reviewer considers that we should further explain another term, we are, of course, happy to do so.

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ROUND 2

2026-06-23

We would like to thank the reviewers for their further comments on our manuscript. Next, we will explain how we have solved the issues raised in this second round of reviews. Again, we have numbered each reviewer's comments to facilitate the revision. After each comment, we have added our reply. It is important to note that in the first round of reviews, Dr. Champollion's comments did not reach us and therefore were not addressed. In the final version of our paper, we have included the relevant amendments to tackle the issues he raised.

Reviewer: Lucas Champollion

Reply: The two issues raised in this comment are addressed in (II) and (III), respectively. See below.

The authors' primary argument for rejecting a universal quantifier analysis is the "multiplication of distributive relations" (nesting). However, Champollion (2016a) explicitly notes that his event-based operators are "stackable" because they maintain a constant input/output type of ⟨v,t⟩. When multiple distributors occur in Champollion's analysis, they combine via intersection (&) rather than functional nesting (∀...∀). To illustrate how this analysis might proceed, start with an even further simplified example:

(31a) (...) k-in-makak in seːsen-meʔ (...) seːsem-aːkal-li t͡ɬaoːlli

(...) O.3-O.PL-give.PFVO DET DISTR~one-PL (...) DISTR~one-canoe-A maize.ABS

"(...) He gave every person one canoe each of maize"

The basic idea of this analysis is that a noun phrase containing a distributive share marker is indexed to a thematic role. In Champollion 2016a, that thematic role is always distinct from the noun phrase in question, but it doesn't need to be. In fact, this is a possibility that Champollion does not exploit.

In (31a), the Key is the people who are beneficiaries of the giving events. Assume that in the theme noun phrase "DISTR~one-canoe-A maize.ABS" "one canoe each_ben of maize", the RUN is indexed to the beneficiary role. Then the theme noun phrase is analyzed as the property of being an event E that is made up of parts e' such that the beneficiary of each e' is an atom and the theme of each e' is a canoe of maize.

As for the beneficiary noun phrase "DET DISTR~one-PL" "every person", assume that the RUN in it is indexed to the beneficiary role. Then the beneficiary noun phrase is analyzed as the property of being an event E that is made up of parts e' such that the beneficiary of each e' is an atom and a person.

The subject is given a straightforward analysis in terms of an agent. The verb "gave" denotes the property of being a possibly plural giving event E. The meaning of (31a) is obtained by conjoining these four properties and then existentially quantifying over the conjunction:

∃E. agent(E)= he & give(E) & E ∈ *λe'[atom(ben(e')) & person(ben(e'))] & E ∈ *λe'[atom(ben(e')) & maize(th(e'))]

Crucially, neither of the RUNs takes scope over the other. In this, they mirror adnominal "each" (see discussion of this point on page 16:27 in Champollion 2016a). This prevents the problem the authors describe in the analysis they reject.

I expect that this analysis can be extended to (31a) itself by adopting an analysis of "and" in terms of pointwise fusion of events as proposed by Lasersohn 1995.

If this alternative analysis is correct, the authors do not need to posit a silent distributivity

operator, and they can rely on an off-the-shelf analysis (and extend it in a crucial way, by taking the thematic role of a distributive share marker to be coindexable with the thematic role of its own host noun phrase).

The main point of the authors, namely that RUNs in Nahuatl are share markers rather than key markers, is unaffected by this move; thus it should be seen as a friendly amendment. Whether the authors choose to adopt this analysis or not, I recommend that they discuss both their own analysis and the one sketched here as alternative options, so that it becomes clearer to the reader to what extent the empirical facts about Classical Nahuatl constrain the theoretical space of analysis.

Reply: Champollion’s (2016a,b) formal proposal

The purpose of our formal analysis is simply to show how we could account for all cases of RUNs by assuming that they are share markers, as expected from a standard distributive numeral. We recognize that the analysis we sketch in our paper could be casted on Champollion’s (2016a: 2016b) terms. In fact, as the reviewer argues, Champollion’s analysis has the advantage of preventing the nesting of multiple distributive relations, which could fit well with the CN data. For this reason, we have modified the wording of section 5 to clearly state that our formal proposal is not conclusive and to acknowledge that Champollion’s (2016a; 2016b) proposal could also work for our data.

Nonetheless, we consider that there is some evidence that favors a syntactic approach to reduplication (like the one we adopt) over the analysis that the reviewer, building on Champollion (2016a; 2016b), suggests for reduplicative morphemes. In short, the reviewer proposes that reduplicant morphemes are overt realizations of a distributive operator taking scope only over their host noun phrase. However, this analysis does not seem to account for the meaning of distributive complex cardinals formed by addition. In CN distributive complex cardinals, all simple cardinals can be reduplicated, as in (1), or only one of them, as in (2).

(1) chinamitl v tetl opa temi apilco hohopohuallhuiac ypan matlatlacmatl

[t͡ʃinaːmi-t͡ɬ maːk^wiːl-te-t͡ɬ]_key oːmpa temi aːpilko

field-abs five-cl-abs there be.together Apilco

[o:~o:m-poːwal-weyaːk iː-pan maʔ~t͡ɬaː~t͡ɬak-ma-t͡ɬ]_share

distr~two-twenty-long posr.3sg-on ten~distr-ten-matl-abs

‘Five fields are together at Apilco, each one fifty matl long’

(Cline; León Portilla, 1984, p. 260, doc. 75, f. 101r)

(2) nochinannemac nauhtetl yn ceçenpohualli ypan macuilmatl

[no-t͡ʃinaːn-nemak naːw-te-t͡ɬ] _key in [seː~sem-poːwal-li iː-pan

posr.1sg-inherited.field four-cl-abs det distr~one-twenty-abs posr.3sg-on

maːk^wiːl-ma-t͡ɬ]_share

five-matl-abs

‘My inherited fields are four, each one twenty five matl long’

(Cline; León Portilla, 1984, p. 198, doc. 55, f. 82r)

Thus, following the reviewer’s suggestion, in (1) there would be two distributive operators, one scoping over the phrase ‘40 (matl) wide’ and the other one scoping over the phrase ‘10 matl’. Nonetheless, this approach might be problematic for the following reasons. First, if the simple numerals in (1) composed by intersection or predicate modification as in Champollion (2016a; 2016b), we would not get the right truth conditions for (1): an individual cannot simultaneously measure 40 matl and 10 matl (4).

(3) ⟦o:o:mpoːwalwejaːk iːpan maʔt͡ɬaːt͡ɬakmat͡ɬ⟧ = λx. x ∈ *λx’ (40-matl(length(x’))) & x ∈ *λy’ (10-matl(length(y’)))

(4) ⟦t͡ʃinaːmit͡ɬ maːk^wiːltet͡ɬ o:o:mpoːwalwejaːk iːpan maʔt͡ɬaːt͡ɬakmat͡ɬ⟧ = ∃x 5-atom(x) & *field(x) & x ∈ *λx’ (40-matl(length(x’))) & x ∈ *λy’ (10-matl(length(y’)))

Even if we assumed some kind of non-Boolean conjunction for (1), we still would not get the right truth-conditions if there were only two distributive operators, each one scoping only over one of the summands, respectively (5–6). This is because the intersection between ⟦5⟧ and ⟦6⟧ is empty: there are no entities exclusively made up of five atoms that also include sums.

(5) ⟦o:o:mpoːwalwejaːk iːpan maʔt͡ɬaːt͡ɬakmat͡ɬ⟧ = λx. ∃a∃b (a ∈ *λa’ (40-matl(length(a’)) & b ∈ *λy’ 10-matl(length(y’))) & x = a∪b)

(6) ⟦t͡ʃinaːmit͡ɬ maːk^wiːltet͡ɬ o:o:mpoːwalwejaːk iːpan maʔt͡ɬaːt͡ɬakmat͡ɬ⟧ = ∃x 5-atom(x) & *field(x) & ∃a∃b (a ∈ *λa’ (40-matl(length(a’)) & b ∈ *λy’ 10-matl(length(y’))) & x = a∪b)

Second, note that to get the right interpretation of the distributive complex numeral in (1), we only need one distributive operator scoping over the whole complex numeral (7).

(7) ⟦t͡ʃinaːmit͡ɬ maːk^wiːltet͡ɬ o:o:mpoːwalwejaːk iːpan maʔt͡ɬaːt͡ɬakmat͡ɬ⟧ = ∃x 5-atom(x) & *field(x) & x ∈ *λx’ (50-matl(length(x’)))

For this reason, it seems to us that the simplest way of accounting for the meaning of (1) only requires a single distributive operator that takes scope over the whole complex numeral. Thus, it appears that, in CN, sometimes there is not a one-to-one correspondence between distributive operators and reduplicant morphemes, and a single distributive operator can be marked by multiple reduplicant morphemes. If this is correct, then the occurrence of multiple reduplicant morphemes in CN could be explained by some kind of agreement or licensing relation between a single distributive operator and the reduplicated numerals. In sum, further research on reduplicated complex numerals is needed to assess if Champollion’s (2016a; 2016b) proposal can fully explain the CN data in (1–2). This is outside the scope of this specific paper, but we will address it in future research on distributive numerals in CN.

Reply: The status of the null distributive operator

It is true that the examples we provide in the paper to argue in favor of the existence of a null distributive operator are all cases in which the share is distributed over non-atomic entities. Thus, those examples would only contribute evidence for a Part-like operator. However, in CN, covert distributivity can also be over atomic individuals, as shown in the following example, which we have also added to the final version of our manuscript (see example 37). In (8), individual portions of amaranth seed dough measuring four mouthfuls are distributed over the atomic members of a plurality made up of slaves.

(8) in quinqualtiaia nauhcamatl

in k-in-k^waltiaː-ja-ʔ naːw-kama-t͡ɬ

det o.3-o.pl-eat.caus-pst.ipfv-s.pl four-mouthful-abs

‘What they (the priests) fed them (the slaves) was four mouthfuls (of amaranth seed dough with honey) each’

∃X slaves(X) & ∀x [x < X & atom(x) → ∃y [4-mouthful(y) & fed(ɩpriests(z), y, x)]]

(Sahagún, 1959, p. 63, f. 45r)

Now, following Champollion (2016b), CN reduplicated numerals seem to overtly code only the Part operator, since they can also mark a share that distributes over non-atomic entities, as in (9). In (9), entities whose monetary value equals 2 tomines are distributed over non-atomic occasions separated by periods of 180 days. Thus, it seems that in CN it is only necessary to assume the existence of the Part operator. Of course, further work is needed to substantiate this claim.

We have added all these clarifications on pages 33–34 (§ 5.1) of the final version of the manuscript.

(9) monequi chicuhnauhpovalticca oume domin macoz yn donjuán

mo-neki [t͡ʃik^wnaːw-poːwal-ti-ka]_key [oː~oːme tomin]_share makoː-s

refl.2/3-want nine-twenty-lig-instr distr~two tomin give.pasiv-fut

in don Xuan

det hon Juan

‘It is required that, every 180 days, two tomines be given to Don Juan’

(Anderson; Berdan; Lockhart., 1976, p. 150, doc. 26)

Reviewer: Fernando Chapa

1) The formal interpretation of the examples is a great addition, but I would recommend adding an atomic restriction in the formulas. Let us take (17) as an example. The formula provided for this sentence is the following: ∀x [x < Ana⊕Francisca → ∃y [LAND(y) & 10-MATL(y) & TAKE(x, y)]]. In this particular case it might seem trivial, because there are only two atoms in the sum, so every x that is a proper part of the sum “Ana⊕Francisca” is atomic. Nevertheless, if the sum consisted of three or more atoms, the proper parts of the sum would include other non-atomic sums e.g. “Ana⊕Francisca⊕Beatriz” by stating that a property applies to ∀x [x < Ana⊕Francisca⊕ Beatriz] the property would apply to Ana, Francisca, Beatriz, Ana⊕Francisca, Francisca⊕Beatriz and Ana⊕Beatriz. This entails nonatomic distributivity, which some authors claim is possible, but I’m not sure that this is what you want (if it is it should be explicitly stated). The problem is easily solvable adding a condition of atomicity to all the formulas.

Reply: We have added the atomic restriction to the relevant examples (for instance, to examples (17) and (25). Note also that we have replaced the parthood relation (<) with the memberhood (∈) relation to ensure that we do not get contradictory propositions in some cases. For instance, we have previously suggested the following formalization for the truth-conditions of example (43):

There is a plurality X of eagles, such that, for every x, if x is a part of X, then the cardinality of x is equal to 1 and there is a plurality y of two eggs and x laid y.

∃X ∀x [x < X → [eagle (x) & 1-atom(x) & ∃y [egg(y) & 2-atom(y) & laid(x, y)]]]

However, these truth-conditions are never fullfilled, since atomicity cannot hold of every proper part of a plurality X (for instance, given a plurality X = {{a}, {b}, {c}}, atomicity would not hold of {{a}, {b}}, which is a part of X). To avoid this problem, we have resorted to the memberhood relation as in the following truth-conditions. Note that the existence of a plurality whose members are all atomic is not a contradictory statement (for instance, a plurality X = {{a}, {b}, {c}} would satisfy atomicity).

There is a plurality X of eagles, such that, for every x, if x is a member of X, then the cardinality of x is equal to 1 and there is a plurality y of two eggs and x laid y.

∃X ∀x [x ∈ X → [eagle (x) & 1-atom(x) & ∃y [egg(y) & 2-atom(y) & laid(x, y)]]]

2) In the first paragraph page 17 it says: “Note that in (17), the distributive key, maʔt͡ɬaːt͡ɬakmat͡ɬ, corresponds to a participant key”. Shouldn’t it be that maʔt͡ɬaːt͡ɬakmat͡ɬ is a distributive share?

Reply: The reviewer is right. There was a mistake: maʔt͡ɬaːt͡ɬakmat͡ɬ is the share. In the text, we are referring to the key which in (17) corresponds to Ana and Francsica. We have corrected the text as follows: “Note that in (17), the distributive key (that is, Ana and Francisca) corresponds to a participant key.”

References

ANDERSON, A. J. O.; BERDAN, F.; LOCKHART, J. Beyond the codices: The Nahua view of Colonial Mexico. Berkeley: University of California Press, 1976.

CHAMPOLLION, L. Covert distributivity in algebraic event semantics. Semantics and Pragmatics, v. 9, n. 15, p. 1–65, 2016a. https://doi.org/10.3765/sp.9.15. Accessed on: 10 November 2025.

CHAMPOLLION, L. Overt distributivity in algebraic event semantics. Semantics and Pragmatics, v. 9, n. 16, p. 1–65, 2016b. https://doi.org/10.3765/sp.9.16. Accessed on: 10 November 2025.

CLINE, S. L.; LEÓN PORTILLA, M. The testaments of Culhuacan. Los Angeles: University of California Press, 1984.

SAHAGÚN, B. de. General History of the Things of New Spain. Book 9: The Merchants. Ed. A. J. O.

Anderson, C. E. Dibble. Santa Fe: The School of American Research/the University of Utah, 1959.