2.1 Semantic notions of number

Starting with Link (1983), work on the semantics of plurality has been dominated by the idea that individuals form a complete join semi-lattice structure. These lattices combine individual entities into sums which can then be predicated upon. Predicates are able to apply to the summation of individual entities as well as to parts of greater wholes. Link provided the mechanisms for probing into these lattices, and he posited sum operations to combine individuals into sums. The plural groups can be ordered into sets or subsets in the lattice.

The mass/count distinction (MCD) refers to the properties and referents of certain nouns and their relationship with the morphological forms available for the noun. Mass nouns (e.g., English water, tea, or stone) are generally unable to be pluralized and refer to substances in English. Meanwhile, count nouns (e.g., dogs, humans, and cards) are able to be pluralized and generally refer to entities that contain atoms. Applying lattices to the distinction between mass and count nouns has also proven fruitful. Link (1983) suggests that the difference between mass nouns and countable plural nouns has to do with the presence of atoms in the lattices of plural count nouns. This has been expanded on in the works of Krifka (1989, 1992). The main properties relevant here are cumulativity and quantization.

Following Krifka (1989), a predicate is cumulative if and only if, whenever the predicate applies to two entities x and y, the predicate applies to the join of those entities (i.e., the set that contains both of the entities). Conversely, a predicate is quantized if and only if for any two entities x and y, if the predicate applies to x and to y, then y cannot be a part of x. Another relevant property is divisivity. A predicate is divisive if and only if, when the predicate applies to any entity, the predicate also applies to any part of that entity. Table 1 displays how these properties relate to different types of nouns.

For example, a mass noun like water has cumulative reference because if you have two separate cups of water, they are still water when one pours them into a bowl together. However, water is divisive and not quantized because if you take some water from the bowl, the part that is taken away is still water. In other words, any entity to which the predicate water applies contains a part to which the predicate also applies, and thus, water is not quantized. Conversely, a count noun like cat is quantized because the cat’s paw is no longer a cat; further, the singular count noun is not cumulative because when you put cat a and cat b together, their join does not fall into the extension of the predicate cat. Instead, count nouns must use plural forms to refer to the join of entities. The plural version of count nouns, like cats, is not quantized because for any entity x such that the predicate cats applies to x, then there is a y such that y is a part of x and cats does not apply to y. In this example, the y would be any atom of cats (i.e., a cat). However, the plural is not divisive because there does exist a part of the set of cats that is no longer the plural noun cats but is instead a singular cat. Looking at Table 1, we can see that no quantized noun is divisive as the two properties are logically incompatible. Quantization relates to semantic singularity. Conversely, divisivity and cumulativity relate to properties of semantic plurality. These properties are important to how scholars have understood the semantic differences between mass and count nouns. Individuation is related to the property of divisivity that underpins the distinction between mass nouns and plural count nouns. Grimm defines individuation as the “propensity of an entity to be construed as an independent individual” (2018: 528).

2.2 Individuation and Aspect

Moving on from nouns, Krifka (1989, 1992) also proposed a relationship between the MCD and verbal aspect. Scholars have noted the similarity between the distinction between mass and count on one hand and the distinction between atelicity and telicity on the other. Telic verbs are those which refer to events that are conceived of as being finite and have clear endpoints, while atelic verbs refer to events that are conceived of as nonfinite. Krifka (1989) provides a framework for unifying these two phenomena. He uses the properties of quantization, divisivity, and cumulativity to characterize nominal reference and temporal reference. In order to do this, he also relies on measure functions, which map entities to numbers. Following Davidson (1967), Krifka assumes that events are entities which are ordered in lattices similar to those of objects. With events structured in the same way as objects, Krifka can extend the properties which apply to predicates over individuals to predicates over events. In this way, telic predicates are quantized event predicates, while atelic predicates can be understood as cumulative event predicates.

For example, an event described by a telic predicate like the cat slept is quantized because if we have two events for which that sentence were true, then the events must be separate. In this example, we know that for the telic predicate to be true, the whole act of sleeping must be completed, and thus the cat must have awoken. Therefore, if two events are described using the same telic predicate, they require the cat to be awake at some point between them. Conversely, if we were to describe two events using an atelic predicate, the cat was sleeping, the join of the events can also be described by the same predicate. If someone walks out of a room and affirms that the cat is sleeping, and then an hour later, someone else affirms the same thing, there is no requirement for the cat to be awake between them. Further, atelic predicates can be shown to be divisive; if a cat was sleeping for an hour, we know that for any subdivision of that time, such as for thirty minutes, the cat was also sleeping.

However, Krifka (1989: 91–92) noted another relationship between aspectual semantics and nominal semantics. Through thematic relationships, the properties of nominal reference can affect temporal reference; see Krifka’s (1989) examples reproduced in (5).

Both of these descriptions can be used to refer to the same event of drinking the same wine; however, in (5a), wine is a cumulative predicate meaning that the sum of two quantities of wine is still wine. This cumulativity is transferred to the event predicate drank wine, which can refer to both the act of drinking the whole glass of wine and any event of drinking part of the glass of wine. Thus, drank wine is atelic. Conversely, a glass of wine is quantized, as no proper part of a glass of wine is still a glass of wine. Similarly, the quantization is transferred to the event predicate drank a glass of wine because it cannot be the case that drinking part of a glass of wine is an event of drinking a glass of wine.

In order to capture these meanings, Krifka proposes a series of properties which connect two-place relations with the properties of single-place predicates. Notably, he proposes summativity as the corollary of cumulativity for two-placed relations. He also defines properties of relations in order to specify things such as the uniqueness of events and the uniqueness of objects, which are the properties which map part of the event to part of the object and vice versa. With these tools, he is able to prove the relationship between the properties of the entity and the properties of the event.

Krifka’s (1989) proposal looks at why the nominal affects event interpretation; however, given the way Krifka sets up this system, there is no reason that the event’s interpretation should not be able to affect the interpretation of the noun, should it be ambiguous. Borer (2005) provides a Czech example where this is the case (6).

In (6a), Borer claims that the prefix na- assigns telicity to the verb. In doing this, it provides a quantity to the nominal interpretation. Contrast this with (6b), in which the prefix u- also provides a telicity value to the verbal phrase but affects nominal interpretation differently.

2.3 Individuation and Arabic

Returning to Arabic agreement, DAgr and SAgr provide different meanings to the sentence. However, the exact semantic contribution is not known. The semantic relationship between verbal aspect and nominal individuation suggests two places that could cause the difference in meaning found under DAgr. The first option is that DAgr is determined by nominal features. Under this interpretation, any aspect-like effects arise as a consequence of the specification of properties of the nominal predicate, and the subject of a sentence with DAgr would carry some covert “collective” feature triggering the agreement. Returning to the example of dogs walking down the street, the noun kilaab, ‘dogs,’ would carry a feature representing that it is non-individuated (i.e., it is a cumulative predicate). This non-individuation feature would then trigger agreement like any number. This option is closest to Kramer & Winchester’s (2017) proposal.

The second option is that DAgr is based on aspectual factors and that the aspect affects the interpretation of the noun. In this case, DAgr would function more like aspectual marking than traditional subject-verb agreement. If the event raaħ-at, ‘walk,’ is a cumulative event predicate, then cumulativity can be transferred to the nominal predicate, kilaab, parallel to what Borer (2005) suggests for Czech.

I argue that Kramer & Winchester’s (2017) herd/clump interpretation has a cumulative interpretation. Kramer & Winchester provide the following Hijazi examples³ to explain the meaning of herd/clump interpretation.

The example in (7a) uses DAgr, while (7b) uses SAgr. According to Kramer & Winchester, if there was only one pan of falafel, speakers preferred to use DAgr. If there are three pans of falafel which burned, then speakers preferred to use SAgr as in (7b), if it was intended that all of the pans of falafel were burned. Following these descriptions, if we have an event with herd/clump interpretation as is described in (7a), subsets of that event could still be considered al-falaafil ʔanħarag-at. Using Krifka’s criteria, this would mean that herd/clump interpretation is cumulative because al-falaafil ʔanħarag-at can refer to both all falafels burning and subsets of that event just like drink wine. Conversely, the event without herd/clump interpretation described in (7b) seems to be non-quantized because a subset of the event cannot necessarily be referred to as al-falaafil ʔanħarag-uu.

Kramer & Winchester (2017) situate a herd/clump head beneath the DP but above a numP. While this head may carry a singular feature, Kramer & Winchester only require that the head carry a feminine feature because the number features can arise from defaulting since the singular is the default, unlike the feminine feature. This head carries a non-human feature that selects a plural noun and, in turn, stops non-human nouns from participating in plural agreement. Crucially, this proposal focuses on the noun as the locus of DAgr. To put it another way, Kramer & Winchester argue that the features of the noun cause the difference in the agreement morphology on the verb.⁴

Fassi Fehri (2018) provides an analysis of the distribution of nominal morphology in Arabic which captures the herd/clump interpretation. Under his analysis, Arabic’s number system exists as a conflation of the syntactic features of atomicity and unicity. Kramer & Winchester’s (2017) herd/clump head is similar to the unicity feature Fassi Fehri (2018) proposes in the nominal domain. However, Fassi Fehri provides environments that cause these features to change, namely counting environments. When counting in many Arabic varieties, including Najdi Arabic, a morphological change occurs in the noun from low numbers to high numbers. If the number is lower than ten, the plural form of the noun is used; however, above ten, the noun appears in its singular form. Fassi Fehri claims that this change in noun form represents a change in the noun’s atomicity and unicity features. The concepts of both herd/clump and unicity refer to the representation of an entity as a single unit, and Kramer & Winchester show that in Saudi Arabic,⁵ this semantic information licenses different morphological forms of agreement. I will use non-individuation to describe both unicity and the herd/clump interpretation.

The historical development of Arabic also provides reasons to think that nominal features like non-individuation might determine DAgr. Bettega (2019) and Bettega & D’Anna (2022) have argued that gender agreement was affected by nominal individuation in earlier forms of Arabic. Under these analyses, when the noun was non-individuated, feminine agreement was more likely. Bettega (2019) claims that as modern Arabic dialects began losing feminine plural verb forms, dialects differed in how they handled individuation. Some varieties shifted the verbal marking of individuation to DAgr. Others lost individuation as a feature, merging it with the newly merged common plural.⁶

Both the nominal proposal and my verbal proposal result in the noun being interpreted collectively; however, they differ in where this information originates. The first option relies on the noun being collective covertly, and DAgr would agree with this covert feature. In the second option, DAgr limits aspectual reference, and this aspectual reference imposes a collective interpretation on the nominal argument.

3.1 Participants

Twenty-nine Najdi Arabic-speaking participants were recruited online through personal connections and advertisements on social networking sites, such as Reddit, Twitter, and WhatsApp. All participants were prescreened for dialect before beginning the experiment. All of the participants came from Saudi Arabia; however, in the demographic survey, participants provided more information on their location. Participants came from across the Najd,⁷ and using this information, I categorized participants into groups based on the sub-dialects reported in Ingham (1994).

Table 2 shows the number of participants in the final analysis. Participants are shown divided into three dialect groups based on the location of their upbringing and Ingham’s (1994) classification of Najdi dialect areas. My participants came from three dialect areas identified by Ingham: Central Najdi, Northern Najdi, and Mixed Northern-Central. Central Najdi speakers are the most represented population (n = 18), with participants coming mainly from Riyadh and surrounding areas. Mixed Northern-Central speakers (n = 8) came from Qasiim. The Northern Najdi speakers (n = 2) came from Hail. An additional participant did not provide any location information. Riyadh is the most populous area of the Najd today, so the dialect spoken there is well represented.

3.3 Results

I analyzed this data using a logistic mixed-effect regression model (Quené & van den Bergh 2008); however, since this is not binomial data, I had to create two models to interpret the data using a hierarchical comparison. The first model predicted the gender of the verb (masculine vs. feminine) with fixed effects of number of actors, nominal gender, and their interaction. Figure 2 shows the data analyzed in this model. The model further included random intercepts for both participants and noun and a random slope for condition on participant.⁸ The maximal random effect structure was attempted, but due to the lack of convergence, random slopes and intercepts were removed from the model by removing the lowest variance (Barr et al. 2013). Both categorical variables (number of actors and nominal gender) were dummy coded. The intercept reflects the reference levels of the number of actors, 25, and the nominal gender, feminine. Significance for simple effects and interactions were then calculated through log-likelihood comparisons. Table 3 provides a summary of the final model’s outcome.

Figure 2

The percentage of responses for both nominal genders and experimental conditions separated by gender

The model found a significant simple effect of nominal gender, suggesting that participants chose the verbal gender based on the gender of the noun. A significant simple effect of intercept suggests that speakers chose feminine forms more than masculine forms; however, this is unsurprising as there are two feminine options as opposed to one masculine option. The number of actors was not found to be significant in this model.

The second model only looked at the data for which participants selected a feminine form (i.e., when participants selected feminine singular or feminine plural forms). Figure 3 shows the data analyzed in this model. It predicted the grammatical number of the verb (singular vs. plural) with fixed effects of number of actors, nominal gender, and their interaction. The model further included random intercepts for both participants and verbs and a random slope for nominal gender on participant. The maximal random effect structure was attempted, but due to the lack of convergence, random slopes and intercepts were removed from the model by removing the lowest variance. Both categorical variables (number of actors and nominal gender) were dummy coded. The intercept reflects the reference levels of the number of actors, 25, and the nominal gender, feminine. Significance for simple effects and interactions were then calculated through log-likelihood comparisons. Table 4 provides a summary of the final model’s outcome.

Figure 3

The percentage of feminine responses for both nominal genders and experimental conditions separated by grammatical number.

The second model found a significant simple effect of nominal gender, suggesting that when participants selected a feminine verb form, they selected verb number based on the gender of the noun, with masculine nouns being more likely to take a feminine singular verb. This model did not have a significant intercept, suggesting that participants were not more likely to select the singular or plural form for feminine nouns. The number of actors was still not found to be significant in this model.

3.4 Discussion

In this experiment, participants selected the gender of the verb based on nominal gender. Participants were relatively more likely to pick the verb conjugated for masculine plural when presented with a masculine noun. Breaking apart the feminine verb forms into singular and plural, masculine nouns also drove participants to select feminine singular verbs over feminine plural ones.

Remember that Kramer & Winchester propose that their herd/clump head exists between the noun and the verb preventing agreement between the two. Crucially, the authors propose this positioning because “it is well-established that the highest-gender feature determines the gender feature of the whole nP” (2017, pg. 49). If this were the case, we would expect that the features of the noun (nominal number and nominal gender) would have no effect on the selection of the verbal number. Instead, the number condition would predict the verbal number. Again, this was not found; in fact, I find that nominal gender is accessed by speakers when selecting agreement. This finding raises the question of why nominal gender is affecting verbal number. I take this to be evidence that feminine plural agreement is dispreferred for masculine nouns. This awareness of nominal gender is a problem for Kramer & Winchester’s head because this head intercepts nominal features and blocks the gender feature of the noun.⁹

These results suggest that nominal gender is central to speakers’ selections of verb form. However, these results do not suggest a cause for the DAgr. In the following experiment, I approach the question of individuation in verbs from an aspectual direction. In experiment 2, I ask if the number of events being discussed can predict DAgr.

4.1 Participants

Twenty-five Najdi Arabic-speaking participants were recruited for an online experiment through personal connections and advertisements on social networking sites, such as Reddit, Twitter, and WhatsApp. All participants were prescreened to ensure that they were Najdi speakers before beginning the experiment. All of the participants came from Saudi Arabia; however, in the demographic survey, participants provided more information on the speakers’ location. Participants came from across the Najd, and using this information, we can place participants into groups based on the sub-dialects reported in Ingham (1994). No participant who participated in Experiment 1 also participated in Experiment 2.

Table 5 shows the number of participants in the final analysis. Participants are shown divided into three dialect groups based on the location of their upbringing and Ingham’s (1994) subdialect groups. As in Experiment 1, participants came from three of the four dialect areas identified by Ingham: Central Najdi, Northern Najdi, and Mixed Northern-Central. Central Najdi speakers are the most represented population (n = 14), with participants coming mainly from Riyadh and surrounding areas. Mixed Northern-Central speakers (n = 8) came from Qasiim. The Northern Najdi speaker (n = 1) came from Hail. Two participants did not provide any location information. As with Experiment 1, given the population distribution in the Najd, the prevalence of Central Najdi speakers is expected.

4.3 Results

I analyzed the data from Experiment 2 using a logistic mixed-effect regression model (Quené & van den Bergh 2008), using the same hierarchical comparison used in Experiment 1. The first model predicted the gender of the verb (masculine vs. feminine) with fixed effects of number of events, nominal gender, and their interaction. Figure 5 shows the data analyzed in this model. The model further included random intercepts for both participants and noun and a random slope for nominal gender on participant. The maximal random effect structure was attempted, but due to the lack of convergence, random slopes and intercepts were removed from the model by removing the lowest variance (Barr et al. 2013). Both categorical variables (number of events and nominal gender) were dummy coded. The intercept reflects the reference levels of the number of events, multiple, and the nominal gender, feminine. Significance for simple effects and interactions were then calculated through log-likelihood comparisons. Table 6 provides a summary of the final model’s outcome.

Figure 5

The percentage of responses for both nominal genders and experimental conditions separated by gender.

The model found no significant simple effects. A significant simple effect of intercept suggests that speakers chose feminine forms more than masculine forms; however, this is unsurprising as there are two feminine options as opposed to one masculine option. The number of events was not found to be significant in this model.

As with Experiment 1, the second model only looked at responses in which participants selected a sentence with a feminine form. Figure 6 shows the data analyzed in this model. It predicted the grammatical number of the verb (singular vs. plural) with fixed effects of number of events, nominal gender, and their interaction. The model further included random intercepts for participants. Table 7 provides a summary of the final model’s outcome.

Figure 6

The percentage of feminine responses for both nominal genders and experimental conditions separated by grammatical number

The second model found a significant simple effect of nominal gender, suggesting that when participants selected a feminine form, they selected number on the verb based on the gender of the noun, with masculine nouns being more likely to a take feminine singular verb. A significant simple effect of the number of events was also found, by which participants used the feminine singular more in the multiple event condition than in the singular event condition. This model did not have a significant intercept, suggesting that for feminine nouns, participants were not more likely to select the singular or plural form of the verb.

4.4 Discussion

In Experiment 2, the results suggest that there is no significant predictor of the gender of the verb. This finding is in direct contrast with the findings of Experiment 1. However, Experiment 2 replicated Experiment 1’s effect of nominal gender on verb form. Participants are still more likely to select a feminine singular verb when shown masculine nouns, suggesting that this nominal gender effect on verbal number is reliable. Speakers do not want to use a feminine plural verb with a masculine noun. Given the response options that were available, it makes sense that, when looking solely at the responses with feminine verbs, masculine nouns would be used with the feminine singular form because there was not a symmetric masculine singular form. Feminine plural verbs in Najdi require a feminine noun, while feminine singular verbs do not have a similar constraint.¹⁰

Importantly, despite the effect of nominal gender on the selection of number within feminine verbs, the model found an additional effect of the number of events the participants saw. When participants saw a single depiction of an event, they were more likely to select a feminine plural verb. Taken with the last analysis, this preference suggests that DAgr is not a shift in gender features as required by Kramer & Winchester (2017) but rather a shift in number features. Here, we find that the semantic facts of event structure predict the grammatical number of the verb.

One problem with the design of this experiment is that the multiple event condition contained five of the same image repeated. It has been pointed out that this condition could be construed as five images of the same event. If participants did have this interpretation of the images, then their selection of the feminine singular could be seen as them selecting a singular form for a singular entity. However, two factors of the design also need to be considered here. First, participants were given sentences with plural subjects. While this cannot prevent participants from interpreting the image as a single event with a single actor, it ideally primed them to create a representation with multiple actors. Secondly, the nouns used to create the sentences were balanced for nominal gender. While plural agreement is complicated in Arabic, singular agreement is more straightforward, and plural agreement is never licit. If this interpretation were common, we would expect a great majority of participants to select the feminine singular verb with feminine nouns, which was not seen. A similar approach cannot be taken with the masculine nouns since the masculine singular verb was not provided.

One question raised in this experiment’s results is why verbal gender is no longer predictable. Experiment 1 was able to predict it, while Experiment 2 was not. In the following section, I discuss an analysis of subsets of Experiment 1 and 2’s data which are directly comparable to determine if other factors are driving the loss of the effect of nominal gender.

5.1 Results

I analyzed this data set in the same way I analyzed the data from Experiments 1 and 2 using a logistic mixed-effect regression model and a hierarchical comparison. The first model predicted the gender of the verb (masculine vs. feminine) with fixed effects of lexical semantics (distributive vs. non-distributive), nominal gender, and their interaction. Figure 7 shows the data analyzed in this model. The model further included random intercepts for participant and noun. The maximal random effect structure was attempted, but due to the lack of convergence, random slopes and intercepts were removed from the model by removing the lowest variance. Both categorical variables (lexical semantics and nominal gender) were dummy coded. The intercept reflects the reference levels of the lexical-semantic type, non-distributive, and the nominal gender, feminine. Significance for simple effects and interactions were then calculated through log-likelihood comparisons. Table 8 provides a summary of the final model’s outcome.

Figure 7

The percentage of responses for both nominal genders and distributivity conditions separated by gender.

The model found a significant interaction of lexical semantics and nominal gender, which suggests that nominal gender has a greater effect on participants’ choices of verb form when a verb is non-distributive. In other words, participants care less about the gender of the noun when they are choosing the gender of a distributive verb. A significant simple effect of intercept suggests that speakers chose feminine forms more than masculine forms; again, this is unsurprising as there are two feminine options as opposed to one masculine option.

The second model only looked at the responses in which participants selected a sentence with a feminine verb form. It predicted the grammatical number of the verb (singular vs. plural) with fixed effects of lexical semantics, nominal gender, and their interaction. Figure 8 shows the data analyzed in this model. The model further included random intercepts for participants and random slopes for nominal gender. The maximal random effect structure was attempted, but due to the lack of convergence, random slopes and intercepts were removed from the model by removing the lowest variance. Significance for simple effects and interactions were then calculated through log-likelihood comparisons. Table 9 provides a summary of the final model’s outcome.

Figure 8

The percentage of feminine responses for both nominal genders and distributivity conditions separated by grammatical number.

The second model found a significant simple effect of nominal gender, suggesting that when participants selected a feminine verb form, they selected number based on the gender of the noun, with masculine nouns being more likely to take feminine singular.

5.2 Discussion

The analysis of this data set supports my claims that the lexical semantics of the verb affects how speakers select the gender of verb forms. When selecting the gender and number of a non-distributive verb, participants pay more attention to nominal gender. However, when participants were given a distributive verb, the effect of nominal gender is neutralized. Psychological understandings of event representation can explain these effects.

I will draw heavily on the psychological idea of mental models. These models were initially proposed by Johnson-Laird (1983) and van Dijk & Kintsch (1983), who suggested that language serves to provide instructions on how to construct these representations. In other words, language allows two people to construct the same or, at least, similar mental representations of an event. In order to facilitate this transmission, people are required to store a lot of information about events, such as information pertaining to the time, space, and participants involved with an event (Altmann & Ekves 2019). I would argue that information about an event’s distributivity would also be modeled because distributivity often involves either distance in space or distance in time.

In Experiment 2, I found that speakers vary the gender and number of the verb depending on how they construct an event based upon a pictorial representation of the event (see Figure 4 for an example). When they saw multiple pictures of an event intended to represent spatiotemporal separation, speakers selected DAgr more often than when they saw an event presented as a single picture, for which they used SAgr. I would argue that this is evidence that speakers are tracking this information about their mental model and changing verbal morphology to be maximally informative. However, for the examples from Experiment 1 which used non-distributive verbs, speakers do not rely on DAgr to be maximally informative because the lexical semantics of the verb requires certain facts about these spatiotemporal properties.

To summarize, if we accept that speakers are cooperative and provide maximally informative information to help a collocutor reconstruct their mental representation, then the difference between distributive and non-distributive verbs can be understood as speakers needing to provide more information for distributive verbs which generally have more possible interpretations. The results of this analysis are consistent with this.

When we look at grammatical number within the feminine forms of verbs, there is no effect of the verbal lexical semantics. Instead, we still find the effect of gender that we have seen in the previous data sets.