Glossa Psycholinguistics

1.1 Theoretical background

The two theoretical positions differ primarily in their predictions about the interpretation of round numerals. Mathematically, roundness is defined by a set of structural properties tied to the base of the number system used in the relevant culture (Sigurd, 1988). The mathematical features contributing to roundness for counting systems using base 10 are simple multiples of the base number and simple fractions of the base number (Lotz, 1955; Sigurd, 1988). These include whether a numeral can be expressed as a single-digit multiple of a power of 10 (10-ness), five times a power of 10 (5-ness), two times a power of 10 (2-ness), or 2.5 times a power of 10 (2.5-ness). These properties do not correspond to ordinary divisibility. For example, 40 has 10-ness, 5-ness, and 2-ness; 50 has 10-ness and 5-ness; 38 has none. The number and type of roundness features predict a numeral’s frequency in use (Woodin et al., 2023), suggesting that these mathematical properties also have cognitive consequences. In domains with different underlying bases, e.g., time, other values count as round accordingly.

With this notion of roundness in place, we turn to the rationale behind the preferred-approximation and the precise-by-default positions. Starting with the preferred-approximation hypothesis, one might think that round numerals tend to convey approximate values because they are much more frequent in discourse than could possibly be justified by actual states of affairs (see the findings in Krifka, 2007, regarding the relative frequency of the rounder 50 versus the less round 40 and 60 in Norwegian versus Danish, and Basque). Speakers obviously actively choose to round numbers, which means that many round numerals in discourse do not correspond to factually true round numbers. But what about the addressee’s interpretation?

The preferred-approximation proposal is based on Bayesian inferences and game-theoretic mechanisms (Bastiaanse, 2011), since, according to this position, addressees infer the intended meaning by engaging with the speakers’ point of view, their motivations, and the alternatives available to them. According to Krifka (2002, 2007), these alternatives are a function of an underlying scale and its degree of coarseness. The degree of coarseness is determined by the minimal units of the scale. Thus, a scale such as [39–40–41–42] is more fine-grained than a scale such as [35–40–45–50], because the distance between the units is smaller. Hence, each unit on the fine-grained scale stands for a smaller scalar range. When the scale that determines the degree of precision of the sentence is fine-grained, a numeral like 40 will receive a precise interpretation. Under the coarse-grained scale, however, it will be imprecise. On this account, round numerals are ambiguous between precise and imprecise interpretations. However, imprecise utterances of round numerals are generally favored, because round numerals are simpler. According to Krifka, a speaker wishing to convey an interval, such as [39–40–41–42], has four alternative numerals to choose from (on the assumption that a speaker may actually convey such an interval by uttering one of its members). However, only forty has an advantage over the other alternatives, having the simplest form. This is recognized by the addressee decoding the speaker’s utterance. Additionally, imprecise utterances offer more values, which increases the likelihood of the statement’s being true: an exact value has a lower probability of being true than an interval. Furthermore, Krifka (2002) suggests that a more coarse-grained representation might be less cognitively costly. Taken together, these assumptions lead to the conclusion that, for round numerals, the imprecise, approximate interpretation is more likely. This is the first theoretical hypothesis this article aims to test.

Complementing Krifka’s motivations for why speakers may choose to use round numerals for imprecise meanings is Van Der Henst et al.’s (2002) Relevance-Theoretic argument that rounding is a hearer-oriented strategy. According to Relevance Theory (Sperber & Wilson, 1986), speakers opt for optimally relevant utterances, i.e., ones that allow the listener to derive a sufficient amount of cognitive effects using only a minimal processing effort. For example, when someone is asked to tell the time, if they are being helpful, they should provide an answer that will be useful for the asker (to the best of their knowledge), while imposing the least amount of cognitive load on the asker. This should lead them to prefer a round time. Thus, using approximative values in contexts where precision is not crucial reduces the cognitive effort imposed on the addressees, while broadly maintaining the contextual implications derived from the utterance. While Van Der Henst and colleagues do not explicitly deal with the addressee’s interpretation, it’s possible to apply the model proposed by Krifka here, as well, by positing that addressees may also consider the speaker’s motivation for an optimally relevant utterance when choosing a numeral.

Moving on to the precise-by-default camp, the starting point for this position is the plausibility of the assumption that there must be a functional motivation behind the extremely rich number system culturally evolved in most languages. Presumably, it complements our species-inherited cognitive propensity to only assess quantities approximately (Feigenson et al., 2004; Odic & Starr, 2018). The number line, based on the successor (+1) principle, makes it possible for speakers to provide precise descriptions regarding numerosities. Since sharp and round numerals figure equally on the number line, both should prompt precise number interpretations by default.

Ariel (2017, 2019, 2021; Ariel & Levshina, 2024) has argued that typical lexemes and numerals belong to different lexical domains, such that the former more easily give rise to contextually-adjusted (nonliteral) interpretations (Carston, 2002). The meaning of nice, for example, can be broadened to mean ‘kind of nice, close to attractive/kind’, etc. According to this account (see especially Ariel & Levshina, 2024), typical lexemes belong to sparse semantic domains, where the relevant meaning options are not continuous over the domain (i.e., the member meanings do not fully cover the whole meaning space). Instead, there are no-man’s-land meaning spaces between adjacent competing members, which broadening can take advantage of.

Unlike the sparse semantic domains characteristic of non-numeral lexemes, the counting number line constitutes a dense lexical domain, according to Ariel. Dense semantic domains are continuous and show no no-man’s-land meaning spaces between adjacent members.³ Therefore, the interpretation of numerals is strict and defaults to precision. This is the second theoretical hypothesis we wish to examine. Contra Bastiaanse (2011), Ariel’s reasoning was that in order to trigger an approximate interpretation, numerals must be explicitly marked (by dedicated approximators, such as about, around, etc.) (see also Gega et al., 2021). In this case, it is actually the modifier, rather than the numeral, that is (compositionally) responsible for the created range, resulting in an approximate reading.

According to this view, it is not impossible, however, to construct a sparse system for numerals, such as 10–20–30 (moving to a coarser scale granularity, according to Krifka, 2007). The spaces between, say, 10 and 20, and between 20 and 30 can then be designated as no-man’s-land meaning spaces. In this way, 20, for example, can broaden towards 10 and towards 30 (say, ‘16–25’). However, according to Ariel & Levshina (2024), interlocutors avoid such a domain shift. They assume that by default, it is the dense, +1 numeral domain that is relevant for interpreting both round and sharp numerals, where the minimally different members (+1, –1) block meaning broadening. Creating a pragmatic halo (Lasersohn, 1991) is not a common step then (especially for counting numerals).⁴ This is contrary to Krifka’s position that coarse scales are less cognitively costly and, therefore, preferred for round numerals.

1.2 Previous findings

This subsection reviews the empirical evidence that supports the two positions above, starting, again, with the preferred-approximation position. One of the earliest studies on numerical use for approximation was Van Der Henst et al. (2002), who found that people were much more likely to provide a round answer (e.g., 6:15 when the actual time was 6:13) when asked for the time. Similar results were obtained by Mühlenbernd & Solt (2022), who found that participants round off the time of a car accident when they know the precise time it happened, and they do that more often in the context of a conversation with a neighbor than in a witness statement in a police station. Using a different methodology but with similar conclusions, Woodin et al. (in prep.) examined the numbers printed on jigsaw puzzle boxes as indicating the number of pieces. They found that these numbers are preferably round and often somewhat imprecise. These studies provide support for the claim that speakers tend to use round numerals for approximation.

The idea that round numerals are easier to process, which serves as a motivation for speakers to use them for approximate values, finds support in Mason et al. (1996), which showed that participants performed better in recall and recognition tasks involving round versus sharp numerals. Similarly, Solt et al. (2017) found that participants who were shown a sequence of clock times were able to determine whether a certain time appeared as part of the sequence with higher speed and accuracy for round times (3:15, 8:30) than for sharp ones (3:21, 8:36). These results indicate that round values have an advantage in short-term memory over non-round values. In an additional time-arithmetic task, participants were more accurate and responded more quickly when the starting time was round than when it was non-round, as well as when the temporal increment was round versus non-round, providing further support for the claim that round numbers are more readily processed.

Kao et al. (2014) present some evidence for the addressee’s recognition of the speaker’s tendency to round numbers off. The study used prices of items to test non-literal uses of numerals (both approximation and hyperbole). Participants read scenarios in which speakers purchased an item and were asked by a friend, “Was it expensive?”. They then replied that the item had cost N, where N was either a round or a sharp number. The participants were then asked to rate the likelihood of the quoted price representing an exact, approximate, or hyperbolic value. The results showed that the average bias for exact interpretation is significantly higher for sharp than for round numerals, indicating that participants recognize that when speakers use round numerals, they are more likely to diverge from the state of affairs than when they use a sharp numeral.

Moving to the evidence in favor of the precise-by-default position, a corpus study by Ariel (2021) showed that whereas typical lexemes (nice, blue, green, red) were hardly ever overtly broadened in her corpus (~3%), a select set of numerals (2, 7, 10, 17, 30) were explicitly broadened 8 times as often (~25% of the time, based on the Santa Barbara Corpus (Du Bois et al., 2000) and the Longman Corpus of Spoken American English. Comparing the co-occurrence of one Hebrew approximator (be.erex ‘approximately’) with numerals and with a maximum standard absolute adjective (rek, ‘empty’), Ariel and Levshina (2024) showed a dramatic difference: there were virtually no ‘approximately’ modifiers among the modifiers of Hebrew ‘empty’ (1.1%). But virtually all of the numeral modifiers involved ‘approximately’ (94.4%). Assuming that default interpretations need not be marked, whereas non-default ones might need to be explicitly marked, these findings suggest that numerals are interpreted as precise numerosities by default. Note also that approximators, such as about, are far more frequent than “precisifiers” (such as exactly) as numeral modifiers.⁵ Furthermore, Jansen and Pollmann (2001) found that round numerals are modified by about and its Dutch (ongeveer), French (environ), and German (etwa) counterparts much more frequently than a chance distribution would suggest.

Aparicio’s (2017) series of self-paced reading studies (Experiments 3a, 3b, 4, 6a, and 6b), which did not find increased reading times for sentences with round numerals in precise contexts, failed to find support for imprecise interpretations of round numerals. These contexts included cases where a question preceding the sentence with the numeral specifically called for either a precise or an approximate interpretation (Experiment 3a), when the numeral was modified by exactly/about/approximately (Experiments 3b, 4), and when the general context was biased for either a precise or an approximate interpretation (Experiments 6a and 6b). The lack of a reading time penalty indicates that there is no evidence that processing approximate interpretations of round numerals is facilitated compared to precise interpretations. In fact, the significant differences that were observed in the experiment showed that precise interpretations were processed faster.

Note that Ferson et al. (2015) also found that bare numerals are overwhelmingly interpreted as precise, but their bare numeral targets included both round and sharp numerals, count and measurement numeral phrases, and integer and decimal number expressions. These different numeral types may very well bias towards more or less precise readings, but Ferson and colleagues do not present separate counts for the different numeral types, so the overall result may not be informative enough. Similarly, Ariel and Levshina (2024) argued for a precise interpretation for all numerals, where again, no systematic attempt was made to distinguish round and sharp numerals. However, a study by Solt and Stevens (2018), in which participants read sentences with numerals modified by about, some or without modification, and were asked to specify the range of numbers conveyed by the sentence, did compare round and sharp numerals directly. The results showed that 80% of the responses to unmodified numerals counted as exact, for both sharp and round numerals. However, due to the lack of a detailed description of the study’s materials, the results are difficult to assess. Hence, the experiments reported below were designed to directly compare round and sharp count numerals of similar magnitudes.

This line of thinking receives additional support from experiments probing the interpretation of maximum standard absolute adjectives, such as empty or straight. Although they correspond to an end point of a scale, they can shift from a boundedness to a graded schematicity even without special marking (Paradis, 2001), and can be more or less precise (Kennedy, 2007). Both Ronderos et al. (2024) and Wu and Aparicio (2025) have argued that, in the absence of contextual prompts for approximation, such adjectives were primarily interpreted as precise. Importantly, interpreting deviating (approximation-inducing) cases incurred a processing cost (see also Aparicio, 2017), which provides another piece of evidence for the assumption that maximum absolute adjectives are interpreted as precise by default. Still, some deviations from the literal denotation were accepted by Ronderos and colleagues’ participants (between 18%–89% in their Experiment 1). Moreover, Wu and Aparicio found participants’ acceptance for both interpretative strategies (at ~20%) when participants’ attention was drawn to an interpretative disagreement between two speakers regarding precise versus approximate interpretations. Additionally, Ronderos et al. (2024) found that a small deviation from a precise interpretation of maximum standard absolute adjectives was acceptable to participants at a rather high rate (68%), even when the experimental context explicitly biased participants to opt for strict interpretations. The picture that emerges from the interpretation of maximum standard absolute adjectives shows a clear bias for preciseness, although approximations are not totally ruled out. According to the precise-by-default position, the interpretation of numerals should be even more strict, since they belong to a dense lexical domain.

All in all, different empirical findings lend support to various components of the two competing proposals. Nevertheless, a systematic experimental investigation that directly tests the predictions of the two competing theoretical accounts has not been conducted, as far as we know. The next subsection presents the current study, which aims to do that.

1.3 The present study

In 1.1, we introduced two theoretical proposals regarding the interpretation of round numerals. The rationales behind both competing proposals seem to offer reasonable guiding principles for how to pragmatically adjust (or not adjust) the baseline precise interpretation of numerals. According to the preferred-approximation position, addressees reason about how states of affairs must be, as well as about the speaker’s motivations and available alternative utterances. Since round numerals are simpler than sharp ones, addressees expect speakers to use them when the precise value is not known or not crucial. This leads them to adopt approximate interpretations for round numerals. On the other hand, according to the precise-by-default position, numerals belong to a dense lexical field, and, therefore, broadening (or approximating) is not common. Thus, addressees are expected to not adjust the precise numeral meaning, which is linguistically derived. The goal of this article is to find out which of these competing proposals better predicts how addressees actually interpret round numerals.

Given the two competing proposals, we derive the following predictions:

• (i) If addressees follow the preferred-approximation reasoning, round and sharp numerals should elicit different responses. Round, but not sharp, numerals would be predominantly interpreted as approximate.

• (ii) If addressees follow the precise-by-default account, based on the fact that all numerals are members of the dense number line, there would be no difference between round and sharp numerals, both receiving precise interpretations predominantly.

• (iii) Addressees might oscillate between the two strategies, in which case a mixed interpretative pattern would emerge (either due to an inconsistent adoption of the two strategies by all or most addressees, or due to consistently different preferences for one or the other strategy by individual addressees).

We recognize that precise and approximate readings are not the only readings potentially associated with numerals. All number expressions receive a variety of context-relevant interpretations: ‘exactly N’, ‘at most N’, ‘at least N’, and ‘approximately N’. Interpreting numerals, addressees start with their linguistic meaning and then adjust it to derive the speaker-intended reading in the specific context. Most linguists assume a lower-bound only lexical meaning for numerals (‘at least N’), which is then strengthened by an upper-bounding (‘no more than N’) implicature. The result is an ‘exactly N’ representation. Originally, these implicatures were considered pragmatic, although the relevant implicature was classified as a generalized conversational implicature, namely, one that goes through unless canceled by context (Horn, 1989; Levinson, 2000). A variant account views the relevant implicature as a grammatical implicature (Chierchia et al., 2012). A third account takes the numerals’ lexical meaning to be lower- and upper-bounded, so no upper-bounding implicature is necessary (Ariel, 2006, and onwards; Breheny, 2008). The differences between these theories are orthogonal to our research question, because all three agree that, prior to the consideration of context-specific inferential processes, numerals are taken to have a precise interpretation. We here ignore ‘at least’ and ‘at most’ readings, and focus on a potential difference between round and sharp numerals with respect to approximate and precise interpretations.

We present three main experiments below. Experiment 1 tests the assumption of the preferred-approximation hypothesis that interlocutors are aware of speakers’ license to diverge from precise numerical descriptions, favoring an imprecise use of round numerals, in particular. Our results support this assumption. Still, the question remains whether addressees actually proceed to then engage in the reasoning outlined by the preferred-approximation account, which should lead them to adopt approximate interpretations.

Experiments 2a and 2b aim to find out whether listeners accept an intended interpretation of a range of values for round numerals more than for sharp ones by comparing bare (unmodified) round and sharp numerals with numerals modified by about (necessarily approximate) and by exactly (necessarily precise).

While Experiments 2a and 2b test the interpretation of numerals, Experiment 3 utilized a truth-value judgment (TVJ) task to test the compatibility of numerals with approximate interpretations. In this task, participants are presented with a sentence containing a numeral, followed by the relevant state of affairs. Their task is to judge whether the sentence is true or false in light of the state of affairs. TVJ is a more “tolerant” diagnostic for interpretation, because it may be enough for the state of affairs to be compatible with (rather than identical to) the statement. If round numerals are more compatible with approximate interpretations than sharp ones, we should see higher acceptance rates for utterances with round numerals in light of a state of affairs in which the actual numeral is not the same one as in the original utterance, but is within the numeral’s range of compatibility.

Based on the results of Experiments 2a, 2b, and 3, we will argue that listeners actually do not translate their awareness of speakers’ rounding practices into an interpretative strategy of approximation. Just like sharp numerals, round ones too are overwhelmingly interpreted as ‘precise N’. In order to reconcile these seemingly contradictory results, we propose that when interlocutors are consciously asked about imprecise numeral descriptions (Experiment 1), they show awareness of the advantage of using round numerals. But left to their own devices, they don’t “overthink”. They don’t engage in reverse engineering, so their default interpretation of both numeral types is as precise.

Decontextualized tasks were used in all the experiments below. Specifically, participants were not offered any cues for why the number information was relevant. Such cues might very well affect participants’ tolerance for imprecision. Participants were also not provided with cues about the speaker’s personality, e.g., as pedantic versus laid-back, which has been identified as relevant for acceptance rates of round numerals diverging from the factual number (Beltrama & Schwarz, 2022, and the references therein). In other words, the design attempted, as much as possible, to strip away the subjective aspects of numeral use and its crucial contribution to argumentation (in the sense of Anscombre & Ducrot, 1989). The only interest is in the most prominent interpretation of round numerals, for which there are competing claims (all agree that round numerals can be interpreted both precisely and approximately). These conflicting predictions have not been tested adequately, i.e., by directly comparing round and sharp numerals across different experimental tasks that tap into interpretation.

2.1 Methods

2.2 Results

The results of Experiment 1 are plotted in Figure 1.

Figure 1: Results of Experiment 1. Error bars represent 95% CI.

The model coefficients are presented in Table 2, and the model’s posterior estimates are visualized in Figure 2.

Figure 2: Posterior predicted probabilities of a ‘Yes’ response by numeral type and condition. Points represent the posterior mean estimates from the Bayesian mixed-effects model: response ~ type * condition + (1 + type * condition | participant) + (1 + type * condition | item). Error bars represent 95% Highest Posterior Density (HPD) credible intervals.

The model revealed that the acceptability of numeral type depended on condition (a credible interaction). While round and sharp numerals were accepted at equal rates in the False and Matching conditions, pairwise comparisons showed that round numerals were accepted significantly more than sharp numerals in the within-range (estimate = 2.48, CI = [1.18, 3.75]) and out-of-range (estimate = 1.81, CI = [0.33, 3.42]) conditions. Full comparisons are presented in Table 3.

Pairwise comparisons between conditions revealed credible differences in acceptability. The matching condition led to the highest acceptance rate of all conditions. False sentences were the least accepted, lower than within-range, out-of-range, and matching (all effects are credible). Within range sentences were more accepted than out of range, but less than matching. The full comparisons are presented in Table 4.

2.3 Discussion

The experiment tested whether addressees are sensitive to the speakers’ license to use round numerals imprecisely, more so than sharp ones. The findings indicated that they are. This is supported by the fact that round numerals received overall higher acceptance rates and, specifically, by the interaction pattern between numeral type and condition, according to which round numerals are accepted more in the within- and out-of-range conditions, but not in the false and matching conditions. These results show that when the divergence is small enough, addressees accept the use of a round, diverging numeral as reasonable. Thus, the idea underlying the preferred-approximation hypothesis, that addressees are aware of speakers’ imprecise use of round numerals more than sharp ones, is supported by the findings.

In addition, the results from the pairwise comparisons of conditions show that participants were sensitive to the compatibility between the speaker’s utterance and the factual information, with graded acceptability: Matching > Within range > Out of range > False. Generally, the greater the distance between the numeral in the utterance and the numeral in the state of affairs, the less participants accepted the utterance as a reasonable description of the state of affairs. That is, across addressees, divergence does not behave as a binary category, but rather as graded, with magnitude constituting the significant factor. Whether this is the case on the individual levels as well requires further examination.

Having tested the assumption that addressees recognize speakers’ imprecise use of specifically round numerals, we now turn to testing the actual interpretation they derive for both types of numerals, in Experiments 2a, 2b, and 3.

3.1 Methods

3.2 Results

Table 7 presents the acceptance rates in the different conditions across both numeral types. Acceptance rates for about N with a range in the target, and bare N and exactly N with a number in the target, are at ceiling (recall that in the number condition, the same number appears in the stimulus and the target). When the stimulus numeral is either modified by exactly or bare and the target is a range, the acceptance rates plummet to around 10%. About N stimuli with a number target are accepted in about 30% of the trials.

Next, we examine whether the response pattern differs between round and sharp numerals. The results by numeral type are presented in Figures 4 and 5. For both numeral types, numerals in the bare condition pattern with the exactly N condition, with high acceptance rates for number targets and extremely low acceptance rates in the range condition. For both numeral types, in the about N condition, responses are acceptable in high proportions for range targets (although less so for sharp numerals) and in medium proportions for number targets.

Figure 4: Results of Experiment 2a – sharp numerals. Error bars represent 95% CI.

Figure 5: Results of Experiment 2a – round numerals. Error bars represent 95% CI.

The model’s coefficients are presented in Table 8, and the model’s predictions are visualized in Figure 6.

Figure 6: Posterior predicted probabilities of acceptance in Experiment 2a for round and sharp numerals, separated by Modifier and Target. Points represent posterior medians; error bars represent 95% HPD credible intervals.

The model showed a credible main effect of modifier, driven by lower acceptance in the about condition (β = –1.61, 95% CI [–3.46, –0.20]). There was also a credible main effect of target type, with number targets yielding higher acceptance than range targets (β = 5.28, 95% CI [3.02, 8.84]). In contrast, the main effect of numeral type was not credible (β = –0.99, 95% CI [–2.48, 0.23]).

However, because both modifier and target appeared in credible two-way interactions between modifier and target and between numeral type and target, the main effects are not interpreted in isolation. The three-way interaction among type, modifier, and target showed no credible evidence (β = 0.73, 95% CI [–0.31, 2.03]; β = 0.12, 95% CI [–1.28, 1.46]).

To understand the interaction between modifier and target, we conducted pairwise comparisons. For targets containing numbers, the acceptance rates were higher in the bare and exactly conditions than in the about condition. For targets containing ranges, the results were reversed, with trials in the about condition leading to higher acceptance rates than in both the exactly and bare conditions. Full pairwise comparisons appear in Table 9.

As for the target type and numeral type interaction, for number targets, acceptance was slightly higher in the sharp condition than in the round condition. For range targets, the opposite pattern was observed: round numerals were accepted more than sharp ones. However, the difference was only credible for number targets. Full pairwise comparisons are presented in Table 10.

3.3 Discussion

Overall, we see that bare numerals, both round and sharp, lead to high acceptance rates of number paraphrases (97%–100% acceptance) and rejection of range paraphrases (9%–10% acceptance). This pattern is similar to that of exactly modified numerals. These results are not consistent with the predictions of the preferred-approximation theory, which expects round numerals to favor range interpretations. They are also inconsistent with the weaker version of the hypothesis, according to which both range and number paraphrases should be moderately accepted. Instead, they are more consistent with the prediction of the precise-by-default proposal, which predicts high acceptance of number paraphrases and high rejection of range paraphrases. Importantly, the same pattern is observed for both numeral types (although there is a small difference in the number condition — see below). We note that participants did not have to choose one type of interpretation over the other, and, in principle, could confirm both number and range paraphrases. Despite that, they completely rejected the range paraphrases for both bare numeral types and, crucially, for round numerals.

At the same time, we also observed a trend in which number targets were slightly more accepted for sharp numerals, and range targets for round numerals. Although the actual differences between round and sharp numerals are small – around 3% across modifiers (see Figures 4, 5), this pattern may suggest that round numerals are slightly more compatible with imprecise interpretations than sharp ones. The trend in the range condition, though not credible, may suggest that sharp numerals resist coercion into imprecise interpretations. These possibilities warrant further testing. Importantly, greater compatibility does not entail that round numerals are, by default, interpreted imprecisely, as the behavioral pattern here shows that, in this task, in both numeral types, participants rejected the range interpretations. Therefore, this finding is consistent with the precise-by-default hypothesis, as it generates predictions about interpretation rather than compatibility.

Now, there are two concerns about the results of Experiment 2a. The first, and more substantial, one is that the ranges in the paraphrases might not correspond to participants’ actual interpretations of round numerals. This is because the ranges were defined quite strictly, and participants may have inferred a different range, either narrower or wider, around the numeral. Moreover, approximate interpretations do not require a specific range to be derived. It is possible that these interpretations come with an unspecified, fuzzy meaning. In both cases, participants may have rejected the range paraphrases not because they interpreted the numerals precisely, but because the paraphrase did not match their inferred approximation.

A second concern raised by a reviewer is that the training phase may have encouraged participants to reject range paraphrases for round numerals, as they may have understood from the training that such paraphrases are only acceptable when approximation is explicitly marked. To address both concerns, we devised Experiment 2b, which was identical to Experiment 2a, except for two changes. First, instead of a specific range, the range paraphrases were replaced with close to N, allowing for an approximate interpretation without specifying numeric boundaries. Second, the training phase was removed entirely.

4.1 Method

4.1.2 Materials

Same as Experiment 2a, only this time, range paraphrases were changed such that instead of the range, the paraphrase included close to N.⁸ Figure 7 presents an example of an experimental item.

Figure 7: A screenshot of an experimental item in the sharp numeral condition in Experiment 2b.

4.2 Results

Table 11 presents the acceptance rates in the different conditions across both numeral types for Experiments 2a and 2b, for comparability. Acceptance rates for about N with close to in the target, and bare N and exactly N with a number in the target, remain at ceiling. In the rest of the trials, acceptance rates in Experiment 2b are higher. This is especially evident in the bare + range/close to conditions, which rose from 10% to 37%, and in the exactly + range/close to conditions, which increased from 6% to 21%. Conversely, participants were much more accepting of number paraphrases for numerals modified by about, as evidenced by the fact that acceptance in the about + number condition was also higher in Experiment 2b (68% vs. 31%).

The results by numeral type are presented in Figures 8 and 9. Here too, for both numeral types, numerals in the bare condition pattern with the exactly N condition, with high acceptance rates for number targets, only now the acceptance rates in the range condition are moderately low. For both numeral types, in the about N condition, responses are acceptable in high proportions for both target types.

Figure 8: Results of Experiment 2b – sharp numerals. Error bars represent 95% CI.

Figure 9: Results of Experiment 2b – round numerals. Error bars represent 95% CI.

The model’s coefficients are presented in Table 12, and the model’s predictions are visualized in Figure 10.

Figure 10: Posterior predicted probabilities of acceptance in Experiment 2b for round and sharp numerals, separated by Modifier and Target. Points represent posterior medians; error bars represent 95% HPD credible intervals.

As in Experiment 2a, the model showed a credible main effect of modifier, driven by lower acceptance in the about condition relative to the grand mean (β = –2.15, 95% CI [–4.56, –0.53]). There was also a credible main effect of target type, with number targets leading to higher acceptance than close to targets (β = 6.54, 95% CI [3.26, 11.88]). The main effect of numeral type (β = –0.23, 95% CI [–1.50, 0.96]) and the three-way interaction among type, modifier, and target were not credible (β = –0.44, 95% CI [–1.54, 0.51]; = –0.17, 95% CI [–1.33, 0.89]).

As for the modifier × target interaction, pairwise comparisons show that sentences containing about led to high acceptance of both number and close to paraphrases. In contrast, for bare numerals and exactly, participants strongly accepted number paraphrases, but rejected close to paraphrases. While the difference between bare numerals and exactly numerals is not credible in the number condition, exactly led to lower acceptance rates than bare in the close to condition. The full pairwise comparisons are presented in Table 13.

Finally, although the type x target interaction term did not reach significance, we conducted a pairwise comparison to see if we could detect a trend similar to the one found in Experiment 2a. In both conditions, round numerals elicited lower acceptance rates (unlike Experiment 2a, where round numerals were accepted more in the range condition), but the contrasts were not credible. The full pairwise comparisons are presented in Table 14.

5.1 Method

5.1.3 Procedure

Same as Experiment 1, but this time participants were asked to make a binary judgment about whether the speaker was right, given the fact. Figure 11 presents an example of a trial.

Figure 11: A screenshot of a trial from Experiment 3.

5.2 Results

The results of Experiment 3 are presented in Figure 12.

Figure 12: Proportion of acceptance of the speaker’s utterance by numeral type and condition in Experiment 3. Error bars stand for 95% CIs.

The model coefficients are presented in Table 15, and the model’s predictions are visualized in Figure 13.

Figure 13: Posterior predicted probabilities of a ‘Yes’ response by numeral type and condition. Points represent the posterior mean estimates from the Bayesian mixed-effects model: response ~ type * condition + (1 + type * condition | participant) + (1 + type * condition | item). Error bars represent 95% Highest Posterior Density (HPD) credible intervals.

The model found a credible effect of condition, with matching receiving higher acceptance rates relative to the grand mean (β = 7.35, 95% CI [6.17, 8.93]). The main effect of numeral type was not credible (β = 0.25, 95% CI [–0.22, 0.72]). The interaction between type and condition showed no credible evidence (β = –0.29, 95% CI [–0.95, 0.41], β = 0.57, 95% CI [–0.36, 1.72], β = –0.20, 95% CI [–0.75, 0.28]).

Table 16 presents the acceptance rates and CIs for each condition. Acceptance was highest in the Matching condition, followed by Within range, with Out of range and False conditions showing much lower acceptance. These results replicate the graded acceptability pattern found in Experiment 1, despite the different task. However, not all contrasts were credible in a pairwise comparison (Table 17).

5.3 Discussion

Not surprisingly, participants overwhelmingly accepted the speaker’s utterance as true in the matching condition. However, given the robust intuition in the literature about round numerals being naturally interpretable as approximate, one might expect participants to be more willing to accept as “right” the use of round numerals when the actual value is close to it, even if it slightly diverges from the numeral. However, the results did not support this expectation, suggesting that even minor discrepancies between the numeral and the actual value led to strong rejections. Our results found substantial rejections of non-matching values in all non-matching conditions. Even the within-range condition showed only a 15% acceptance rate.

We note that our findings differ from previous findings regarding the potential association of round numerals with approximate values. For example, as many as 35% (or even 45%) of Beltrama and Schwarz’s (2022) participants confirmed as “Right” the use of a round numeral (e.g., 200) for a state of affairs of a sharp numeral (e.g., 207.06) (the different rates are a function involving the persona of the speaker). Several methodological and conceptual differences between the studies may contribute to this contrast. Most importantly, our experiments tested counting numerals, whereas Beltrama and Schwarz examined measurement numerals, which have been argued to license approximate construals more readily (Ariel & Levshina, 2024; Cummins, 2015). Their stimuli also included decimal values, which are quite marked and fall outside the basic number counting scale, potentially encouraging approximate interpretations. Taken together, these differences may help explain why our results show lower overall acceptance of approximate readings.

Now, we have so far assumed that the basis for judging true a stimulus numeral describing a state of affairs where an in-range number diverges from the stimulus number points to the acceptance of a compatibility between the actual state of affairs and the stimulus numeral. The expectation is that, if this is the case, we should perhaps see this more often with round numerals. However, our results found no evidence for that being the case.⁹ However, three potential issues remain. The first concerns the asymmetry between trials in which the numeral in the fact is below the sentence numeral as opposed to above it. The second concerns the decision to include all conditions in the analysis. These issues are addressed in 5.4 and 5.5, respectively. The third concerns the epistemic state attributed by the participants to the speakers. This is addressed in the supplementary files available at the OSF site for this article.

5.4 Position of the target numeral with respect to the sentence numeral

Two theoretical accounts predict an asymmetry between higher-number and lower-number divergences, i.e., when 30 is said to describe a state of affairs of ‘33’ versus ‘27’. Both suggest higher acceptance rates for higher than for lower values. The originally consensual account posits a lower-bound only (‘at least’) semantic meaning for numerals, with exact interpretations derived via a quantity implicature (Gazdar, 1979; Grice, 1975; Horn, 1972). According to this analysis, since higher values are part of the numeral’s meaning, using a numeral when the actual number is higher should be more acceptable than when the actual number is lower.

Others have argued that the meaning (or the “what-is-said” interpretation) of numerals is circumbounded, i.e., lower and upper-bounded, and is, therefore, exact (Ariel, 2004; Breheny, 2008; Horn, 1992; Koenig, 1991). Still, such an analysis is not incompatible with higher confirmations for higher-number divergences. Ariel (2004) predicts that a truth-compatible inference can bridge the gap, between e.g., ‘30’ and a state of affairs of ‘33’, because if there are, e.g., 33 chairs, then there are 30 chairs. Higher values do not contradict the speaker’s commitment to ‘30’ (the speaker might have been under-informative), but lower values (e.g., ‘27’) do, making it more difficult to infer compatibility.

We should, then, check this hypothesis. Note that the prediction here applies equally to round and sharp numerals. If, however, we find higher confirmations of cases in which the numeral in the fact is below the uttered numeral for round numerals than for sharp numerals, this could be an argument in favor of approximation, because neither account can explain such a difference. 5.4.1 reports our post hoc analysis, which investigates whether rounding is influenced by whether the number in the fact was below or above the numeral in the sentence.

5.4.2 Results

Figures 14 and 15 present the results of Experiment 3 again, this time with acceptance rates for non-matching conditions separated into fact numerals that are higher than the utterance numeral and fact numerals that are lower than the utterance numeral.

Figure 14: Results of Experiment 3, including acceptance rates by the type of numeral in the fact, for sharp numerals.

Figure 15: Results of Experiment 3, including acceptance rates by the type of numeral in the fact, for round numerals.

The model’s coefficients are presented in Table 18, and the model’s predictions are presented in Figure 16.

Table 18: Bayesian mixed-effects logistic model, formula: response ~ type * condition * position + (1 + type + condition + position | participant) + (1 + type + condition + position | item). Factors are sum-coded. Therefore, the intercept reflects the grand mean.

Predictor	Estimate	Std. Error	95% CI	$\widehat{R}$
Intercept (Grand Mean)	–11.14	2.15	[–16.00, –7.68]	1.00
Type: Round	0.68	0.61	[–0.48, 1.96]	1.00
Condition: False	0.10	0.76	[–1.42, 1.62]	1.00
Condition: Out of Range	0.19	0.61	[–1.04, 1.42]	1.00
Position: Over	1.45	0.75	[0.09, 3.01]	1.00
Round × False	–0.47	0.52	[–1.45, 0.62]	1.00
Round × Out of Range	–0.17	0.41	[–1.05, 0.58]	1.00
Round × Over	–0.81	0.49	[–1.91, 0.02]	1.00
False × Over	–0.80	0.44	[–1.74, –0.02]	1.00
Out of Range × Over	0.33	0.34	[–0.31, 1.04]	1.00
Round × False × Over	–0.01	0.31	[–0.64, 0.58]	1.00
Round × Out of Range × Over	0.24	0.27	[–0.27, 0.79]	1.00

Figure 16: Posterior predicted probabilities of a ‘Yes’ response by numeral type and condition. Points represent the posterior mean estimates from the Bayesian mixed-effects model: response ~ type * condition * position + (1 + type + condition + position | participant) + (1 + type + condition + position | item). Error bars represent 95% Highest Posterior Density (HPD) credible intervals.

The analysis revealed a credible main effect of position (β = 1.45, 95% CI [0.09, 3.01]), with higher acceptance rates when the fact N was larger than the utterance N. There was no credible main effect of numeral type (β = 0.68, 95% CI [–0.48, 1.96]), nor evidence of a credible interaction between type and position or condition, or for the triple interaction.

The model also found a credible interaction between numeral condition and position (β = –0.80, 95% CI [–1.74, –0.02]). However, pairwise comparisons found no credible differences (see Table 19).

Table 19: Pairwise comparisons of condition by position.

Position	Contrast	Estimate	95% CI
Above	False – Out of range	–1.16	[–3.59, 1.05]
	False – Within range	–0.86	[–3.87, 2.00]
	Out of range – Within range	0.29	[–1.98, 2.83]
Below	False – Out of range	0.98	[–1.66, 4.05]
	False – Within range	1.56	[–1.81, 5.25]
	Out of range – Within range	0.58	[–2.37, 3.70]

5.5 Focusing on within-range trials

A possible concern raised by a reviewer is that the predicted effect of numeral type on acceptance of approximate trials should be assessed specifically within the within range condition, rather than across all conditions, as this is where differences in interpretation are most likely to surface. Additionally, acceptability might vary with numerical magnitude: because the range of acceptable values was fixed across all numerals, the proportion of divergence differed. For example, a 3-unit difference from 20 represents about 15%, whereas the same difference from 60 is only 5%. If participants are sensitive to proportional divergence, we might expect lower acceptance for numerals with larger relative deviations.

We therefore conducted a focused analysis including only within-range trials. We fit a Bayesian logistic regression model using brms’s package’s default weakly informative priors: Normal(0, 1) priors for fixed effects, a Student-t(3, 0, 2.5) prior for the intercept, and a Student-t(3, 0, 2.5) prior for the random intercept standard deviation. We modeled response as a function of numeral type (round/sharp), position (above/below), scaled numeral magnitude (z-scored), and their interactions as fixed effects.

A model with a maximal random-effects structure failed to converge. We therefore applied the same stepwise simplification procedure as in the previous analysis. Because an additive random-effects structure did not resolve the convergence problems, we progressively removed random-effects terms with the smallest estimated variances and refit the model at each step. This procedure continued until convergence was achieved, resulting in a model that included only random intercepts for participants. More complex random-effects structures consistently exhibited convergence problems, manifested in inflated intercept estimates (greater than 20). The model’s coefficients are presented in Table 20.

The analysis revealed credible main effects of type (β = 0.47, 95% CI [0.03, 0.93]), with higher acceptance rates for round numerals, and position (β = 0.68, 95% CI [0.38, 1.01]), with higher acceptance rates when the fact N was larger than the utterance N. No credible main effect of numerical magnitude was found (β = 0.24, 95% CI [–0.08, 0.57]).

The interaction between type and position was credible (β = –0.32, 95% CI [–0.63, –0.03]), with follow-up contrasts showing that round numerals were more acceptable than sharp ones only in the under condition (estimate = 1.58, 95% CI [0.41, 2.78]), with no difference in the above condition (estimate = 0.29, 95% CI [–0.66, 1.26]).

The interactions of Magnitude × Type, Magnitude × Position, and the three-way interaction were not credible.

Unlike the analysis in 5.4.2, this analysis found credible evidence that round numerals are slightly more compatible with approximation than sharp numerals, specifically when the fact numeral was smaller than the sentence numeral (‘below’ condition). However, despite this relative difference, acceptance rates for round numerals remained low overall, a finding which does not indicate that they are inherently compatible with approximation in this task, and that they are only less incompatible than sharp numerals. Finally, the magnitude of the numerals had no measurable effect on the acceptability of approximate interpretations. But in view of the magnitude effect on preference for round numerals reported in Woodin et al. (in prep.), it may very well be that the magnitude difference between our small and large numbers is just not large enough.