INTRODUCTION

Virtually every culture in the world uses numbers for a wide range of purposes, from very basic usages such as counting, to the most sophisticated enterprises of humankind as understanding the Universe. Not surprisingly, they have their own linguistic equivalent: number terms¹, which are elements distinct from and not to be confused with the mathematical terms known as numbers². A number term is a linguistic element that denotes cardinal quantities and is used in various ways, such as when we learn to add or to subtract numbers as shown in (1a), or when referring to a label or proper name as in (1b), when we quantify things as in (1c), or when we delimitate nouns as in (1d). Number terms in language often have multiple functions: arithmetical, as labels, as quantifiers, as adjectives, among others (^{Geurts, 2006}).

(1) a. Two plus two equals four.

b. Chanel number five.

c. Four students passed the exam.

d. The four children over there.

Not only can a number term exhibit different functions, but it may also have different interpretations within the same function. Take for example the utterances illustrated in (2).

(2) a. You need to make three mistakes to be allowed to take the test again. (^{Musolino, 2004}, p. 3).

b. You can make three mistakes and still pass this test. (^idem).

c. You must take three cards from the pile to start the game.

In (2a-c) we have the number term “three” functioning as a quantifier, but note that the readings one could derive could be very different in each case, at least, prima facie: (2a) is compatible with you having made three mistakes or more in order to repeat the test, so the interpretation that becomes available seems to be that you need to make “at least three” mistakes to repeat the test, while (2b-c) suggest a different reading. In fact, (2b) will receive the preferred reading of “at most three” whilst (2c) will receive an exact interpretation or an “exactly three” reading. In other words, (2a), (2b) and (2c) seem to be interpreted as having the same truth conditions as (2*a-c). These interpretations are cases of what are called intervalar or bound readings, namely the “at least” or lower-bound reading as in (2*a), the “at most” or upper-bound reading as in (2*b) in contrast to the exact bilateral, or doubly-bounded reading as in (2*c).

(2*) a. You need to make (at least) three mistakes to be allowed to take the test again. (^{Musolino, 2004}, p. 3).

b. You can make (at most) three mistakes and still pass this test. (^idem).

c. You must take (exactly) three cards from the pile to start the game.

Several discussions in the field of semantics and pragmatics have arisen from these observations, where “bare numerals present an interesting challenge to formal semantics and pragmatics: they seem to be ambiguous between various readings, and the choice of a particular reading seems to depend on complex interactions between contextual factors and linguistic structure” (^{Spector, 2013}). The goal of a theory of interpretation would be to give a systematic account of whether, when and how each type of bound reading arises. Although no such theory is available as yet, we have made significant advances in this direction.

Theoretical discussions have long revolved around the semantic content of number terms, and three main accounts have been formulated (^{Carston, 1998}; ^{Geurts, 2006}; ^{Horn, 1972}). The neo-Gricean account sustained that number terms have a lower-bound semantics, and depending on the context, their exact interpretation would be derived via a scalar implicature (^{Horn, 1972}). Other scholars have defended the naïve view of number terms as having an exact semantic content. In this line of reasoning, ^{Geurts (2006)} explains the derivation of the lower-bound reading via semantic operations, while ^{Breheny (2008}) maintains that non-exact interpretations are derived due to background knowledge and pragmatic mechanisms. A third account, defended by ^{Carston (1998)}, suggests that number terms have a neutral, underspecified lexical meaning and, depending on the context of emission, their interpretation would adopt one of the various construals discussed above.

The vast majority of studies have provided both theoretical and experimental evidence that contradicts the neo-Gricean account (^{Breheny, 2008}; ^{Papafragou & Musolino, 2003}; ^{Huang et al., 2013}, ^{Villaseñor, 2018}; ^{Villaseñor et al., 2021}) in favor of an exact semantics of number terms. However, we still need to explain the reasons why and under which contexts number terms sometimes receive intervalar interpretations. These reasons could be categorized as either grammatical or pragmatic phenomena.

Grammatical factors that affect the interpretation of number terms

As previously noted, it has been observed that under certain linguistic contexts, number terms can receive bounded readings, while in others the only possible reading will be the exact.

Predicate and quantifier semantic function

It has been noted that depending on the grammatical environment in which it appears, a number term can have either a predicate or a quantifier function in a sentence (^{Geurts, 2006}; ^{Curcó, 2015}; ^{Villaseñor, 2018}) and depending on this function, a lower-bound reading could be available. In certain languages such as Spanish, the predicate function will be given when the term appears after a linking verb such as “ser” (to be), in purely grammatical terms referring to the property or quality attributed to that entity. In this case, the only possible reading of the term will be the exact, as observed in the example in (3). The reason why the only possible reading is the exact is because under this function the number term will represent a unique set with cardinality ǀnǀ which the entity must be linked with. It is due to this characteristic of unicity in the set which leads to all intervalar readings being excluded.

(3) Los detenidos fueron tres. (exactly three)

‘The detainees were three.’

In contrast, when the number term has a quantifying function, it will not predicate a property of the entity but will rather denote the cardinality of a set about which something else is predicated. This set will not necessarily be defined as a unique set, but the existence of several other sets could be possible. For example, in (4), it is predicated that there is a set of three girls who wear glasses, but it is compatible with the possibility of there being more than three girls wearing glasses.

(4) Tres niñas usan lentes. (exactly / at least three)

‘Three girls wear glasses.’

As illustrated in Figure 1, if there actually were, for example, exactly four girls wearing glasses (set A), it would also be compatible with the existence of at least two different subsets of three girls wearing glasses (subsets B and C). In this way, uttering ‘three girls wear glasses’ will be logically true even in the case where the total cardinality of the referred set were larger than 3. In other words, it is true about set A that there are 3 girls wearing glasses. In this case and for that reason, when a number term has a quantifying function, a lower-bound reading will be logically available.

Figure 1 Quantifier function compatibility with larger cardinality sets 

In sum, when a number term has a predicate function, the only possible reading will be the exact, whereas when it has a quantifier function, a lower-bound reading will be acceptable as well as the exact. This has been observed by ^{Geurts (2006}) but, to our knowledge, no empirical studies have been conducted other than ^{Villaseñor et al. (2021)} in order to explore this specific observation within an experimental method.

Number terms preceded by a definite article

Definite articles such as “the” in English or “el / la / los / las” in Spanish express a characteristic of maximality and uniqueness of the referent (^{Von Fintel et al., 2014}). In this sense, when a number term is preceded by a definite article it presupposes that the referent is the only and maximum set. For example, the predicate in (5) states that there is a unique set with a maximum cardinality of 3 students who passed the test. The uniqueness feature is presupposed because of the definite determiner. Accordingly, the only possible interpretation in this linguistic context will be the exact.

(5) Los tres estudiantes pasaron el examen.

‘The three students passed the test.’

When there is no definite article preceding the number term, the interpretation will be equal to the case in which the number term functions as a quantifier (‘Three students passed the test’), giving rise to an acceptable reading of “at least three / three or more students”.That is, it is compatible with either an exact or a lower-bound interpretation for the reasons previously discussed.

METHODOLOGY

Procedure

The study consisted of two versions of a questionnaire created in Google Forms, versions A and B. Respondents were invited to participate voluntarily and they were assigned to one of the questionnaire, which they answered remotely.

Participants were asked to respond to the whole questionnaire in a single session and to try to avoid being exposed to any distractions during the task. They were also reassured that there were no correct or incorrect answers, so they were expected to select answers based on which they considered more natural, with the option of choosing more than one possible interpretation. An example of the task was provided on the instructions screen (Figures 2 and 3).

Figure 2 Instructions as seen by participants 

Figure 3 Instructions (English translation, not seen by participants) 

Each item was shown individually, as illustrated in Figure 4. Once respondents chose the option(s) that they thought could naturally arise from the utterance, they moved on to the next item and could not go back to the previous items in the task.

Figure 4 Example of stimuli used in the experiment 

Within the options available, the possible interpretations of the number term in the stimuli (exact and intervalar bindings) were explicitly phrased, by using one of the expressions in 8.

(8). a. Exact expression in Spanish

(exactly n)

exactamente n

b. Upper-bound expressions in Spanish

a lo mucho n (at most)

a lo más n

máximo n

c. Lower-bound expressions in Spanish

al menos n (at least n)

mínimo n

RESULTS

Firstly, our findings showed, as predicted, that the exact interpretation was the most salient in all the explored variables when contrasting the pragmatic variables versus the possible readings of the number terms (with all linguistic variables grouped). Additionally, contrary to the findings of the previous experiment in which bound readings were not accepted, we found that the pragmatic background was strong enough to bias interpretation towards the corresponding intervalar readings. These two findings were in line with our initial predictions and we found statistical significance when contrasting the two pragmatic variables with the type of accepted interpretation of the number term (chi sq. p<0.0001), as shown in Table 2.

An interesting result was that, in general terms, even with strong pragmatic contexts, the exact reading still appears to be preferred over the corresponding intervalar interpretations, as seen in Table 2. That is, in both pragmatic contexts we expected to find higher frequencies in the corresponding intervalar option rather than in the exact one but instead found that even when the pragmatic variable does make the intervalar readings accessible, it does not overwrite the exact semantics. This can be taken as additional indirect evidence in favor of the exact semantics view of number terms (^{Geurts, 2006}; ^{Villaseñor, 2017}; ^{Villaseñor et al., 2021}), contradicting the contextualist view of them having underspecified semantics (^{Carston, 1998}).

Moreover, when looking into the results per condition, some interesting observations can be made. As we will show next, we found an acceptance of lower-bound readings in the LPC even under conditions that only accept exact interpretations from speakers for grammatical reasons (predicative and definite conditions). Likewise, we found a high acceptance of upper-bound readings in the UPC even when the three linguistic conditions under study do not grammatically accept them.

Table 3 and Figure 5 show the frequencies of interpretations per linguistic condition and per pragmatic variable. The two italicized columns in the table correspond to the expected choice of interpretation given the pragmatic context, and the numbers in bold are the highest frequencies obtained. When conducting a chi-square statistical test contrasting the linguistic conditions and the possible readings of the number terms per pragmatic condition, we found significance in both pragmatic variables. In Table 3, standardized residuals are included, where the values that most contributed to the level of significance are in bold⁴.

Figure 5 Frequencies of interpretations per linguistic condition and per pragmatic variable 

From to the previous table and figures, it can be seen that the pragmatic context generally affects the acceptance of intervalar interpretations of number terms. This holds true even when the linguistic condition does not accept any non-exact readings, as noted in the predicate condition where we found a high acceptance of “at most” readings in the UPC and “at least” readings in the LPC. Nevertheless, in this condition the exact interpretation was preferred over the intervalar reading in both pragmatic contexts.

Additionally, it stands out that in the definite determiner condition, respondents did not fully accept the intervalar readings in any of the pragmatic variables, and there was a marked preference for the exact interpretation. This suggests, is that in this particular linguistic condition the pragmatic context has almost no influence on the interpretation, and the grammatical context is the strongest factor in determining possible interpretations.

Lastly, it seems that in the quantifier condition, the pragmatic context does have an influence on the interpretation of the number term, especially in the case of the UPC where the most salient reading was the “at most” option. In this case, it seems that the pragmatic factor is even stronger than the semantic reading of the number term. On the other hand, in the LPC we expected to find a very high frequency of “at least” choices, since such reading would be in line with both the pragmatic and the compatible lower-bound semantic reading, assuming that in this condition the core meaning is the exact but is also logically compatible with the “at least” interpretation . This was not the case though, as we found that in this pragmatic context both the exact and the lower-bound readings were equally salient, and the frequency of exact choices was even a little higher than the intervalar one.

The following question was whether there is a significant correlation between the types of linguistic conditions and the preferred interpretation of the number term independent of the pragmatic context. When grouping both pragmatic variables and conducting a chi-sq statistical analysis contrasting the types of linguistic conditions against the type of interpretation, we found a significant statistical result (p<0.0001), as shown in Table 4 and Figure 6.

Figure 6 Frequencies of number term interpretations per linguistic condition (pragmatic variables grouped) 

In both the predicative and the definite determiner conditions, the preferred reading was the exact, especially in the latter, where there does not seem to be a high acceptance of the intervalar readings. This preference for the exact interpretation does not seem to hold as strongly for the quantifier condition, where the frequencies are more or less equally distributed across the three types of interpretation, even when the number in the exact reading is slightly higher. This is worth noting as it aligns with the findings of the previous study, in which lower-bound linguistic conditions, such as the quantifier function, do not seem to favor the corresponding lower-bound interpretation of number terms.

DISCUSSION

In general, we found that a strong pragmatic context does seem to affect the interpretation of number terms and makes non-exact readings accessible. This does not come as a surprise but rather confirms that a given pragmatic context related to background knowledge can be strong enough to bias the interpretation of these terms to the corresponding intervalar reading, no matter the linguistic condition in which they appear.

In the condition that accepts lower-bound readings (a number term as a quantifier), we expected to find significantly higher frequencies of the “at least” paraphrase in the LPC, but we really did not find a strong bias towards the lower-bound reading compared to the exact interpretation. That is, even when the quantifier condition logically allows for a lower-bound reading of number terms, results showed that the frequency of acceptance of exact readings (n=73) was even higher than the lower-bound readings (n=63) in the LPC, contrary to the prediction. In this sense, we started out assuming that pragmatic background knowledge would be strong enough to bias the interpretation of any given linguistic structure, and so we expected to find in the LPC that the corresponding lower-bound reading would be overwhelmingly higher than the exact option in this grammatical condition, which was not the case. This result should be further explored but indirectly adds to the evidence that number terms have an exact lexical meaning, contradicting the assumption that number terms have a lower-bound semantics, as first suggested in the neo-Gricean view (^{Horn, 1972}).

Additionally, in the UPC we found high frequencies of “at most” readings and very low frequencies in the “at least” in this same linguistic condition, suggesting that this pragmatic context was even stronger than the grammatical environment in which the number term appears.

Under the exact linguistic conditions (number terms as a predicate and preceded by a definite determiner), we found a higher rate of exact interpretations in both the UPC and the LPC, no matter the pragmatic condition. Overall, even when there was some acceptance of upper-bound readings in the UPC and vice versa, the exact interpretation seems to be the most salient, which aligns with the linguistic predictions assuming that those grammatical contexts do not accept intervalar readings.

However, only in the predicative condition did we obtain a moderate acceptance of the intervalar readings according to the pragmatic context. In the UPC, the frequency of accepted “at most'' readings was moderately lower (n=71) than the exact readings (n=111), whereas in the LPC the lower-bound reading (n=63) was slightly lower than its exact (n=89) counterpart. What this suggests is that in this condition, the pragmatic factor does have an influence on the reading of the number terms, even when grammatical context is still the strongest factor that guiding the interpretation.

However, in the definite determiner condition we did not find a marked acceptance of any intervalar readings of the number terms. In the UPC, the acceptance of upper-bound interpretations was significantly lower (n=29) than the exact counterpart (n=113), while in the LPC the same pattern holds, where the the lower-bound (n=16) reading was significantly lower than the exact interpretation (n=103). What this suggests is that this grammatical condition effectively blocks any possible intervalar readings, even when the utterance is inserted in a strongly bias-raising, pragmatic context.

CONCLUSION

The findings of this study add to evidence in favour of number terms having an exact core meaning, as suggested in previous literature and studies (^{Geurts, 2006}; ^{Villaseñor, 2017}; ^{Villaseñor et al., 2021}) and against the contextualist view of them having an underspecified semantic content which is fully dependent on pragmatic factors (^{Carston, 1998}), as well as against the neo-Gricean account which suggests that they have a lower-bounded semantic content.

Additionally, contrary to the findings of previous studies (^{Villaseñor, 2017}; ^{Villaseñor et al., 2021}) where intervalar readings were not accepted at all in any of the linguistic conditions explored here when minimizing pragmatic factors, we found that under a strong pragmatic context related to background and world knowledge, these terms can receive non-exact interpretations in both the quantifier and predicative conditions. In the definite determiner condition, however, it seems that the pragmatic context does not have an important influence on the interpretation of these terms, since in neither of the pragmatic conditions did we find an acceptance of non-exact readings. This suggests that in this linguistic condition, the grammatical context is not only the strongest but likely the only factor guiding interpretation.

We can conclude that even when the pragmatic context does play a significant role in the derivation of the possible interpretations that a number term can receive, this pragmatic factor is not as strong as to overwrite the semantic core meaning of number terms. This is especially true under certain grammatical environments, as in the case of definite constructions, where no matter the background assumption under consideration, an intervalar reading is not accessible.

Finally, our findings also call for further study to explain why the upper-bound readings seem to be more accepted than the lower-bound interpretations, as observed in the results of this experiment. Moreover, we need to further explore what other grammatical environments make intervalar interpretations available, as in the case of number terms being under the scope of modal operators, as shown in some of the examples in 2(c) at the beginning of this paper.