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PSYCHOLOGICAL AND COGNITIVE SCIENCES observations about, e.g., the cognitive complexity of quantity vals of their GQT meanings. It has been argued that this type of words (19), their order of acquisition (20), the aptitude of gradience and focality is fundamentally incongruous with GQT people’s reasoning with quantified sentences (21), and the dis- (25–27)—or even with a bivalent truth-conditional approach to tribution of negative polarity items like “any” and “ever” (22). At meaning more generally (4, 5). Instead, it has been argued that the same time, however, it has been observed that there is no quantity words have a prototype-based semantics (25, 27); e.g., straightforward connection between the set-theoretic meanings the prototype for “most” may be situated at about 75% of the postulated by GQT and the way people use quantity words, as total set size, with the typicality of a situation decreasing with the we will show presently. distance from that prototype.

Using Quantity Words. We conducted a large-scale study on the Outline. The goal of this paper is twofold. First, we show how production of quantity words in English (Experiment [Exp.] 1a). abstract semantic theories can be compared empirically with Each of 600 participants described 10 displays showing 432 cir- data-driven statistical modeling. Second, we show that GQT’s cles, which were either red or black. To describe these displays, truth-conditional account of quantity-word meaning offers an participants freely completed the sentence frame “— of the cir- equally compelling account of quantity-word production as a cles are red.” Each display varied the intersection set size, i.e., semantics based on prototypes, but only when it is complemented the number of red circles. To elicit quantity words that are by a probabilistic theory of language use. potentially vague and thus relevant to our purposes, the dis- The next section formalizes GQ-theoretic and PT-based plays had a large total set size, thereby likely triggering only semantics of quantity words. We then describe four speaker approximate representations of the true intersection set size. models that make probabilistic predictions about quantity-word Additionally, experimental instructions discouraged the use of choices as a function of the underlying semantic theory. numerical expressions like “fifty-two.” Fig. 1 A and B visualizes the production probabilities of the Semantics of Quantity Words 15 most frequently produced quantity words, all of which were A lexical meaning function L: M × T → [0, 1] maps each pair of produced 50 times or more, as well as “all” and “none.” “All” quantity word m and state t to a truth value in the unit interval. and “none” were included because they are crucial for commu- Participants in Exp. 1 were presented with displays consisting of nication and historically salient, even though they were not very 432 circles, which were either red or black. There were thus 433 frequently produced in the experiment for the obvious reason possible intersection set sizes T = {t0, ... , t432}, i.e., relevant that displays with uniformly colored dots occurred infrequently. states of the world. Taken together, these 17 quantity words made up 87% of the production data. GQT Semantics. According to GQT, many quantity words—and In line with earlier interpretation studies (23, 24), the results all of the ones that we are interested in—express thresholds of Exp. 1 suggest that the ranges in which quantity words are on the intersection set size |S ∩ P| (18). Whether a quantity produced lack clear boundaries and peak in restricted subinter- word expresses a lower bound, an upper bound, or both can be

(other) A B Training data (Exp. 1a) Test data (Exp. 1b) a few 1.00 a lot about half all all almost all 0.75 almost all few most half majority 0.50 hardly any less than half a lot majority many more than half 0.25 more than half

many Production probability most about half none 0.00 several 0 100 200 300 400 0 100 200 300 400 half some several Number of red circles very few some C GQ−lit PT−lit GQ−prag PT−prag less than half 1.00

few 0.75 a few 0.50 very few 0.25 hardly any none 0.00 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 Production probability Number of red circles Number of red circles

Fig. 1. (A) Density plot showing the distribution of each quantity word over intersection set sizes (Exp. 1a). The height of the distribution indicates the production frequency. (B) Bar plot showing the production probabilities in each bin of intersection set sizes. Each bar shows which quantity words were produced in that bin and their production probability. (C) Bar plot showing the predicted production probabilities for each of the four computational models.

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PSYCHOLOGICAL AND COGNITIVE SCIENCES marked preferences for some quantity words over others. For The evaluatively neutral nature of our experimental task example, “most,” “the majority,” and “more than half” are often makes it intuitively unlikely that factors such as argumentativity viewed as truth-conditionally equivalent, despite having different played an important role in our production study. conditions of use. Nonetheless, participants used “most” much more often than “more than half” or “the majority.” We assume Imprecise Number Representation. Participants in Exp. 1 were not that this difference in frequency is due to the fact that “most” told how many circles were red, but, rather, had to estimate the was more salient, i.e., accessible in the task at hand. (See SI intersection set size based on the displays. The cognitive module Appendix for further discussion.) To capture effects of differen- used to estimate the numerosity of large sets is the Approxi- tial salience, we paired each quantity word with a salience value mate Number System (ANS) (38). The accuracy of ANS estimates PSal(m), which is treated as a free variable. decreases as the numerosity to be estimated increases. This effect Combining truth-based and salience-based production prefer- is a corollary of Weber’s law, which states that the probability of ences, we consider a simple model of a literal speaker, which confusing two stimuli increases with their relative, rather than serves as a baseline for later comparison: absolute, difference (39). In order to model the accuracy of participants’ estimates of PS (m | t, L) ∝ PSal(m) L(m, t). the intersection set size t, we consider the following simple def- lit 0 inition of the confusion probability PCf(t | t) of representing 0 Pragmatic Speakers. One of the central insights of modern prag- the true intersection set size t as t . Since the visual displays in 0 matics is that speakers’ behavior can be understood, at least Exp. 1 were upper-bounded, PCf(t | t) is defined as the prod- 0 in substantial part, as optimal or near-optimal goal-directed uct of the probability PANS(t | t) of maintaining an approximate 0 action for the purpose of communicating information (7). On representation of the number t as t and the inverse proba- 0 this assumption, speakers are predicted to preferably produce bility PANS(432 − t | 432 − t ). These probabilities, in turn, are utterances not just based on how true they are, but also based on specified following common assumptions in the literature on how likely they are to receive the intended interpretation (9, 10). imprecise representation of number: Consequently, a pragmatic speaker produces utterances propor- 0 0 0 tionally to how likely they are to convey the intended meaning PCf(t | t) ∝ PANS(t | t) PANS(432 − t | 432 − t) to a literally interpreting listener (L ), while also factoring in Z t0+0.5 lit 0 salience as before: PANS(t | t) = Gaussian(x, µ = t, σ = w t)dx. t0−0.5 P (m | t, L) ∝ P (m) P (t | m, L)α, Sprag Sal Llit where The parameter w stands for Weber’s fraction, which represents participants’ sensitivity to relative differences between intersec- PLlit (t | m, L) ∝ L(t, m). tion set sizes. To prevent the possibility that different production This model implies that speakers prefer, all else equal, 1) true models leverage the Weber fraction’s power as a free parameter descriptions over false ones, 2) informative descriptions over unduly, we collected independent data to estimate w based on less informative ones, and 3) more salient descriptions over the same kind of stimuli we used in Exp. 1. To this end, Exp. 3 less salient ones. The higher the “rationality parameter” α, asked 20 participants to estimate the proportion of red circles in the more pronounced is the preference for the optimization of the displays used in Exp. 1. Each participant provided estimates communicative success. (See SI Appendix for more details.) for 24 displays using a continuous slider. Based on the results The pragmatic speaker rule is rooted in standard Gricean of this experiment, we determined the maximum-likelihood esti- pragmatic theory, but also goes beyond. The preference for mate of the Weber fraction, resulting in wˆ = 0.576, and used this informative utterances is reminiscent of how scalar inferences, value for all production models reported here. (See SI Appendix such as the inference from “some” to “some but not all,” are for more information.) usually explained (36). The central idea underlying scalar infer- Adding imprecise number representation to the previously ences is that words that stand in an entailment relation may defined speaker models helps explain gradience in language use. form lexical scales. For example, “all” unilaterally entails “some,” There are certainly other factors that generate gradience (29), which delivers the lexical scale hsome, alli. Since efficient speak- but adding imprecise number representation is motivated given ers are expected to provide as much information as relevant, an the stimuli used in Exp. 1 and sufficient to capture probabilis- utterance with the weaker scalar word “some” may imply that tic gradience in production behavior. If PS (m | t, L) is a speaker the speaker believes that the corresponding sentence with the production rule, either literal or pragmatic, the production prob- stronger scalar word “all” is false. abilities under an approximate representation of the true world The pragmatic speaker model formulated here generalizes this state are: type of reasoning by assuming that words may compete with each CF X 0 other in spontaneous language use, even if they do not stand PS (m | t, L) ∝ PCf(t | t) PS (m | t, L). in a proper entailment relation. For example, consider “some” t0∈T and “few.” These two quantity words do not stand in an entail- Fig. 2B illustrates the effect of factoring in approximate num- ment relation because both may be true when the other is false ber representation on the production of the quantity words. The (e.g., “few” is true, but “some” is false when t is zero; and ANS component asymmetrically increases the amount of gradi- “some” is, true but “few” is false when t equals the total set size). ence present in the semantics, reflecting the fact that estimates Nonetheless, we propose that pragmatic speakers may, for some of the intersection set size are less accurate in the middle range intersection set sizes t, consider both “few” and “some” as viable than around the two extremes. Note that, implemented in this utterances and that they prefer whichever utterance promises to way, speakers factor in effects of imprecise number represen- be more reliable in communicating t. tation in an essentially egocentric way, i.e., they do not take Here, we abstract away from other considerations that may into consideration potential biases in the listener’s estimates of influence speakers’ choice of quantity words. For example, numerosity. “some” and “few” also differ in their argumentative direction, as shown by the following felicity pattern (37): Model Comparison • Some people liked the food, which is {good / ∗bad}. All four models were implemented in Stan (40) to obtain sam- • Few people liked the food, which is {∗good / bad}. ples from the posterior distribution over free parameter values

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PSYCHOLOGICAL AND COGNITIVE SCIENCES (DGfS August 1, 2014). Participants received information about their rights Exp. 3: ANS. For Exp. 3, a total of 20 participants were drafted on AMT. and about the purpose of the study. Afterward, they gave informed consent. Displays were as in Exp. 1. Participants saw a random display from each of 24 bins, each including 18 intersection set sizes, and estimated the proportion Exp. 1: Production. For Exp. 1, there were 600 (Exp. 1a) and 200 (Exp. 1b) of red circles by moving a slider. participants drafted on Amazon’s Mechanical Turk (AMT). Only workers with an internet protocol address from the United States were eligible for partic- Exp. 4: Evaluation. For Exp. 4, a total of 200 participants were drafted on AMT. ipation. In addition, participants were asked in which language they usually Each saw 20 random displays from Exp. 1. Displays were described by sen- counted and were excluded if they did not answer English. (These constraints tences of the form “Q of the circles are red,” where Q was obtained by first hold for all experiments.) Items showed displays containing 432 black or red sampling an arbitrary number 0, ... , 432 and then sampling from the poste- circles, which were randomly scattered across a 25×36 grid. Participants saw rior predictive distribution of each model, as well as from the data observed a random selection of 10 displays. At the bottom of each display, participants in Exp. 1. We only tested items where at least two of the sampled expressions were asked: “How many of the circles are red?” Participants had to complete differed. Participants rated the adequacy of descriptions by adjusting a slider a sentence of the form “—are red” by typing freely into the blank. For the bar (with endpoints labeled “bad” and “good”). Ratings were recalibrated as analyses, we grouped together synonyms. For the purpose of data visualiza- differences from the ratings for the quantity words sampled from the data. tion (Fig. 1 B and C), we binned the data into bins consisting of 10 intersection A mixed-effects linear regression model predicting differential rating on the set sizes, except for the first and last bins, which only contained data for inter- basis of the source of the quantity word (using the data as reference category) section set sizes 0 and 432. Moreover, the second-to-last bin contained 11 with random intercepts for participants, quantity words, and intersection set intersection set sizes, also including data for intersection set size 431. sizes was the maximal converging model.

Exp. 2: Monotonicity. For Exp. 2, a total of 120 participants were drafted on Data Availability. Anonymized experimental data and model code have AMT. Based on a pretest (Exp. 2a), 17 predicate pairs hP1, P2i were selected, been deposited at Open Science Framework (DOI: 10.17605/OSF.IO/HSYTK) such that participants agreed that P1 entailed P2, e.g., hplay poker, play (50). cardsi. Using these predicate pairs, two types of arguments were randomly generated for each quantity word: ACKNOWLEDGMENTS. We thank R. Ba˚ ath˚ for contributing to the design of Exp. 1 and giving permission to make use of the data. We thank B. Geurts, • Q of the people P1. → Q of the people P2. G. Jager,¨ G. Scontras, and K. Syrett for valuable feedback. This research was • Q of the people P2. → Q of the people P1. supported by German Research Council Grants FR 3482/2-1, KR 951/14-1, SA 925/4-1, SA 925/11-1, and SA 925/17-1, in part within Grant SPP 1727 Participants indicated whether these arguments were valid. Two anno- (Xprag.de); and by Dutch Science Organisation Gravitation Grant “Language tated examples illustrated the notion of validity. in Interaction” 024.001.006.

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