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BY 4.0 license Open Access Published by De Gruyter Mouton December 22, 2022

Modeling (im)precision in context

  • Roland Mühlenbernd EMAIL logo and Stephanie Solt
From the journal Linguistics Vanguard

Abstract

Speakers’ choice between linguistic alternatives often depends on the situation, a prime example involving level of precision at which numerical information is communicated. We report on a production study in which participants report the time of an event in two different situations, and demonstrate that the results can be reproduced by a probabilistic game-theoretical model in which the speaker’s choice reflects a tradeoff between informativity, accuracy and hearer-oriented simplification. These findings shed light on the pragmatics of (im)precision, and the dynamics of situationally driven pragmatic variation more generally.

1 Introduction

It is well known that speakers’ choice of linguistic forms varies according to the utterance situation. This is particularly well documented in the case of classic sociolinguistic variables such as the velar versus alveolar realization of ING, e.g. working versus workin’ (Labov 1966, and ff.). But situationally driven variation is also observed for alternatives that differ in their semantic content, a domain more usually investigated in semantics and pragmatics rather than sociolinguistics.

A nice example of such pragmatic variation involves the level of precision at which numerical information is conveyed. A speaker who wishes to communicate the time at which a certain event occurred might do so in at least three ways: with a precise non-round number (1a); by rounding off to a multiple of 5 (1b); or by using an explicit approximator such as about or roughly (1c).

(1)
a.
The accident happened at 8:31.
b.
The accident happened at 8:30.
c.
The accident happened at about 8:30.

Intuitively, the choice between such alternatives depends on the situation: in a casual conversation, (1b) or (1c) would seem most appropriate, whereas when giving a statement to a police officer, (1a) might be a better choice. Indeed, Van der Henst et al. (2002) demonstrate empirically that speakers often round off when telling the time, but they do so less frequently when they perceive their interlocutor to need a more precise value.

In the theoretical literature on the pragmatics of imprecision (e.g. Krifka 2002, 2009; Lasersohn 1999), it is widely accepted that contexts differ with respect to the degree of precision that is required. But a puzzling question remains. Clearly, a speaker who has only imprecise knowledge of the relevant value must choose an approximate expression to convey that. But why would a speaker with precise knowledge choose to round off in communicating their knowledge, given that this is both less informative than reporting the precise value and seemingly requires greater effort? Two main sorts of explanations have been put forward. First, by giving up some precision, the speaker potentially gains in accuracy: by choosing a more approximate form, the speaker may avoid negative consequences of saying something false if their information turns out to be incorrect, e.g. if their wristwatch is off by a few minutes (Krifka 2002; Pinkal 1995). Secondly, it is proposed that round numerical expressions are in some way easier or less costly than precise ones. This might involve lower effort required on the part of the speaker, round numbers being on average shorter than non-round ones (Kao et al. 2014; Krifka 2002); alternately, it might reflect greater ease of comprehension or recall on the part of the hearer (Van der Henst et al. 2002); such an advantage for round numbers is demonstrated experimentally by Solt et al. (2017) for temporal expressions and by Nguyen et al. (2022) for other types of numerical expressions. Either way, the consequence is that in situations in which precise detail is not relevant, a round expression will be preferred.

Yet while the factors that underlie the choice of precision level have been investigated, there is to date no explicit formal model of how speakers weigh factors such as informativity, truthfulness, simplicity and processing ease to arrive at a choice between precise and approximate numerical forms, and how this choice depends on the context. Recently, the methods of probabilistic game-theoretic pragmatics have been successfully applied to model other instances of situationally driven linguistically variation. Two notable examples of such work are Burnett’s (2019) analysis of the variable realization of ING via the novel Social Meaning Games framework, and Yoon et al.’s (2020) modeling of the choice between evaluative adjectives – for example, whether a performance is characterized as terrible, not bad or amazing – as depending on whether the speaker’s goal is to be polite, informative or both. What these two analyses have in common is that they feature interlocutors who reason about each other’s beliefs and goals, with the speaker’s choice between forms optimized to meet certain potentially conflicting criteria. These characteristics are similar to those proposed to underlie the phenomenon of rounding, suggesting a similar approach to the present topic. A simple non-probabilistic model of this sort is sketched out by Jäger (2013), but he does not test the model against actual speaker data. Somewhat relatedly, Egré et al. (2021) develop a Rational Speech Act model of the choice between around n and between n and m in cases of imprecise knowledge, but do not address situational variation, and likewise do not attempt an experimental validation.

In this paper, we extend this body of work. We first present a production study illustrating how speakers’ choice of precision level varies by context, and then fit a game-theoretic probabilistic choice model to the results. Through this modeling process we gain insights into the pragmatics of imprecision, and the dynamics of situationally driven pragmatic variation more generally.

2 The imprecision experiment

2.1 Design and materials

A production experiment was conducted to obtain robust empirical data on the choice of precision level by context. Participants read a scenario in which they had witnessed an automobile accident; the time it occurred (as recalled by the participant) was displayed visually. Participants then read one of two continuations in which an interlocutor asks what time the accident took place, and typed in their answer in a text box. Subsequently, participants were asked the reasons for their answer.

Seven information states were tested: the precise states 8:30, 8:30 ± 1, 8:30 ± 2, 8:30 ± 3, 8:30 ± 4, 8:30 ± 5 (where 8:30 ± 1 etc. means that half of respondents saw 8:29 while half saw 8:31); and the approximate state 8:26–8:34. These were tested in two contexts: a police station, where a police officer takes a witness statement; and a party, where a neighbor asks about the accident. The first of these was intended to exemplify a situation where both precision and accuracy are important, and was predicted to elicit a high level of precise/non-round responses. The second was designed to represent a situation where precise detail and absolute accuracy are less important than facilitating hearer comprehension, and was predicted to elicit a higher level of approximate/rounded responses. This resulted in 14 (2 × 7) conditions in total, which were tested in a single-item fully between-subjects design. See Figure 1 for the full text of both contexts and Figure A.1 in the appendix for depictions of all information states.

Figure 1: 
Screenshot of experimental tasks. Above: police context and information state 8:30 ± 3 (here 8.27); below: neighbor context and information state 8.30 ± 5 (here 8:35) (images from Freepik.com).
Figure 1:

Screenshot of experimental tasks. Above: police context and information state 8:30 ± 3 (here 8.27); below: neighbor context and information state 8.30 ± 5 (here 8:35) (images from Freepik.com).

The experiment was programmed in LabVanced; links are listed in Appendix Table A.1.

2.2 Participants

499 self-declared native speakers of English were recruited via Prolific, comprising approximately 30 participants/condition in each of the 12 precise information state conditions and 60 participants/condition in the 2 approximate information state conditions (see Appendix Table A.2 for details). Participants completed an informed consent approved by the Ethics Committee of the German Linguistic Society (DGfS) and were paid £0.50 for their participation.

2.3 Results

Responses to the primary ‘what time?’ question were coded according to the time value (e.g. 8:30, 8:34) and the presence of an approximator (e.g. about, approximately). Truth-conditionally equivalent responses were collapsed together (e.g. eight thirty and half past eight were both coded as 8:30).[1] Interval-denoting expressions (e.g. between 8:26 am and 8:34 am) were coded separately. Data from 24 respondents could not be successfully coded; this consisted of cases where the participant had misread the clock representation (which occurred primarily in the approximate information state condition) or otherwise misunderstood the experimental task. These data points were excluded from further analysis and model fitting. Normalized results matrices are shown in Figure 2 (Matrices with absolute numbers are given in Appendix Figure A.2).

Figure 2: 
Normalized color-coded matrices representing speaker behavior, left for the police context, right for the neighbor context. Each entry represents proportion of participants in a particular information state (row units) using a particular (categorized) response expression (column units). Note that response expressions with subscript ‘a’ (e.g. 




v

a
30





${v}_{a30}$



) involve an approximator term, and 




v

I
n





${v}_{In}$



 represents interval-denoting expressions.
Figure 2:

Normalized color-coded matrices representing speaker behavior, left for the police context, right for the neighbor context. Each entry represents proportion of participants in a particular information state (row units) using a particular (categorized) response expression (column units). Note that response expressions with subscript ‘a’ (e.g. v a 30 ) involve an approximator term, and v I n represents interval-denoting expressions.

The primary findings can be summarized as follows:

  1. Most importantly, participants in precise non-round information states (8:30 ± 1, 8:30 ± 2, 8:30 ± 3, 8:30 ± 4) frequently ‘rounded off’ their answers to values divisible by 15 (e.g. [about] eight thirty) or divisible by 5 (e.g. [approximately] 8:25). But as predicted, a difference in the frequency of rounding was observed between the two contexts (χ 2 = 13.198, p < 0.001), with more rounded answers in the neighbor context than the police context. Both bare round numbers and approximator-modified numbers were used in rounding (see Appendix Figure A.3 for details on choice of approximator by context).

  2. Secondly, participants in the precise round information states 8:30 and 8:30 ± 5 almost universally used bare round numbers.

  3. Finally, participants in the approximate information state 8:26–8:34 gave three types of answers: the bare round value 8:30; approximator +8:30; and interval expressions such as between 8:26 and 8:34; the latter expression type is not used in the other conditions. There was a near-significant difference in answer type between context conditions (χ 2 = 5.121, p = 0.0773), with interval expressions tending to be used more frequently in the police context, and bare round numbers more frequently in the neighbor context.

Participants’ justifications for their answers were likewise coded into broad categories; results are summarized in Appendix Table A.3. Here we observe that participants’ own reasoning reflects factors posited in the theoretical literature to underlie the choice between precise and approximate numerical expressions, including the level of detail required by the context and the interlocutor, the lack of precise information or possibility of incorrect information, the importance of accuracy or truthfulness, and considerations of what is easier to say and/or easier for the hearer to understand and remember. Furthermore, typical justifications tended to differ by context, with mentions of accuracy and possible information lack/misinformation more frequent in the police context, and mentions of habit/convention, contextual appropriateness and speaker/hearer ease more frequent in the neighbor context.

Complete participant-level data are part or the supplementary material and also available at: https://www.doi.org/10.6084/m9.figshare.21629531.

3 The imprecision model

In this section we introduce the main aspects of the Imprecision Model in a semiformal manner. A complete formal definition including the full implementation is given in the Technical Report. The Python implementation of the model is available at: https://osf.io/ur5fp/?view_only=7d646308e246437caee7015a96650da3.

3.1 Model structure

The Imprecision Model predicts pragmatic speaker behavior, more concretely, the probability that a pragmatic speaker will use an utterance v given an information state s. In the present implementation, both information states and utterances make reference to the temporal domain T N , with the set of information states S corresponding to the 7 states tested in the production experiment, and the set V of utterances reflecting answers provided by participants.[2] Formally, each s S is a (possibly singleton) set of time points t T ; for details see Technical Report, Table 1. Each v V is an utterance, which is assigned a semantic (or literal) interpretation by a denotation function , which returns a (possibly singleton) set of time points t T ; for details see Appendix, Table A.4.

We can then model a speaker’s probability P S ( v | s , P H , w ) to produce an utterance v , given (i) her information state s, (ii) an assumption about the hearer’s interpretation, defined by a probability function P H (details in Section 3.2.1), and (iii) a vector of goal weights w, that determines how much the speaker prioritizes particular goals (details in Section 3.2). Following standard practice in models of probabilistic pragmatics, the speaker’s production decision is defined by a so-called softmax function, which interpolates between choosing the maximum-utility utterance and probability matching via optimality parameter λ (cf. Franke and Jäger 2016; Goodman and Stuhlmüller 2013).[3] Formally, it is defined as follows:

P S ( v | s , P H , w ) exp [ λ U t o t ( v , s , P H , w ) ]

whereby U tot is the speaker’s total utility for using utterance v given information state s, assumed hearer’s interpretation P H and goal weights w.

3.2 Speaker utilities

We posit that a speaker’s total utility contains distinct components that represent different goals that speakers may entertain. Formally, the speaker’s total utility U tot for using utterance v given information state s, assumed hearer interpretation P H and goal weights w = ( w R , w A ) is defined as follows:

U t o t ( v , s , P H , w ) = U inf ( v , s , P H ) + w R U r n d ( v ) + w A U a c c ( v , s ) C ( v )

U tot is a combination of the informational utility U inf, and the weighted addends roundness utility U rnd and accuracy utility U acc, minus the utterance cost C ( v ) , which is used to capture the general pressure toward economy in speech (e.g. longer utterances are more costly).[4] The different speaker utilities U inf, U rnd and U acc are manifestations of different goals that the speaker entertains in situation where numerical information is communicated. In a nutshell, U inf represents the speaker’s goal of informativity, U rnd represents the speaker’s goal of hearer-oriented simplification, and U acc represents the speaker’s goal of accuracy.[5]

We assume that a cooperative speaker follows the goal of informativity to a full degree. The rationale is that we see informativity as the most basic and general speaker goal, which should be present across situations; therefore, U inf is assumed to be fully weighted with factor 1 in both contexts.[6] The goals of hearer-oriented simplification and accuracy may be only followed to some degree. Therefore, U rnd and U acc are weighted with the weight factors w R and w A , respectively, whereby 0 w R , w A 1 . Both weights represent particular pressures in the speaker’s language use: w R represents the pressure towards hearer-oriented simplification and w A represents the pressure towards accuracy.

3.2.1 Informational utility

The informational utility U inf represents the speaker’s goal of informativity: it expresses how informative an utterance v is assumed to be for the hearer, given information state s and an assumption about the hearer’s interpretation P H . As it is standard in game-theoretic pragmatics, U inf is defined as follows:

U inf ( v , s , P H ) = s S P H ( s | v ) × π ( s , s )

whereby P H ( s | v ) determines the probability that the hearer construes utterance v with s, and π ( s , s ) is a payoff function that determines how useful it is to obtain interpretation s when the actual information state is s.

More concretely, P H ( s | v ) represents the speaker’s belief regarding how likely it would be for the hearer to guess information state s upon receiving utterance v . In our model, we stipulate that the speaker believes that the hearer believes that the speaker’s language use follows from a so-called truthful speaker strategy P S 0 (details further down). Formally, the speaker believes that P H ( s | v ) is defined as a Bayesian update to a truthful speaker strategy P S 0 , as follows:

P H ( s | v ) Pr ( s ) P S 0 ( v | s )

whereby Pr ( s ) is the prior probability of being in information state s. As already mentioned, P S 0 ( v | s ) is a truthful speaker strategy, on which a speaker uses utterances with respect to their core-semantic meaning. More precisely, according to a truthful speaker strategy P S 0 , the sender uses utterance v in information state s if and only if v is truthful in s, where an utterance v is truthful in an information state s iff s is a subset of its core-semantic meaning. More details and a formal definition can be found in the Technical Report.

In many models, the standard assumption for the payoff function is π ( s , s ) = 1 if s = s , else 0. In other words, it is assumed that an interpretation is only useful when it perfectly fits to the information state. In our model, the payoff function π captures the idea that communication is successful in proportion to how closely the hearer’s interpretation matches the actual information state. With respect to the given model, this means that interpretations that are less distant from an information state can be more informative than interpretations that are more distant from it. We can formalize this idea with a perceptual similarity function π, as defined by Nosofsky (1986), and as similarly used in diverse studies that apply related game-theoretic model approaches towards language use (cf. Franke 2014; Franke and Correia 2018; Jäger and van Rooij 2007). π is defined as follows:

π ( s , s ) = exp ( d i s ( s , s ) 2 α 2 )

whereby d i s ( s , s ) is a function that returns a distance value between s and s . More concretely, d i s ( s , s ) computes the average minimal distance between all items in s and s . The formal definition can be found in the Technical Report. Moreover, α is an imprecision parameter, which represents the degree to which being imprecise has an impact on the payoff. The higher α, the less is the impact of being imprecise on the utilities.[7] For more details, see the Technical Report.

3.2.2 Roundness utility

The roundness utility U rnd represents the speaker’s goal of hearer-oriented simplification: the roundness utility of the utterance accommodates the hearer’s task to better perceive and memorize the utterance, reflecting the processing advantage for round numbers (see above).

We distinguish different levels of roundness with respect to a roundness hierarchy. We define a function rnd(n) that returns a roundness level from a numeric utterance (for details see the Technical Report). All utterances’ roundness levels are listed in Table A.4. The roundness utility U r n d ( v ) of an utterance v is defined as follows:

U r n d ( v ) = r n d ( v ) max v V r n d ( v )

where the roundness level of utterance v is divided by the maximal roundness level over all utterances that are relevant in the given scenario (in other words, all v V of the given model). This number is always between 0 (no roundness) and 1 (maximal roundness). In the present model we assigned the highest roundness utility to multiples of 15, an intermediate utility to multiples of 5 (but not 15), and 0 to non-round values.

3.2.3 Accuracy utility

The accuracy utility U acc represents the speaker’s goal of accuracy: it rewards the chance of the literal interpretation [8] of utterance v being true given information state s.

In the situation where the speaker’s knowledge is imprecise, there can be utterances which have a better chance to communicate the true state of the world (or short: to be accurate) than others. Let’s give an example: imagine that the speaker’s information state is s I n = { 26,27 33,34 } , that is, that she knows that the event happened at a time in that set but does not know which one. When she uses the utterance v 32 , then there is a likelihood of 1 / 9 that the literal interpretation of her utterance is true. However, when she uses utterance v a 30 , and we assume that v a 30 = { 27,28,29,30,31,32,33 } , then there is a likelihood of 7 / 9 that the literal interpretation of her utterance is true. Here, by using the utterance v a 30 , the speaker has a better chance to be accurate than by using v 32 . Formally, this is because the core-semantic meaning of v a 30 covers a greater part of the information state s I n than the core-semantic meaning of v 32 .

Following the line of thought above, we define the accuracy utility U a c c ( v , s ) as the likelihood for the literal interpretation of utterance v being true given information state s, as follows:

U a c c ( v , s ) = | v s | | s |

U a c c ( v , s ) is a value between 0 and 1. Value 0 means that there is no overlap between the core-semantic meaning v and information state s and the literal interpretation of the utterance v is definitely false. Value 1 means that the core-semantic meaning v totally covers information state s and the literal interpretation of the utterance v is definitely true.[9]

4 Reproduction of empirical data

From the experiment we obtained empirical data on speaker behavior, where proportions (e.g. matrix entries in the matrices of Figure 2) can be interpreted as probabilities. The goal of the Imprecision Model is to reconstruct such probabilistic speaker behavior. Here, the probability of using utterance v for information state s is given by the probability function P S ( v | s , P H , w ) . Moreover, we assume that the difference in probabilities across contexts is mainly due to the fact that speakers follow particular goals with different strengths in different situations. More concretely, we assume that the difference in speaker behavior in the police versus neighbor contexts is mainly due to the fact that they weight the goals of speaker-oriented simplification (weighted by w R ) and accuracy (weighted by w A ) differently. By calibrating the weight parameters w R and w A , we are aiming to find the optimal reconstruction matrix for each context: we are searching for the optimal parameters w R * and w A * that minimize the so-called mean square error (MSE) between empirical data and model predictions (for more details see the Technical Report).

The results of this analysis are as follows: the ‘police context’ matrix is best reconstructed with the parameter values w R * = 0.46 , w A * = 0.22 , achieving an MSE of 0.00139. The ‘neighbor context’ matrix is best reconstructed with the parameter values w R * = 0.66 , w A * = 0.02 , achieving an MSE of 0.00114. These parameter values are global optima, as illustrated in Appendix Figure A.4, which shows the MSE heat map over the ( w R , w A ) -space for each context.

Figure 3 displays the juxtaposition of optimal parameter values across both matrices. These finding show that speaker goals are differently weighted across contexts: the pressure towards hearer-oriented simplification (weighted by w R ) is stronger in the neighbor context than in the police context, whereas for the pressure towards accuracy (weighted by w A ), it is exactly the other way around. Further model checks showed that this cross-contextual difference of optimal weights w R * and w A * is very robust across alternative model settings.[10]

Figure 3: 
Optimal parameter values for 




w
R




${w}_{R}$



 and 




w
A




${w}_{A}$



 that minimize the mean square error between empirical data and model prediction.
Figure 3:

Optimal parameter values for w R and w A that minimize the mean square error between empirical data and model prediction.

As a next step we want to evaluate the quality of the matrix reconstructions. In Figure 4 we juxtapose the empirical matrices (top) with the optimal reconstructions (bottom). This comparison gives a first visual impression. In a next step we conducted a statistical analysis of the reconstruction data. We computed the Pearson correlation coefficient r and the coefficient of determination r 2 between the empirical data (independent variable) and the reconstructed data (dependent variable), taking the identity function y = x as linear regression. The r 2 value with identity function as linear regression is also known as the ‘goodness-of-fit’ value. The results over all 63 data points were: r = 0.985 , r 2 = 0.971 for the police context, and r = 0.985 , r 2 = 0.967 for the neighbor context.[11] The presentation of data points and regression line, including an overview of the statistical results, is given in Figure 5.

Figure 4: 
The empirical data (top) and the best reconstructions of the model (bottom) of the police context (left) and the neighbor context (right).
Figure 4:

The empirical data (top) and the best reconstructions of the model (bottom) of the police context (left) and the neighbor context (right).

Figure 5: 
Representation of all data points of the police context matrix (left) and the neighbor context matrix (right), positioned with respect to the empirical data from the experiment (x-axis) and the model prediction data from the best reconstruction (y-axis). The red line represents the identity function (with x=y), used as linear regression for r
2 calculations. The top left corner lifts the Pearson correlation coefficient r, the coefficient of determination (goodness-of-fit) for all data points 




r
2


(
a
l
l
)




${r}^{2}\left(all\right)$



, and the goodness-of-fit for ’non-near-zero’ values 




r
2


(
n
n
z
)




${r}^{2}\left(nnz\right)$



, where x and y both are not below 0.01.
Figure 5:

Representation of all data points of the police context matrix (left) and the neighbor context matrix (right), positioned with respect to the empirical data from the experiment (x-axis) and the model prediction data from the best reconstruction (y-axis). The red line represents the identity function (with x=y), used as linear regression for r 2 calculations. The top left corner lifts the Pearson correlation coefficient r, the coefficient of determination (goodness-of-fit) for all data points r 2 ( a l l ) , and the goodness-of-fit for ’non-near-zero’ values r 2 ( n n z ) , where x and y both are not below 0.01.

Finally, we tested our model against a second version of the Imprecision model where both weights w R and w A are assumed to be invariant across contexts. For this second version, the best reconstruction has a clearly worse mean square error (around 0.00351, in comparison to 0.00139/0.00114), and achieves a lower Pearson correlation (0.969 vs 0.985) and a lower r-square value (0.935 vs 0.967/0.971). The results are juxtaposed in the Appendix, Figure A.5; more details are given in the Technical Report.

All in all, the statistical analysis of our model yields correlation values and goodness-of-fit values that are relatively close to 1, thus close to a perfect reconstruction. Moreover, with respect to mean square error, Pearson correlation coefficient and goodness-of-fit-value, our model outperforms a second version of the Imprecision model where weights w R and w A are assumed to be invariant across contexts. This shows that the central underlying assumption, namely that cross-situational variation in a speaker’s precision level is to a great extent due to different weighting of the two speaker goals of hearer-oriented simplification and accuracy, is strongly supported by the model comparison analysis.

5 Conclusions

In this paper, we have demonstrated that speakers’ choice of how precisely to communicate numerical information depends on the utterance situation, and shown that these patterns of behavior can be reproduced by a probabilistic game-theoretic model featuring a multi-component utility function. Through this modeling approach, we have found support for the view that situational variation in precision level derives from variation in the goals that the speaker pursues in different situations, with the relative importance of accuracy versus hearer-oriented simplification playing a crucial role in the two scenarios tested.

We see these findings as important in their own right, numerical expressions being frequent in language. More generally, we take these findings to demonstrate the value of probabilistic game-theoretic modeling as a route to understanding the dynamics of situationally driven pragmatic variation. In future work we plan to extend this approach to additional phenomena and parameters of situational variation.


Corresponding author: Roland Mühlenbernd, Leibniz-Zentrum Allgemeine Sprachwissenschaft (ZAS), Berlin, Germany, E-mail:

Award Identifier / Grant number: SFB 1412, 416591334

Acknowledgments

We thank Heather Burnett, Manfred Krifka, Uli Sauerland and the audiences at the ZAS and Humboldt University for helpful discussion, and Alexandra Fossa and Hadewych Versteegh for assistance with the data analysis of the experimental data.

  1. Research Funding: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB 1412, 416591334.

Appendix A: Supplementary tables and figures

Table A.1:

The experimental sessions 1 and 2, the date when each took place, the number of participants, and links to the experiment, designed with LabVanced and stored at the LabVanced server. Of the 502 participants, 3 did not complete the experiment, and were excluded for analysis. Participants took on average 4.38 min for the whole session.

ID Date Participants Link
1 May 3rd 2021 250 www.labvanced.com/player.html?id=22972
2 May 17th 2021 252 www.labvanced.com/player.html?id=23753
Table A.2:

Distribution of participants over 14 different conditions.

Situation Information Participants
Police context 8:30 32
Police context 8:30 ± 1 27
Police context 8:30 ± 2 28
Police context 8:30 ± 3 35
Police context 8:30 ± 4 29
Police context 8:30 ± 5 30
Police context 8:26–8:34 65
Neighbor context 8:30 31
Neighbor context 8:30 ± 1 32
Neighbor context 8:30 ± 2 35
Neighbor context 8:30 ± 3 31
Neighbor context 8:30 ± 4 32
Neighbor context 8:30 ± 5 32
Neighbor context 8:26 − 8:34 60

499
Table A.3:

Categorization of reason(s) for choice (# of respondents; multiple categories possible).

Answer category Police Neighbor
Level of precision/detail 51 61
Accuracy/truthfulness 28 20
Possible lack of information 17 13
Possible misinformation 9 3
Safe choice 2 2
Hearer needs 19 13
Appropriateness for context 15 33
Speaker ease 7 22
Hearer ease 6 14
Habit/convention 35 52
How it sounds 2 7
Other/irrelevant 108 105

Total # of respondents 231 244
Table A.4:

Utterances v V , each with its corresponding core semantic meaning v as a subset of the temporal domain T, its roundness level r n d ( v ) , and a sample utterance. Here, utterances with value that are completely divisible by 15 have the highest roundness level of 2, remaining utterances with values that are divisible by 5 have a roundness level of 1, and all remaining utterances have a roundness level of 0; these correspond to distinct granularity levels in the temporal domain discussed in Krifka (2009).

v V Core semantic meaning 〚v rnd(v) Sample utterance
v 25 { 25 } 1 ‘ … at 8:25.’
v 26 { 26 } 0 ‘ … at 8:26.’
v 27 { 27 } 0 ‘ … at 8:27.’
v 28 { 28 } 0 ‘ … at 8:28.’
v 29 { 29 } 0 ‘ … at 8:29.’
v 30 { 30 } 2 ‘ … at 8:30.’
v 31 { 31 } 0 ‘ … at 8:31.’
v 32 { 32 } 0 ‘ … at 8:32.’
v 33 { 33 } 0 ‘ … at 8:33.’
v 34 { 34 } 0 ‘ … at 8:34.’
v 35 { 35 } 1 ‘ … at 8:35.’
v a 25 { 22,23,24,25,26,27,28 } 1 ‘ … around 8:25.’
v a 30 { 27,28,29,30,31,32,33 } 2 ‘ … about 8:30.’
v a 35 { 32,33,34,35,36,37,38 } 1 ‘ … approximately at 8:35.’
v I n { 25,26,27,28,29,30,31,32,33,34,35 } 0 ‘ … between 8:25 and 8:35.’
Figure A.1: 
Images of the different clocks presenting the different information states.
Figure A.1:

Images of the different clocks presenting the different information states.

Figure A.2: 
Participant responses (absolute numbers) by information state in (a) police context and (b) neighbor context. ‘Other’ responses are those for which participants’ justifications indicated they had misread the clock face (e.g. reading 8:32 as 8:37 or incorrectly interpreting the approximate state 8:26–8:34 as representing an interval around 6 o’clock) or otherwise misunderstood the experimental task. These responses were excluded from further analysis and model fitting.
Figure A.2:

Participant responses (absolute numbers) by information state in (a) police context and (b) neighbor context. ‘Other’ responses are those for which participants’ justifications indicated they had misread the clock face (e.g. reading 8:32 as 8:37 or incorrectly interpreting the approximate state 8:26–8:34 as representing an interval around 6 o’clock) or otherwise misunderstood the experimental task. These responses were excluded from further analysis and model fitting.

Figure A.3: 
Normalized distribution of approximator term choices in each context. Here, ‘approx’ stands for ‘approximately’, and ‘just b/a’ stands for ‘just before’ and ‘just after’.
Figure A.3:

Normalized distribution of approximator term choices in each context. Here, ‘approx’ stands for ‘approximately’, and ‘just b/a’ stands for ‘just before’ and ‘just after’.

Figure A.4: 
Mean square errors between an empirical production matrix and computationally reconstructed matrices over different parameter values for 




w
R




${w}_{R}$



 (x axis) and 




w
A




${w}_{A}$



 (y axis), left for police context, right for neighbor context. Mean square error values are coded from high value 



≥
0.0035



$\ge 0.0035$



 (dark blue) down to 0.001 (dark red); color bar on the right. The juxtaposition of both heat maps shows that the optimal reconstruction of the ‘police context’ matrix involves a lower 




w
R




${w}_{R}$



 value and a higher 




w
A




${w}_{A}$



 value than the optimal reconstruction of the ‘neighbor context’ matrix.
Figure A.4:

Mean square errors between an empirical production matrix and computationally reconstructed matrices over different parameter values for w R (x axis) and w A (y axis), left for police context, right for neighbor context. Mean square error values are coded from high value 0.0035 (dark blue) down to 0.001 (dark red); color bar on the right. The juxtaposition of both heat maps shows that the optimal reconstruction of the ‘police context’ matrix involves a lower w R value and a higher w A value than the optimal reconstruction of the ‘neighbor context’ matrix.

Figure A.5: 
Juxtaposition of mean square error (left), Pearson correlation and goodness-of-fit value (right) for the optimal reconstruction of the police context matrix (blue bar), the neighbor context matrix (red bar) and for all data points of both matrices combined (across contexts, beige bar). The ’across context’ model assumes no difference of weights 




w
R




${w}_{R}$



 and 




w
A




${w}_{A}$



 across both contexts. It is optimal reconstruction achieves a clearly worse mean square error (about triple as high), as well as a worse Pearson correlation 



r



$r$



 and a worse goodness-of-fir value 




r
2




${r}^{2}$



.
Figure A.5:

Juxtaposition of mean square error (left), Pearson correlation and goodness-of-fit value (right) for the optimal reconstruction of the police context matrix (blue bar), the neighbor context matrix (red bar) and for all data points of both matrices combined (across contexts, beige bar). The ’across context’ model assumes no difference of weights w R and w A across both contexts. It is optimal reconstruction achieves a clearly worse mean square error (about triple as high), as well as a worse Pearson correlation r and a worse goodness-of-fir value r 2 .

References

Burnett, Heather. 2019. Signalling games, sociolinguistic variation and the construction of style. Linguistics and Philosophy 42. 419–450. https://doi.org/10.1007/s10988-018-9254-y.Search in Google Scholar

Egré, Paul, Benjamin Spector, Adèle Mortier & Steven Verheyen. 2021. On the optimality of vagueness: “around”, “between”, and the Gricean maxims. Available at: https://arxiv.org/pdf/2008.11841.pdf.Search in Google Scholar

Franke, Michael. 2014. Typical use of quantifiers: a probabilistic speaker model. In Proceedings of the Annual Meeting of the Cognitive Science Society, Vol. 36.Search in Google Scholar

Franke, Michael & José Pedro Correia. 2018. Vagueness and imprecise imitation in signalling games. The British Journal for the Philosophy of Science 69(4). 1037–1067. https://doi.org/10.1093/bjps/axx002.Search in Google Scholar

Franke, Michael & Gerhard Jäger. 2016. Probabilistic pragmatics, or why Bayes’ rule is probably important for pragmatics. Zeitschrift für Sprachwissenschaft 35(1). 3–44. https://doi.org/10.1515/zfs-2016-0002.Search in Google Scholar

Goodman, Noah & Andreas Stuhlmüller. 2013. Knowledge and implicature: modeling language understanding as social cognition. Topics in Cognitive Science 5. 173–184. https://doi.org/10.1111/tops.12007.Search in Google Scholar

Jäger, Gerhard. 2013. Game theory in semantics and pragmatics. In C. Maienborn, P. Portner & K. von Heusinger (eds.), Semantics. An International Handbook of Natural Language Meaning, vol. 3, 2487–2516. Berlin: De Gruyter.Search in Google Scholar

Jäger, Gerhard & Robert van Rooij. 2007. Language structure: psychological and social constraints. Synthese 159(1). 99–113. https://doi.org/10.1007/s11229-006-9073-5.Search in Google Scholar

Kao, Justine T., Jean Y. Wu, Leon Bergen & Noah D. Goodman. 2014. Nonliteral understanding of number words. Proceedings of the National Academy of Sciences 33. 12002–12007. https://doi.org/10.1073/pnas.1407479111.Search in Google Scholar

Krifka, Manfred. 2002. Be brief and be vague! and how bidirectional optimality theory allows for verbosity and precision. In David Restle & Dietmar Zaefferer (eds.), Sounds and Systems. Studies in Structure and Change: A Festschrift for Theo Vennemann, 439–458. Berlin: Mouton De Gruyter.10.1515/9783110894653.439Search in Google Scholar

Krifka, Manfred. 2009. Approximate interpretations of number words: a case for strategic communication. In Erhard W. Hinrichs & John Nerbonne (eds.), Theory and Evidence in Semantics, 109–132. Stanford: CSLI Publications.Search in Google Scholar

Labov, William. 1966. The social stratification of English in New York city. Cambridge: Cambridge University Press.Search in Google Scholar

Lasersohn, Peter. 1999. Pragmatic halos. Language 75(3). 522–551. https://doi.org/10.2307/417059.Search in Google Scholar

Nguyen, Huy Anh, Jake M. Hofman & Daniel G. Goldstein. 2022. Round numbers can sharpen cognition. In CHI Conference on Human Factors in Computing Systems (CHI ’22). New York: ACM.10.1145/3491102.3501852Search in Google Scholar

Nosofsky, Robert M. 1986. Attention, similarity, and the identification-categorization relationship. Journal of Experimental Psychology: General 115(1). 39–57. https://doi.org/10.1037/0096-3445.115.1.39.Search in Google Scholar

Pinkal, Manfred. 1995. Logic and Lexicon: The Semantics of the Indefinite. Dordrecht: Kluwer.10.1007/978-94-015-8445-6Search in Google Scholar

Solt, Stephanie, Chris Cummins & Marijan Palmović. 2017. The preference for approximation. International Review of Pragmatics 9(2). 248–268. https://doi.org/10.1163/18773109-00901010.Search in Google Scholar

Van der Henst, Jean-Baptisete, Laure Carles & Dan Sperber. 2002. Truthfulness and relevance in telling the time. Mind & Language 81(17). 457–466. https://doi.org/10.1111/1468-0017.00207.Search in Google Scholar

Yoon, Erica J., Michael Henry Tessler, Noah D. Goodman & Michael C. Frank. 2020. Polite speech emerges from competing social goals. Open Mind 4. 71–87. https://doi.org/10.1162/opmi_a_00035.Search in Google Scholar


Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/lingvan-2022-0035).


Received: 2022-03-27
Accepted: 2022-09-26
Published Online: 2022-12-22

© 2022 the author(s), published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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