Lost your marbles? The puzzle of dependent measures in experimental pragmatics
Judith Degen (jdegen@stanford.edu), Noah D. Goodman (ngoodman@stanford.edu)
Department of Psychology, Stanford University
Stanford, CA USA 94350
Abstract
A rarely discussed but important issue in research on pragmatic
inference is the choice of dependent measure for estimating
the robustness of pragmatic inferences and their sensitivity tocontextual manipulations. Here we present the results fromthree studies exploring the effect of contextual manipulations
on scalar implicature. In all three studies we manipulate thesalient question under discussion and the perceptual availability of relevant set sizes. The studies differ only in the dependent measure used: Exp. 1 uses truth judgements, Exp. 2 uses
word probability ratings, and Exp. 3 uses a direct measure ofsentence interpretation. We argue that the first two are effectively measures of production, and find they are sensitive to ourcontextual manipulations. In contrast the interpretation measure shows no effect of context. We argue that this methodologically troubling finding can be understood and predictedby using the framework of probabilistic pragmatics.
Keywords: pragmatics; psycholinguistics; scalar implicature;
QUD; methodology

Introduction
Context affects language understanding in complex and profound ways. Relatively small changes in the context of anutterance can radically change the interpreted meaning—or
fail to entirely. For instance, take a scalar implicature: “someof the candies in that box contained nuts” will often be pragmatically strengthened to “and not all of them did”. However, finding out that the speaker ate only two candies (thuslacked knowledge of the stronger situation) or has a nut allergy (thus cares only if any candies contain nuts) can decrease the strength of this implicature (i.e., make it more plausible that all the candies had nuts).
Pragmatic judgments, e.g., about scalar implicatures, arenotoriously volatile, responding not only to minor aspectsof the cover story, but also to the dependent measure used
to probe linguistic intuitions (Geurts & Pouscoulous, 2009;
Zondervan, 2010). Contradictory results from different dependent measures are particularly troubling to progress in theempirical study of pragmatics. Here we systematically explore three dependent measures of scalar implicature, varying
the context, aiming to relate them to each other and to recent
formal models of pragmatics. We offer a potential explanation for why some measures are more sensitive to context manipulation than others.
Most experimental studies on scalar implicature haveused either sentence verification or metalinguistic judgment
paradigms to probe participants’ interpretation of utterances
containing scalar items (Bott & Noveck, 2004; Geurts &Pouscoulous, 2009; Geurts, 2010; Zondervan, 2010; Degen& Tanenhaus, to appear). In sentence verification studies,participants are often shown a visual display or asked to reada story, thus establishing the facts about the world. They arethen asked to provide a binary judgment about the truth of anutterance containing a scalar item, such as Some of the B’s arein the box on the left. In metalinguistic judgment paradigms,
participants are shown an utterance like Some of the B’s arein the box on the left and are then asked explicitly whether
it follows that not all of the B’s are in the box on the left
(Geurts & Pouscoulous, 2009). Yet neither of these dependent measures seems to directly measure the natural modesof language: production and interpretation. In everyday life
people constantly produce and interpret language, but much
more rarely adjudicate the truth of a sentence. Occasionallyresearchers have used measures that more directly evaluate
interpretation (e.g. Goodman & Stuhlmüller, 2013) but someevidence—as well as much anecdotal experience—suggeststhat these measures may be less sensitive and more unstable.
Methodologically, it is unsettling that implicature rates differ depending on the dependent measure, and that different
dependent measures are sensitive to contextual manipulations
to different degrees. How is one to choose a dependent measure that adequately reflects the underlying contextual effect,
if there is one? We will not be able to answer this question
fully here. Rather, we will present results from three experiments that provide the first step towards a full investigation
into the sensitivity of different dependent measures to manipulations of context and the underlying pragmatic inference
process. We manipulate the dependent measure that is used
to evaluate listeners’ sensitivity to two contextual cues when
they interpret utterances containing scalar items: a) the implicit contextual Question Under Discussion (QUD, Roberts,2004) that interlocutors are trying to address; and b) the sizeof the set that the quantifier some is being used to describe.
The experiments we present here are identical with the exception of participants’ task.
Exp. 1 uses sentence verification, the most widely employed measure in the literature. We will speculate that sentence verification is closely related to production; to evaluate this idea we perform Exp. 2, which uses word probability
ratings as a more direct test of listeners’ expectations about
speakers’ production. Finally, Exp. 3 uses sentence interpretation—a measure that constitutes arguably the most direct and natural test of understanding by allowing participants
to distribute confidence over different potential states of theworld in response to an observed utterance.
We delay discussion of the differences between these measures to the General Discussion, where we provide an argument that the differences in the results of Exps. 1 and 2
vs. those of Exp. 3 are expected under a formal, probabilistic
model of pragmatics.

Experiment 1 - sentence verification
The two contextual features we investigate are the implicit
question that interlocutors are trying to address (the QUD)
and the total size of the set of objects, known by the speaker,
that is contextually available. We elaborate briefly on each.
Previous studies have found, using sentence verification,that a contextually evoked QUD like Did I get all of the gumballs? yields higher implicature rates than a QUD like Did I
get any of the gumballs? when participants are presented witha scenario in which they got all of the gumballs but are told
You got some of the gumballs (Degen, 2013). A similar effectwas obtained by Zondervan (2010). This effect is presumably
due to the former QUD making the stronger scalar alternative
You got all of the gumballs more relevant than the latter QUD
does, which in turn has the effect of listeners inferring thatthe speaker must have really meant that they did not get all of
the gumballs if she didn’t use the relevant all alternative. In
the following we will refer to a QUD that makes the utterancewith all relevant as the all? QUD and a QUD that make only
the lower bound (at least one) relevant as the any? QUD.
The second contextual feature of interest is the total number of objects in the domain. Previous studies have foundthat some is more natural to describe sets of sizes that are notsubitizable (Degen & Tanenhaus, to appear; van Tiel, 2013).
This is presumably due to the increased effort it takes speakers to establish exact set size for larger sets—using a vague
quantifier to describe a larger set allows speakers to hold thefloor without saying untrue things. This predicts that listeners
should expect speakers to be less likely to use some to refer
to all the objects in a set when that set is subitizable (e.g. of
cardinality 4) than when it is not (e.g. of cardinality 16).
Exp. 1 tests the effect of QUD and set size using sentence
verification, the most widely used measure of scalar implicature derivation in the literature (Bott & Noveck, 2004; Degen& Tanenhaus, to appear; Zondervan, 2010; Geurts & Pouscoulous, 2009). Participants are given the facts about theworld and a speaker’s utterance and are asked to judge thetruth of the utterance.

MethodParticipants We recruited 48 participants over Amazon’s
crowd-sourcing platform Mechanical Turk.
Procedure and materials Participants read a brief twoparagraph story about a character (henceforth, the speaker)
who lost her marbles in shoe boxes when her nephew cameover to play. The story introduced a number of marbles (either 4 or 16) that the speaker owned and evoked an implicit
QUD that made it relevant either that the speaker find all of
the lost marbles (all? condition) vs. at least one of them (any?
condition). An example of an all? context is the following:
Ann is really into collecting marbles. Recently, herfriends gave her a special edition of 16 marbles, which
she loves. Yesterday, her five-year-old nephew came tovisit and found her set of marbles in a drawer. He also

found some shoe boxes. He played with the marbles fora long time and moved them from one box to another until they were all hidden and he didn’t remember where he
put them.
When Ann later entered the room, she saw that all themarbles were gone and there was a pile of shoe boxes on
the floor. She was upset and complained bitterly to herhusband. She was determined to find every last one ofher marbles. She started opening one box after another,looking for marbles.
In contrast, contexts like the following were employed to
evoke the any? question:Ann’s five-year-old nephew loves playing with marbles.
For when he comes to visit, Ann keeps a set of 16 marbles in a drawer. Yesterday, he came to visit and found
her marbles in the drawer. He also found some shoeboxes. He played with the marbles for a long time andmoved them from one box to another until they were allhidden and he didn’t remember where he put them.
When Ann later entered the room, she saw that all themarbles were gone and there was a pile of shoe boxeson the floor. Her nephew was upset because he wanted amarble to play with. He started to cry and Ann’s husband
tried to console him while Ann started opening one box
after another, looking for marbles.
In order to ensure that participants paid attention to thestory, they were then asked two questions: How many marbles are there in Ann’s set? and When will Ann be satisfied?
They were only allowed to proceed once they correctly answered the questions. The correct answers were either 4 or
16 (to the first question), and if she finds all of the marbles
or if she finds at least one of the marbles (to the second question), respectively. They were then shown a target display
containing a box with the complete set of marbles and weretold that this was the box that the speaker found. They werealso shown the speaker’s utterance I found some of the marbles and were asked to judge whether the statement was trueby clicking either a ‘Yes’ or a ‘No’ button.1 If the speakeris interpreted as saying that she did not find all of the marbles, reflecting an implicature was drawn, participants should
respond ‘No’. If instead they interpret the utterance literally,
they should respond ‘Yes’.
QUD and total number of marbles were betweenparticipant manipulations. Speaker name and gender, genderof the visiting relative, and marble color were randomized.
Marble location was determined by adding noise to fixed initial positions within the box.
Prediction If listeners are sensitive to the QUD, participants should be more likely to respond ‘No’, reflecting theimplicature, when the QUD is all? than when it is any?.
1 The same experiment was run where the task was not to judge
truth, but rather whether participants agreed or disagreed with the
speaker. The results were qualitatively the same.

Proportion of ‘No' responses

Experiment 2 - word probability rating
Exp. 2 used a more direct measure of participants’ expectations about production.

0.75

QUD

0.50

all?
any?
0.25

0.00
4

16

Total marbles

Figure 1: Proportion of pragmatic ‘No’ responses in Experiment 2 by QUD and total number of contextual marbles.
If listeners further expect speakers to use all when it is easy
to verify that all the marbles are present (i.e., when the set issubitizable) but less so when the speaker would have to invest
effort to determine set size (i.e., when the set is not subitizable), participants should be more likely to respond ‘No’ forthe small than for the big set.

Results and discussion
Proportion of pragmatic ‘No’ responses are shown in Figure
1. In a logistic regression predicting ‘Yes’ responses from
centered predictors of QUD, S ET S IZE, and their interactions,
there were main effects of QUD and S ET S IZE such that participants were more likely to respond pragmatically (‘No’)
when the question was all? (β = 1.37, SE = 0.68, p < .05)
and when set size was small (β = -1.6, SE = 0.68, p < .02).
This suggests that listeners, despite being asked about thetruth of the utterance, display pragmatic effects: when thestronger alternative I found all of the marbles matters to thecontextual QUD they are more likely to reject the some statement than when it is not. In addition, the main effect of S ETS IZE suggests that listeners took into account the uncertainty
that the speaker had about the actual size of the larger set andwere aware of the presumably extra counting effort that the
speaker would have had to invest in order to verify the precise size of the set.
Using sentence verification, we thus uncovered two contextual effects on scalar inference. If participants had been
judging literal truth, we would not have expected any context
effects. Why do we nevertheless observe these effects? One
possibility is that participants are reinterpreting the task to answer a question like Could a speaker have said this if he had
intended to communicate this particular state of the world?,
suggesting that sentence verification is a measure that probes
listeners’ intuitions about production rather than measuring
literal truth or comprehension directly. If this is right, a more
direct measure of production should replicate the contextual
effects.

MethodParticipants We recruited 52 participants over Mechanical
Turk.
Materials and procedure Exp. 2 differed from Exp. 1 onlyin the speaker’s utterance and participants’ task. Again, they
saw the box with the complete set of marbles and were toldthat that was the box the speaker found, but rather than the
speaker’s utterance being I found some of the marbles, it was Iof the marbles. Participants were then shown four
foundutterance alternatives—some, all, none, and the number termfour or sixteen, respectively—and asked to adjust a slider foreach word to indicate how likely they thought it was that the
speaker used that word. The endpoints of the sliders weremarked as very unlikely and very likely.
Prediction If speakers are less likely to use some and more
likely to use all when the question is all? and listeners are
sensitive to this, higher slider values for some should be obtained when the question is all? than when it is any?.
If listeners further expect speakers to not go to the effort ofcounting the big set in order to be able to use the stronger all
alternative, higher slider values for some should be obtained
for large than small sets.

Results and discussion
Participants’ ratings were normalized such that for each participant, their slider values summed to 1. Mean ratings foreach utterance are shown in Figure 2. In a linear regression predicting ratings for some from centered predictorsfor QUD, S ET S IZE, and their interactions, the main effects
of QUD and S ET S IZE and the lack of interaction found in
Exp. 2 replicated. Participants were more likely to indicate
lower probability of use for some when the question was all?
(β = .16, SE = .06, t = 2.84, p < .007) and when set size wassmall (β = -.16, SE = .06, t = 2.81, p < .008). The results
were qualitatively the same when the analysis was performedon unnormalized slider values. These results mimic the results from Exp. 1 that used sentence verification.
Interestingly, these pragmatic effects are also reflected inparticipants’ expectations of use for all: ratings were higher
for all when set size was small (β = .14, SE = .07, t = 2.18,
p < .04) and marginally higher when the question was all?
(β = -.12, SE = .07, t = -1.76, p < .04), suggesting that eventhe use of all, a quantifier that is typically treated as having
a fixed semantics that does not allow for pragmatic slack, iscontext-dependent. There were no pragmatic effects on expectation of use for number terms and none.

Experiment 3 - sentence interpretation
Exp. 3 used a measure of utterance comprehension that is arguably more natural than sentence verification: rather than

all

none

some

numberProbability rating

0.8

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QUD

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16

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● any?

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16

4

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Total marbles

Figure 2: Normalized probability ratings for the four utterance options, by QUD and total number of contextual marbles. Error
bars indicate bootstrapped 95% confidence intervals.
being given the facts about the world and an utterance describing those facts, we present participants with the morenaturally occurring situation of observing an utterance that aspeaker produced—I found some of the marbles—and having
to infer what the world is that the speaker is trying to communicate.

small (subitizable).
It is plausible to expect that effects of pragmatics would belargest with this dependent measure, certainly compared totruth judgement, since it is for comprehension that pragmatic
inference is most important. However previous research has
indicated that interpretation is a particularly fragile measure.

MethodResults and discussion

Participants We recruited 48 participants over Mechanical
Turk.

Participants’ ratings were normalized such that for each participant, slider values summed to 1. Mean ratings for eachbox are shown in Figure 3. Mean ratings for the complete set(containing 4 or 16 marbles, respectively) were close to floor
and ranged from 0.09 to 0.13. In a linear regression predicting ratings for the complete set from centered predictors forwhether the QUD was any?, whether the total set size wassmall, and their interactions, no effects reached significance(all ps > .39). Results were qualitatively the same when theanalysis was performed on unnormalized slider values.
We thus find no evidence for effects of QUD and set sizeon the probability of generating a scalar implicature with thesentence interpretation measure, despite this measure being
the most direct measure of comprehension. We discuss thissurprising result and the asymmetry with Exps. 1 and 2 in thefollowing.

Materials and procedure Exp. 3 was identical to Exps. 1
and 2 with the exception of the target display and participants’task. Rather than seeing just the box with the complete set,
participants instead saw a target display as in Figure 3, whichcontained the speaker’s utterance—I found some of the marbles—and five hypothetical boxes of marbles that the speakercould have found. Boxes contained (in order) 0, 25, 50, 75,
and 100% of the total number of marbles introduced in thecontext.
Participants’ task was to adjust a slider for each box to indicate how likely they thought it was that the speaker had foundthat box. Once all the sliders were adjusted they proceeded tothe post-experiment questionnaire.
Prediction The slider adjustment task allows us to obtaina probability distribution over situations that participants aretaking the speaker to convey. If the QUD modulates this distribution, we should observe that when the QUD is all?, thedistribution is shifted to the left, compared to when the QUD
is any?. This should become most apparent in the ratings forthe box with the complete set: ratings should be higher whenthe question is any? than when it is all?.
For the set size manipulation, participants should be more
likely to expect some to be used with the complete set whenset size is not subitizable and determining whether all of themarbles were found would require counting. This should bereflected in higher ratings for the complete set when the total
number of marbles is large (not subitizable) than when it isGeneral discussion
The empirical pattern of obtained results is the following: aneffect of QUD and set size was observed when using the sentence verification and word probability rating measures (inthe same and predicted direction, Exps. 1 and 2), but notwhen using the sentence interpretation measure (Exp. 3). It
is surprising that sentence interpretation, which is the taskthat most directly mimics the task that listeners are confronted with every day—inferring the state of the world that aspeaker is most likely intending to communicate—is the onethat does not show sensitivity to context manipulations; whileat the same time the less natural measures of interpretation—
judging whether a sentence can be uttered in a given context

Normalized slider value

Total
marbles

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●●●●● any?

1

Proportion of total marbles in box

Figure 3: Left: Target display in Exp. 3, small set condition (4 total marbles). Right: Normalized slider values indicating
probability of each box of marbles being the actual box in Exp. 3. Error bars indicate bootstrapped 95% confidence intervals.
or explicitly evaluating truth of utterance alternatives—do exhibit sensitivity to the contextual manipulations. Why is this?
In the following we offer some speculative remarks about thenature of the different measures, their likely connection, andideas for future work.
One possibility for why the contextual effects do not show
up in the comprehension measure is that comprehension is
simply more difficult or more noisy than production. This byitself is not a satisfying explanation without an argument for
why comprehension should be more difficult—particularly
in light of the ecological importance of comprehension andintrinsic relation between comprehension and production (itwould not be communicatively useful for a speaker’s production to be sensitive to context if a listener ignores context).
Another possibility is suggested by recent developments
in computational modeling of pragmatic inference (Frank &Goodman, 2012; Goodman & Stuhlmüller, 2013; Franke,
2009; Franke & Jäger, 2013), which provides an intuitive but
precise way to characterize the points in participants’ reasoning that different measures probe. These models are based on
the idea that listeners arrive at an interpretation by reasoning
about what a speaker who is trying to be informative would
have said. That is, listeners are modeled as having a model ofthe speaker’s contextual utterance probabilities (in our casethe probability of uttering some, all, etc.) that they invert using Bayes’ rule (Equation 1 below) to recover the most likely
states of the world (in our case, different possible boxes ofmarbles) from the speaker’s actual utterance.
This perspective suggests that sentence interpretation tasks
measure listeners’ ultimate interpretation, while sentence verification and word probability rating measure listeners’ underlying speaker models. If we take seriously the idea thatthe final listener interpretations are derived from Bayesian
reasoning about the speaker model, the context effects observed at the speaker level may simply be too small to survive the final step of reasoning, leading to a null effect at thelistener level. To validate this argument we next simulate theexpected interpretation effect size (Exp. 3) from the production probabilities obtained in Exp. 2, the probabilistic model,
and different assumptions about prior probabilities.

A simulation: the importance of the prior
For the sake of simplicity, we simulate only the effect of the
QUD and ignore set size differences.2 We follow previous
probabilistic models of pragmatic inference (Frank & Goodman, 2012; Goodman & Stuhlmüller, 2013) in modeling listener behavior by assuming that listeners can use Bayesianinference to recover the state of the world b (a box) that the
speaker intended to convey, given that he produced utterance
w. We further extend previous models by assuming that listeners condition their inference on the QUD. By Bayes’ rule:
Plistener (b|w, QUD) ∝ Pspeaker (w|b, QUD)P(b)(1)

where P(b) captures the listener’s prior beliefs about the box
and Pspeaker (w|b, QUD) describes the listener’s model of thewords a speaker will choose. We assume only two box types
b = {b∃ , b∀ }, where b∃ is a box that contains at least one but
not all of the marbles and b∀ is the box containing the complete set of marbles.3 We further assume there are four possible utterances w = {wsome , wall , wnone , wfour }, which are
the utterance alternatives participants saw in Exp. 2.
We take the empirical normalized mean slider values fromthe Exp. 2 small set condition as Pspeaker (w|b∀ , QUD).
Since we did not include an incomplete set target display inExp. 2, we instead set values for b∃ for both QUDs at some:
0.94, all: 0.02, none: 0.02, four: 0.02. We vary the priorprobability p(b∀ ) from 0.1 to 0.9 in steps of 0.1 and generate
the listener posterior Plistener (b|w, QUD).
2 We focus only on the small set condition here but the results are
qualitatively the same for the big set condition.
3 The set sizes 1 - 3, which form an equivalence class, are included in b∃ . Alternately, one could model each set size independently.

Conclusion
We have shown that the choice of dependent measure in experimental pragmatics research can greatly affect the conclusions drawn. Focusing only on the sentence interpretation
measure would have led to the conclusion that there is no evidence to support contextual QUD and subitizing effects on
scalar implicature. However, the production-based dependentmeasures did reveal contextual effects. We then showed that
this asymmetry is predicted by models of pragmatic inference
that assume listeners reason about informative speakers.
This work has both methodological and theoretical implications. Methodologically, that different ways of probing theunderlying process are sensitive to these effects in different
ways suggests that the choice of dependent measure is a serious issue that researchers doing experimental pragmatic research should take into account. Theoretically, this paper
makes two contributions. First, it adds to a growing body ofwork showing that the strength of scalar inferences is modulated by an implicit Question Under Discussion and the estimated speaker effort of reducing uncertainty about set size.
4 Confidence intervals on model predictions were obtained by fitting the model to the maximum and minimum of the bootstrapped
95% confidence intervals obtained from the empirical data.
5 A preliminary pilot study, a minor variant of Exp. 3 in which
participants were not shown the speaker’s utterance, revealed thatparticipants’ prior p(b∀ ) was between .04 and .24, consistent withthe predicted value. For the time being, the empirical prior and therobustness of the predicted prior to varying model assumptions await
a more systematic investigation.

Posterior probability of complete set

Results are shown in Figure 4.4 For the two different
QUDs, the difference in the posterior probability of believingthat the speaker intended to communicate the box containing
the complete set is negligible for very small prior probabilities of b∀ . With low p(b∀ ), the predicted p(b∀ |wsome , QUD)
is small for either QUD, resulting in a floor effect. However,the predicted QUD effect increases as p(b∀ ) increases. This
provides a potential explanation for the observed lack of QUD
effect on interpretation (Exp. 3): the prior probability of thecomplete set may have been too small to detect an effect.
Given the empirical comprehension means and confidence
intervals from Exp. 3, the range of predicted p(b∀ ) can beread off directly from Figure 4: the predicted prior range isthe area where the model prediction confidence intervals foreach QUD intersect the confidence intervals from the empirical data. This yields a predicted p(b∀ ) between 0.22 and 0.66.
This result makes two predictions: a) measuring b∀ explicitly should yield values in the predicted range;5 and b) increasing p(b∀ ) experimentally should lead to an increase in
the size of the QUD effect. We leave this step for future work.
Note that in one of the few reported cases of the successful
use of the interpretation measure to detect an effect of context(in this case, of speaker knowledge) on scalar implicature,
the prior for the complete set was explicitly manipulated tobe large (Goodman & Stuhlmüller, 2013). This is consistent
with the simulation reported here that predicts that a large
prior makes it more likely to detect a contextual effect.

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Figure 4: Model predictions for p(b∀ |wsome , QUD) as afunction of p(b∀ ). Dashed lines/shaded areas indicate empirical means/confidence intervals from Exp. 3. Vertical lines
indicate range of p(b∀ ) consistent with observed data.
Second, it provides evidence for the utility of using formal,
probabilistic models of pragmatics to understand the effects
of choosing a dependent measure to probe effects of context
on pragmatic inference.

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