Cognition 168 (2017) 294–311

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Cognition
journal homepage: www.elsevier.com/locate/COGNIT

Original Articles

Resolving uncertainty in plural predication q
Gregory Scontras a,⇑, Noah D. Goodman bab

University of California, Irvine, United States
Stanford University, United Statesa r t i c l e

i n f oArticle history:
Received 19 October 2015
Revised 30 June 2017
Accepted 5 July 2017

Keywords:
Plurality
Distributive vs. collective interpretations
Ambiguity resolution
Rational Speech Act models

a b s t r a c t
Plural predications (e.g., ‘‘the boxes are heavy”) are common sources of ambiguity in everyday language,
allowing both distributive and collective interpretations (e.g., the boxes each are heavy vs. the boxes
together are heavy). This paper investigates the role of context in the disambiguation of plural predication. We address the key phenomenon of ‘‘stubborn distributivity,” whereby certain predicates (e.g., big,tall) are claimed to lack collective interpretations altogether. We first validate a new methodology formeasuring the interpretation of plural predications. Using this method, we then analyze naturallyoccurring plural predications from corpora. We find a role of context, but no evidence of a distinct classof predicates that resists collective interpretations. We further explore the role of context in our final
experiments, showing that both the predictability of properties and the knowledgeability of the speakeraffect disambiguation. This suggests a pragmatic account of how ambiguous plural predications are interpreted. In particular, stubbornly distributive predicates are so because the collective properties they
name are unpredictable, or unstable, in most contexts; this unpredictability results in a noisy collective
interpretation, something speakers and listeners recognize as ineffective for communicating efficiently
about their world. We formalize the pragmatics of utterance disambiguation within the Bayesian
Rational Speech Act framework.
Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND
license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction
The semantics and pragmatics of predicating properties to singular individuals (e.g., ‘‘The box is big”) is already a complicatedendeavor. What does it mean for a box to be big, or tall, or heavy?
How do we represent its size or height or weight? Recent answers
to these sorts of questions highlight the central role of context inthe calculus of meaning: what properties get ascribed depends crucially on the context of predication (cf. tall for a boy vs. tall for abasketball player; e.g., Kamp & Partee, 1995; Kennedy, 1999;
Lassiter & Goodman, 2013). The complexity associated withanswering these questions increases once we introduce pluralities
into the mix. Particularly puzzling is the distributive vs. collective
ambiguity of plural predication (e.g., Link, 1983, 1987, 1998; Scha,
1984; Landman, 1989a, 1989b, 1996; Lasersohn, 1988, 1990, 1995,
1998; Schwarzschild, 1994, 1996):qThis work was supported in part by a James S. McDonnell Foundation ScholarAward (to N.D.G.); Office of Naval Research Grant N000141310788 (to N.D.G.); andthe DARPA Probabilistic Programming for Advanced Machine Learning (PPAML)
program, agreement number FA8750-14-2-0009 (to N.D.G.).
⇑ Corresponding author.
E-mail address: g.scontras@uci.edu (G. Scontras).
URL: http://linguistics.uci.edu/scontras/ (G. Scontras).

(1)

The boxes are heavy.
a. The boxes each are heavy.
b. The boxes together are heavy.

DISTRIBUTIVE
COLLECTIVE

When hearing (1), we may infer that each box on its own counts asheavy, as in (1a), or only that their total weight is heavy, as in (1b).
Under a collective interpretation, many light boxes may together
count as heavy (Scha, 1984). How is the interpretation determined?
To help answer this question, we consider the well-documented
yet poorly-understood phenomenon of ‘‘stubborn distributivity”(Quine, 1960; Schwarzschild, 2011; Syrett, 2015; Vázquez Rojas
Maldonado, 2012; Zhang, 2013), whereby some gradable predicates do not admit collective construals (e.g., big, tall, round), in
contrast to those that do (e.g., heavy, expensive, voluminous). For
instance, a speaker is hard-pressed to use the sentence in (2) tosay of the boxes that their total size is large.
(2)

The boxes are big.
a. The boxes each are big.
b. The boxes together are big.

http://dx.doi.org/10.1016/j.cognition.2017.07.002
0010-0277/Ó 2017 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

DISTRIBUTIVE
COLLECTIVE

G. Scontras, N.D. Goodman / Cognition 168 (2017) 294–311

Size and shape adjectives resist collective interpretations.
Schwarzschild (2011) terms them ‘‘stubbornly distributive,” signaling the lack of collective interpretations. Zhang (2013) labels thisclass of predicates ‘‘delimitable,” highlighting their usage todescribe physical extents. Supporting the robustness of this phenomenon, Syrett (2015) finds evidence of stubborn distributivity
in children as young as 3 years of age.
Collective predication arises when a property is ascribed
directly to a plurality, rather than distributed among its members(Link, 1983). Semanticists treat stubborn distributivity as a lexicaldistinction: stubbornly distributive predicates behave as such
because they cannot contain pluralities in their basic denotations(Schwarzschild, 2011; Vázquez Rojas Maldonado, 2012; Zhang,2013).1 This explanation suffers two flaws: First, it hypothesizes acategorical distinction that may not hold; in Expt. 2, we find a gradient of collectivity with no clear boundary. Second, it leaves unexplained why some predicates would contain pluralities and others
not—is there a deeper explanation in terms of the properties named?
Note that stubbornly distributive predicates like big or tall contrast with truly distributive predicates like be a man or have blueeyes, whose meaning requires a distributive interpretation (Link,
1983). Having blue eyes or being a man is a property that necessarily holds of individuals, hence their mandatory distributivity; (collective) size is not such a property, at least not transparently so.2
There is no conceptual barrier to attributing some size directly to aplurality of individuals. Collective nouns bring this point into focus:
if a group of men has blue eyes, then each member must; if a pile ofboxes counts as tall, we make no direct claim about individual box
heights.
(3)

a.
b.

The group of men has blue eyes.
The pile of boxes is tall.

Rather than roaming the lexicon with name tags specifying theirattitudes toward collective interpretations (a state awfully difficult
for a child to acquire), something about these predicates or theproperties they name must influence or determine their behavior
in plural predication. The task at hand is to determine what thatthing is.
We suggest that the key underlying notion is the stability orcontextual predictability of a property when applied to a collection.3
That is, there are aspects of context that affect the computation ofcollective properties, over and above those that affect the individual
properties; when there is more uncertainty about these aspects ofcontext, the collective property is less stable. For instance, collective
tall names the vertical extent of a set of objects, which can vary dramatically with their arrangement. In contrast, collective heavy
names the total weight of a set of objects, which does not vary witharrangement. If the precise arrangement is not in common ground,then the collective use of tall would be more likely to mislead a listener than that of heavy—the listener and the speaker could be
encountering different arrangements. In contrast, the distributive
meaning of each predicate is fully specified, independent of arrange1
In his characterization of stubborn distributivity, Schwarzschild (2011) first
assumes that gradable adjectives are predicates of events (e.g., Higginbotham &Schein, 1989). Thus, heavy predicates over HEAVY events, big over BIG events, etc. For
Schwarzschild, stubbornly distributive predicates are those that require their events
to have only single participants. The details are not directly relevant for our purposes;
what is relevant is the lexical distinction Schwarzschild and others make between
stubbornly distributive and complaisantly collective predicates (i.e., those predicatesthat allow collective construals).
2
Similarly, truly collective predicates like gather or meet name properties that
require a collective interpretation by virtue of their meaning (Schwarzschild, 1994).
3
See Schein (2017, chap. 4.1.1) for a similar line of reasoning; Schein frames his
observation in a Neo-Davidsonian event-semantic framework.

295

ment. A listener would thus not expect a cooperative speaker tointend the collective meaning of tall.
Here is the general hypothesis: computing the meaning of acollective interpretation is susceptible to a varying amount ofnoise depending on the contextual predictability of the property
at issue. To communicate effectively and avoid possible confusion,collective interpretations will be avoided by speakers in the caseof such noisy predicates; distributive interpretations, being lessvariable, are the safe bet. Listeners know this, hence collective
interpretations will become more likely as contextual predictability of the collective property increases. In Section 6, we formalize
the notion of contextual predictability of the collective property
and its role in the pragmatic disambiguation of the meaning ofa plural predication. This account explains the phenomenon of
stubborn distributivity as a result of the relative unpredictabilityof the collective property for predicates like big and tall. It also
predicts that context should matter: in a context where the collective property is more stable, the collective reading should be
more felicitous.
For physical properties, the stability of physical arrangements isa key element of context. If arrangements are less likely to change,
collective properties become more predictable, and so collective
interpretations become more useful and more likely. We test thisprediction in Expt. 3, measuring whether collective big and tallindeed feel more natural for boxes that are always stacked regularly than for jumbled piles of boxes. More generally, some typesof objects may have more predictable collective properties thanothers by virtue of how regularly they actualize in the world. If
we could find objects that actualize in regular physical arrangements (e.g., shelves or waves) or with no physical arrangement
at all (e.g., numbers or ideas), we might expect stubbornly distributive predicates to more readily admit collective interpretations
when the nouns that name these objects serve as subject. This reasoning leads to the prediction that the nouns which serve as subjects in a plural predication should also affect the likelihood ofcollective interpretation. In Expt. 2, we test whether this holds
for a variety of predicates.
Beyond contextual predictability of collective properties, thepragmatic view of plural predication entails that other standard
pragmatic pressures should come to bear on the reasoning processes of speakers and listeners, affecting the felicity and thereby
likelihood of the collective (and distributive) interpretation. To
whit, changing the speaker’s epistemic state ought to influence
the resulting disambiguation between distributive and collective
interpretations. If a speaker lacks evidence that would verify a
specific interpretation, and if a listener knows this about the
speaker, then that unsupported interpretation should be particularly improbable. This intuition derives straightforwardly from
the Maxim of Quality from Grice (1975): ‘‘Do not say that for which
you lack adequate evidence.” We test this prediction first in Expt. 1
with a reference task and then with a more sensitive measure in
Expt. 3.
Before testing the predictions of our pragmatic account, wemust establish a way of studying plural predication experimentally; that is, a way to unambiguously access distributive vs. collective interpretations. To that end, in Expt. 1, we show that theparticles each and together reliably disambiguate plural
predication.

2. Experiment 1: Validating the paraphrase methodology
One way to access interpretations of potentially ambiguous plural predications is to elicit ratings of unambiguous paraphrases. We
construct these paraphrases using distributive each and collective
together, as in (4).

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(4)

G. Scontras, N.D. Goodman / Cognition 168 (2017) 294–311

The boxes were big/heavy/tall.
a. The boxes each were big/heavy/tall.
b. The boxes together were big/heavy/tall.

Although many theorists assume the success of these disambiguating particles (e.g., Schwarzschild, 1994), diligence is due: we must
establish that these are in fact unambiguous paraphrases in thecontext of our experimental paradigm.4 We do so by asking participants to choose between two collections of boxes as the referent ofeach definite description in (4), (4a), and (4b); the objects in one collection hold the property individually (but not collectively), while
the other collection holds the property (but the individuals in it donot). We expect the two disambiguated sentences to lead to robust
selection of the corresponding collection, while selections for theambiguous sentence will indicate preferences for its interpretation.
In addition, we begin to explore the role of context by manipulating
the plausible knowledge of the speaker: whether he is likely to haveaccess to the relevant information about each object (by, e.g.,
inspecting each box), or only about the collection (by, e.g., movethe boxes all together).
2.1. Participants
We recruited 50 participants (21 female, 29 male; mean age:31) with U.S. IP addresses through Amazon.com’s Mechanical Turk
crowd-sourcing service. Participants received $0.25 for their
participation.
2.2. Design and methods
Participants were introduced to Cubert, a worker in a box factory.5 To manipulate Cubert’s access to knowledge about the boxes,
between participants, the context scenario either had Cubert ‘‘move”
or ‘‘inspect” the boxes that he received from a dispenser. In ‘‘move”
scenarios, Cubert appeared with a cart positioned under the dispenser (Fig. 1, left); in ‘‘inspect” scenarios, there was no cart
(Fig. 1, right). Participants were told that after moving/inspecting ashipment of boxes, Cubert told his friend Dot about the boxes hemoved/inspected. Their task was to help Dot decide which boxes
Cubert was referring to with his utterance. Participants chose
between two sets of boxes, one that implied a collective interpretation of the utterance (five small boxes, Fig. 2, left) and one that
implied a distributive interpretation (two large boxes which were
together smaller than the five small boxes, Fig. 2, right).
Cubert used three predicates, big, heavy, and tall, in three sentence frames, or utterances: ‘‘bare” (4), ‘‘each” (4a), and ‘‘together”(4b). Participants saw each version of each predicate in a random
order, completing nine trials. All 50 participants indicated thatthey were native speakers of English, so we included all of the datain the analyses reported below.
2.3. Predictions
If each and together are clear disambiguators for plural predication, we should find a split in the choice of referent depending on
which of these words appears. Encountering distributive each, as in(4a), participants should choose the distributive referent (Fig. 2,
4
Previous experimental work has investigated the disambiguating potential ofeach and so-called ‘‘post-VP” together (i.e., the boxes were big together). Syrett andMusolino (2013, 2016) found that these particles reliably disambiguate between
distributive and collective interpretations for adults. However, we are aware of noempirical work on post-NP together, which we employ in our experimental materials.
5
The full experiment is viewable online at http://cocolab.stanford.edu/experiments/collective/expt1/expt1.html.

right); with collective together, (4b), participants should choose
the collective referent (Fig. 2, left).
For the potentially ambiguous bare utterances, stubbornly distributive big ought to always prefer distributive referents, while
complaisantly collective heavy should be split. If predicates of sizeand shape are always stubbornly distributivity, tall should patternwith big. Given that size and height are visually assessable properties, the scenario manipulation should have no effect on Cubert’s
epistemic state, and therefore no effect on the resulting interpretations for bare big and tall. However, given that Cubert would needto lift individual boxes to assess their individual weight, we shouldfind an effect of scenario for heavy: moving all of the boxes (asopposed to inspecting each box) would yield higher rates of collective referent choice; without lifting each box, Cubert would lackthe knowledge needed to be justified in using a distributive
interpretation.

2.4. Results
Results were coded in terms of whether participants chose thecollective referent, and therefore accessed the collective interpretation for Cubert’s statement to Dot. Fig. 3 displays the proportionof collective choices, with bootstrapped 95% confidence intervals
drawn from 10,000 samples of the data (DiCiccio & Efron, 1996);
a value of 1 indicates that participants chose the collective referent
100% of the time.
To evaluate the disambiguating potential of our paraphrases,
we first look at responses to these paraphrases, excluding the bareutterance. Mean proportion of collective referent choice for collective paraphrases with together was 85% (i.e., near ceiling); for distributive paraphrases with each this proportion was 10% (i.e., nearfloor). We fit a mixed effects logistic regression model (Baayen,Davidson, & Bates, 2008) using the lme4 package (Bates,
Maechler, Bolker, & Walker, 2014) in R, predicting referent choice
by UTTERANCE (‘‘each,” ‘‘together”) and its interaction with PREDICATE(big, heavy, tall; big was coded as the reference level), as well as
TRIAL order. The model included random intercepts for participants
and scenarios (i.e., the maximal random effects structure supported by the data). The full regression model appears in Appendix
A. The model finds a main effect of UTTERANCE (b = 4.99, SE = 0.82,
z = 6.08, p < 0.001), confirming that collective together yielded
greater rates of collective referent choice than distributive each;the interactions with PREDICATE were not significant (heavy:b = 1.39, SE = 0.91, z = 1.52, p ¼ 0.13; tall: b = 0.63,
SE = 0.95, z = 0.66, p ¼ 0.51). The effect of TRIAL was also not significant (b = 0.07, SE = 0.07, z = 1.00, p ¼ 0.32). Thus, the ‘‘together”
utterances yielded collective referent choices, and the ‘‘each”
utterances did not, for each of the three predicates.
To evaluate the behavior of the predicates in the absence of thedisambiguating particles, together with the effect of our scenario
manipulation, we next consider responses to the bare, potentiallyambiguous utterance. We fit a mixed effects logistic regression
model predicting referent choice by PREDICATE (big, heavy, tall),SCENARIO (‘‘move,” ‘‘inspect”), and TRIAL order. We dummy coded
the PREDICATE predictor, with big as the reference level. The model
included random intercepts and slopes for participants (groupedby TRIAL; the maximal random effects structure supported by thedata). The full regression model appears in Appendix A. The model
finds main effects of PREDICATE for both the heavy (b = 3.05, SE = 1.23,
z = 2.48, p < 0.05) and the tall (b = 3.63, SE = 1.29, z = 2.82, p < 0.01)
contrasts. Compared to big, the other predicates yielded greater
rates of collective referent choice in ‘‘bare” utterances. While theeffect of SCENARIO did not reach significance, we note the trend in
Fig. 3 whereby ‘‘move” scenarios tended to have higher rates of collective referent choice for bare heavy utterances (34% ‘‘move” vs.

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297

Fig. 1. Example contexts from Expt. 1. Left: ‘‘move” scenario with dolly. Right: ‘‘inspect” scenario without dolly.

Fig. 2. Example trial from Expt. 1 with the predicate big in the ‘‘each” utterance.

14% ‘‘inspect”); a post hoc test of this difference approaches significance (t = 1.70, df = 47.96, p = 0.096).
2.5. Discussion
The disambiguating paraphrases behave as expected: eachdelivers a distributive interpretation and together delivers a collective one. We also find that the bare form of big patterns differently
from heavy and tall—it is more distributive. This finding is expectedwith respect to big vs. heavy (the former should be stubbornly distributive), but surprising given the behavior of tall: both tall and big
are predicates of size and shape. If naming properties of physicalextent mediates the choice between distributive and collective

interpretations of plural predications, we should find that tall patterns with big. Equally damning is the comparison between tall andheavy: if anything, tall more consistently delivers collective interpretations. What allows tall to readily receive collective interpretations—while precluding them for big—in our experimental
context?
As discussed above, one candidate factor is the contextual predictability of the collective property: how easy it is for speakers
and listeners to arrive at the same collective property for a given
set of objects. We have seen that collective weight is a stable property of sets, which stands to explain the complaisance of heavy
toward collective interpretations. In the context of the current
experiment, collective height is also predictable: boxes always
appeared stacked one on top of the other, yielding stable total
set heights. Collective size, however, could vary depending on
the strategy used to evaluate it (height? width? area? volume?)
as well as on the arrangement of the objects said to hold it, thus
accounting for the lack of collective interpretations for bare big.
In Expt. 3, we use the paraphrase methodology to directly investigate the role of contextual predictability in plural predication.
The results of the current experiment also leave open the possibility of the influence of speaker knowledge in plural predication:
based solely on whether the context scenario had Cubert move vs.
inspect the boxes he received, we see a trend whereby bare heavy
received more or less collective interpretations, respectively. The
absence of the effect for big and tall suggests an explanation interms of the epistemic state of the speaker: in ‘‘move” scenarios,
the speaker is less likely to have access to individual box weights(having plausibly moved them en masse); the listener is therefore
less likely to assume that the speaker intended a distributive interpretation for which he lacks evidence. We attempt to confirm thiseffect with a similar scenario manipulation in Expt. 3.
Before manipulating contextual predictability and speakerknowledge directly, we extend the paraphrase methodology in

Fig. 3. Proportion of collective choices in the results of Expt. 1.

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Expt. 2 to naturally-occurring examples of plural predication,
investigating the role of the local linguistic context.
3. Experiment 2: Naturally-occurring examples
In Expt. 1, we used a reference task to validate disambiguating
paraphrases as a probe of interpretations of plural predications. We
next use the paraphrase methodology to evaluate natural plural
predications gathered from corpora. We aim to both extend ourunderstanding of distributive vs. collective preferences to predicates beyond big/tall/heavy, and explore the effect of linguistic context: how the subject noun affects this preference. We thereby test
the null hypothesis of a simple lexical distinction between stubbornly distributive and complaisantly collective predicates: if stubbornly distributive predicates are truly stubborn, we should find aclear split in behavior between them and the other predicates, andwe should find no effect of the subject noun on their resistance tocollective interpretations.
3.1. Experiment 2a: Frequent plural predications
We begin with naturally-occurring examples of plural predications extracted from the British National Corpus.6
3.1.1. Participants
We recruited 90 participants (46 female, 44 male; mean age:36) with U.S. IP addresses through Amazon.com’s Mechanical Turk.
Participants received $0.30 for their participation.
3.1.2. Design and methods
In a search of the British National Corpus, we identified the 40
most frequent subject-predicate combinations in the plural predication frame in (5).
(5)

The NOUNs were ADJECTIVE.

Each participant was presented with 30 sentences; 15 of these sentences were drawn randomly from the original set of the 40 mostfrequent instances. The other sentences were constructed by randomly pairing subjects and predicates from the original set of 40
to create 15 novel sentences for each participant.
Each sentence was presented as an utterance by a speaker, andparticipants were tasked with determining what the speaker
meant.7 Participants rated two possible paraphrases on a sliding
scale. The paraphrases were distributive (with ‘‘each”, (6a)) andcollective (with ‘‘together”, (6b)).
(6)

a.
b.

The NOUNs each were ADJECTIVE.
The NOUNs together were ADJECTIVE.

utterance. Before rating its paraphrases, participants judged
whether the utterance made sense (choosing between ‘‘Yes” and‘‘No”). We analyzed data from the 85 participants who indicatedthat their native language was English.
3.1.3. Results
Of the original 40 most frequent sentences, participants indicated that they ‘‘made sense” 95% of the time.8 To avoid possible
confusion that nonsensical interpretations might introduce, in theanalyses reported below we only look at responses to utterances thatparticipants said made sense.
We begin with a look at the range of collective endorsement
ratings.9 Fig. 4 plots average collective endorsement ratings for eachof the 40 most frequent sentences with bootstrapped 95% confidence
intervals. Descriptively, we see a wide range of ratings, spanningfrom approximately 25% to 85% endorsement; participants appear
to be comfortable using the full scale. An important observation
about the distribution of ratings in Fig. 4 is its relatively smooth gradient: we fail to find clear groupings of predicates, such that they
behave in one way or another with respect to collective interpretations. Stubbornly distributive small indeed occurs toward the low
end of collective endorsement ratings, but not markedly so.
There are 8 predicates that occur with multiple nouns in oursmall corpus. We plot their collective endorsement ratings in
Fig. 5, shifting to violin plots to give a better picture of the ratings
distribution. The plots also include mean collective endorsement
ratings and bootstrapped 95% confidence intervals. For at least 3
out of these 8 predicates, rates of collective endorsement differ significantly by subject noun (bright, closed, small; see Appendix B for
details). We draw the reader’s attention to the case of stubbornly
distributive small (bottom right of Fig. 5). If stubborn distributivity
were a binary, predicate-level phenomenon hard-coded into thesemantics, we should find no effect of the subject noun in thedetermination of small’s distributive vs. collective interpretations:stubbornly distributive predicates, regardless of their subject,should always be maximally distributive. But this is not the casefor small in Fig. 5. A linear mixed effects model predicting collective
endorsement rates by subject noun for small, with random byparticipant intercepts, finds that sentences with the subject numbers received greater rates of collective endorsement than sentences with children (b ¼ 0:15; SE ¼ 0:07; t ¼ 2:24; p < 0:05),
rooms (b ¼ 0:19; SE ¼ 0:07; t ¼ 2:73; p < 0:01), or classes(b ¼ 0:27; SE ¼ 0:07; t ¼ 3:87; p < 0:01).10 Curiously, numbers
names objects that do not instantiate physically, which plausibly
increases the predictability of their collective size (i.e., their summation) and thus increases rates of collective interpretations.
This analysis relies on the chance occurrence of multiple subject
nouns in our small corpus to test the effects of local linguistic context. In Expt. 2b, we follow up on this result by systematically testing the effect of subject nouns on a narrower set of predicates.
3.2. Experiment 2b: big, heavy, tall

Given our interest in rates of collective interpretations, we are primarily concerned with collective paraphrase endorsement rates; we
expect distributive paraphrase endorsement rates to give redundant information. We included both distributive and collective
paraphrases to highlight the potential ambiguity for participants.
Scale endpoints corresponded to whether the paraphrase was ‘‘definitely” or ‘‘definitely not” what the speaker intended by his or her
6
Sentences were extracted from the British National Corpus Online service,
managed by Oxford University Computing Services on behalf of the BNC Consortium.
All rights in the texts cited are reserved.
7
The full experiment is viewable online at http://cocolab.stanford.edu/experiments/collective/expt2a/expt2a.html.

We then narrowed our sights to the predicates tested in Expt. 1:
big, heavy, and tall. These predicates are of particular interest
because they come from both supposed lexical categories (stubbornly distributive and complaisantly collective) and the physical8
Of the novel sentences, participants indicated that they ‘‘made sense” 67% of thetime.
9
A look at distributive endorsement ratings, or normalized collective ratings (i.e.,the difference between collective and distributive ratings) reveals similar patterns ofresults, owing to the high negative correlation between distributive and collective
endorsement ratings (r ¼ 0:78).
10
We find the same pattern of results if instead we compare each subject noun withthe grand mean (i.e., through effect, or deviation, coding of our variables): numbers
deviates significantly, while the other nouns do not.

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299

Fig. 4. Collective paraphrase endorsement rates for different noun-predicate combinations in Expt. 2a (i.e., for different sentences).

Fig. 5. Collective paraphrase endorsement ratings for different subject nouns grouped by predicate (Expt. 2a).

properties they name lend themselves to direct manipulation. Dueto the limited number of instances of plural predication with thesepredicates in the British National Corpus, we use instead the muchlarger Google Books Corpus (Davies, 2011) to look at their most frequent subject nouns.
3.2.1. Participants
We recruited 30 participants (25 female, 5 male; mean age: 38)
with U.S. IP addresses through Amazon.com’s Mechanical Turk.
Participants received $0.30 for their participation.
3.2.2. Design and methods
The design and methods for this experiment were identical tothose of Expt. 2a, with the exception that our naturally-occurring
examples feature the predicates big, heavy, and tall, and derivefrom the Google Books Corpus. We identified the five most fre-quent subject nouns in the plural predication frame (i.e., the NOUNs
were ADJECTIVE) for each of the predicates big, heavy, and tall.11 The
subject-predicate pairs appear in (7).
(7)

Noun-predicate pairings:
big:boys, children, houses, rooms, waves
heavy:
bags, lids, loads, men, trees
tall:buildings, offspring, plants, trees, windows

11
For heavy, many of the most frequent subject nouns resulted in a non-canonical,
abstract interpretation. For example, heavy rains or heavy casualties. We excluded
these cases from the current experiment. In a separate experiment, we included justthese abstract instances of plural predication with heavy and found no effect of thesubject noun on interpretation.

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Fig. 6. Collective paraphrase endorsement ratings for different nouns serving as the subject to the predicates big, heavy, and tall (Expt. 2b).

As with Expt. 2a, participants judged whether the resulting
sentences made sense and rated disambiguating paraphrases of
each.12 We analyzed data from the 26 participants who indicatedthat their native language was English.
3.2.3. Results
Of the original 15 most frequent sentences, participants
indicated that they made sense 98% of the time.13 We again lookonly at responses to the original most frequent utterances thatparticipants indicated made sense to them.
Fig. 6 plots collective endorsement ratings for each of the three
predicates grouped by their subjects.14 To compare the predicateswith each other, we fit a linear mixed effects model predicting
collective endorsement by predicate, with random intercepts forparticipants and nouns, and random slopes for participants. The
predicate predictor was dummy coded with heavy as the reference
level. There were main effects for both contrasts, such that big(b ¼ 0:22; SE ¼ 0:07; t ¼ 3:24; p < 0:01) and tall (b ¼ 0:20;
SE ¼ 0:06; t ¼ 3:20; p < 0:01) led to lower rates of collective
endorsement than heavy.
We might take the split in behavior between heavy on the one
hand and big and tall on the other as evidence for a categorical
divide between complaisantly collective and stubbornly distributive predicates. However, comparing the predicates at the level ofindividual subject nouns suggests otherwise: the subject noun
with the highest rates of collective endorsement for big, waves,yields rates of collective endorsement that do not differ significantly from heavy’s least collective subject noun, men (42% big
waves vs. 38% heavy men; b ¼ 0:04; SE ¼ 0:09; t ¼ 0:51;
p ¼ 0:62). The same holds of the comparison between tall’s mostcollective subject noun, trees, and heavy men (30% tall trees;b ¼ 0:07; SE ¼ 0:08; t ¼ 0:87; p ¼ 0:39).
Next, we compare the behavior of the subject nouns withineach predicate. Starting with tall, we fit a linear mixed effects
model predicting collective endorsement by subject noun, withrandom intercepts for participants; full regression models appear
in Appendix B. The subject noun predictor was dummy coded withtrees as the reference level (i.e., the subject with the highest
average collective endorsement rating). Compared to trees, we do
not find an effect of the contrast with plants (b ¼ 0:00; SE ¼ 0:06;
t ¼ 0:07; p ¼ 0:95), offspring (b ¼ 0:03; SE ¼ 0:06; t ¼ 0:55;
p ¼ 0:59), buildings (b ¼ 0:06; SE ¼ 0:06; t ¼ 1:00; p ¼ 0:32),
12
The full experiment is viewable online at http://cocolab.stanford.edu/experiments/collective/expt2b/expt2b.html.
13
Of the novel sentences, participants indicated that they made sense 87% of thetime.
14
As in Expt. 2a, distributive endorsement ratings provide redundant information,owing to their high negative correlation with the collective ratings (r ¼ :98).

or windows (b ¼ 0:09; SE ¼ 0:06; t ¼ 1:52; p ¼ 0:13). In otherwords, tall’s subjects behave similarly in their general resistanceto collective interpretation.
Turning to big, we performed a similar analysis, thistime dummy coding waves (the subject with the greatest averagecollective endorsement rating) as the reference level. There weresignificant effects of each contrast: rooms (b ¼ 0:18; SE ¼ 0:06;
t ¼ 2:97; p < 0:01),
boys(b ¼ 0:22; SE ¼ 0:06; t ¼ 3:53;
p < 0:01), children (b ¼ 0:21; SE ¼ 0:06; t ¼ 3:54; p < 0:01),
and houses (b ¼ 0:23; SE ¼ 0:06; t ¼ 3:68; p < 0:01); waves
stands out with its relatively high levels of collective endorsement
for big.
Finally, heavy; here, bags received the highest collective
endorsement ratings, so we coded it as the reference level. The
contrast with lids was not significant (b ¼ 0:08; SE ¼ 0:08;
t ¼ 1:04; p ¼ 0:30), the contrast with trees was marginallysignificant (b ¼ 0:15; SE ¼ 0:08; t ¼ 1:93; p ¼ 0:06), and thecontrasts with loads (b ¼ 0:24; SE ¼ 0:08; t ¼ 3:15; p < 0:01)
and men (b ¼ 0:25; SE ¼ 0:08; t ¼ 3:56; p < 0:01) were significant. In other words, we find similarly high ratings for bags and lids,
intermediate ratings for trees, and similarly low ratings for loads
and men.

3.3. Discussion
We have extended our paraphrase methodology beyond a simple reference task, using naturally-occurring examples of plural
predications. The pattern of ratings sheds some light on the statusof stubborn distributivity. Perhaps most striking is the absence ofclear groupings of predicates in the results of Expt. 2a: one would
be hard-pressed to read stubborn distributivity (or complaisantcollectivity) off the gradient of ratings in Fig. 4. Still, one might
wonder about the precise meanings of the collective interpretations of some of our sentences, for example ‘‘the boys together
were smiling” or ‘‘the natives together were friendly.” To obviate
the murky evaluation criteria of collective action statements likethese, we have chosen to consider in detail a specific class of predicates with (relatively) objective evaluation strategies: gradable
adjectives like heavy or big.15 The measurement inherent to thesemantics of these predicates delivers scalar interpretations thatare more amenable to quantitative study. By examining intuitions
about the use of these predicates, together with the predication context that delivers them, we stand to better understand the mechanism of plural predication: when collective interpretations are
15
This is not to say that we know nothing about collective action statements. For
example, Margaret Gilbert’s Plural Subject Theory makes great gains on this ground in
the domain of social philosophy (for a recent overview, see Gilbert, 2013).

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appropriate, what they communicate, and what mediates the choice
between distributive and collective interpretations in the first place.
We therefore narrowed our focus to the predicates big, heavy,
and tall in Expt. 2b, finding that complaisantly collective heavy
indeed does yield greater rates of collective paraphrase endorsement, and that stubbornly distributive big and tall pattern together
in their resistance to collective interpretations. However, once weconsider the effect of local linguistic context by comparing
responses to specific subject nouns, the division between the twoclasses of predicates disappears. Big and tall are no longer so stubborn with the appropriate context (cf. the relatively high rate ofcollective interpretations participants demonstrated for tall in
Expt. 1).
By comparing specific subject nouns within predicates, we
begin to find evidence for the context-dependent nature of thesepredications. With big, four out of the five subject nouns strongly
resisted collective interpretations. However, with waves, participants much more readily endorsed collective paraphrases (aswas the case with small numbers in Expt. 2a). Several interpretations of this finding are possible; one is that limiting the noise ofa collective interpretation—by reducing the variability in the physical instantiation of the subject—increases the rates of that interpretation. We follow up on this hypothesis with a direct
manipulation of predictability in Expt. 3.
With heavy, we also found that different subject nouns yielded
different rates of collective interpretations. At first blush, these differences appear to track the probability that a speaker would haveaccess to the information necessary to verify a distributive interpretation. Lids and bags are probably picked up together, certainly
more so than loads or men. If speakers experience only the collective weight of a plurality, not the individual weights of the members, they lack knowledge which would license the distributive
meaning. If listeners are aware of this, then the alternative, collective reading becomes more likely for collections (such as lids) thatare likely to be lifted together. As with contextual predictability,
we follow up on this result with a direct manipulation of speaker
access to knowledge in Expt. 3.

4. Experiment 3: Manipulating context
Physical properties of collections will tend to be more predictable when the physical arrangement is more predictable.
According to our hypothesis, greater predictability should lead tomore collective interpretations. For instance, if collections of certain boxes are always stacked neatly, the collective height of a
stack can be stably predicted (as the sum of individual heights);
if these boxes come in jumbled-up piles the collective height is lesspredictable. In Expt. 3, we manipulate expectations about thearrangement of boxes, using the paraphrase endorsement methodology to test the effect on the interpretation of plural predications.
We expect to see an effect on tall, whose collective property is the
direct target of our manipulation; we also expect to see a more
moderate effect on big to the extent that big can refer to height;
we expect no effect on heavy, which is stable regardless ofarrangement.
In Expt. 3, we also return to the effect of speaker’s knowledge.
Recall the prediction of our pragmatic hypothesis, that more collective interpretations should attain when the speaker lacks
knowledge about individual object properties (but has knowledge
about collective properties). We conjectured that the effect of heavy’s subject nouns in Expt. 2 was due to knowledge in this way: forsmaller objects the likelihood of interacting with them individuallyis higher, making the likelihood of individual weight knowledge
higher. In Expt. 1, we attempted to manipulate the likelihood thatthe speaker had interacted with the boxes individually, via the301‘‘inspect” scenario, versus only collectively, in the ‘‘move” scenario.
We found a trend in the predicted direction that did not reach significance. It is possible that this effect was real, but small becauseof a bias for the distributive referent (leading to a floor effect). In
Expt. 3, we repeat the move/inspect manipulation using the pureparaphrase endorsement dependent measure of Expt. 2, which
may have greater sensitivity than a binary choice between possiblereferents.
4.1. Participants
We recruited 80 participants (26 female, 54 male; mean age:32) with U.S. IP addresses through Amazon’s Mechanical Turk. Participants received $0.25 for their participation.
4.2. Design and methods
The experimental context was similar to that of Expt. 1. Participants were first introduced to an agent, Cubert, who works in afactory with boxes. Cubert receives boxes from a dispenser in theceiling. Participants began by observing the dispenser in action.16
Between subjects, we manipulated the SCENARIO story (‘‘move” vs.
‘‘inspect”): Cubert’s job was either to move shipments of boxes, orto inspect them. As in Expt. 1, in ‘‘move” scenarios, Cubertappeared with a dolly with which to carry the boxes. In ‘‘inspect”
scenarios, Cubert appeared without a dolly.
Also between subjects, we manipulated the variability of the
CONTEXT (‘‘regular” vs. ‘‘random”). In regular contexts, the dispenser

consistently stacked boxes on top of each other (Fig. 7, left). In random contexts, boxes were dispensed without any consistent physical arrangement (Fig. 7, right). Participants saw a series of fourcontext priming scenarios; in each scenario, they saw a different
set of boxes dispensed.
After observing the dispenser in action via the context scenarios, participants were then introduced to a second agent, Dot,
whom Cubert told about some boxes he moved/inspected. Participants were tasked with helping Dot understand what Cubertmeant by his utterance. To do so, they saw an utterance and rated
potential paraphrases on a sliding scale (as in Expt. 2). The paraphrases were distributive (with ‘‘each”) and collective (with ‘‘together”). Scale endpoints corresponded to whether theparaphrase was ‘‘definitely” or ‘‘definitely not” what Cubert meantby his utterance (Fig. 8). (Note that, unlike Expt. 1, there was no
explicit referent visible.) Participants completed three trials in arandom order with three different predicates: big, heavy, and tall.
For the analyses reported below, we included data from the 77 participants who indicated that their native language was English.
4.3. Results
Fig. 9 displays violin plots of the raw paraphrase endorsement
ratings, with mean endorsement ratings and bootstrapped 95%
confidence intervals. For the purpose of analysis, we analyze
responses to the collective paraphrases (i.e., the red bars in
Fig. 9); higher rates of endorsement for the collective paraphrase
signal greater rates of collective interpretation. First note the qualitative effects of our two manipulations: collective interpretations
do appear to be greater for tall and big in the ‘‘regular” conditions,and greater for heavy in the ‘‘move” conditions.
For each predicate, we fit a linear regression model predicting
collective endorsement ratings by CONTEXT (‘‘random” vs. ‘‘regular”),
SCENARIO(‘‘inspect” vs. ‘‘move”), and TRIAL order; the model also

16
The full experiment is viewable online at http://cocolab.stanford.edu/experiments/collective/expt3/expt3.html.

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Fig. 7. Example context scenarios from Expt. 3. Left: the outcome of a ‘‘regular,” ‘‘move” context scenario. Right: the outcome of a ‘‘random,” ‘‘inspect” context scenario.

Fig. 8. Example ‘‘big” trial.

included the interaction between CONTEXT and SCENARIO.17 The full
regression models appear in Appendix C. As is apparent in Fig. 9,
there was a main effect of CONTEXT for both big (b = 0.18, SE = 0.07,
t = 2.52, p < 0.05) and tall (b = 0.30, SE = 0.07, t = 4.58, p < 0.01), suchthat regular contexts had more collective interpretations than did
random contexts for these predicates. The effect of CONTEXT was not
significant for heavy (b = 0.11, SE = 0.07, t = 1.43, p ¼ 0.16), but
the effect of SCENARIO was (b = 0.19, SE = 0.08, t = 2.54, p < 0.05): heavy
received more collective interpretations in ‘‘move” scenarios. No
other effects reached significance.

4.4. Discussion
As we would expect if contextual predictability of collective
properties influences the viability of collective interpretations,
regular contexts had higher ratings for collective paraphrases(and lower ratings for distributive ones). The predicate tall
was most affected by our contextual manipulation, which
directly targeted the height dimension, stacking boxes on topof each other. But big, also claimed to be stubbornly distributive,17
All reported results hold if instead we look at normalized collective paraphrase
endorsement ratings (i.e., the difference between collective and distributive endorsements for each trial), given the strong negative correlation between the twoendorsement ratings (r ¼ 0:62). The results also hold if we analyze the resultsfrom all three predicates in a single mixed effects regression model; we present thethree separate models here for ease of interpretation.

is similarly affected: regular contexts yield more collective
interpretations.
For the predicate heavy, our contextual predictability manipulation had no measurable effect, presumably because collective
weight is already maximally predictable. However, heavy was sensitive to the ‘‘move” vs. ‘‘inspect” scenario manipulation: as was
the trend in the results of Expt. 1, ‘‘move” scenarios yielded muchgreater collective endorsement ratings for heavy. This manipulation privileges collective interpretations by limiting speakers’
access to the information they would need to verify a distributive
interpretation; Cubert lacked access to individual box weights in‘‘move” scenarios. The predicates big and tall were not affectedby the speaker knowledge scenario manipulation, owing to the factthat they name properties that are visually accessible, and thus
visually assessable.
An alternative interpretation of the contextual predictability
effect is that in regular, stacked contexts, participants reanalyzed
the plural definite description the boxes as referring to a single individual (e.g., to a single pile, and not to a set of boxes, perhaps byimagining a silent collectivizing noun); if the boxes referred to asingle individual, there never was any collective predication. Sucha representational coercion story raises serious problems for thesemantics of plural definite descriptions (for discussion, see Link,
1983; Landman, 1989a; Schwarzschild, 1996; Link, 1998). More
importantly it is inconsistent with the lack of effect for heavy: if
the pile of boxes was simply represented as an individual whenstacked, we would expect greater collective interpretation rates
for all predicates in regular contexts.
Taken together, our results demonstrate the central role of context in plural predication: as collective properties become morepredictable, collective interpretations become more likely; andwithout epistemic support, distributive interpretations becomeless likely. These findings support a pragmatic account of stubborn
distributivity according to which ostensibly stubbornly distributive predicates are such because the properties they name areunpredictable, or unstable in most contexts. In the next experiment we probe the generalizability of our findings.

5. Experiment 4: Generalizing our findings
In Expt. 3, we looked at the role of contextual predictability ininterpretation choice for three adjectives of English: big, heavy,
and tall. One might worry that the observed success of our contextual manipulation is an artifact of the small set of adjectives wetested, and might not generalize to a broader set of adjectives.
Therefore, to test the generalizability of the findings from Expt. 3,
we used the design on a much broader set of 25 dimensional
adjectives.

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Fig. 9. Paraphrase endorsement ratings from Expt. 3.

5.1. Participants
We recruited 54 participants (21 female, 32 male; mean age:35) with U.S. IP addresses through Amazon’s Mechanical Turk.18
Participants received $0.50 for their participation.

5.2. Design and methods
The experimental design was near-identical to that of Expt. 3.
Two aspects of the design differed: there was no ‘‘move” vs. ‘‘inspect” scenario manipulation, and participants provided paraphrase endorsements for 25 rather than three predicates. At theoutset of the experiment, participants were introduced to Cubert,
who was tasked with ‘‘moving” boxes that he received from a dispenser in the ceiling (there was no ‘‘inspect” condition). Between
subjects, we manipulated the variability of the CONTEXT (‘‘regular”
vs. ‘‘random”) by having the dispenser either stack boxes on topof each other or dispense them in a random physical arrangement.
Participants saw a series of four context priming scenarios todemonstrate the behavior of the dispenser; we used the same context priming scenarios from Expt. 3.
After observing the dispenser, participants were instructed tohelp Cubert’s friend, Dot, understand Cubert’s statements about
the boxes he moved. To do so, they saw an utterance and rated
potential paraphrases on a sliding scale; the paraphrases were
either distributive (with ‘‘each”) or collective (with ‘‘together”).
Participants provided paraphrase endorsements for each of theadjectives in Table 1. We arrived at the set of adjectives in Table 1
by extracting every unique adjective appearing in an ‘‘A A N” NP
from the Penn Treebank subset of the Switchboard corpus; there
were 350 unique adjectives. Then, adjectives were independently
coded according to semantic classes from the literature (e.g.,
Dixon, 1982). Among this set there were 24 unique dimensional
adjectives; to those 24 we added the weight adjective heavy.
Participants completed 25 trials in a random order, one trial foreach adjective. For the analyses reported below, we included datafrom the 51 participants who indicated that their native language
was English.

Table 1
Adjectives, modified dimensions, and polarities tested in Expt. 4.
ADJECTIVEDIMENSION

POLARITY

Full
Deep
Flat
Low
Short
High
Tall
LengthyLong
LittleMini
Slight
Small
Tiny
Big
Huge
Humongous
Large
Heavy
Narrow
Skinny
Thin
Fat
Thick
Wide

Capacity
Depth
HeightHeightHeightHeightHeightLengthLength
Size
Size
Size
Size
Size
Size
Size
Size
Size
Weight
Width
Width
Width
Width
Width
Width

Positive
Positive
Negative
Negative
Negative
Positive
Positive
Positive
Positive
Negative
Negative
Negative
Negative
Negative
Positive
Positive
Positive
Positive
Positive
Negative
Negative
Negative
Positive
Positive
Positiveinterpretation. We grouped data by the dimension modified,
together with adjective polarity.
For each dimension, we fit a linear regression model predicting
collective endorsement ratings by CONTEXT (‘‘random” vs. ‘‘regular”)
and TRIAL order; for those dimensions with both positive and negative adjectives (i.e., the height, size, and width dimensions), models
additionally included the POLARITY predictor and the interaction
between CONTEXT and POLARITY.19 The full regression models appear
in Appendix D.
We begin with a look at the dimensions with only positivepolarity instances. For capacity adjectives, the model found a marginally significant effect of CONTEXT (b = 0.16, SE = 0.08, t = 1.95,
p < 0.06), such that regular contexts tended to have more collective interpretations than did random contexts. However, repeating

5.3. Results
Fig. 10 displays the paraphrase endorsement ratings withmeans and bootstrapped 95% confidence intervals; higher paraphrase endorsement ratings indicate greater rates of the relevant
18

One participant chose not to provide a gender.

19
All reported results hold if instead we look at normalized collective paraphrase
endorsement ratings (i.e., the difference between collective and distributive endorsements for each trial), given the strong negative correlation between the twoendorsement ratings (r ¼ 0:49). The single exception is the marginally significant
effect of CONTEXT for the capacity adjective full, which holds marginally for thecollective endorsement ratings but not for the normalized ratings. We return to thispoint below.

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Fig. 10. Paraphrase endorsement ratings from Expt. 4.

the analysis for the normalized endorsement ratings, we fail to findany effect of CONTEXT (b = 0.14, SE = 0.13, t = 1.04, p < 0.31). There
was also no significant effect of CONTEXT for depth (b = 0.07, SE =0.09 t = 0.85, p < 0.40), length (b = 0.01, SE = 0.06, t = 0.15, p <0.89), or weight (b = 0.04, SE = 0.07, t = 0.51, p < 0.62)
adjectives.
We turn next to those dimensions with both negative- andpositive-polarity instances. For height adjectives, the model founda main effect of polarity (b = 0.28, SE = 0.04, t = 7.82, p < 0.01), suchthat positive-polarity height adjectives received more collective
interpretations than negative-polarity height adjectives. The model
also found a marginally significant interaction between CONTEXT and
POLARITY (b = 0.13, SE = 0.07, t = 1.82, p < 0.07); we found the same

marginally significant interaction in an analysis of the normalized
endorsement ratings (b = 0.25, SE = 0.13, t = 1.96, p < 0.06). A
post hoc analysis revealed that this interaction was driven by a significant effect of CONTEXT within the positive-polarity height adjectives (b = 0.12, SE = 0.06, t = 2.11, p < 0.05): regular contexts
yielded more collective interpretations. For size adjectives, themodel found a main effect of polarity (b = 0.20, SE = 0.03,
t = 7.60, p < 0.01), such that positive-polarity size adjectives
received more collective interpretations than negative-polarity
size adjectives. The model also found a significant interaction
between CONTEXT and POLARITY (b = 0.13, SE = 0.05, t = 1.82, p <0.05). A post hoc analysis revealed that this interaction was driven
by a significant effect of CONTEXT within the positive-polarity size
adjectives (b = 0.10, SE = 0.04, t = 2.21, p < 0.05): regular contexts
yielded more collective interpretations. For width adjectives, themodel found a significant interaction between CONTEXT and POLARITY(b = 0.14, SE = 0.07, t = 2.02, p < 0.05). A post hoc analysis
revealed that this interaction was driven by a significant effect of
CONTEXT within the negative-polarity width adjectives (b = 0.13,
SE = 0.05, t = 2.66, p < 0.01): regular contexts yielded morecollective interpretations.

5.4. Discussion
We have replicated the findings from Expt. 3 with a broader setof adjectives: tall and the three other positive-polarity heightadjectives are affected by the contextual manipulation such that
more predictable, ‘‘regular” contexts yield greater rates of collective interpretations; big and the four other positive-polarity size
adjectives show the same effect of context. Moreover, the complaisantly collective weight adjective heavy continues to be unaffected by our contextual manipulation, which does little toincrease the contextual predictability of the already-predictablecollective weight. This replication serves double duty, first as a sanity check for comparing the results of Expt. 3 with the current
experiment, and second as confirmation that the context effectobserved in Expt. 3 generalizes beyond the specific lexical items
that we tested.
In addition to replicating the effect of contextual predictability
with a broader set of adjectives, we tested four more physicaldimensions, as well as adjective polarity. Among those additional
dimensions with only positive-polarity instances (i.e., capacity,
depth, and length adjectives), none showed a reliable effect ofour contextual manipulation. Thus, stacking boxes on top of eachother failed to deliver greater rates of collective interpretations
for adjectives modifying box capacity (i.e., full), depth (i.e., deep),or length (i.e., long, lengthy).20 From this it would appear that generally regularizing physical arrangements is insufficient to delivercollective interpretations; rather, the dimension that the interpretation would target must be regularized. Given that stacking boxes
plausibly regularizes collective height, size (to the extent that heightfactors into size), and width, it stands to reason that only these
dimensions showed effects of our contextual manipulation. However, there is more to this story, because these dimensions showedsensitivity to our contextual manipulation according to theirpolarity.
Within the dimensions that showed clear effects of our contextual manipulation, positive—but not negative—height and size
adjectives, and negative—but not positive—width adjectives had
increased collective interpretations with increased contextual predictability. The strong influence of polarity highlights an important
aspect of the pragmatics of utterance disambiguation that we haveso far ignored: interpretations that are unlikely to be true are unlikely to be chosen—cooperative speakers do not intentionally utter
false statements (Grice, 1975). Take the case of negative-polarity
height adjectives like short or flat. It seems unlikely that Cubertwould intend to communicate that a stack of boxes taller thanhim is collectively short when the distributive alternative is available, namely that each box is short. Stacking boxes may remove
barriers to the collective interpretation by increasing the predictability of the collective property, but it creates its own barriers
by highlighting that the collective property is unlikely to satisfy thetruth conditions of the predicate.
20
One might worry that lengthy in fact modifies a temporal dimension (e.g., theconversations were lengthy) that cannot felicitously be attributed to boxes. Still, all of
the reported results hold if instead we analyze long and lengthy separately: neither isaffected by our contextual manipulation.

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It could be the case that negative-polarity dimensional adjectives are truly stubbornly distributive; the lack of a contextual
effect for negative height and size adjectives would follow fromthis peculiarity of their lexical semantics. But then we should findthe same pattern for the width adjectives, when in fact we findexactly the opposite pattern: an effect of context for the negative—but not positive—width adjectives like narrow. Again, themost straightforward interpretation of this pattern relies on the
avoidance of false interpretations. Cubert most likely would nothave intended to describe a tall, thin set of stacked boxes as collectively wide (or fat or thick) when the distributive interpretation
stands as a ready alternative. However, the reverse holds for narrow and the other negative width adjectives: stacking the boxes
on top of each other both increases the predictability of their collective width and focuses attention on their narrowness, that is,on the truth of the collective interpretation.
The results of the current experiment thus add nuance to ourclaims about contextual predictability. We continue to find thatcollective interpretations depend on the contextual predictabilityof collective properties, such that increasing predictability by regularizing arrangements increases rates of collective interpretations. However, to increase predictability one must regularize thespecific dimension under discussion (e.g., the height dimension
for tall but not long, or the width dimension for narrow but notdeep). Moreover, we have observed that contextual predictability
interacts with yet another aspect of the pragmatic calculus that
influences interpretation choice, namely the probability that a
specific interpretation would truthfully describe a state of affairs.
When an interpretation appears unlikely to be true (e.g., describing
a tall stack of boxes as collectively short), listeners are unlikely toattribute that interpretation to speakers’ utterances. We next formalize the role of context in the interpretation of plural predication, using tools from probabilistic modeling.

inform the listener about the true state of the world; he accounts
for his uncertainty about this true state by choosing an utterance
according to the expected informativity given his belief distribution.
The pragmatic listener knows this, but does not know what private
observations the speaker has had—she takes this uncertainty into
account when evaluating an utterance.
There are two significant innovations in the current model.
First, we treat ambiguity resolution via a lifted variable: the pragmatic reasoner infers which sense (collective or distributive) of theplural predication was intended by the speaker. Second, we
explore the effects of contextual noise that differentially affects
the two interpretations. That is, we allow that there are uncertain
contextual factors that affect evaluation of the collective property
but not the distributive. In the formal exposition that follows, wewill be brief about the pre-existing model features (lifted thresholdvariables, speaker knowledgeability) and more expansive about
the new aspects.
6.1. The model
We take states of the world, s 2 S, to consist of a collection ofentities, each of which has some individual degree: for eachx 2 s; dðxÞ 2 R is the relevant property. We assume a simple
truth-functional literal semantics, wherein a plural predicationutterance u (e.g., the boxes are big) denotes a mapping from statesof the world to truth values.21 We parameterize the truth function,

sutv ;c;hc ;hd : S ! Bool, so that it depends on context, c, on utteranceinterpretation, v, and on two gradable thresholds, hc and hd . We willconsider three alternative utterances: an unambiguously distributive(e.g., the boxes each are big), an unambiguously collective (e.g., theboxes together are big), and an ambiguous utterance (e.g., the boxes
are big).
(8)

6. Modeling plural predication
We aim to formalize the pragmatic account described so far,
especially the role of contextual predictability and speaker knowledge in ambiguity resolution. We will base our analysis on the
Rational Speech Act (RSA) approach (Frank & Goodman, 2012;
Goodman & Stuhlmüller, 2013; Goodman & Frank, 2016). Within
the RSA framework, language understanding is modeled as a social
reasoning process: the listener interprets an utterance by reasoning about a cooperative speaker who is trying to inform a naive listener about some state of affairs.
In several recent RSA models, the pragmatic listener is extended
to reason over factors needed to fully fix meaning, in addition to
the situation itself. That is, some aspect of the literal interpretation
model is ‘‘lifted” into the inference performed by the pragmatic
reasoner. This sort of lifted variable RSA model has proven usefulin accounts of gradable adjective semantics (where the lifted variable is the adjective’s degree threshold; Lassiter & Goodman,2013), specificity implicatures (where the lifted variable determines the choice of lexica; Bergen, Goodman, & Levy, 2012), andnon-literal meaning (where the lifted variable concerns the dimension along which the speaker intends to communicate, or the question under discussion; Kao, Wu, Bergen, & Goodman, 2014); seealso Goodman and Lassiter (2015) for an overview. Our model
treats the two underlying senses (collective and distributive) as
vague expressions that have an underspecified threshold semantics—following Lassiter and Goodman (2013), we lift the corresponding threshold variables to be resolved by the pragmatic
listener.
From Goodman and Stuhlmüller (2013), we borrow a treatmentof the speaker’s knowledge. The speaker has a basic motivation to305

Literal semantics:
a.
b.
c.

sdistribthd = k s. 8 x 2 s ½dðxÞ > hd Pscolltc;hc = k s. ½c þ x2s dðxÞ > hc sambtv ;c;hd ;hc = if v sdistribthd , else scolltc;hc

The unambiguous distributive utterance, (8a), is a universal quantification over a vague scalar meaning: each object’s degree must
exceed the distributive threshold hd . The unambiguous collective
utterance, (8b), is again a vague scalar meaning: the collective
degree must exceed the collective threshold hc . However, the collective degree captures two assumptions. First, we follow the work oncollective semantics in assuming that the appropriate aggregation isbased on a sum (e.g., Scha, 1984): the total degree of the collection.
Second, we assume that the computation of this total degree may
depend on contextual factors c (such as the particular arrangementof the objects); we treat c as a simple additive variable that can distort the total degree of the state—an approximation of the many
ways that context could affect the collective property (e.g., the variability in collective height introduced by mentally stacking boxes indifferent arrangements). Finally, the ambiguous utterance, (8c), isgoverned by the variable v: depending on v it is either collective
or distributive.22
The literal listener L0 has prior uncertainty about the context,
PðcÞ, and the true state, PðsÞ, and otherwise updates beliefs about
s by conditioning on the meaning of u:21
We assume truth values are coerced to numbers where appropriate: true is 1 andfalse is 0.
22
We ignore so-called ‘‘cumulative” construals of plural predication, which arise forsentences with sequences of noun phrases (e.g., 600 Dutch firms use 5,000 Americancomputers; Scha, 1984).

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PL0 ðsju; v ; hd ; hc Þ / sutv ;c;hd ;hc ðsÞ  PðcÞ  PðsÞThe distribution on collective-context, PðcÞ, will determine how
predictable the collective property is compared to the distributive:
higher entropy PðcÞ will correspond to noisier collective interpretations. The other interpretation variables (v ; hd ; hc ) are lifted so thatthey will be actively reasoned about by the pragmatic listener. Put
simply, the pragmatic listener resolves the interpretation of an
ambiguous utterance and fixes the appropriate thresholds for vague
gradable predicates.
The speaker, S1 , chooses an utterance u that would most effectively communicate some state s to a literal listener L0, increasing
utterance informativity by decreasing surprisal. In other words, thespeaker’s utility function U S1 minimizes the effort L0 would need toarrive at s from u, while being efficient (i.e., minimizing utterance
cost, CðuÞ):U S1 ðu; s; v ; hd ; hc Þ ¼ logðL0 ðsju; v ; hd ; hc ÞÞ  CðuÞTo capture the speaker’s epistemic state, we follow Goodman andStuhlmüller (2013) by assuming the speaker has a private belief distribution about the state of the world, PðsjoÞ, that depends on someobservations o of the true world. The speaker then selects an utterance u to convey information about the likely state s that generated
the observation o:PS1 ðujo; v ; hd ; hc Þ / expðaEPa ðsjoÞ ½U S1 ðu; s; v ; hd ; hc ÞÞWe will assume, based on our experimental setup, that the speakerhas one of two modes of access to the world state: if a ¼ full, thenthe speaker has observed the full state, in which case
Pfull ðsjoÞ ¼ ds¼o is a delta distribution on the true (observed) state;
if a ¼ sum, then the speaker has only observed the sum (e.g., observing the total weight of some boxes by lifting them together), in
which case the speaker considers all of the possible states that
could have led to the sum observation, Psum ðsjoÞ / PðsÞdP dðxÞ¼o .
x2s

At the top level of inference, the pragmatic listener L1 interprets
the speaker’s utterance u to jointly infer the state s and the freeinterpretation variables: the ambiguous utterance’s intended
interpretation v, and the thresholds hd ; hc . By Bayes’ rule:PL1 ðs; v ; hd ; hc ju; aÞ / P S1 ðujoða; sÞ; v ; hd ; hc Þ  PðsÞ  Pðv Þ  Pðhd ; hc ÞHere oða; sÞ is a deterministic function that reads off the observation
the speaker would have had given an access mode and a hypothesized true state.
The plural predication is both ambiguous (v) and vague (hc ; hd );the listener reasons about likely interpretations given an intuitive
understanding of speakers and prior knowledge of the world (i.e.,
which states and interpretations are a priori likely). This inferenceis guided by reasoning about how informative each utterance
would have been with different interpretations. Informativity isin turn influenced by semantic factors including the variability ofcontext, c, that may differentially affect the different
interpretations.
6.2. Model results
To generate predictions, we need to fix various parameter settings within the model; we aim to make the simplest choices possible, focussing on qualitative predictions.23 Each state, s, consists ofthree objects, each of which has degree 2; 3, or 4—these degrees correspond to object size, weight, or height. The prior on states PðsÞ was
taken to be uniform. The distributive threshold hd was uniformly
2; 3, or 4 (the possible object degrees). The collective threshold hc
23
The full model is viewable online at http://forestdb.org/models/plural-predication.html.

was uniformly drawn from the possible individual and total degrees,f2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12g.24 We take Pðv Þ to be Bernoulli with asingle free parameter (i.e., the prior bias toward collective interpretations); this parameter was set to 0:8. We take the cost CðuÞ to be 2
for the ambiguous utterance and 3 for both of the unambiguous
utterances—a reasonable assumption from the point of view of production, given that unambiguous utterances contain an extra word
(i.e., the ‘‘surface-level” interpretation cues each or together; Syrett& Musolino, 2016). We also include a null utterance with cost 1;
the null utterance corresponds to saying nothing at all and signaling
the listener to rely wholly on prior beliefs. The parameter a > 0 controls the speaker’s optimality; we set a to 10. Importantly, the qualitative patterns reported below are robust to changes in theseparameters and assumptions.
Our central claim surrounds the effect of PðcÞ, which controls
the predictability of the collective interpretation. To evaluate itsrole, we generated model predictions for distributions of varying
entropy; c was drawn from a normal distribution centered around
0. The variance of this distribution controls the amount of collective interpretation noise. We begin by comparing the effects ofno (r ¼ 0:01), low (r ¼ 1), mid (r ¼ 2), and high (r ¼ 3) collective noise. In Fig. 11, we plot the probability that theinterpretation-resolving variable v is collective after hearing
the ambiguous utterance, marginalizing the other variables fromthe pragmatic listener L1 . In other words, we plot the probabilityof a collective interpretation depending on collective noise andspeaker access.
The qualitative predictions of our plural predication model
match the hypotheses and experimental results that motivatedthe model. Most striking is the monotonic decrease in probabilityof collective interpretations as collective noise increases. We now
have a formal understanding as to why this effect holds: the noisierthe collective interpretation, the less likely the listener L0 is to correctly resolve the question under discussion, namely the propertiesof the named boxes s. Given that the speaker S1 ’s goal is to successfully communicate s to L0 , the noisier an interpretation, the lessuseful it is at achieving this goal; the speaker is therefore less likely
to use the ambiguous utterance, and as a result the pragmatic listener L1 is less likely to infer that the ambiguous utterance, whenused, meant the noisy collective meaning.
The model also successfully predicts the qualitative increase in
the probability of a collective interpretation when the speaker’s
access to individual object properties is limited (red vs. blue bars
in Fig. 11).25 L1 tracks S1 ’s epistemic state, and therefore knows
whether the speaker has full or partial knowledge. In the case of partial (i.e., sum) knowledge, a distributive interpretation lacks epistemic support and therefore becomes less likely; as a result,collective interpretations become more likely.
To compare the quantitative predictions of our model with theresults of Expt. 3, we would need to fit the variance of PðcÞ for eachpredicate and each arrangement condition (i.e., each of the red bars
in Fig. 9). But with only 12 data points by which to fit these 12
parameters, we would not want to claim that the model uniquely,
quantitatively predicts the data. Future work will need to experimentally measure the appropriate noise values and ideally doparametric manipulations of the conditions.
7. General discussion
We began with an old observation about collective interpretations for gradable predicates with plural subjects: some predicates
24
Model predictions remain qualitatively the same if hc is drawn instead from only
the possible total degrees, f6; 7; 8; 9; 10; 11; 12g.
25
For interpretation of color in ‘Fig. 11’, the reader is referred to the web version ofthis article.

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307

Fig. 11. Model predictions for the probability of the ambiguous utterance receiving a collective interpretation as a function of collective interpretation noise and speakerknowledge access.

admit collective interpretations, while others strongly (or totally)
resist them. The puzzle is made even more pronounced by the lackof conceptual barriers to the collective construals of these predicates (cf. the truly distributive predicates like be a man or have blueeyes). This phenomenon, dubbed ‘‘stubborn distributivity” after
those gradable predicates that ostensibly refuse collective interpretations, has been documented and described, even characterized in terms of semantic constraints. Still, an explanation of thephenomenon has proven elusive. Here, we offer an explanation:
stubbornly distributive predicates are so because the collective
properties they name are unpredictable, or unstable, in most contexts; this unpredictability results in a noisy collective interpretation, something speakers and listeners recognize as ineffective forcommunicating efficiently about their world. On this pragmatic
account, stubborn distributivity does not form a distinct category;rather, there is a gradient of stubbornness. Moreover, the extent
that collective interpretations are drawn should be sensitive tocontextual factors.
We found evidence for this pragmatic explanation over thecourse of four experiments. Perhaps most problematic for a categorical account of stubborn distributivity were the results of Expt.
2, where we showed that predicates do not form neat groupings
according to whether they are stubbornly distributive; stubbornness indeed is gradient. It further depends on many aspects ofthe context of predication. Here we focused on two such aspects:
the contextual predictability of collective properties and speakers’
access to knowledge.
After manipulating the regularity of the predication context
directly in Expt. 3, we found that more regular contexts yielded
greater rates of collective paraphrase endorsement. We thus confirmed our hypothesis: increasing contextual predictability of collective properties—by increasing the regularity of the contexts
themselves—increases rates of collective interpretations. In otherwords, rates of collective interpretation were shown to be malleable, in this case influenced by the predictability of the property
named in the specific context of predication.
We similarly confirmed our hypothesis regarding speakerknowledge: in Expts. 1 and 3, we manipulated the discourse context in such a way as to plausibly limit the speaker’s access toknowledge of individual properties. We found that when speakers
lacked epistemic support for a distributive interpretation, collective interpretations became more likely; speakers are less likely
to intend an interpretation for which they lack epistemic support.
In Section 6, we formalized our hypotheses with a computational model of pragmatic dissambiguation that gives a precise rolefor contextual noise and speaker knowledge. This pragmatic model
indeed predicts that the acceptability of collective readings should
depend on predictability and speaker knowledge. Formalizing thepragmatic approach is an important step in validating it as a coherent theory of plural predication and stubborn distributivity.
7.1. Implications for semantic theories of plural predication
Recall that current theories of stubborn distributivity describe it
as a constraint on the sort of subjects certain predicates may compose with. Given the prohibition on plural subjects in their basic
denotations, stubbornly distributive predicates simply cannot deliver collective interpretations. At the very least, the results of ourexperiments demonstrate that stubborn distributivity should not(indeed, cannot) manifest in terms of an all-out prohibition against
collective interpretations. Even unabashedly stubbornly distributive predicates like big yield greater rates of collective interpretations when context supports them; we claim it is the way that
context supports collective interpretations that provides the key
to understanding where stubborn distributivity comes from in
the first place: contextual predictability.
Complaisantly collective predicates like heavy or expensive do
not require support from context to ensure that their collective
properties are predictable. The properties of collective weightand price are inherently stable in context; there exists but one
way to evaluate these properties, and they persist despite changing
physical arrangement, speaker perspective, etc. Not so for the collective properties named by stubbornly distributive predicates likebig or tall, which name properties of physical extent. Owing to
uncertainty in evaluation strategy, compounded by definitional
dependence on physical arrangement, these collective propertiesare relatively unpredictable, unstable, and as a result difficult for
speakers and listeners to coordinate on in the absence of additional
contextual or linguistic cues. Grammar might encode stubborn distributivity as a preference against plural arguments for certain
predicates (or as a preference for plural arguments with fixedphysical arrangements), but here we suggest a simpler explanation
for this preference: stubbornly distributive predicates name collective properties that are difficult to accurately communicate.
Increasing the contextual predictability of these properties
decreases this difficulty and thereby increases the utility and likelihood of collective interpretations.
Further evidence for the non-trivial role of contextual predictability in collective predication comes from data demonstrating that one and the same predicate receives varying rates ofcollective interpretations depending on the noun that serves as
its subject. Looking at naturally-occurring examples of plural predication, we identified the most frequent subject nouns for the stubbornly distributive predicates big and small in a plural predication
sentence frame. These predicates generally resisted collective
interpretations, as we would expect if they were stubbornly

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distributive, but certain subject nouns yielded greater rates of collective interpretations. Crucially, these seemed to be subject nouns
that suggested relatively specific collective properties. Again, contextual predictability of the collective property determines availability of collective interpretations.
Note that subject nouns already played a central role in the disambiguation of plural predications. Schwarzschild (2011) observedthat physically aggregating a plurality using an atomizing noun
like pile or stack greatly increases the availability of collective
interpretations. He observed a similar, though opposite effect formass nouns, which lack any information about physical arrangement. Again, semantics might attend to the atomic status of theresulting noun phrase in its evaluation of stubborn distributivity(e.g., the pile of boxes names a single atom, while the boxes most
likely does not), but pragmatics provides us with an explanation
for why these collectivizing nouns seem to unlock the collective
interpretation. Just as our contextual manipulation increased contextual predictability of collective size or height by increasing the
likelihood of encountering a plurality of boxes in a specific physicalarrangement, so too do atomizing nouns. The arrangement of a
stack or pile of boxes is much more regular across contexts thanan abstract collection thereof. It is this regularity introduced by
the atomizing noun that permits an otherwise elusive collective
interpretation; the pragmatic mechanism of determining the reading remains the same.
To understand this mechanism, namely how contextual predictability affects the calculus of interpretation choice, we modeled
its contribution explicitly within a computational model of plural
predication. Both the formal analysis and experimental results
operate at the level of whole utterances. They leave open several
questions of compositional semantics. Most notably, we assume
that a plural predication is ambiguous between collective and distributive readings. The locus of this ambiguity could be in the predicate, in the noun phrase, or only in the composition of the two. For
instance, it may be that the noun phrase can be construed as either
a plurality (e.g., some boxes) or an aggregate (e.g., a single collection), giving rise to the ambiguity in plural predication. Future
research will be needed to pin down the compositional source of
the ambiguity; having a theory of how the ambiguity is resolvedin context is likely to help.
7.2. Implications for formal models of pragmatics
Our explanation for stubborn distributivity relies on a sophisticated process of recursive reasoning involving listeners and speakers. Thanks to recent advances in computational cognitive science(in our case, simulation-based probabilistic programs; Goodman,
Tenenbaum, & Gerstenberg, 2015), we are able to move beyond
simply describing this reasoning process by implementing a formal
pragmatic model that makes quantitative predictions. By considering utterance semantics within an articulated model of pragmatics,
what began as a description of the phenomenon—certain predicates resist collective interpretations—now has an explanation:
speakers avoid interpretations that are unlikely to correctly resolve
the question under discussion (e.g., collective interpretations of
stubbornly distributive predicates), and listeners know this about
speakers, so they avoid inferring interpretations that speakers
avoid using.
Our model was formulated within the Rational Speech Act
framework, and explores two extensions that may have broader
consequences. First, we used the ‘‘lifted variable” approach to formulate ambiguity resolution as an active pragmatic process. Ambiguity is endemic to natural language (for a recent discussion, see
Piantadosi, Tily, & Gibson, 2012); taken out of context, nearly
everything we say lacks its intended meaning. Applying the pragmatic approach to other cases of ambiguity may be a fruitful way

to understand complex interactions between semantics, context,
and world-knowledge in cases of multiple potential meanings.
Second, we highlighted the role that context-specific predictability, or noise, in computing meaning can have on pragmaticinterpretation. Interlocutors are finely tuned to the pragmatic usefulness of expressions in language; unavoidable noise in interpretation can make expressions less useful, and hence dis-preferred.
We have applied this mechanism to disambiguition in plural predication, but it is likely to have an impact throughout language.
8. Conclusion
We have argued that understanding plural predication, andnotably the phenomenon of stubborn distributivity, arises froman interaction between the rational reasoning processes of language users and qualities of the collective properties they name.
Speakers and listeners require more coordination to align theirbeliefs about properties that are less predictable across contexts(e.g., collective size or shape, which might change as arrangementschange). As a result, they require more support from context toaccurately discuss these potentially variable, uncertain properties.
This support might come in the form of words that increase contextual predictability (e.g., stack, which regularizes otherwise variable physical arrangements) or from increasing the regularity of
the context itself (e.g., by ensuring that sets realize a common
physical arrangement). As collective properties become more predictable, collective interpretations become more informative, more
useful, and therefore more likely.
Appendix A. Full mixed effects logistic regression models fromExpt. 1
Table A.1 presents model coefficients for the full mixed logistic
regression model used in the analysis of disambiguating paraphrases in Expt. 1. The model predicts collective referent choicefrom fixed effects for disambiguating utterance and its interaction
with predicate, together with trial order. The model includes random by-participant and by-scenario intercepts. The fixed effects
predictors were centered before analysis.
Table A.2 presents model coefficients for the full mixed logistic
regression model used in the analysis of bare utterances in Expt. 1.
The model predicts collective referent choice from fixed effects forpredicate, scenario, and trial, as well as random by-participant
intercepts and slopes grouped by trial. Fixed effects predictors
were centered before analysis.

Table A.1
Full logistic regression model from paraphrase analysis of Expt. 1.

Intercept
Utterance
Trial
Utterance:Pred-heavy
Utterance:Pred-tall

Coef b

SE(b)

z

p

0.25
4.99
0.07
1.39
0.63

0.25
0.82
0.07
0.91
0.95

0.97
6.08
1.00
1.52
0.66

0.33
<0.01
0.32
0.13
0.51

Table A.2
Full logistic regression model from bare utterance analysis of Expt. 1.

Intercept
Pred-heavy
Pred-tall
Trial

Coef b

SE(b)

z

p

4.65
3.05
3.63
0.17

1.49
1.23
1.29
0.20

3.12
2.48
2.82
0.84

<0.01
<0.05
<0.01
0.40

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Appendix B. Full mixed effects linear regression models fromExpt. 2
Table B.1 presents model coefficients for the full mixed linear
regression models used in the analyses of the 8 predicates occurring with more than one subject noun in Expt. 2a. The models predict collective endorsement ratings from fixed effects for subject
noun, as well as random by-participant intercepts. For each predicate, the subject noun with the highest mean collective endorsement rating served as the reference level.
Table B.2 presents model coefficients for the full mixed linear
regression models used in the analyses of subject nouns in Expt.
Table B.1
Full linear regression models from subject noun analysis of Expt. 2a.
Coef b

SE(b)

t

p

BrightIntercept-eyesNoun-colors

0.75
0.19

0.07
0.08

11.46
2.26

<0.01
<0.05

ClosedIntercept-doors
Noun-shops

0.60
0.16

0.07
0.09

8.61
1.87

<0.01
0.07

FriendlyIntercept-natives
Noun-people

0.73
0.05

0.05
0.06

13.41
0.93

<0.01
0.37

Full
Intercept-streets
Noun-papers
Noun-pubs

0.68
0.11
0.23

0.06
0.10
0.08

11.53
1.19
2.91

<0.01
0.24
<0.01

GuiltyIntercept-defendants
Noun-men

0.68
0.02

0.05
0.08

12.51
0.21

<0.01
0.84

Open
Intercept-eyesNoun-gates
Noun-doors
Noun-curtains
Noun-pubs
Noun-windows

0.65
0.01
0.03
0.00
0.11
0.15

0.07
0.09
0.08
0.09
0.09
0.08

9.73
0.08
0.40
0.03
1.29
1.77

<0.01
0.94
0.69
0.98
0.20
0.08

Quiet
Intercept-dogs
Noun-streets

0.71
0.06

0.07
0.09

10.72
0.65

<0.01
0.52

Small
Intercept-numbers
Noun-children
Noun-rooms
Noun-classes

0.57
0.16
0.20
0.28

0.06
0.07
0.07
0.07

9.65
2.34
2.78
3.88

<0.01
<0.05
<0.01
<0.01

Table B.2
Full linear regression models from subject noun analysis of Expt. 2b.
Coef b

SE(b)

t

p

Big
Intercept-waves
Noun-rooms
Noun-boys
Noun-children
Noun-houses

0.42
0.18
0.22
0.21
0.23

0.06
0.06
0.06
0.06
0.06

6.73
2.96
3.53
3.54
3.68

<0.01
<0.01
<0.01
<0.01
<0.01

Heavy
Intercept-bags
Noun-lids
Noun-trees
Noun-loads
Noun-men

0.63
0.08
0.15
0.24
0.25

0.07
0.08
0.08
0.08
0.08

9.63
1.04
1.93
3.15
3.36

<0.01
0.30
0.06
<0.01
<0.01

Tall
Intercept-trees
Noun-plants
Noun-offspring
Noun-buildings
Noun-windows

0.30
0.00
0.03
0.06
0.09

0.06
0.06
0.06
0.06
0.06

4.65
0.07
0.55
1.00
1.52

<0.01
0.95
0.59
0.32
0.13

2b. The models predict collective endorsement ratings for each of
the three predicates (big, heavy, tall) from fixed effects for subject
noun, as well as random by-participant intercepts. For each predicate, the subject noun with the highest mean collective endorsement rating served as the reference level.
Appendix C. Full linear regression models from Expt. 3
Table C.1 presents model coefficients for the full linear regression models used in the analysis of Expt. 3. For each predicate,
the models predict collective endorsement ratings from fixed
effects for context, scenario, trial, and the interaction between context and scenario. The fixed effects predictors were centered before
analysis.
Table C.1
Full linear regression models from analysis of Expt. 3.
Coef b

SE(b)

t

p

Big
Intercept
Context
Scenario
Trial
Context:Scenario

0.57
0.18
0.00
0.03
0.21

0.04
0.07
0.07
0.04
0.15

15.69
2.52
0.03
0.67
1.43

<0.01
<0.05
0.97
0.51
0.16

Heavy
Intercept
Context
Scenario
Trial
Context:Scenario

0.61
0.11
0.19
0.02
0.12

0.04
0.07
0.08
0.05
0.15

16.47
1.43
2.54
0.42
0.83

<0.01
0.16
<0.05
0.67
0.41

Tall
Intercept
Context
Scenario
Trial
Context:Scenario

0.66
0.30
0.07
0.01
0.02

0.03
0.07
0.07
0.04
0.13

19.99
4.58
1.11
0.25
0.17

<0.01
<0.01
0.27
0.80
0.86

Appendix D. Full linear regression models from Expt. 4
Tables D.1 and D.2 present model coefficients for the full linear
regression models used in the analysis of Expt. 4. For each dimension, the models predict collective endorsement ratings from fixed
effects for context and trial; for those dimensions with both
positive- and negative-polarity adjectives, the models also
included fixed effects of polarity and the interaction between
polarity and context. The fixed effects predictors were centered
before analysis.

Table D.1
Full linear regression models from the positive-polarity dimension analyses of Expt. 4.
Coef b

SE(b)

t

p

Capacity
Intercept
Context
Trial

0.38
0.16
0.00

0.08
0.08
0.01

4.61
1.95
0.75

<0.01
<0.06
0.46

Depth
Intercept
Context
Trial

0.58
0.07
0.01

0.08
0.09
0.01

7.27
0.85
1.71

<0.01
0.40
0.09

LengthIntercept
Context
Trial

0.47
0.01
0.00

0.07
0.06
0.00

7.07
0.15
0.28

<0.01
0.89
0.78

WeightIntercept
Context
Trial

0.78
0.04
0.01

0.08
0.07
0.01

10.05
0.51
1.86

<0.01
0.62
<0.07

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Table D.2
Full linear regression models from the multi-polarity dimension analyses of Expt. 4.
Coef b

SE(b)

t

p

HeightIntercept
Context
Polarity
Trial
Context:Polarity

0.44
0.04
0.28
0.00
0.13

0.04
0.04
0.04
0.00
0.07

11.53
1.12
7.82
0.30
1.82

<0.01
0.26
<0.01
0.76
<0.07

Size
Intercept
Context
Polarity
Trial
Context:Polarity

0.42
0.03
0.20
0.00
0.13

0.03
0.03
0.03
0.00
0.05

15.52
0.94
7.60
0.16
2.39

<0.01
0.35
<0.01
0.87
<0.05

Width
Intercept
Context
Polarity
Trial
Context:Polarity

0.45
0.06
0.06
0.00
0.14

0.03
0.03
0.03
0.00
0.07

12.95
1.74
1.79
0.98
2.02

<0.01
0.08
0.07
0.33
<0.05

Appendix E. Materials from Experiment 2
Tables E.1 and E.2 present the full set of attested sentences usedin Expts. 2a and 2b, respectively.

Table E.1
Full set of attested sentences from Expt. 2a.
Item

Sentence

Item

Sentence

1
2
3
4
5
6
7
8
9
10
11
12

21
22
23
24
25
26
27
28
29
30
31
32

The men were guilty
The items were unreasonable
The injuries were consistent
The guests were due
The gates were open
The facts were similar
The eyes were brightThe eyes were open
The doors were closed
The doors were open
The dogs were quiet
The defendants were liable

13
14
15

The windows were dark
The windows were open
The walls were bare
The walls were white
The streets were full
The streets were quiet
The shops were closed
The shares were unchanged
The rooms were small
The roads were busy
The results were encouraging
The results weredisappointing
The results were gratifying
The pubs were full
The pubs were open

33
34
35

16
17
18
19
20

The people were friendly
The papers were full
The omens were good
The numbers were small
The natives were friendly

36
37
38
39
40

The defendants were guilty
The curtains were open
The conditions wereappalling
The conditions were difficult
The colors were brightThe classes were small
The children were small
The children were young

Table E.2
Full set of attested sentences from Expt. 2b.
Item

Sentence

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

The houses were big
The waves were big
The rooms were big
The boys were big
The children were big
The trees were heavy
The men were heavy
The loads were heavy
The lids were heavy
The bags were heavy
The trees were tall
The offspring were tall
The buildings were tall
The windows were tall
The plants were tallAppendix F. Supplementary material
Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.cognition.2017.
07.002.

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