How Tall Is Tall? Compositionality, Statistics, and Gradable Adjectives
Lauren A. Schmidt1 (lschmidt@mit.edu), Noah D. Goodman1 (ndg@mit.edu),David Barner2 (barner@ucsd.edu), and Joshua B. Tenenbaum1 (jbt@mit.edu)
1 Department of Brain and Cognitive Sciences, MIT;2 Department of Psychology, University of California, San Diego

Abstract
What is tall? Like many words, tall is fundamentally compositional in meaning — whether an item is tall depends on
the statistical properties of the set of items it is being compared to. Despite preliminary evidence to this effect, no mathematical models of how tall operates in a given context haveever been empirically evaluated. We compare a number of statistical models to adults and children in judging which items
are tall in various contexts, including both threshold-based andcategorization-based models. We find that non-parametric statistical models of gradable adjectives best describes the judgments of people across a wide variety of contexts.
Keywords: psychology; language understanding; Bayesianmodeling

Introduction
Skyscrapers, giraffes, and basketball players all belong to asingle category, despite having almost nothing in common.
In fact, the one property that they do share — being tall —is not true of all three when considered together. Basketball
players are short relative to skyscrapers, and even comparedto giraffes. So how do we – or better yet, 3-year-old children– ever figure out what these things have in common? For thatmatter, what do tall things have in common?
The answer to this problem has two components, the firstof which is old, and was most famously defended by Frege(1892). This is the observation that noun phrases – and expressions of language more generally – are subject to compositionality: the meanings of complex expressions are a function of their syntax and the meanings of their constituent parts(see Fodor & Lepore, 2002, for discussion). In the case ofan expression like tall boy, the interpretation of tall is determined in part by the meaning of the noun it modifies – i.e.,boy. Tall, a subsective adjective, picks out a subset of things
referred to by the noun it modifies (ktallk ⊆ kboyk) unlikeintersective adjectives, like Californian, which pick out theintersection between two sets (kCali f orniank ∩ kboyk). As aresult, it makes sense to ask whether someone is tall for a boy,but not whether he is Californian for a boy. Tall is interpreted
relative to the class of things denoted by the noun it modifies,or the comparison set.
The compositional nature of tall explains why it can betrue of both humans and towers. However, it does not explain
how we know which humans and which towers are tall for
their respective kinds. What, precisely, does it mean to be tallfor a building?
Past work from both psychology and linguistics falls shortof answering this question. Whereas theories of concepts
have traditionally been concerned with determining howmental representations encode properties of things in theworld (e.g., via definitions, prototypes, inferential roles, orcausal histories), previous studies have restricted these investigations almost entirely to nominal categories (e.g., natural kinds, artifact kinds), or to stable perceptual properties(e.g., color, shape, texture). Although previous studies suggest that young children are sensitive to the statistics of sets
when using gradable adjectives like tall and high (e.g., Smith,Cooney, & McCord, 1986; Barner & Snedeker, 2008) none ofthese studies has characterized what type of statistical function is used by children in understanding these terms. Considerable attention has been paid to the formal semantics ofgradable adjectives (Cresswell, 1976; Kennedy, 1999; Klein,1991), but these accounts do not specify how speakers identify things in the world as tall, short, big, or small.
Although no previous study has investigated how classes ofthings are divided into gradable categories like big and tall,some suggestions have been made. For example, things may
count as tall if they are taller than average, or taller than mostthings in a class (Barner & Snedeker, 2008).
In this paper we compare the performance of a numberof possible models of tall to the tallness judgments of people. These models include both models with an absolute standard of comparison (included as a baseline for model performance), and those which perform simple statistical functions
to determine the standard of comparison for a given set ofitems. We additionally explore an alternative model of tallthat combines statistics and categorization, using Bayesian
methods to probabilistically cluster items (e.g., buildings)
into subcategories (e.g., tall buildings) based on their heights.
The standard of comparison for such a model is the boundary
between the tallest subcategory of items and all shorter subcategories. Experiments 1 and 2 explore the performance of
all of these models in a number of different contexts.

Models of tall
We considered several possible models of gradable adjectives
based on the context. We describe the models with regards tothe meaning of tall, but each of the models applies directly to
any gradeable adjective.
For all model definitions, let C be the set of objects in thecontext, and h : C → R be the height function from objects totheir height1 .
Each model below defines a probability P(x is tall|C). Thatis, each gives an alternative way to answer the question “isobject x tall, given the context?”
1 To apply to another gradeable adjective, replace the height function h with an appropriate function mapping from items to the scaleof the adjective.

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CLUS

Figure 1: A distribution of rectangular items of different heights isshown at the top. The graphs below show the probability that each
item is tall, according to each model. Model name and parametersetting specified to right of each graph. (Cluster model shown uses
typical hyperparameter settings.)

We considered two families of models. The first family isa wide class in which a height threshold is computed from thecontext, and tallness decisions are made based on this threshold. To the best of our knowledge, all previous proposals forthe meaning of gradeable adjectives fall into this family.
The second class treats tallness judgement as a categorization problem and invokes the machinery of probabilistic clustering.

Threshold-based models
Threshold-based models compute a threshold statistic fromthe heights of the objects in the context, and make tallness
judgements by comparing to this threshold.
We include normally-distrubuted noise on the threshold2 inorder to permit vague use of “tall.” Let T be a threshold function mapping from context sets C to a positive real number.
The probability of item x being tall is the cumulative probability that a normal random variate with mean T (C) is lessthan h(x):·µ¶¸1
h(x) − T (C)√P(x is tall|C) =1 + erf,2
ε 2• Unique percent number (UAN): as in AN, but computed
based on only one object of each height.
• Percent number (PN): the tallest k% of the items are tall:
T (C) = h(xbk·Nc ).
• Unique percent number (UPN): as in PN, but computed
based on only one object of each height.
• Relative height by range (RH-R): any item within the top
k% of the range of heights is tall. If we write Mx =maxx∈C h(x) and Mn = minx∈C h(x), then: T (C) = Mx − k ·(Mx − Mn).
• Relative height by standard deviation (RH-SD): any itemwith a height greater than k standard deviations above themean is tall. If we write x̄ and σ for the mean and standard
deviation of heights of object in C, then: T (C) = x̄ + k · σ.
• Maximum
margin(MM)3 :
items
tallerthan
the biggest gap in heights are tall:
T (C) =h(argmaxx∈C miny∈Cs.t. h(y)<h(x) |h(x) − h(y)|).

Category-based models
Category-based models classify an element as tall if it is in
the “tall category.” The same psychological mechanism used
for other categorization tasks can be used to determine which
subset of the items are in the tall category, but subject to theconstraint that the tallest item (i.e. argmaxx∈C h(x)) is in thetall category. For example, the max-margin threshold model,
described above, can be seen as a primitive category-basedmodel: it first separates the items into clusters (determinedby the largest gap in heights), then sets the threshold in order
to divide these clusters.
Next consider a more sophisticated category-based model,
related to the infinite Gaussian mixture model (Rasmussen,
2000), a modern version of Anderson’s rational model of categorization (Anderson, 1991); we’ll refer to this as the Cluster model (CLUS). Let Q be a partition of C into clusters ofcontiguous items based on height, and write qx for the partition containing x. First, an item is tall if it is in the samecluster as the tallest item (call this cluster qtall ). The shortest
item is required to be in a separate category from the tallest
item, based on the idea that people expect the items in any
given context to range from “short” to “tall” within that context.
Thus the probability that a particular item is tall is givenby:P(x is tall|C) = ∑ P(x ∈ qtall |Q)P(Q|C)
Q

The posterior probability of a particular partition is:where erf is the error function, and ε a noise-width parameter.
We consider a number of possible threshold functions:

P(Q|C) ∝ P(C|Q)P(Q)Ã• Absolute height (AH): all items taller than a fixed reference
height are tall: T (C) = k, with k a fixed parameter.
• Absolute number (AN): the tallest k items are tall. Let
x1 , ..., xN be an ordering of C by height (i.e. h(xi ) ≤ h(x j )if i < j). Then: T (C) = h(xk ).

= P(Q)

2 This noise may result from within- or between-subjects factors.

!Ã∏ P(x|µqx , σqx )x∈C!

∏ P(µq )P(σq ) ,q∈Qwith the remaining probabilities determined by the model
setup (see Rasmussen, 2000, for details): P(x|µqx , σqx ) is
3 The noise parameter for the MM model is set to 0 to serve as asimple heuristic clustering model.

Gaussian, P(µq ) is Gaussian, P(σq ) is Gamma, and P(Q) is
given by the Chinese restaurant process.
Intuitively, Bayesian inference using this model looks forclusters of items based on height. The model parameters set
prior expectations about the mean and variance of these clusters, as well as the number of items in the clusters, but strong
evidence can overcome these prior expectations.4
For each possible partition of items into clusters, only thetallest cluster of items is considered the ”tall category”; allother items are not tall. To find the overall probability thatan item is tall, we weight whether it is tall in each possible
partition of items by the posterior probability of items being
partitioned in that manner. The posterior probability is basedon both prior expectations and the heights of the items.

Model predictions
Figure 1 shows several different sample contexts C, and examples of model probability that each item is tall, for all of
the models described above.
In the case of an approximately uniform distribution of
items, Figure 1(a), all models show a sigmoid-like drop-off
in probability in approximately the same place (with the exception of the MM model, which is a step function for reasonsdescribed above). Though all the items apparently fall into asingle cluster here, the prior expectations of the CLUS model
about the size of clusters, together with the requirement thatthe shortest item not be tall, lead the Cluster-based model toalso show a sigmoid-like drop-off in determining which items
are tall.
Figure 1(b) and (c) show two more distributions, with 2 or
3 apparent clusters of items respectively. Some of the threshold models separate the two clusters and make the same predictions in Figure 1(b) as the CLUS model, but others show
the same sigmoid-like drop-off as before. In Figure 1(c),the CLUS model treats the items in each of the three clusters uniformly and distinctly from each of the other clusters, while all the other models show very different predictions. Note that varying either the parameter settings (whichchanges the location of the drop-off) or the noise parametersetting (which changes how gradual or sudden the drop-off is)
for the threshold-based models can cause their predictions tochange greatly for individual items, though they will always
have a sigmoidal drop-off.
The major differences in threshold models are also apparent across the distributions, including some limitations: AH
is entirely insensitive to context; AN is sensitive only to theheight-ordering, not the actual heights; MM is determinedsolely by the largest gap in heights. Meanwhile the relative
height models are more flexible, extracting useful statistics ofthe height distribution. In Experiment 1 we will use these differences to seek double-dissociations ruling out the less flexible models as an explanation of human judgements.
4 Four model hyperparameters were varied in the following stud-ies, one controlling the CRP prior, and three controlling the NormalGamma conjugate prior distribution for the cluster mean and variance.

Experiment 1
In Experiment 1, we compared the performance of the models and people in judging which items were tall given distributions of items of different heights.
Adult subjects judged which items were tall in a wide variety of distributions of items. We generated distributions bysampling randomly or at regular intervals from one or two
Gaussian, uniform, or other statistical distributions (such as
an exponential distribution). The means, variances, and number of items within each of the resulting clusters of items varied. Representative distributions are shown in Figure 2(a).
181 adults participated. Each adult saw one distributionof items, all presented at the same time. Items were shownin a frame, like one of those in Figure 2(a), but they were
shuffled into a pseudo-random order instead of being sorted
by height. We labeled each set of items with a novel namelike “pimwits,” and then asked adults to specify which ones
were the tall pimwits.
All models ran on the distributions with a wide variety ofparameter settings.
Results For each distribution, we compared a given
model’s probability that each item was tall to the percent
of adults who labeled that item tall using a measure called
mean difference, which measured the average error per item.
The method of calculating the mean difference is illustrated
in Figure 2(c).
We selected the best parameter setting for each model by
averaging the mean differences between the adult judgments
and the model probabilities across all distributions. Figure 2
shows a histogram of the MD across all distributions for eachof these best models. The models are ranked in order of average performance.
The Relative Height by Standard Deviation, Relative
Height by Range, and Cluster models all perform approximately equally well. The Unique Percent Number and Percent Number models perform only slightly worse overall;however, there are some distributions where both of thesemodels perform very poorly, as shown in Figure 3(a). For
these two distributions we can see a strong double dissociation in performance; on the left, the PN model predicts far
too many items are tall, and on the right, far too few (UPN has
the same problem in many cases, since it reduces to the samemodel as PN whenever the distribution values are unique).
Because of this double dissociation, the PN threshold parameter cannot be adjusted in either direction to better predict all
the adult results, and it systematically fails to capture humanjudgments about what is tall.
Figure 3(b)-(d) also shows similar double dissociations for
the AN, MM, and AH models. Figure 2 demonstrates thesemodels’ increasingly poor performance on the overall distribution set. Because all of these models fail in both directions – sometimes calling too few items tall and sometimes
too many – these models also cannot be account for humanjudgments of tallness. This failure of the two absolute standard of comparison models (AN and AH) is consistent with

(b) RH−SD: 0.094 RH−R: 0.098 CLUS: 0.098

UPN: 0.116

AN: 0.181

PN: 0.124

MM: 0.251

UAN: 0.162

AH: 0.283(c)1
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Figure 2: (a) A number of representative distributions used as stimuli in Experiment 1 (subjects saw these and other sets of objects in
randomized order instead of sorted by height, as shown here). (b) Histograms demonstrating the performance of each model in Experiment 1,
where success is measured along the x-axis in terms of mean difference (see (c) for explanation of measure), with a performance of 0 being
best and 1 being worst (for these histograms, all results worse than 0.5 are binned at 0.5). Each histogram represents the mean difference
measure of one model across all distributions, with the number of distributions receiving each score measured on the y-axis. The model
name and the average mean difference score for that model is shown at the top of each histogram. The models are sorted from best to
worst performance overall. (c) An illustration of how mean difference is calculated for one distribution. For each item in the distribution,
the distance between people’s judgments (blue) and the model’s predictions (red) is calculated (black lines) and the mean difference is theaverage of all these distances.

previous empirical evidence that people are sensitive to context in making judgments of tallness. The failure of the Maximum Margin model indicates that people are not performing
this simple clustering heuristic in judging the tall items.
Also of interest is the one distribution where the RH-SD
model did significantly more poorly than its general performance (see Figure 3(e)). Here, there are many items that are
taller than 0.4 standard deviations above the mean height (theoptimal RH-SD threshold overall), and the RH-SD model labels them all tall. Adult judgments fall off far more quickly,
and there are many items that very few people think are talland the RH-SD model predicts are tall with 100% probability.
Despite this large divergence between model and human performance, however, this model does not show a similar but
opposite pattern of picking far too few items on any of the
distributions in Experiment 1. Experiment 2 further investigates this model’s performance given similar distributions inExperiment 2.
Overall, in Experiment 1, many of the models failed tomatch human judgments of tallness. However, all of the models that depend on the height of objects in a given distribution(RH-R, RH-SD, and CLUS) performed well overall. Thissuggests that, in assessing which items in a given context are
tall, people are juding tallness in a way that can be modeledby a statistical function based on the heights of the objects inthe category. In Experiment 2, we further explore which ofthese three models best describes human judgments.

Experiment 2
We attempted to find distributions where the three best models from Experiment 1 made significantly different predictions.
We designed a set of distributions with a large number of
items taller than the best previous RH-SD threshold (0.4 stan-

dard deviations above the mean), as in the distribution in Figure 3(e), such that the best previous RH-SD model would
make different predictions from the best RH-R and CLUS
models. Additionally, this set has several distributions withvery distinct clusters of objects, causing the CLUS model tomake somewhat different predictions from the more gradualsigmoidal fall-off of the threshold models.
We created six distributions, three of which were designedwith a mid-height cluster of objects that people found to be
ambiguous in terms of tallness, and three which were not.
Two of the distributions used are pictured in Figure 4. The
distributions were each divided two “clumps” of items – theclearly short items, divided into 1–3 clusters, and the taller
items, divided into either one broad cluster or two very distinct clusters (the shorter of which was the ambiguously tallcluster). All the clusters were sampled from a Gaussian distribution. Across all six distributions, the mean and varianceof the short clump remained the same, and likewise for thetall clump. All the items in the tall clump were greater than
0.4 standard deviations above the mean of the overall distribution.
A total of 107 adults judged the tallness of the items inthese distributions, each adult viewing only 1 distribution.
They viewed the distributions on a computer screen and wereasked to identify each of the tall pimwits.
Results Figure 4(a) compares the performance of the threemodels to human judgments for two of the distributions. The
performance of both the best parameter fit for just the six Experiment 2 disttributions is shown, as well as the performanceof the best models overall (for the combination of all Experiment 1 and Experiment 2 distributions).
The best parameters for the RH-R and CLUS models are
similar for the Experiment 2 distributions and for all the
distributions together, and the performances of these mod-

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Figure 3: Several models failed to predict the tallness judgmentsof people in Experiment 1. (a)-(d) show four models exhibiting adouble dissociation – they predict too many items being tall given
the distribution on the left, and too few items being tall given the
distribution on the right. In (e) we see that RH-SD also performed
poorly in one of these directions for one distribution.

els are very similar in both cases; the best RH-R threshold is k = 27% for Experiment 2 (with a noise parameter of
ε = 0.01) and k = 29% overall (with a noise parameter of
ε = 0.05). The CLUS hyperparameters were also similar forthe best Experiment 2 and best overall performance.
In strong contrast, the parameter values and the performance for the RH-SD model are far different for the distributions in Experiment 2 than the combined distribution set. The
best overall model (k = 0.4, ε = 2.5) predicts with certainty
that all the items in the tall clump are tall, whereas people
are uncertain about the middle cluster of items. The RH-SD
parameters can be adjusted to better match the human judgments for Experiment 2 (k = 1.0, ε = 75), but these parameter
results do poorly on the overall set of distributions, as shownin the histograms in Figure 4(b). These histograms also show
the good performance overall of the best RH-R and CLUS
models from Experiment 2.
Both the RH-R and the CLUS models fit the human judgments reasonably closely, and it is not clear which model pro-vides the best fit. There are, however, differences in theirpredictions. The CLUS model does show an almost equal
set of probabilities for the mid-height cluster items being tall;the RH-R model shows a more sigmoidal fall-off for these
items. Additionally, the RH-R model predicts a sharper falloff in tallness judgments for the tallest cluster of items thanthe CLUS model does. And, with the best overall parameters, the RH-R model predicts a greater-than-zero probability of some of the short clump of items being labeled tall,though no humans rated any of these items tall. If the RH-R
noise parameter is adjusted such that the predictions for the
mid-cluster items are even flatter, then the sigmoidal curve
becomes more apparent in both the shorter and taller items.
The CLUS model, by contrast, does not predict any of theshort clump items as tall without drastic parameter variation.
Though these results are suggestive about differences in themodels, we do not have a definitive answer as to which of
the two models best fit human judgments, which, given thecurrent amount of data, are well approximated by both.
Overall, the results of Experiment 2 show clearly that the
RH-SD model, an intuitive parametric model based on themean and standard deviation of a distribution, cannot account for how people make judgments about gradable adjectives. The qualitatively different predictions of the two
non-parametric models, RH-R and CLUS, suggest that future
work will help us understand which of these two models is
the strategy used by people, though on average both models
predict the Experiment 2 data well.

Conclusion
The success of the RH-R model and the CLUS model in Experiments 1 and 2 suggests that people do perform a statisticalcomputation upon the comparison set when using the wordtall. However, this computation is not based on a simple parametric statistical criterion as in the RH-SD model, but somemore nonparametric function that is meaningful for a wider
range of distributions.
Though a parametric model such as RH-SD may seem intuitively appealing, people regularly encounter sets of items
that do not follow a parametric distribution. Such contexts
include cases where the objects under consideration do notbelong to one natural type, e.g., “the things on the table”.
In these situations, the distribution of object heights may behighly nonuniform or multimodal, and non-parametric models can accomodate this in ways that parametric models cannot.
The RH-R model is simpler than the CLUS model, in thesense that it has two free parameters where the CLUS model
has four. However, there are two reasons why the CLUS
model is a compelling model from a cognitive perspective.
First, it seems to capture a crucial idea of linguistic compositionality: we use noun phrases to pick out complex subcategories that are not lexicalized as single nouns but that nevertheless correspond to interesting chunks of the world. Second, this model relies on well-established, domain-general

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Figure 4: (a) Two sample distributions from Experiment 2 are shown at the top, with the performance of the three models shown below. For
each model, the best parameter settings for just Experiment 2 and the best parameter settings for all the distributions of both experiments arecompared. The best RH-R and CLUS models are very similar for overall results vs. for just the Experiment 2 distributions. The best RH-SD
model overall, however, is very different from the best RH-SD model for just the Expt 2 distributions – so much so that the best overall model
does not capture adult judgments for the Experiment 2 distributions well at all. (b) The best Experiment 2 RH-R and CLUS models also
perform very well overall, but that is not the case for the best Experiment 2 RH-SD model. The histograms show the mean differences for thebest Experiment 2 models across all distributions from Experiments 1 and 2.

mechanisms for categorization. If these strategies also apply to understanding gradable adjectives, this would not entail
positing any truly new cognitive complexity.
Future work will further explore the different predictions of
the RH-R model and the CLUS model to answer the questionof which non-parametric method people are employing when
they use gradable adjectives. Once the best model has beenfound for tall, this work can be extended to related questionsof language compositionality — does the best model also apply to how people use other gradable adjectives, such as big,small, short, and loud, and are the best parameters similar?
What about intensifiers like very — can the very tall boys
be identified by applying either the Cluster model or rangebased thresholding to the class of tall boys? Additionally,
there are quantifiers such as most or several that apply to anordered scale of set-sizes, and like gradable adjectives, canhave flexible, context-sensitive meanings that seem to draw
on both compositionality and statistics (Halberda, Taing, &Lidz, 2008). Can a model of gradable adjectives give insight
into how those words are used? Further work remains in orderto determine the extent to which the model applies to otherwords with compositional meaning besides tall.
Also of interest is the developmental progression of gradable adjective understanding. While work by Barner andSnedeker (2008) suggests that children use tall as a statistical categorization function from an early age, it remains to be
seen whether they start out with the same model of gradable
adjectives that they eventually use as adults, or whether theylearn the structure of the meaning of tall based on experience.
While much work remains to be done, the knowledge that astatistical, non-parametric model predicts human usage of tallmarks a step forward in our understanding of how people use
gradable adjectives and, more broadly, how word meanings
can compose to form phrases.

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