Intuitions about magic track the development of intuitive physics.

Cognition 214 (2021) 104762
Available online 26 May 2021
0010-0277/© 2021 Elsevier B.V. All rights reserved.Intuitions about magic track the development of intuitive physics
Casey Lewrya,*, Kaley Curtisb, Nadya Vasilyevaa, Fei Xub, Thomas L. Griffithsa,c
aDepartment of Psychology, Princeton University, USA
bDepartment of Psychology, University of California, Berkeley, USA
cDepartment of Computer Science, Princeton University, USA
ARTICLE INFO
Keywords:
Magic
Intuitive physics
Ontological commitments
Violation-of-expectation ABSTRACT
Many successful magic tricks violate our assumptions about how physical objects behave, but some magic tricks
are better than others. We examined whether the interest adults express in a magic trick is predicted by the age at
which infants first respond to violation of the corresponding physical principle. In Experiment 1, adults (N
319) rated their interest in magic tricks mimicking stimuli from violation-of-expectation experiments with in-
fants. We found a clear correlation between how interesting a trick is and the age at which infants demonstrate a
sensitivity to its underlying principle. In a second experiment (N 350), we replicated this finding and also used
three additional tricks for which there is no established age of acquisition to predict the age at which those
physical principles might be acquired. A third experiment (N 368) replicated these findings measuring adults ’
surprise at physical violations rather than their interest in magic tricks. Our results suggest that adults ’ intuitions
reflect the development of physical knowledge and show how magic can reveal our expectations about the
physical world.
1.Introduction
Psychology and conjuring are natural partners, being the science and
the art, respectively, of understanding the limits of the human mind. The
two disciplines have a shared history that goes back more than a century
(Binet, 1894 ; Triplett, 1900 ). Over the last few years, this relationship
has been reinvigorated through a significant increase in research (for a
review, see Kuhn, 2019 ; Rensink & Kuhn, 2015 ) and a number of col-
laborations between psychologists and magicians (e.g., Macknik, King,
Randi, Teller, & J., & Martinez-Conde, S., 2008 ; Mohr, & Koutrakis &
Kuhn, G., 2015 ; Phillips, Natter, & Egan, 2015 ). However, much of this
research is focused on the mechanisms by which human perception and
thinking can be deceived, rather than what it is that makes that
deception seem magical (for exceptions, see Griffiths, 2015 ; McCoy &
Ullman, 2019 ; Parris, Kuhn, Mizon, Benattayallah, & Hodgson, 2009 ;
Shtulman & Morgan, 2017 ).
For something to seem like magic, it has to defy a mundane expla -
nation (Ortiz, 2006 ). To that end, people ’s intuitions about magic must
reflect their intuitions about the mundane. Intuitions about magic can
thus be a guide to the implicit notions we have about how the world
around us works. Two recent studies have supported this idea. First,
Griffiths (2015) showed that people ’s intuitions about magical transformations revealed an implicit ontological hierarchy among ob-
jects, with transformations moving in the direction of allowing more
predicates (reflecting animacy and intelligence) being more compelling
as magic tricks. For example, transforming a glass of milk into a white
dove is more compelling than the reverse. Second, McCoy and Ullman
(2019) showed that magical acts perceived as more substantial viola-
tions of physical principles were also believed to require more effort on
the part of the magician. These studies raise a new question: what is it
that makes some physical principles seem harder to violate than others?
In this article we explore a potential answer to this question, drawing
on the substantial literature on the development of physical intuitions
about object solidity, continuity, contact causality, support, etc. in in-
fancy (for reviews, see Baillargeon, 2004 ; Xu, 2019 ). The grasp of a
principle is typically assessed in a violation-of-expectation paradigm,
measuring infants ’ looking times (Sim & Xu, 2019 ; Spelke, Breinlinger,
Macomber, & Jacobson, 1992 ; Wang, Baillargeon, & Brueckner, 2004 ).
In violation-of-expectation paradigms, infants are first given a set of
habituation trials in which they become familiar with the experimental
stimuli and events (e.g., a toy car rolls down a ramp and goes behind an
occluder). Then, infants view a test event which is consistent with the
physical principle being studied (e.g., the car stops when it hits a solid
wall – a demonstration of solidity) and a test event which is inconsistent
*Corresponding author at: Department of Psychology, Princeton University, Peretsman Scully Hall, Princeton, NJ 08540, USA.
E-mail address: lewry@princeton.edu (C. Lewry).
Contents lists available at ScienceDirect
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https://doi.org/10.1016/j.cognition.2021.104762
Received 25 September 2020; Received in revised form 29 April 2021; Accepted 30 April 2021
Cognition 214 (2021) 104762
2with the physical principle (e.g., the car stops at the far end of the solid
wall as if it has passed through it – a violation of solidity). In these
studies in the infant literature, longer looking time during inconsistent
trials is commonly interpreted to provide evidence for three things:
“first, infants possess the expectation being examined; second, they have
detected a violation of that expectation; and third, they are surprised by
the violation, measured as increased attention or interest ” (Sim & Xu,
2019 , p. 155).
However, there is some debate surrounding the extent to which
looking time is an accurate measure of surprise. Wang et al. (2004) note
that when they refer to violation-of-expectation paradigms as measuring
infants ’ surprise, “surprise ” is shorthand for a state of attention or in-
terest. While there is wide consensus that a difference in looking time
indicates detection of a difference between the two events, some have
argued that this attention or interest could be caused by familiarity with
the event or prediction of an event, rather than by surprise at a violation
of an expectation, thus providing no evidence for an understanding of
the physical principle in question (e.g., Bogartz, Shinskey, & Speaker,
1997 ; Jackson & Sirois, 2009 ). However, as Hamlin (2014) explains, the
evidence for infants ’ surprise at an event is distinct from evidence for
infants ’ prediction of an event, and well-designed research can distin -
guish between these two interpretations. The studies used in the ex-
periments reviewed here were specifically selected because they
conform to best practices in violation-of-expectation research, thus
minimizing the concern that they fail to accurately measure infants ’
surprise through looking times (Cohen, 2004 ). While our findings should
be considered in light of this debate about looking time as a measure of
infants ’ surprise and physical reasoning, further discussion is beyond the
scope of this paper (for further reading, see Aslin, 2007 ; Bogartz et al.,
1997 ; Hamlin, 2014 ; Tafreshi, Thompson, & Racine, 2014 ; Sim & Xu,
2019 ; Stahl & Feigenson, 2015 ).
Violation-of-expectation experiments suggest a timeline for the
acquisition of physical principles based on careful testing of failures (i.e.,
infants not distinguishing between the expected and unexpected out-
comes, measured by their looking time; this lack of difference has been
interpreted as the infants not being surprised to see a violation and
therefore not yet understanding the physical principle in question) and
successes (i.e., infants looking longer at the unexpected outcome than
the expected outcome; this difference has been interpreted as surprise to
see a violation and therefore understanding of the physical principle)
with infants of different ages, these experiments suggest development
trajectories in physical intuitions, with infants being sensitive to some
physical principles earlier than others and a clear progression in the ages
at which sensitivity to different principles emerges (e.g., Baillargeon,
1999 ; Spelke et al., 1992 ). Furthermore, there is evidence that alterna -
tive explanations, such as development of the visual system, are insuf-
ficient to explain why infants appear to “succeed ” in demonstrating
understanding of physical concepts at certain ages, as infants still “fail”
these paradigms even as they demonstrate sensitivity to visual features
of the task (e.g., Wang & Baillargeon, 2008 ).
Prior work has demonstrated similarities between infants ’ and
adults ’ physical reasoning. For example, Turk-Browne, Scholl, and Chun
(2008) argue that habituation trials in infant studies are analogous to
habituation trials in adult functional neuroimaging studies since both
depend on attenuating interest after repetition and both measure pref-
erence for novelty. Along a similar vein, Strickland and Scholl (2015)
argue that both infants ’ and adults ’ visual processing is structured by
representations of event-types (e.g., occlusion, containment). These
studies provide evidence that in terms of habituation and visual pro-
cessing, infants and adults may reason about the physical world simi-
larly. This continuity between results with infants and adults leads us to
the hypothesis that the ages at which infants show violation of expec -
tation reactions to an anomalous physical event may provide a clue as to
why some magic tricks are better than others. Specifically, we predict
that violating principles that infants become sensitive to earlier will
make for stronger magic. We can imagine different causal mechanisms by which the age of
acquisition of physical principles could come to be related to adult re-
sponses to the violation of those principles. Some principles may be
more fundamental because of the evolved architecture of the human
mind: we may be innately endowed with a small set of principles about
the physical world (e.g., Spelke & Kinzler, 2007 ) with those principles
that are most fundamental to making sense of physical events being
those that emerge earliest. Another possibility is that some physical
principles may be more fundamental because abundant statistical evi-
dence in the environment makes those principles easier to learn (e.g.,
Baillargeon ’s model of adding physical variables to event types over
time; see Baillargeon, 1998 ) and these principles are strengthened by
more statistical evidence from the real world over the course of devel -
opment. Under either of these mechanisms, or some combination of
them, the more entrenched our beliefs are about the physical world –
measured by age of acquisition – the more surprising and interesting it is
to witness a violation of such beliefs, as in magic tricks. Showing a
relationship between age of acquisition and the strength of a magic trick
would thus provide further support for the idea that our intuitions about
magic reflect the unconscious commitments we have about the world
around us.
To evaluate this hypothesis, we conducted three experiments in
which people viewed magic tricks and judged how interesting they
found each trick to be. The magic tricks mimicked stimuli shown to
infants in classic violation-of-expectation experiments (e.g., Baillargeon,
1995 ; Spelke et al., 1992 ), which suggest ages of acquisition for the
physical principles underlying each trick. In the first experiment, we
compared these ages of acquisition to participants ’ interest ratings to
determine whether there is a relationship. In the second experiment, we
replicated the first experiment and also collected interest ratings of three
magic tricks for which there is no established age of acquisition. These
ratings allowed us to predict ages at which infants might become sen-
sitive to the physical principles underlying these tricks.1 Finally, these
first two studies operationalized infants ’ looking time as an adult mea-
sure by asking for interest ratings. However, as we have discussed, there
is some debate over how exactly infants ’ looking times should be
interpreted. For this reason, we ran a third experiment measuring adults ’
“surprise ” at physical events. Using this alternative measure and
removing the magical context, we confirmed that adults ’ interest in
magic tricks tracks their surprise at the corresponding anomalous
physical events, providing additional evidence for continuity between
our approach and the violation-of-expectation paradigm used with
infants.
2.Experiment 1
2.1. Method
2.1.1. Participants
Participants in Experiment 1 were 319 adults recruited via Amazon
Mechanical Turk. An additional 86 participants were excluded for
failing an attention check (described below). Participants were paid at a
rate of $7.50 an hour (i.e., $1.25 for a 10-min experiment) and partic -
ipation was restricted to workers in the United States who had
completed at least 500 prior tasks with a 95% approval rating. The
sample size was selected so as to reach a power greater than 0.95 for the
main correlation analysis on the basis of the results of a pilot study with
participants on Mechanical Turk.
1 The reported experiments were approved by the Institutional Review Board
at Princeton University under protocol #10859 (Computational Cognitive Sci-
ence). Experiments 1 and 2 were preregistered and are available at https://as
predicted.org/blind.php?x 6b6u84 (Experiment 1) and https://aspredicted.
org/blind.php?x6dv58v (Experiment 2). C. Lewry et al.
Cognition 214 (2021) 104762
32.1.2. Materials and procedure
Participants in all studies completed an online task on Qualtrics.
Magic tricks were presented one at a time in a random order, each fol-
lowed by an interest rating question. For each trick, participants viewed
a three-panel picture of the trick accompanied by text describing each
panel. Each trick was displayed for ten seconds before an interest rating
question – “How interesting is this magic trick? ” – was presented. Par-
ticipants responded on a “1 - Not interesting ” to “10 - Very interesting ”
scale.
The tricks were selected by reviewing the literature on infants ’
acquisition of physical principles. We carefully selected a set of physical
principles and their corresponding suggested ages of acquisition from
violation-of-expectation studies based on several selection criteria. We
aimed to select studies which: (1) found evidence of an age of acquisi -
tion, (2) adhered to best practices in infant research, (3) had similar
levels of visual simplicity (e.g., all tricks were presented in grayscale and
featured spheres, cylinders, and cubes as the primary object), (4)
employed a specific version of the looking-time method consistently,
therefore allowing us to test the developmental trajectory in adults
without having to delve into the controversy surrounding different ways
of employing the looking time methods (e.g., Bogartz et al., 1997 ), (5)
are representative of work standardly reviewed in discussions of phys-
ical reasoning in infancy (e.g., Carey, 2009 ) and textbooks on develop -
mental psychology (e.g., Siegler, Saffran, Eisenberg, & Gershoff, 2020 ).
All ages of acquisition were determined using the earliest age at which
there was evidence suggesting that infants could comprehend the
physical principle as measured in violation-of-expectation paradigms.
For most of the tricks, this age was reported in a published paper, but for
two of the tricks (3 and 5), we used ages cited in unpublished manu -
scripts that were referenced in published work. For some tricks we used
three-panel illustrations from the original publications; new illustrations
were made for the remaining tricks. Each picture panel was accompa -
nied by text describing a magician ’s actions (see Fig. 1). We also
included an attention check, which had a three-panel picture-and-text
format, but replaced some text in the middle panel with instructions to
select option 10 as the interest rating. In total, participants in Experi -
ment 1 viewed ten magic tricks and one attention check (see Table 1 and
supplemental material).
2.2. Results
Supporting our main hypothesis, we found a significant negative
correlation between individual participants ’ interest ratings and age of
acquisition for each trick (r(317) 0.16, p D.001). To account for
variation across participants, we normalized each participant ’s interest
rating and also found a significant negative correlation between the
individual normalized ratings and age of acquisition (r(317) 0.21, p
D.001). As a more rigorous way of capturing individual variation, we
used the lmerTest library in RStudio (Kuznetsova, Brockhoff, & Chris -
tensen, 2017 ; RStudio, 2020 ) to fit a multilevel model with age of
acquisition as a predictor of interest rating, with random intercepts and
slopes for each participant. The model showed that age of acquisition
was a significant predictor (β 0.16, t 10.91, p D.001), providing
evidence that the age at which a physical principle was acquired offers
information about how interesting a magic trick is. As an additional
analysis using the same multilevel model, we re-ran the regression by
item, predicting mean ratings for each trick from the acquisition age (see
Fig. 2). To give a sense for the effect size, the correlation between the
average ratings and the age of acquisition was r(8) 0.59.2 Inspection of Experiment 1 in Fig. 2 shows that Tricks 3 and 5, the
tricks for which we used ages cited in unpublished manuscripts, deviate
most significantly from the negative linear relationship. For this reason,
we conducted follow-up exploratory analyses using only published ages.
This resulted in changing the age of acquisition for Trick 3 from 3
months to 4.5 months and removing Trick 5 from analyses. With these
changes, we again found a significant negative correlation between raw
interest ratings and age of acquisition (r(317) 0.16, p D.001), and
between normalized interest ratings and age of acquisition (r(317)
0.21, p D.001). In addition, the magnitude of the correlation between
averaged ratings and age of acquisition increased (r(7) 0.63), and
age of acquisition remained a significant predictor of rating when ages
cited in manuscripts were removed from the multilevel model (β
0.15, t 10.23, p D.001). Taken together, these results suggest that
the earlier a physical principle is learned, the more interesting its vio-
lations are in the context of magic tricks. We aimed to replicate this
effect in Experiment 2.
3.Experiment 2
3.1. Method
3.1.1. Participants
Participants in Experiment 2 were 350 adults recruited via Prolific.3
An additional 50 participants were excluded for failing an attention
check. Participants were paid at a rate of $7.50 an hour and participa -
tion was restricted to workers in the United States who had completed at
least 100 prior tasks with a 95% approval rating.
3.1.2. Materials and procedure
Experiment 2 materials and procedures were the same as Experiment
1, with two changes. First, the age acquisition of Trick 3 was changed
from 3 months to 4.5 months, consistent with the earliest published re-
sults (Needham & Baillargeon, 1993 ). Second, the stimuli included three
exploratory tricks that were presented alongside the other tricks, but
analyzed separately. These included Trick 5, which was removed from
our main analyses because there was no published evidence of an age of
acquisition for the relevant principle, and two additional tricks for
which there was no published age of acquisition (see Table 1 and sup-
plemental material).
3.2. Results
By only using tricks for which there is a published age of acquisition,
we were able to replicate the findings of Experiment 1 in which we
removed manuscript ages (see Fig. 3). For the nine tricks earmarked for
our main analyses, we found a significant negative correlation between
participants ’ raw interest ratings and age of acquisition (r(348) 0.19,
p D.001) and normalized interest ratings and age of acquisition (r(348)
0.25, p D.001). A multilevel model with age of acquisition as a
predictor of rating and random intercepts and slopes for each participant
revealed a significant main effect of age of acquisition (β 0.16, t
12.88, p D.001). The observed relationship between average ratings
and age of acquisition was still strong (r(7) 0.63).
Next, we calculated the means of the three tricks for which there is no
published age of acquisition. Trick 5 received a mean rating of 3.56 (CI
[3.33, 3.78]), Trick 11 received a mean rating of 4.35 (CI [4.12,
4.59]), and Trick 12 received a mean rating of 6.43 (CI [6.19, 6.67]).
2 We focus on effect size rather than statistical significance here because there
are only ten observations, an intrinsic result of the limited number of studies on
physical intuitions in infants. With ten observations, the Pearson correlation
would need to exceed 0.8 for an experiment to have a power of 0.95 for
detecting it. 3 Recruitment for Experiment 2 was changed to Prolific due to data quality
issues experienced on Mechanical Turk. In piloting Experiment 2 on Mechanical
Turk, we found that only 61% of participants passed the attention check used in
Experiment 1, suggesting a higher incidence of bots or unfocused workers at the
time of the study. This change has the added benefit of replicating with a
slightly different population. C. Lewry et al.
Cognition 214 (2021) 104762
4The linear regression model based on the main set of nine tricks was used
to predict the ages of acquisition of these additional tricks from their
average interest ratings. Assuming that the relationship between age of
acquisition and ratings is linear, our model predicts that sensitivity to
Trick 5 would be acquired at 9.61 months (CI [5.18, 14.05]), Trick 11
at 7.57 months (CI [4.91, 10.23]), and Trick 12 at 2.26 months (CI
[2.29, 6.82]).
4.Experiment 3
4.1. Method
4.1.1. Participants
Participants in Experiment 3 were 368 adults recruited via Prolific.
An additional 32 participants were excluded for failing an attention
check. Participants were paid at a rate of $7.50 an hour and participa -
tion was restricted to workers in the United States who had completed at
least 100 prior tasks with a 95% approval rating.
4.1.2. Materials and procedure
Experiment 3 comprised two between-subjects conditions. In the
interest condition (N 180), materials and procedures were the same as
Experiment 2. In the surprise condition (N 188), participants were
asked “How surprising is this event? ” and responded on a “1 - Not sur-
prising ” to “10 - Very surprising ” scale. The surprise condition contained
no references to magic and instructed participants that they would be
Fig. 1.Example of a trick shown to participants. Aguiar and Baillargeon (1999) report that the physical principle underlying this trick is understood by 2.5-
month-olds.
Table 1
Ages of acquisition and sources of stimuli used in Experiment 1.
Trick
number Age of acquisition
(months) Source
1 2.5 Spelke et al., 1992
2 2.5 Aguiar & Baillargeon, 1999
3* 3 (Exp. 1); 4.5
(Exps. 2, 3) Needham & Baillargeon, 1992 (m.s. cited in
Needham & Baillargeon, 1993 ); Needham &
Baillargeon, 1993
4 3.5 Baillargeon & DeVos, 1991
5* 5 (Exp. 1); N/A
(Exps. 2, 3) Baillargeon, Raschke, & Needham, 1994 (m.s.
cited in Baillargeon, 1995 )
6 6.5 Leslie & Keeble, 1987 ; Oakes, 1994
7] 6.5 Baillargeon, Needham, & DeVos, 1992
8 7.5 Hespos & Baillargeon, 2001
9 8.5 Aguiar & Baillargeon, 1998
10 12 Wang, Baillargeon, & Paterson, 2005
11* N/A Spelke et al., 1992
12* N/A Spelke et al., 1992
*In Experiments 2 and 3, the age of acquisition for Trick 3 was adjusted to 4.5
months, Trick 5 was removed from main analyses, and Tricks 11 and 12 were
added.
]In the original study, a finger pushes a smiley-faced box to the edge of a
striped platform. In our version (see Appendix), a hand lowers a plain gray box
to the edge of a plain platform. These features were changed so they would not
be more visually salient than other tricks.
Fig. 2.Results of Experiments 1 (left) and 2 (right): mean normalized participant interest ratings plotted against the age of acquisition of each magic trick, and the
best fit line from Pearson regression analysis. Error bars indicate 95% CI. C. Lewry et al.
Cognition 214 (2021) 104762
5presented with a series of physical events and that we would like to
know how surprising it would be to see each of these events actually
occur.
4.2. Results
By including surprise ratings as a dependent variable in addition to
interest ratings, we were able to directly replicate the findings of
Experiment 2 and demonstrate that they extend to ratings of surprise, a
term which may better reflect infant violation-of-expectation measures.
For the nine tricks earmarked for our main analyses, we found a sig-
nificant negative correlation between participants ’ raw interest ratings
and age of acquisition (interest : r(178) 0.19, p D.001; surprise : r
(186) 0.18, p D.001) and normalized interest ratings and age of
acquisition (interest : r(178) 0.26, p D.001; surprise : r(186) 0.26,
p D.001). A multilevel model with age of acquisition as a predictor of
rating and random intercepts and slopes for each participant revealed a
significant main effect of age of acquisition (interest: β 0.17, t
9.82, p D.001; surprise: β 0.17, t 9.09, p D.001). The observed
relationship between average ratings and age of acquisition was still
strong (interest: r(7) 0.61; surprise: r(7) 0.66).
Next, we calculated the means of the three tricks for which there is no
published age of acquisition. In the surprise condition, Trick 5 received a
mean rating of 5.54 (CI [5.13, 5.94]; interest: M 3.34, CI [3.01,
3.68]), Trick 11 received a mean rating of 5.82 (CI [5.43, 6.22]; in-
terest: M 4.03, CI [3.69, 4.36]), and Trick 12 received a mean rating
of 7.27 (CI [6.89, 7.66]; interest: M 6.38, CI [6.03, 6.72]). The
linear regression model based on the main set of nine tricks was used to
predict the ages of acquisition of these additional tricks from their
average interest ratings. Assuming that the relationship between age of
acquisition and ratings is linear, our model for the surprise condition
predicts that sensitivity to Trick 5 would be acquired at 8.44 months (CI
[5.27, 11.61]; interest: 9.37 months, CI [4.92, 13.82]), Trick 11 at
7.70 months (CI [5.07, 10.32]; interest: 7.87 months, CI [4.85,
10.89]), and Trick 12 at 3.96 months (CI [1.10, 6.83]; interest: 2.71
months, CI [1.65, 7.07]).
5.Discussion
We explored whether adults ’ interest in magic tricks is related to the age at which children become sensitive to the physical principles which
underlie the tricks, as suggested by violation-of-expectation studies. In
Experiment 1, we found evidence of a relationship between age of
acquisition and interest ratings such that the most interesting magic
tricks were those which violate the physical principles learned earliest in
infancy, and vice versa. In Experiment 2, we replicated these findings
with published ages of acquisition and were able to predict the ages of
physical principles for which there is not yet evidence in the literature.
In Experiment 3, we again replicated these findings and found that they
extend beyond the magical context to ratings of surprisingness of
physical events, which may more closely parallel infant violation-of-
expectation paradigms.
Our results contribute to the growing literature demonstrating the
ways in which magic can help us understand aspects of cognition beyond
perception. Asking participants a simple question about how interesting
they find a magic trick recapitulates the time course of the development
of physical expectations. Although the literature on infant physical
reasoning suggests that we commit to physical expectations rapidly – for
example, all of the tricks we tested are thought to be typically under -
stood within the first year of life4 – the order in which we likely acquire
them can still be parsed using adults ’ intuitions. This finding is consis -
tent with other results showing that early intuitions about the world
persist into adulthood in many domains (e.g., Keil, 2011 ; Lombrozo,
Kelemen, & Zaitchik, 2007 ; Shtulman & Harrington, 2016 ). One tanta -
lizing implication is that we may be able to use adults ’ intuitions to
guide infant research.
Of course, our predictions for the ages of acquisition of the additional
tricks in Experiments 2 and 3 are estimates based on an assumed linear
relationship and are most informative when considered relative to the
other principles tested. Future work could provide more evidence for
these predictions and the relationship between age of acquisition and
magic trick interest by testing, using an infant violation-of-expectation
paradigm, at what age infants acquire the physical principles underly -
ing our additional tricks. This paradigm would provide additional sup-
port that magic is a useful tool for furthering our understanding of infant
cognition.
In order to connect distant dots across development we have bridged
different experimental paradigms: infant studies assessing looking time,
on the one hand, and adult interest and surprise ratings, on the other
hand. In both infant violation-of-expectation tasks and in the present
experiments, the aim is to measure the clearest expression of interest
that each participant group can provide. Since infants cannot verbally
communicate, researchers rely on whatever indications of engagement
with a task that infants can provide; in younger infants and most
violation-of-expectation publications, this is looking time. In some
studies with older infants, this is their physical exploration of an object
that has violated a physical principle (e.g., Sim & Xu, 2017 ). These
dependent variables – looking time or engagement with an object – are
taken by researchers to be an indication of infants ’ interest in and sur-
prise at an expectation-violating event. Adults, on the other hand, are
capable of providing a more direct measure of interest: self-reporting.
We have therefore used participants ’ reported interest in magic tricks
and surprise at physical violations.
As mentioned in the introduction, we can imagine different mecha -
nisms by which age of acquisition and adult intuitions might be related.
The first is consistent with the view that the development of physical
intuitions is strongly shaped by innate constraints (e.g., Spelke et al.,
1992 ; Spelke & Kinzler, 2007 ). Under this view, the age of acquisition
might reflect the strength of those constraints, a factor that has enduring
influences even into adulthood. The second mechanism instead em-
phasizes the role of learning. Under this view, the reason some physical
Fig. 3.Results of Experiments 3 for the interest and surprise conditions: mean
normalized participant interest ratings plotted against the age of acquisition of
each magic trick, and the best fit line from Pearson regression analysis. Error
bars indicate 95% CI.
4 However, as noted in the Introduction, there is debate surrounding
violation-of-expectation paradigms which test understanding of these phe-
nomena. Our results should be considered in light of this debate. C. Lewry et al.
Cognition 214 (2021) 104762
6principles are acquired earlier is the abundance of statistical evidence in
the world in favor of those principles. Age of acquisition thus tracks a
property of the world – the extent to which the principle is manifest,
something that we might expect to correlate with the extent to which it
seems immutable. Teasing apart these two possible mechanisms is an
intriguing direction for future research.
Beginning in infancy, our observations of the world around us
quickly turn into expectations. We expect that solid objects cannot pass
through one another, that things cannot teleport to another location,
and so on. As researchers, we can learn when infants develop these ex-
pectations by measuring whether they look longer at events in which
physical laws are violated. In a similar way, as we have shown, adults ’
interest in magic tricks allows us to measure their assumptions about
physical objects. Since magic tricks, like stimuli in developmental
research, go against what we typically see in everyday life, they allow us
to discover our ontological commitments.
Open practices
All data and stimuli have been made publicly available via the Open
Science Framework and can be accessed at https://osf.io/pj3uh/?
view_only57a7ce2eca9c434fbd5bdb814df25278 . Both experiments
were preregistered and can be accessed at https://aspredicted.
org/blind.php?x6b6u84 (Experiment 1) and https://aspredicted.
org/blind.php?x6dv58v (Experiment 2).
Funding
This research did not receive any specific grant from funding
agencies in the public, commercial, or not-for-profit sectors.
Declaration of Competing Interest
None.
Appendix A.Supplementary data
Supplementary data to this article can be found online at https://doi.
org/10.1016/j.cognition.2021.104762 .
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Casey Lewry
Kaley Curtis
Nadya Vasilyeva
Fei Xu
Thomas L
Griffiths


The Dynamics of Exemplar and Prototype Representations Depend on Environmental Statistics.

The Dynamics of Exemplar and Prototype Representations
Depend on Environmental Statistics
Arjun Devraj (adevraj@princeton.edu)
Department of Computer Science, Princeton University
Qiong Zhang (qiongz@princeton.edu)
Princeton Neuroscience Institute, Princeton University
Thomas L. Grifﬁths (tomg@princeton.edu)
Departments of Psychology and Computer Science, Princeton University
Abstract
How people represent categories—and how those representa-
tions change over time—is a basic question about human cog-
nition. Previous research has suggested that people catego-
rize objects by comparing them to category prototypes in early
stages of learning but use strategies that consider the individ-
ual exemplars within each category in later stages. However,
many category learning experiments do not accurately reﬂect
the environmental statistics of the real world, where the proba-
bility that we encounter an object changes over time. Our goal
in this study was to introduce memory constraints by present-
ing each stimulus at intervals corresponding to the power-law
function of memory decay. Since the exemplar model relies
on the individual’s ability to store and retrieve previously seen
exemplars, we hypothesized that adding memory constraints
that better reﬂect real environments would favor the exemplar
model more early on compared with later. Conﬁrming our hy-
pothesis, the results illustrate that under realistic environmen-
tal statistics with memory constraints, the exemplar model’s
advantage over the the prototype model decreases over time.
Keywords: category learning; prototype model; exemplar
model; memory decay; environmental statistics
Introduction
Computational models of categorization have played a key
role in understanding the representations people use when
learning categories. Two prominent models of categorization
include the prototype model, which posits that a stimulus is
categorized by comparing it to solely the “mean” object, or
prototype, of each category (Posner & Keele, 1968; Reed,
1972), and the exemplar model, which asserts that a stimulus
is categorized by comparing it with all of the objects for each
category (Medin & Schaffer, 1978). Earlier ﬁndings strongly
favored the exemplar model (McKninley & Nosofsky, 1996;
Medin & Schaffer, 1978; Shin & Nosofsky, 1992), but these
did not consider the progression of category learning over
time. Smith & Minda (1998) studied the ﬁts of the proto-
type and exemplar models of categorization to human perfor-
mance on a word categorization task over time and found that
although the exemplar model dominates in the end, the pro-
totype model has a strong advantage in early stages of learn-
ing, particularly with non-linearly separable (NLS) category
structures. This result is consistent with the proposal that in-
dividuals categorize using a simple rule-based approach in
early epochs and more exemplar-speciﬁc strategies in later
epochs (Nosofsky et al., 1994).
While prototype-based categorization vs. exemplar-based
categorization has been extensively studied in the litera-ture, less attention has been paid to the constraints of mem-
ory on categorization; namely, how much one is able to
utilize prototype-based or exemplar-based categorization is
constrained by how accessible these representations are in
their memory. Prototype-based categorization requires re-
trieving generalized knowledge about the category, whereas
exemplar-based categorization requires retrieving memory
about speciﬁc instances. It has been shown that one cat-
egorizes using generalized knowledge in early epochs and
switches to an exemplar-based model in later stages (Smith &
Minda, 1998), which does not seem consistent with ﬁndings
in the memory literature showing that it becomes increasingly
easier to access representations of general knowledge than
representations of speciﬁc items over time—i.e., memory of
speciﬁc items decays much faster than memory of the gist, or
general knowledge (Posner & Keele, 1970; Zeng et al., 2021).
How would one rely more on exemplar-based representations
in late stages of categorization when it is difﬁcult to access
these exemplars in memory?
Memory of speciﬁc items decays quickly for a reason.
From a rational perspective, there is no need to retain infor-
mation in memory if it is no longer needed, as determined
by the statistics of the environment (Anderson & Schooler,
1991). Therefore, memory decay obeys a power law function
because the relationship between stimulus recency and the
need odds, the odds that the stimulus will be required in the
future, often obeys the power law in realistic environments,
as examined from environmental sources such as The New
York Times , parental speech, and electronic mail (Schooler
& Anderson, 1997; Figure 1a). In contrast, categorization
experiments, including Smith & Minda (1998), usually use
stimuli that appear with the same frequency regardless of the
last time the stimulus was seen (Figure 1b). This provides
an opportunity for overlearning individual stimuli, which is
unrealistic given the environmental statistics encountered in
the real world (Figure 1a). To ﬁll this gap, we designed an
experiment that better reﬂects the power-law relationship be-
tween stimulus recency and the need odds (Figures 1c, 1d).
We hypothesize that setting up the categorization experiment
with realistic environmental statistics will reduce the accessi-
bility of exemplar-based representations in the memory over
time, therefore reducing or inverting the trend previously ob-
served in favoring prototype-based representation early on
(a) From Schooler & Anderson (1997)
 (b) Smith & Minda (1998)
(c) Our Experiment
 (d) Our Experiment, in Log-Log Coordinates
Figure 1: The environmental recency function, displaying the relationship between stimulus recency and its need odds, cal-
culated as P(stim)=(1P(stim))where P(stim)is the probability of seeing the stimulus. (a) is based on New York Times
headlines (1986 and 1987), reproduced from Figure 4a in Schooler & Anderson (1997). (b) is calculated based on the experi-
mental design of Smith & Minda (1998). (c) and (d) display average data from the 60 sequences of stimuli generated for the
experimental condition of our experiment. In (c) and (d), P(stim)was calculated using a window size of 15 trials. The linear
log-log relationship in (d) indicates that (c) is a power-law function.
and exemplar-based representations at late stages.
We expand on the work of Smith & Minda (1998) using
Anderson & Schooler’s (1991) observations about realistic
environmental structures and memory retention to redesign
the experiment. In the control condition, we replicated Smith
& Minda’s second experiment (with the NLS category struc-
tures), in which stimuli are presented with uniform frequency
over time. In the experimental condition, however, we con-
strained the presentation of stimuli to reﬂect the power-law
function in order to better simulate a real-world environment
involving memory constraints (Anderson & Schooler, 1991).
Since the exemplar model requires the storage and retrieval of
all exemplars, later trials would involve heightened memory
constraints.
The plan of the paper is as follows. We ﬁrst explain the
formulations of the prototype and exemplar models of cate-
gorization. Next, we delineate how we constructed the pre-
sentation of stimuli in the experimental condition in order to
reﬂect the power-law model encountered in the real world. Fi-
nally, we discuss the behavioral and model results from bothour replication of Smith & Minda (1998) in the control con-
dition and our manipulations of the environmental statistics
in the experimental condition.
Background
Prototype Model
Various versions of the prototype model exist (Medin &
Schaffer, 1978; Medin & Smith, 1981; Reed, 1972); we
use here the formulation speciﬁed by Smith & Minda (1998,
2011). The prototype model compares the observed stimulus
to the prototype for each category. First, the distance di;Pkbe-
tween stimulus iand the prototype for category k2f1;2g,Pk,
is computed as
di;Pk=m
å
j=1wjjijPk;jj; (1)
where ijis the stimulus’ value for dimension j, and Pk;jis
the the value of the prototype of category kfor dimension
j, and wjis the attentional weight assigned to dimension j.
Each attentional weight wjis constrained to take on a value
in[0;1], and the weights together sum to 1 ( åm
j=1wj=1).
Once the raw distance di;Pkhas been calculated, stimulus i’s
similarity, hi;Pk, to the category kprototype Pkis computed as
hi;Pk=ecdi;Pk; (2)
where cis a sensitivity parameter constrained to [0;20]. The
sensitivity parameter has the effect of amplifying or shrink-
ing psychological space (Smith & Minda, 1998). Finally, the
probability that stimulus i(Si) will be categorized into cate-
gory 1 ( R1) is given by
P(R1jSi) =hi;P1
hi;P1+hi;P2: (3)
Exemplar Model
The exemplar model has been developed and generalized
from the original context model (Medin, 1975; Nofosky
1984, 1986, 1987, 1988; Palmeri & Nofosky, 1995; McKin-
ley & Nosofsky, 1995). This formulation of the exemplar
model was used and thoroughly described in Smith & Minda
(1998). The exemplar model compares the observed stimu-
lus to all of the previously seen exemplars in each category in
order to generate a category prediction for the observed stim-
ulus. The distance di;xand similarity measure hi;xbetween
stimulus iand exemplar xare calculated identically as in the
prototype model. However, the exemplar model considers all
exemplars in order to generate a prediction, so the probability
that stimulus i(Si) will be categorized into category 1 ( R1) is
P(R1jSi) =åx2C1hi;x
åx2C1hi;x+åx2C2hi;x; (4)
where C1is the set of exemplars for category 1 and C2is the
set of exemplars for category 2.
While Nosofsky & Zaki (2002) criticized the lack of a
response-scaling parameter in the exemplar model used in
Smith & Minda (1998), extensive debate in the literature
(Myung, Pitt, & Navarro, 2007; Smith & Minda, 1998, 2002)
led us to exclude this parameter in order to reduce the com-
plexity of the model and enable direct comparison to the orig-
inal experiment. Because our experiment used stimuli with
six dimensions, both models had a total of seven parameters:
the six attentional weights ( w1;:::;w6) and the sensitivity pa-
rameter ( c). The two models were then ﬁt to participant data
from the control and experimental conditions of the experi-
ment.
Methods
Our experiment aims to replicate Experiment 2 (NLS cate-
gories) from Smith & Minda (1998) in the control condition
and made modiﬁcations to introduce memory constraints in
the experimental condition. The task required participants
to repeatedly categorize 14 stimuli into two categories. The
stimuli were six-letter nonsensical words taken from Ap-
pendix A of Smith & Minda (1998). Each stimulus can bethought of as a six digit string of bits, such that the prototype
for category 1 was 000000 and the prototype for category 2
was111111 . Seven stimuli belonged to category 1 [ 000000 ,
100000 ,010000 ,001000 ,000010 ,000001 ,111101 ], and the
other seven stimuli belonged to category 2 [ 111111 ,011111 ,
101111 ,110111 ,111011 ,111110 ,000100 ]. Each digit and
position in the binary string corresponds to a unique letter. In
each trial, the participant was shown one of the 14 stimuli and
had unlimited time to select a response of 1for category 1 and
2for category 2. For exactly one second after each trial, the
participant was shown Correct orIncorrect accordingly.
We collected data from 60 participants for each condi-
tion. (For reference, Smith & Minda (1998) had 32 par-
ticipants.) Participants were recruited from the 18-23 age
range and English-speaking population using Proliﬁc. The
hypothesis, design, sample size, and analysis of this experi-
ment have been preregistered ( https://aspredicted.org/
yq984.pdf ). Scripts for the exemplar and prototype models
were derived from the open source code ( https://github
.com/jpminda/Categorization Models ).
Control Condition
The control condition, similar to Smith & Minda’s (1998) ex-
periment (the only difference being 616 trials in our experi-
ment vs. 560 trials in the original experiment), consisted of
616 trials divided into 44 blocks, each consisting of 14 tri-
als. Each block consisted of a random permutation of the 14
stimuli such that each stimulus was shown exactly once per
block.
Experimental Condition
To create the environmental statistics in Figure 1a, we de-
veloped an algorithm to generate a sequence of stimuli for
the experiment. The speciﬁc pattern of stimulus presentation
should, in turn, inﬂuence the need odds and hence the par-
ticipant’s retention of objects in memory. The experimental
design ﬁrst involved assigning each of the fourteen stimuli its
own power-law function, generically deﬁned as
f(x) =axk: (5)
We added several new parameters to the power-law function
to generate the necessary experimental conditions:
f(t) =8
>>><
>>>:0 t<t0
s t =t0
a(tt0
r)kqt0<tt0+n
0 t>t0+n(6)
where tis the trial number, t0is the start trial for the stimulus
to be introduced, ris the range, nis the number of trials the
object’s power-law function should be sampled from, sis the
starting value for the power-law function, and qis a calibra-
tion parameter. Power-law functions for consecutive stimuli
were introduced in intervals of 35 trials. As in the control
condition, each participant participated in a total of 616 tri-
als. Figure 2a displays the resulting power-law functions for
(a) Theoretical Power-Law Functions
 (b) Stimuli Assignments after Binning and Sampling
(c) Example Stimulus Sequence for a Single Participant
Figure 2: (a) Each curve corresponds to the power-law function for a particular stimulus. (b) Since data is aggregated, each
curve corresponds to the i-th stimulus seen across all participants. For example, the ﬁrst curve illustrates the average proportion
of each bin allocated to the ﬁrst stimulus (regardless of its identity) shown to the 60 participants. (c) Each color corresponds to
a unique stimulus i2f1;14g, and the ﬁgure highlights the signiﬁcant temporal clustering and subsequent decay caused by our
stimulus generation algorithm based on power-law functions. Parameter values of a=1,k=0:3,r=1000, n=350, s=10,
andq=a(n
r)kwere chosen to adequately reﬂected the power-law function in Figure 1a.
each stimulus over the entirety of the experiment. The order
of the stimuli was randomly assigned for each participant.
To assign a stimulus to each trial based on the continuous
power-law functions, we binned trials into discrete bins of 35
trials, generated a probability distribution for the bin based
on the relative values of the various stimuli power-law func-
tions over the trials in the bin, and generated the stimuli for
trials in the bin by sampling from this distribution. Figure 2b
shows how the resulting sequences of stimuli adequately re-
ﬂected the original power-law functions. Figure 2c shows an
example of stimuli sequence for a single participant. Finally,
we veriﬁed that the sequences of stimuli generated by this
method reﬂected the environmental statistics in Figure 1a.
Model Fitting
Trials were split and grouped into 11 trial segments, each con-
taining1
11th of the trials for each stimulus based on when the
trial occurred in the experiment. Conceptually, for the exper-
imental condition, the ﬁrst trial segment corresponds to the
earliest point along all stimuli power-law functions (highest
retention), and the last trial segment corresponds to the lat-
est point (lowest retention); for the control condition, trial
segments correspond to chronological groupings of trials in
absolute time, exactly as in Smith & Minda (1998).
For the comparison of the control condition to Smith and
Minda’s (1998) results, we used the sum of squared er-
rors (SSE) between the observed and predicted probabilities
as the measure of ﬁt, which we sought to minimize. We
ran Scipy’s Sequential Least Squares Programming (SLSQP)method with 10 initial random conﬁgurations to obtain the
best-ﬁt parameters. The model was ﬁt for each participant
receiving the control condition over each trial segment, as
in Smith & Minda (1998). The resulting best-ﬁt parameters
were used to calculate the SSE that was ultimately used as the
measure of ﬁt in Figure 4.
However, the same model-ﬁtting procedure and deﬁnition
of ﬁt are not germane for the experimental condition because
trial segments did not represent chronological groupings of
trials in absolute time; consequently, we could not use the
models to generate predicted probabilities for entire trial seg-
ments, as these contained many trials distributed over time,
and not the exact sequences that were seen by the partici-
pant. (For the exemplar model, this is especially problematic
because each trial within a given trial segment was seen at
a different point in time, thereby involving a different num-
ber of previously seen exemplars. Any probabilities gener-
ated by the model for an entire trial segment could not possi-
bly account for these differences.) Hence, we used the mean
squared error (MSE) within a trial segment, calculated on a
trial-by-trial basis (automatically setting the predicted prob-
abilities of trial-irrelevant stimuli to zero), as the measure of
ﬁt, and ﬁt the model for each participant over the entire exper-
iment (rather than per trial segment) using the SLSQP method
with 10 initial random conﬁgurations. We ensured that the ex-
emplar model only considered exemplars seen thus far since
not all exemplars were seen from the beginning. The new
model-ﬁtting procedure and MSE ﬁt calculation were neces-
(a) Control Condition
 (b) Experimental Condition
Figure 3: Average categorization accuracy over trial segments. Error bars denote the standard error of the mean.
sary so that we could apply the models to an individual trials
(since it would be inappropriate to apply the model to non-
contiguous sets of trials constituting a trial segment and gen-
erate meaningless predicted probabilities based on a set of
data that was not actually observed by the participant, partic-
ularly when different trials within a given trial segment might
involve a different set of previously seen exemplars) and then
average the squared per-trial residuals to derive a metric of
the “ﬁt” for a given trial segment. For ease of comparison,
the same model-ﬁtting procedure was applied to the control
condition, and the comparison is shown in Figure 5.
Results
Behavioral Results
The categorization accuracies per trial segment in each con-
dition were averaged over all participants and graphed overtime, resulting in Figure 3. In the control condition, signiﬁ-
cant learning occurred between the ﬁrst ( M=.62,SD=.11)
and last ( M=.73, SD=.18) trial segment, t(59) =5:08,
p< :001. In the experimental condition, memory constraints
effectively caused participant classiﬁcation accuracy to de-
crease from the second ( M=.92,SD=.08) to last ( M=.81,
SD=.15) trial segment, t(59) =7:99,p< :001. The second
trial segment was compared in the experimental condition be-
cause this was the point that best reﬂected the beginning of
the power-law function—when the participant has repeatedly
seen the stimulus and is past the very initial stage of learning.
Model Results
The model results over trial segments from the control con-
dition are compared with the results from Smith & Minda
(1998) in Figure 4. The model results over trial segments
(a) Replication
 (b) Original
Figure 4: Comparison of (a) the results from our approximate replication in the control condition with (b) the results from the
original experiment, reproduced from Smith & Minda (1998). Models were ﬁt for each participant’s performance over a trial
segment and then averaged over all participants for the condition, with ﬁt calculated as the SSE in a trial segment. Error bars
denote the standard error of the mean.
(a) Control Condition
 (b) Experimental Condition
Figure 5: Comparison of the prototype and exemplar models in the (a) control and (b) experimental conditions. Models were
ﬁt for each participant’s performance over the entire experiment and then averaged over all participants for the condition, with
ﬁt calculated as the MSE of all per-trial residuals in a trial segment. Error bars denote the standard error of the mean. While
the MSE ﬁt was required for the experimental condition due to trial segments being non-contiguous groupings of trials, it was
also done for the control condition for ease of comparison.
from the experimental condition are compared with the re-
sults from the control condition in Figure 5. As shown in
Figures 4 and 5, the advantage of the exemplar model over
the prototype model is increasing in the control condition but
decreasing in the experimental condition.
To formally test if the difference in model ﬁts between the
prototype and exemplar models changed over the course of
the experiment, we ran a two-way ANOV A on the model
ﬁts with the trial segment (ﬁrst/second, last) and model type
(prototype, exemplar) as within-subject variables. For the
analysis of the control condition using the ﬁt measure and
model-ﬁtting method derived from Smith & Minda’s (1998)
experiment, the interaction between model type and trial
segment was still signiﬁcant, F(1;59) =19:72,p< :001.
The interaction between model type and trial segment was
also signiﬁcant when using our modiﬁed model-ﬁtting proce-
dure and calculation of ﬁt for both the experimental condi-
tion, F(1;59) =80:75,p< :001, and the control condition,
F(1;59) =21:55,p< :001.
General Discussion
How well do the prototype and exemplar models of cate-
gorization reﬂect human performance over different stages
of learning under realistic environmental statistics? In this
study, we replicated the experiment by Smith & Minda (1998)
in our control condition, in which the frequency of stimulus
presentation remains ﬁxed regardless of the last occurrence
of the stimulus. We modiﬁed this design to present stimuli
according to the power-law model of memory decay in or-
der to better reﬂect realistic environmental statistics (Ander-
son & Schooler, 1991) in our experimental condition. We
hypothesized that setting up the categorization experiment
with realistic environmental statistics would reduce the ac-cessibility of exemplar-based representations in the memory
over time, therefore reducing or inverting the trend in Smith
& Minda’s (1998) ﬁndings. Our ﬁndings from the control
condition mostly align with Smith & Minda’s (1998) ﬁndings
that when stimuli are presented with consistent frequency, the
exemplar model’s advantage over the prototype model grows
over time. The results from the experimental condition con-
ﬁrm our hypothesis: when stimulus presentation more accu-
rately reﬂects the power-law property of the environment, the
exemplar model’s advantage over the prototype model gener-
ally wanes over time.
Average categorization accuracy improved over time in the
control condition; this pattern is expected, as participants
were shown stimuli with equal frequency throughout the ex-
periment and thus could learn from more information over
time. In the experimental condition, however, categoriza-
tion accuracy steadily declined after the second trial segment.
(The ﬁrst trial segment represents the initial stage of learn-
ing, during which accuracy is still expected to be lower.)
Therefore, our experimental design was successful at intro-
ducing memory constraints that inhibited performance over
time. Since later trial segments contain trials further along
each object’s memory decay curve, when the object is pre-
sented less frequently, the behavioral results follow the im-
plications of the recency effect on memory—that individuals
have more difﬁculty retaining objects that were not presented
as recently (Anderson & Schooler, 1991).
Our ﬁndings help to reconcile incongruities between the
results from category learning and memory experiments.
Indeed, unlike traditional experimental scenarios involving
constant environmental recency functions (Smith & Minda,
1998), memory constraints encountered in realistic environ-
ments reduce the advantage of the exemplar model over the
prototype model over time. Under such memory constraints,
the faster decay of individual exemplars in memory relative to
the decay of the prototype (Posner & Keele, 1970; Zeng et al.,
2021) more strongly inhibits the performance of the exemplar
model over the prototype model later in learning.
Acknowledgements
This work was supported by a C.V . Starr Fellowship awarded
to Q.Z.
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Arjun Devraj
Qiong Zhang
Thomas L
Griffiths