REPORT
“Impossible” Somatosensation and the
(Ir)rationality of Perception
Isabel Won1,2, Steven Gross1,2,3, and Chaz Firestone1,2,3
1Department of Psychological and Brain Sciences, Université Johns Hopkins, Baltimore, MARYLAND, Etats-Unis
2Department of Cognitive Science, Université Johns Hopkins, Baltimore, MARYLAND, Etats-Unis
3Department of Philosophy, Université Johns Hopkins, Baltimore, MARYLAND, Etats-Unis
un accès ouvert
journal
Mots clés: perception, somatosensation, impossible figures, magic, rationality
ABSTRAIT
Impossible figures represent the world in ways it cannot be. From the work of M. C. Escher to
any popular perception textbook, such experiences show how some principles of mental
processing can be so entrenched and inflexible as to produce absurd and even incoherent
outcomes that could not occur in reality. Cependant, impossible experiences of this sort are
mostly limited to visual perception; are there “impossible figures” for other sensory modalities?
Ici, we import a known magic trick into the laboratory to report and investigate an impossible
experience for somatosensation—one that can be physically felt. We show that, even under
full-cue conditions with objects that can be freely inspected, subjects can be made to experience
a single object alone as feeling heavier than a group of objects that includes the single object
as a member—an impossible and phenomenologically striking experience of weight. De plus,
we suggest that this phenomenon—a special case of the size-weight illusion—reflects a kind
of “anti-Bayesian” perceptual updating that amplifies a challenge to rational models of perception
and cognition.
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
e
d
toi
o
p
m
je
/
je
un
r
t
je
c
e
–
p
d
F
/
d
o
je
/
je
/
.
/
1
0
1
1
6
2
o
p
m
_
un
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
un
_
0
0
0
4
0
p
d
.
/
je
F
b
oui
g
toi
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
INTRODUCTION
One of the most striking and puzzling aspects of our minds is that they permit “impossible”
experiences: perceptions of the world that are physically, geometrically, or even conceptually
incoherent. Par exemple, when an image carefully exploits patterns of shading and layout, nous
may see it as a triangle with three 90° angles, or as a closed staircase that descends in every
direction (Chiffre 1), even though such objects could never actually exist. Impossible experi-
ences are intuitively compelling, but they are also theoretically significant: They go beyond
ordinary visual illusions (par exemple., when stimuli appear larger, faster, or darker than they really are)
in revealing how some principles of mental processing can be so entrenched and inflexible as
to produce absurd and even self-contradictory outcomes that could not occur in reality.
De plus, they do so in a way that is intrinsic to the perceiver’s experience itself, such that
they also go beyond (un) inconsistencies in the underlying perceptual processing (par exemple., Smeets &
Brenner, 2008, 2019; Sousa et al., 2011) et (b) conflicts between perception and higher level
connaissance (Firestone & Scholl, 2016un). In many impossible figures, a single experience rep-
resents the world in a way it cannot be.
Citation: Won, JE., Gross, S., &
Firestone, C. (2021). “Impossible”
Somatosensation and the (Ir)rationality
of Perception. Open Mind: Discoveries
in Cognitive Science, 5, 30–41. https://
doi.org/10.1162/opmi_a_00040
EST CE QUE JE:
https://doi.org/10.1162/opmi_a_00040
Supplemental Materials:
https://osf.io/g7kb8
Reçu: 8 Août 2020
Accepté: 20 Avril 2021
Intérêts concurrents: The authors
declare no conflict of interest.
Auteur correspondant:
Chaz Firestone
chaz@jhu.edu
droits d'auteur: © 2021
Massachusetts Institute of Technology
Publié sous Creative Commons
Attribution 4.0 International
(CC PAR 4.0) Licence
La presse du MIT
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
Chiffre 1. Examples of images that produce “impossible” perceptual experiences. Such figures represent the world in ways it could not be
(par exemple., a triangle with three 90° angles).
Despite their ubiquity and popularity, the scope and impact of impossible experiences have
been limited in at least two important ways. D'abord, whereas more conventional perceptual
illusions are found in all sensory modalities, impossible figures arise mostly or only within
visual perception (with perhaps an exception in audition; Shepard, 1964); en effet, it may be
difficult to even fathom impossible experiences for other senses (par exemple., an impossible taste or
smell). Deuxième, all known impossible figures require unnatural or impoverished viewing con-
ditions, such as stimuli drawn on two-dimensional surfaces, or accidental views of precisely
arranged 3D scenes (Deregowski, 1969; Macpherson, 2010; Penrose & Penrose, 1958).
En effet, these factors may have diminished the perceived scientific value of such phenomena,
casting them as ecologically invalid tricks rather than theoretically significant data for under-
standing the mind.
Impossible Feelings?
In contrast to these classical examples, here we report an impossible perceptual experience
that can be physically felt (rather than seen or heard), using ordinary real-world objects that
perceivers can freely inspect (rather than restrictive presentation conditions). We suggest that
the space of impossible experiences is larger than has been appreciated, extending into another
sensory modality. Surtout, we further suggest that the present phenomenon is not just a
phenomenological curiosity, but rather that it interacts with discussions about “rational” pro-
cessing in the mind by amplifying a prominent challenge to Bayesian models of perception
and cognition. Enfin, we connect this phenomenon to a research trend where principles
known to professional magicians can inform psychological investigation (Ekroll et al., 2017;
Rensink & Kuhn, 2015).
The Present Experiments
Our work here exploits the logic of the century-old size-weight illusion (Charpentier, 1891). Dans
the classical incarnation of this illusion, subjects are shown two objects whose sizes differ but
whose weights are identical; surprisingly, subjects who lift both objects find that the smaller
object feels heavier than the larger object. Though the cognitive mechanisms underlying this
illusion have remained surprisingly difficult to pin down, the conditions under which it occurs
are extensively catalogued, including variants of the illusion arising even in blind or blind-
folded subjects who sense the objects’ sizes using only touch (Élise & Lederman, 1993) or even
echolocation (Buckingham et al., 2015). In these and many other cases, a smaller object feels
heavier than an equally weighted larger object (for a review, see Buckingham, 2014).
OPEN MIND: Discoveries in Cognitive Science
31
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
e
d
toi
o
p
m
je
/
je
un
r
t
je
c
e
–
p
d
F
/
d
o
je
/
je
.
/
/
1
0
1
1
6
2
o
p
m
_
un
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
un
_
0
0
0
4
0
p
d
.
/
je
F
b
oui
g
toi
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
The experience elicited by the classical size-weight illusion is certainly odd (and even im-
probable), but it is not quite “impossible”; après tout, the smaller object could be (and usually is)
made of a denser material than the larger object. But might there be a way to modify this
illusion in ways that truly do produce an impossible—and even conceptually incoherent—
percept? En effet, a lesser-known “bar trick” variant of the illusion might do just that.
Instead of comparing one object to an unrelated object, this variant asks subjects to compare
one object to a group of objects that includes the first object as a member. The thought is that,
under these circumstances, subjects might perceive the single object alone as heavier than a
group including the single object. Bien sûr, a group of objects could never weigh less than a
member of that group; and so if this is indeed what subjects experience, they will have had an
impossible experience of weight.1
To our knowledge, this latter phenomenon, if it occurs at all, has never been reported in the
scientific literature.2,3 We thus (un) introduce it here, (b) run several new experiments exploring
it (and controlling for alternative explanations that have never been discussed or tested), et
(c) suggest that it poses a unique challenge to rational models of perception and cognition, due
to the chain of updating that seems to occur in producing it. We expand on this final possibility
in the General Discussion, since the nature of this challenge is clearest only after the exper-
iments are described.
EXPERIMENT 1: IMPOSSIBLE SOMATOSENSATION
Methods
All of the raw data supporting the experiments in this article are available in a Materials
Archive (see https://osf.io/g7kb8). We report how we determined our sample size, all data ex-
clusions, all manipulations, and all measures in the study.
Participants Thirty subjects were recruited from the Johns Hopkins University (JHU) community.
Given how subjectively apparent the effect was to each author of this article, we chose this
sample size because it would produce a statistically reliable effect if more than two-thirds of
subjects experienced the illusion. All other experiments here used this sample size.
1 Philosophers distinguish various kinds of impossibility. Par exemple, logical impossibility is formal inconsis-
tency (of the form “P and not-P”); conceptual impossibility is inconsistency with the meaning of the concepts
deployed (“He is a married bachelor”); nomological impossibility is inconsistency with natural law (“It traveled
faster than the speed of light”) (Kment, 2021). It is often controversial what kind of impossibility is at issue in a
particular case. For our purposes, it suffices that it is at least nomologically impossible, at any given time, for a
part to weigh more than the whole of which it is a part—and, further, nomologically impossible, across times, pour
a part to weigh more than the whole of which it is a part if neither has changed in any relevant way.
2 The closest reference, peut-être, is Koseleff (1937), who reported that a small heavy block lifted alone felt
heavier than when lifted with the addition of a much larger (but lighter) block on top—such that subjects de-
scribed their experience as “unlogisch” in debriefing. Cependant, this older study included significant visual dif-
ferences between the objects (unlike in our studies here), did not counterbalance lift order (with the two objects
always lifted together before the single object, which could produce effects of adaptation or hysteresis), et
failed to equate grasp posture and force across the lifts. (En effet, the same paper reported that the effect was
reduced or eliminated when subjects pushed the boxes up from below vs. when they were grasped from above.)
Enfin, this previous report of course did not connect the findings up with contemporary issues around Bayesian
models of perception and cognition (as we do; see General Discussion). Toujours, this result is encouraging enough
to suggest that a modernized version might produce similar or even more striking results, and in ways that could
justify the weightier theoretical consequences we attach to it here. We thank Robert Volcic for bringing this
paper to our attention.
3 A variant of it, cependant, is discussed (and occasionally sold) as a “magic trick” (https://www.grand-illusions
.com/three-card-box-illusion-c2x21140225).
OPEN MIND: Discoveries in Cognitive Science
32
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
e
d
toi
o
p
m
je
/
je
un
r
t
je
c
e
–
p
d
F
/
d
o
je
/
je
/
/
.
1
0
1
1
6
2
o
p
m
_
un
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
un
_
0
0
0
4
0
p
d
.
/
je
F
b
oui
g
toi
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
Chiffre 2.
Schematic depiction of the three-boxes illusion. In the present experiments, sujets
performed lifts of identical-looking boxes. Though the boxes were identical in appearance, one of
the boxes weighed much more than the others. Subjects lifted either the heavy box alone, or all
three together.
Stimuli and Procedure Subjects saw three identical-looking opaque boxes in a stack, which we
refer to here as Boxes A, B, and C. All three boxes were 6.7 cm × 8.7 cm × 1.6 cm, 3D-printed
in blue ABS plastic. Instructions for creating these boxes are available in our materials ar-
chive.4 Unbeknownst to subjects, Box A (located on top of the stack) was filled with zinc
(250 g), while Boxes B and C were empty (weighing only 30 g; Chiffre 2).
Subjects performed two lifts, one immediately after the other. In one case, they lifted Boxes
UN, B, and C together; in another case, they lifted Box A alone. Here in Experiment 1, sujets
lifted the boxes simply by grasping them with their hands, in whatever posture felt natural;
later experiments varied this grasp posture.
After the two lifts (order counterbalanced across subjects), subjects were asked which lift
felt heavier (ou, for half of subjects, which lift felt lighter), and the experimenter recorded the
subject’s response.
Results
Subjects overwhelmingly reported that Box A alone felt heavier than Boxes A, B, and C together
(90% of subjects reporting A > ABC, binomial probability test, p < .001; Figure 3).5 However,
this result should be “impossible,” because the sum of weights over a set of objects could never
be less than the sum of weights over a subset of those objects: Unless the boxes somehow
changed between lifts, Box A couldn’t weigh more than a group of weighted objects that
includes Box A as a member.
Indeed, the experience was so striking that subjects often spontaneously and astoundedly
commented on its impossibility, and even requested to lift the objects again after the session
was over. Anecdotally, those subjects reported that the illusion persisted even during these
repeated lifts, including when subjects placed all three boxes on their palm and then suddenly
removed the two lighter boxes—distilling the phenomenon into a single impossible “moment”
wherein removing weight caused the sensation of adding weight.
4 We thank Mark Fuller for assistance designing and creating these objects.
5 Two subjects insisted that the two lifts felt the “same.” When this happened, we coded their response as 0.5
for the purposes of the binomial probability test (rather than 1 or 0). However, no result here—both in this ex-
periment and later experiments—depended on this coding scheme; in other words, all effects remain statistically
reliable even if those data are simply excluded, and even if they are counted against our hypothesis.
OPEN MIND: Discoveries in Cognitive Science
33
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
e
d
u
o
p
m
i
/
l
a
r
t
i
c
e
-
p
d
f
/
d
o
i
/
i
.
/
/
1
0
1
1
6
2
o
p
m
_
a
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
a
_
0
0
0
4
0
p
d
/
.
i
f
b
y
g
u
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
e
d
u
o
p
m
i
/
l
a
r
t
i
c
e
-
p
d
f
/
d
o
i
/
Figure 3. Results from Experiments 1–3. No matter how subjects lifted the boxes, they over-
whelmingly reported that the single heavy box seem to weigh more than all three boxes together—
an “impossible” experience of weight. (Though the image for Experiment 3 shows a “floating” hand,
subjects in that experiment in fact passively rested their hands on a flat table.)
These results thus confirmed experimentally what is also just subjectively apparent upon
casually lifting the boxes: Under these conditions, a single box alone feels impossibly heavier
than a group of boxes that includes the single box as a member.
i
/
/
.
1
0
1
1
6
2
o
p
m
_
a
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
a
_
0
0
0
4
0
p
d
/
.
i
EXPERIMENT 2: EQUATING GRASP POSTURE
An important assumption for interpreting this effect as an impossible experience is that it not be
driven by differences in how subjects perform the two lifts. (This is also important for interpreting
other instances of the size-weight illusion.) For example, subjects in Experiment 1 inevitably
used a different grasp posture when lifting all three boxes (ABC) vs. when lifting just one (A),
since the ABC lift required fitting three times as much material in their hands, and so required
a wider grasp, more extended fingers, and other such adjustments. Could this sort of difference
explain the difference in perceived weight? Experiment 2 ruled out this possibility by attaching
handle-like “tabs” to the boxes, such that each lift was performed by pinching a single tab
between one’s fingers. If the effect persists under these circumstances, then it is unlikely to be
explained by differences in grasp posture, since the same posture was assumed for both lifts.
f
b
y
g
u
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
Methods
Experiment 2 was identical to Experiment 1 except as noted here. Thirty new subjects partic-
ipated. This time, two loops of clear fishing line were attached to the boxes—one loop around
all three boxes (Boxes ABC), and one loop around just the heavier box (Box A), with each
OPEN MIND: Discoveries in Cognitive Science
34
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
ending in a small “tab” secured with electrical tape. Subjects performed the same lifts as in
Experiment 1, but here they simply pinched the tabs between their thumb and index fingers—
and did so in the same way across the two lifts.
Results
As in Experiment 1, subjects reported that A felt heavier than ABC (80%, p = .002; Figure 3),
even when equating grasp posture across the lifts. This suggests that differences in hand shape
or finger extension (etc.) could not explain the impossible experience of one object feeling
heavier than a group that includes it as a member.
EXPERIMENT 3: EQUATING LIFT FORCE
Despite the phenomenon appearing under naturalistic grasping conditions and under condi-
tions of equated grasp posture, a further possibility in the previous two experiments is that
subjects exerted a greater lifting force for the ABC lift than the A lift, perhaps because they
expected ABC to be heavier than A. In that case, ABC might have seemed lighter than A if
subjects used much more force on ABC than on A (though see Flanagan & Beltzner, 2000).
Could this explain the results observed so far?
On one hand, we noted in Experiment 1 that subjects observed (anecdotally) that the illu-
sion persisted after multiple lifting attempts; in that case, subjects should have then had the
correct expectations about the boxes, which in turn should have caused them not to badly
miscalibrate their lifting forces on repeated lifts. Moreover, the classical size-weight illusion
is known not to depend on differences in lifting force, even if such differences are indeed ob-
served in some cases (Buckingham, 2014).
On the other hand, even under such conditions, implicit or automatic expectations might
have caused subjects to exert a greater force in one case than in another. Experiment 3 thus
ruled out even this possibility, by not requiring any “lifting” at all but rather asking subjects to
passively feel the weight of the boxes while their hands rested on a table.
Methods
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
e
d
u
o
p
m
i
/
l
a
r
t
i
c
e
-
p
d
f
/
d
o
i
/
i
/
.
/
1
0
1
1
6
2
o
p
m
_
a
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
a
_
0
0
0
4
0
p
d
/
.
i
Experiment 3 was identical to Experiment 1 except as noted here. Thirty new subjects
participated. This time, rather than grasp the boxes, the subjects placed their hands palm up
on a flat table; then, the experimenter herself placed the boxes (either ABC, or A, in a counter-
balanced order) onto subjects’ passively open palms. This method not only controlled for
grasp posture in yet another way (since the subjects’ hands were in the same posture for both
lifts), but also for lifting force, since any force exerted by subjects was small or nonexistent.
f
b
y
g
u
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
Results
As in Experiment 1, subjects reported that A was heavier than ABC (90%, p < .001; Figure 3),
even when they didn’t “lift” the boxes at all but simply felt their pressure against their passively
open palms. This suggests that neither differences in grasp posture nor in lifting force explain
the impossible experience of one object feeling heavier than a group that includes it.
GENERAL DISCUSSION
In three experiments, subjects perceived a single object alone as heavier than a group contain-
ing that object. We interpret this experience as a somatosensory analog to “impossible figures“
that arise in visual perception.
OPEN MIND: Discoveries in Cognitive Science
35
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
An “Impossible Figure” for Somatosensation
Importantly, the present phenomenon goes beyond known somatosensory illusions or unusual
bodily experiences. For example, in “Aristotle’s Illusion,” touching one’s nose with crossed
index and middle fingers produces the bizarre feeling of having two noses (Hayward, 2008;
Lackner, 1988). Similarly, stimulation of the biceps tendon can make one feel as though one’s
arm is extending (Goodwin et al., 1972)—so much so that, after enough stimulation, one’s arm
would have to be so extended as to imply a break at the elbow.6 However, haptic illusions
such as these (see also Guterstam et al., 2011; Guterstam et al., 2018), while striking and un-
usual, arguably lack the genuinely “impossible” character of the present phenomenon (and
tend to involve only strange bodily sensations rather than interactions with other objects). In
other words, sprouting a new nose or extending one’s arm past its breaking point are certainly
biologically implausible experiences; but the present phenomenon is not merely strange or
unusual—it is physically or even conceptually incoherent.
It is this incoherence that we emphasize here, since in our view it is what makes this phe-
nomenon a somatosensory analog to the impossible visual figures in Figure 1. Just as those
images show a kind of global incoherence (even though many regions of the images are lo-
cally coherent), the present experience has a similar character. And although the present phe-
nomenon does rely on being temporally extended (with the lifts occurring one after another
but not literally at the same time), this is arguably the case even for impossible visual figures,
which are difficult to take in all at once. For example, in the “impossible staircase” (Figure 1,
center), one may find one’s attention flitting around the staircase until one notices that the
local transitions add up to an impossible global figure. Moreover, given that it is possible in
the present phenomenon to visibly reduce the weight of the stack (by removing B and C) and
yet cause a perceived increase in weight, it may well be that the present phenomenon can be
distilled into a single impossible “moment,” and perhaps thus a single impossible experience.7
Note further that it was not a foregone conclusion that the size-weight illusion would
extend to the present case. An alternative possibility, for example, was that somatosensory
processing would have access to part/whole relations in held objects (in ways analogous to
part-based decomposition in visual perception; Hoffman & Richards, 1984; Lowet et al., 2018;
Singh et al., 1999), and that heaviness perception would respect various physical constraints
on such part-whole relations. The fact that this did not occur may suggest that haptic process-
ing does not segment objects into parts in the same way as visual processing does (or at least
6 There are also variants of the size-weight illusion that approach the present result but do not imply the same
consequences. For example, a single barbell-shaped object whose width (but not weight) can be physically
adjusted may feel differently light or heavy depending on its size (Plaisier & Smeets, 2015), even though the
same amount of material is seen. This phenomenon too may be viewed as “impossible” in some sense, since it is
not physically possible for an object to get heavier simply by changing its shape (though it is possible for shape
changes to produce differences in leverage and required lifting effort). However, this result still lacks the kind of
conceptual incoherence of the phenomenon we explore here, and also doesn’t evoke the “conjunction fallacy”
(nor is it presented as such) in the way the present phenomenon does. See the main text for an expansion on this
latter point.
7 Still, it may nonetheless be objected that, insofar as this experience is temporally extended, it is not an ex-
perience of something impossible. According to this objection, what is impossible is that, at a specific time t, A
weigh more than ABC; but what’s not impossible is that ABC at t1 weigh less than A at t2, since weights can
change over time. A reply to this objection, however, is that it is indeed impossible that ABC at t1 weigh less than
A at t2 if A has not changed in any relevant way—which is the case here (and is believed by the subject to be the
case).
OPEN MIND: Discoveries in Cognitive Science
36
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
e
d
u
o
p
m
i
/
l
a
r
t
i
c
e
-
p
d
f
/
d
o
i
/
i
/
/
.
1
0
1
1
6
2
o
p
m
_
a
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
a
_
0
0
0
4
0
p
d
.
/
i
f
b
y
g
u
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
may not incorporate physical constraints into whatever part-based segmentation it does carry
out, or cannot carry forward such information for use even across extremely short time-
scales, etc.).
More Than a Trick
Beyond adding to the inventory of impossible perceptual experiences, the fact that our minds
generate such an impossible or incoherent outcome in the present case may interact with clas-
sical and contemporary discussions about the “rationality” of mental processing, in at least two
ways.
“Anti-Bayesian” Updating First, the classical size-weight illusion—where smaller objects feel
heavier than equally-weighted larger objects—has sometimes been described as defying
Bayesian norms of inference (Brayanov & Smith, 2010; Buckingham & Goodale, 2013).
Typically, the mind’s interpretation of uncertain data is attracted toward its priors, in line with
a broadly Bayesian recommendation. For example, if one encounters an object whose circular
retinal image is equally consistent with it being (a) a sphere or (b) an elongated ellipsoid
viewed at just the right angle to project a circle to the viewer, one typically experiences that
object as a sphere—arguably because this experience reflects the visual system’s prior assump-
tions about which shapes and views are most likely. In other words, ambiguity in the data is
resolved “in favor” of the assumptions one had prior to encountering those data. By contrast,
the size-weight illusion instead seems to involve repulsion from such priors: One comes in
expecting the larger object to be heavier than the smaller object, and then one receives
equivocal or ambiguous sensory evidence about which is truly heavier (since they really
weigh the same); but then one somehow experiences the larger object as lighter than the
equally weighted smaller object. In other words, the mind seems to resolve the ambiguous
sensory evidence “against” the larger-is-heavier prior, rather than toward it—an apparent
counterexample to notions that perception and cognition implement or approximate
Bayesian inference.
The present results, it seems to us, amplify this challenge further, and make the “irrational-
ity” of this pattern of updating all the more stark. Evidently, the norm-defying bias in the size-
weight illusion is so powerful that it can generate not only improbable outcomes (as in the
classical size-weight illusion, as well as in other phenomena sometimes described as as
anti-Bayesian; Wei & Stocker, 2015; see also Rahnev & Denison, 2018) but also impossible
outcomes whose probability should be zero and that should therefore be unacceptable to a
rational updater—since no chain of updating should end with A being heavier than ABC (see
also Mandelbaum, 2019; Mandelbaum et al., 2020).
Indeed, the astonishment that subjects express upon experiencing this phenomenon—both
here in our study, as well as in Koseleff (1937), where subjects described their experience as
“unlogisch”—seems to tell against other rationality-preserving accounts of this illusion. For ex-
ample, one classical account of the size-weight illusion suggests that subjects are substituting
or integrating density with weight (Ross & Di Lollo, 1970; perhaps also Stevens & Rubin, 1970).
In that case, it would be natural (and perfectly in line with Bayesian norms) to experience the
larger object as lighter, if “lighter” here roughly means “less dense” rather than “less heavy.” But
such an account seems less plausible for the present phenomenon: If subjects were reporting
the objects’ densities, rather than their weights, then there should be no reason for subjects to
find their experiences here “impossible” or “illogical”—since there is nothing impossible about
a single object being denser than other members of its group, or a stack of objects feeling less
dense together than one of its members feels alone. By contrast, it is indeed impossible for a
OPEN MIND: Discoveries in Cognitive Science
37
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
e
d
u
o
p
m
i
/
l
a
r
t
i
c
e
-
p
d
f
/
d
o
i
/
i
.
/
/
1
0
1
1
6
2
o
p
m
_
a
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
a
_
0
0
0
4
0
p
d
.
/
i
f
b
y
g
u
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
single object to be heavier than a group of which it is a member—and so the present results
provide a new kind of evidence for the (anti-Bayesian) weight-based account of the size-weight
illusion, by better aligning with the explicit reports of subjects who experience it.8
Of course, the nature of this challenge is controversial, and a full account of it is beyond the
scope of the present discussion. Indeed, there are now more sophisticated density-based ac-
counts exploring how the mind might rationally estimate the relative weight of two objects by
first estimating their densities (about which the perceiver might have prior hypotheses) and
then integrating that estimate with the objects’ perceived sizes. For example, Peters et al.
(2016) explore such an account, and argue that the size-weight illusion is not anti-Bayesian
after all. (It is unclear, however, how their model—which directly generates judgments of
heaviness ratios—applies to sequential liftings with temporarily separated haptic signals and
heaviness percepts, as in the present experiments.) More generally, for any such model
(including the more recent approach proposed by Lieder & Griffiths, 2020; see also Wei &
Stocker, 2015), the crux of the present challenge is for it to recast as “rational” not only
improbable outcomes, but also impossible ones.
A Perceptual “Conjunction Fallacy”? Another intriguing aspect of the present results is that they
are reminiscent of the “conjunction fallacy” from the heuristics and biases tradition, wherein
two propositions jointly seem more probable than one of the propositions alone (Tversky &
Kahneman, 1983). For example, when told a story about a young woman (“Linda”) who ma-
jored in philosophy and is concerned with social justice, subjects judge it less likely that Linda
is a bank teller than that she is a bank teller and active in the feminist movement. But this is
impossible, since a conjunction of propositions could never be more probable than one of the
propositions making up the conjunction. The present phenomenon, wherein a “conjunction”
of objects feels lighter than one “conjunct,” has a similar flavor, since it is also true that a
collection of objects could not be less massive than one of the objects making up the collec-
tion. In that case, these results could suggest that it is not only higher level cognition but also
perception itself that can systematically fail when considering together entities that are usually
considered separately (and in a way that goes beyond even the visual images in Figure 1,
which are incoherent in their own way but do not readily evoke the conjunction fallacy—
since they are not cases where “adding” or “joining” one thing to another moves the relevant
representation in the “wrong” direction).
This interpretation of the present phenomenon would not only be theoretically interesting
in its own right (since it was not previously thought that perception itself might “commit” this
type of error), but it could also matter for other questions about the perceptual or cognitive
nature of various psychological phenomena. For example, Ludwin-Peery et al. (2020) recently
discovered that intuitive physical reasoning (e.g., about the movement of physically interacting
objects) exhibits conjunction-fallacy-like behavior, and interpreted this as evidence that phys-
ical intuitions must have a cognitive basis rather than a perceptual one (cf. Firestone & Scholl,
2016b, 2017; Hafri & Firestone, 2021; Little & Firestone, 2021)—because perception isn’t the
sort of process that could arrive at such a fallacious outcome. The present work suggests that
this may not be a secure inference, if perception can indeed show conjunction-fallacy-like
behavior after all. Of course, the phenomenon we explore here replaces probability (in the
8 A related account suggests that heaviness impressions in the size-weight illusion are better understood as
impressions of how easily an object can be thrown (Zhu & Bingham, 2011) since smaller objects are often easier
to throw than larger objects when weight is held constant. Still, even this account permits that the judgments
made here are genuinely heaviness judgments, and so arguably preserves the impossible character of the
experience.
OPEN MIND: Discoveries in Cognitive Science
38
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
e
d
u
o
p
m
i
/
l
a
r
t
i
c
e
-
p
d
f
/
d
o
i
/
i
/
/
.
1
0
1
1
6
2
o
p
m
_
a
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
a
_
0
0
0
4
0
p
d
/
.
i
f
b
y
g
u
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
Linda case) with weight (in the present case), but nevertheless the structural similarity of these
two cases seems at least to leave open the question of whether perception is immune to such
fallacious patterns of updating.
One might object to the analogy between the present phenomenon and the conjunction
fallacy by noting that the present phenomenon is simply a natural extension of whatever per-
ceptual heuristics are operative in the size-weight illusion, rather than the discovery of a new
phenomenon unto itself. However, in our view such an analysis only increases the appropri-
ateness of the analogy. The conjunction fallacy, after all, is itself not typically considered a
single phenomenon but rather a special case arising from the application of heuristics that
are operative in many other circumstances—especially the representativeness heuristic, ac-
cording to which the probability of an event is judged based on its similarity to the process
that generated it, or to a family of related events. On a standard account of the conjunction
fallacy, it arises because “Linda the feminist bank teller” seems more representative of her
biography than “Linda the bank teller”; and this sense of representativeness is so powerful a
heuristic that it can lead people to draw flatly impossible conclusions when just the right sce-
nario is set up. Normally, of course, representativeness-based reasoning doesn’t lead one to
reach fallacious conclusions, just as normally whatever heuristics give rise to the size-weight
illusions don’t produce impossible perceptual experiences. In other words, both the conjunc-
tion fallacy and the present phenomenon of impossible somatosensation arise from heuristics
that are unproblematic when applied locally (e.g., to one proposition, or to one object) but can
become “impossible” when applied globally (e.g., when comparing a conjunction of propo-
sitions against one of its conjuncts, or when comparing a group of objects to one of its mem-
bers) in certain conditions.
From Magic to Mind
Finally, our results add to a growing literature that has taken inspiration from professional mag-
ic to study and reveal principles of psychological processing. For example, previous work
along these lines has applied specific insights from magicians’ knowledge of misdirection to
shed light on the operation of visual attention, leading to discoveries of new phenomena of
change blindness (Yao et al., 2019). The same is true of more specific magical “demonstra-
tions,” including powerful and previously unknown consequences of amodal completion
(Ekroll et al., 2016). Our work here is another example of this growing trend, and so further
suggests that magic can be a source of meaningful insight into how our minds work (for re-
views, see Ekroll et al., 2017; Macknik et al., 2008).
More generally, our results show how impossibility can be not only seen and heard, but
also felt—and in ways that matter for core questions about mental processing.
ACKNOWLEDGMENTS
For helpful discussion and/or comments on previous drafts, we thank Simon Brown, Ernie
Davis, Jorge Morales, Ian Phillips, Jeroen Smeets, and members of the JHU Perception and
Mind Laboratory. For technical support, we thank Mark Fuller.
FUNDING INFORMATION
This work was supported by NSF BCS-2021053 awarded to CF, as well as a JHU Science of
Learning Institute Grant to SG and CF.
OPEN MIND: Discoveries in Cognitive Science
39
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
e
d
u
o
p
m
i
/
l
a
r
t
i
c
e
-
p
d
f
/
d
o
i
/
i
.
/
/
1
0
1
1
6
2
o
p
m
_
a
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
a
_
0
0
0
4
0
p
d
/
.
i
f
b
y
g
u
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
AUTHOR CONTRIBUTIONS
IW: Conceptualization: Equal. Data curation: Lead; Formal Analysis; Lead; Investigation: Lead;
Methodology: Lead; Visualization: Lead Writing – original draft: Equal Writing – review &
editing: Equal. SG: Conceptualization: Equal; Funding acquisition: Supporting; Supervision:
Supporting; Writing – original draft: Supporting. Writing – review & editing: Supporting. CF:
Conceptualization: Equal; Data curation: Supporting; Formal Analysis: Supporting; Funding
acquisition: Lead; Investigation: Supporting; Methodology: Supporting; Supervision: Lead;
Visualization: Supporting; Writing – original draft: Equal; Writing – review & editing: Equal.
REFERENCES
Brayanov, J. B., & Smith, M. A. (2010). Bayesian and “anti-
Bayesian” biases in sensory integration for action and perception
in the size–weight illusion. Journal of Neurophysiology, 103(3),
1518–1531. https://doi.org/10.1152/jn.00814.2009, PubMed:
20089821
Buckingham, G. (2014). Getting a grip on heaviness perception: A
review of weight
illusions and their probable causes.
Experimental Brain Research, 232(6), 1623–1629. https://doi
.org/10.1007/s00221-014-3926-9, PubMed: 24691760
Buckingham, G., & Goodale, M. A. (2013). When the predictive
brain gets it really wrong. Behavioral and Brain Sciences, 36(3),
208. https://doi.org/10.1017/S0140525X12002233, PubMed:
23663693
Buckingham, G., Milne, J. L., Byrne, C. M., & Goodale, M. A.
(2015). The size-weight illusion induced through human echolo-
cation. Psychological Science, 26(2), 237–242. https://doi.org/10
.1177/0956797614561267, PubMed: 25526909
Charpentier, A. (1891). Analyse experimentale: De quelques ele-
ments de la sensation de poids [Experimental analysis: On some
of the elements of sensations of weight]. Archives de Physiologie
Normale et Pathologique, 2, 122–135.
Deregowski, J. B. (1969). Perception of the two-pronged trident by
two- and three-dimensional perceivers. Journal of Experimental
Psychology, 82(1), 9–13. https://doi.org/10.1037/ h0028027,
PubMed: 5371998
Ekroll, V., Sayim, B., Van der Hallen, R., & Wagemans, J. (2016).
Illusory visual completion of an object’s invisible backside
can make your finger feel shorter. Current Biology, 26(8),
1029–1033. https://doi.org/10.1016/j.cub.2016.02.001,
PubMed: 27040774
Ekroll, V., Sayim, B., & Wagemans, J. (2017). The other side of mag-
ic: The psychology of perceiving hidden things. Perspectives on
Psychological Science, 12(1), 91–106. https://doi.org/10.1177
/1745691616654676, PubMed: 28073329
Ellis, R. R., & Lederman, S. J. (1993). The role of haptic versus visual
volume cues in the size-weight illusion. Perception &
Psychophysics, 53(3), 315–324. https://doi.org/10.3758
/BF03205186, PubMed: 8483695
Firestone, C., & Scholl, B. J. (2016a). Cognition does not affect per-
ception: Evaluating the evidence for “top-down” effects.
Behavioral and Brain Sciences, 39, Article e229. https://doi.org
/10.1017/S0140525X15000965, PubMed: 26189677
Firestone, C., & Scholl, B. (2016b). Seeing stability: Intuitive physics
automatically guides selective attention. Journal of Vision, 16(12),
Article 689. https://doi.org/10.1167/16.12.689
Firestone, C., & Scholl, B. (2017). Seeing physics in the blink of an
eye. Journal of Vision, 17(10), Article 203. https://doi.org/10
.1167/17.10.203
Flanagan, J. R., & Beltzner, M. A. (2000). Independence of percep-
tual and sensorimotor predictions in the size–weight illusion.
Nature Neuroscience, 3(7), 737–741. https://doi.org/10.1038
/76701, PubMed: 10862708
Goodwin, G. M., McCloskey, D. I., & Matthews, P. B. (1972).
Proprioceptive illusions induced by muscle vibration: Contribution
by muscle spindles to perception? Science, 175(4028), 1382–1384.
https://doi.org/10.1126/science.175.4028.1382, PubMed:
4258209
Guterstam, A., Petkova, V. I., & Ehrsson, H. H. (2011). The illusion
of owning a third arm. PLoS One, 6, Article e17208. https://doi
.org/10.1371/journal.pone.0017208, PubMed: 21383847
Guterstam, A., Szczotka, J., Zeberg, H., & Ehrsson, H. H. (2018).
Tool use changes the spatial extension of the magnetic touch il-
lusion. Journal of Experimental Psychology: General, 147(2),
298–303. https://doi.org/10.1037/xge0000390, PubMed:
29369683
Hafri, A., & Firestone, C. (2021). The perception of relations.
Trends in Cognitive Sciences, 25(6), 475–492. https://doi.org/10
.1016/j.tics.2021.01.006, PubMed: 33812770
Hayward, V. (2008). A brief taxonomy of tactile illusions and dem-
onstrations that can be done in a hardware store. Brain Research
Bulletin, 75(6), 742–752. https://doi.org/10.1016/j.brainresbull
.2008.01.008, PubMed: 18394520
Hoffman, D. D., & Richards, W. A. (1984). Parts of recognition.
Cognition, 18(1–3), 65–96. https://doi.org/10.1016/0010-0277
(84)90022-2, PubMed: 6543164
Kment, B. (2021). Varieties of modality. In E. N. Zalta (Ed.), The
Stanford encyclopedia of philosophy (Spring 2021 Ed.).
Metaphysics Research Lab, Stanford University. https://plato
.stanford.edu/archives/spr2021/entries/modality-varieties/
Koseleff, P. (1937). Eine modifikation des “charpentier-effektes.”
Psychologische Forschung, 21, 142–145. https://doi.org/10
.1007/BF02441205
Lackner, J. R. (1988). Some proprioceptive influences on the per-
ceptual representation of body shape and orientation. Brain,
111(2), 281–297. https://doi.org/10.1093/ brain/111.2.281,
PubMed: 3378137
Lieder, F., & Griffiths, T. L. (2020). Advancing rational analysis to the
algorithmic level. Behavioral and Brain Sciences, 43, Article e27.
https://doi.org/10.1017/S0140525X19002012, PubMed: 32159501
Little, P. C., & Firestone, C. (2021). Physically implied surfaces.
Psychological Science, 32(5), 799–808. https://doi.org/10.1177
/0956797620939942, PubMed: 33792421
Lowet, A. S., Firestone, C., & Scholl, B. J. (2018). Seeing structure:
Shape skeletons modulate perceived similarity. Attention,
Perception, & Psychophysics, 80(5), 1278–1289. https://doi.org
/10.3758/s13414-017-1457-8, PubMed: 29546555
OPEN MIND: Discoveries in Cognitive Science
40
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
e
d
u
o
p
m
i
/
l
a
r
t
i
c
e
-
p
d
f
/
d
o
i
/
i
.
/
/
1
0
1
1
6
2
o
p
m
_
a
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
a
_
0
0
0
4
0
p
d
/
.
i
f
b
y
g
u
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
“Impossible” Somatosensation and the (Ir)rationality of Perception Won et al.
Ludwin-Peery, E., Bramley, N. R., Davis, E., & Gureckis, T. M.
(2020). Broken physics: A conjunction-fallacy effect in intuitive
physical reasoning. Psychological Science, 31(12), 1602–1611.
https://doi.org/10.1177/0956797620957610, PubMed:
33137265
Macknik, S. L., King, M., Randi, J., Robbins, A., Thompson, J.,
Martinez-Conde, S., et al. (2008). Attention and awareness in
stage magic: Turning tricks into research. Nature Reviews
Neuroscience, 9(11), 871–879. https://doi.org/10.1038
/nrn2473, PubMed: 18949833
Macpherson, F. (2010). Impossible figures. In E. B. Goldstein (Ed.),
The Sage encyclopedia of perception. Sage. https://doi.org/10
.4135/9781412972000.n155
Mandelbaum, E. (2019). Troubles with Bayesianism: An introduction
to the psychological immune system. Mind & Language, 34(2),
141–157. https://doi.org/10.1111/mila.12205
Mandelbaum, E., Won, I., Gross, S., & Firestone, C. (2020). Can
resources save rationality? “Anti-Bayesian” updating in cognition
and perception. Behavioral and Brain Sciences, 43, Article e16.
https://doi.org/10.1017/S0140525X19001717, PubMed:
32159502
Penrose, L. S., & Penrose, R. (1958). Impossible objects: A spe-
cial type of visual illusion. British Journal of Psychology, 49(1),
31–33. https://doi.org/10.1111/j.2044-8295.1958.tb00634.x,
PubMed: 13536303
Peters, M. A., Ma, W. J., & Shams, L. (2016). The size-weight illu-
sion is not anti-Bayesian after all: A unifying Bayesian account.
PeerJ, 4, Article e2124. https://doi.org/10.7717/peerj.2124,
PubMed: 27350899
Plaisier, M. A., & Smeets, J. B. J. (2015). Object size can influence
perceived weight independent of visual estimates of the volume
of material. Scientific Reports, 5, 17719. https://doi.org/10.1038
/srep17719, PubMed: 26626051
Rahnev, D., & Denison, R. N. (2018). Suboptimality in perceptual
decision making. Behavioral and Brain Sciences, 41, Article
e229. https://doi.org/10.1017/S0140525X18000936, PubMed:
29485020
Rensink, R. A., & Kuhn, G. (2015). A framework for using magic to
study the mind. Frontiers in Psychology, 5(1508), 1–14. https://
doi.org/10.3389/fpsyg.2014.01508, PubMed: 25698983
Ross, J., & Di Lollo, V. (1970). Differences in heaviness in relation to
density and weight. Perception & Psychophysics, 7(3), 161–162.
https://doi.org/10.3758/BF03208648
Shepard, R. N. (1964). Circularity in judgments of relative pitch.
The Journal of the Acoustical Society of America, 36(12),
2346–2353. https://doi.org/10.1121/1.1919362
Singh, M., Seyranian, G. D., & Hoffman, D. D. (1999). Parsing sil-
houettes: The short-cut rule. Perception & Psychophysics, 61(4),
636–660. https://doi.org/10.3758/ BF03205536, PubMed:
10370334
Smeets, J. B., & Brenner, E. (2008). Why we don’t mind to be in-
consistent. In P. Calvo & T. Gomila (Eds.), Handbook of cognitive
science (pp. 207–217). Elsevier. https://doi.org/10.1016/B978-0
-08-046616-3.00011-6
Smeets, J. B., & Brenner, E. (2019). Some illusions are more incon-
sistent than others. Perception, 48(7), 638–641. https://doi.org/10
.1177/0301006619853147, PubMed: 31126210
Sousa, R., Brenner, E., & Smeets, J. B. J. (2011). Objects can be
localized at positions that are inconsistent with the relative
disparity between them. Journal of Vision, 11(2), 18. https://doi
.org/10.1167/11.2.18, PubMed: 21350127
Stevens, J. C., & Rubin, L. L. (1970). Psychophysical scales of apparent
heaviness and the size-weight illusion. Perception & Psychophysics,
8(4), 225–230. https://doi.org/10.3758/BF03210210
Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive
reasoning: The conjunction fallacy in probability judgment.
Psychological Review, 90(4), 293–315. https://doi.org/10.1037
/0033-295X.90.4.293
Wei, X.-X., & Stocker, A. A. (2015). A Bayesian observer model
constrained by efficient coding can explain “anti-Bayesian” per-
cepts. Nature Neuroscience, 18(10), 1509–1517. https://doi.org
/10.1038/nn.4105, PubMed: 26343249
Yao, R., Wood, K., & Simons, D. J. (2019). As if by magic: An
abrupt change in motion direction induces change blindness.
Psychological Science, 30(3), 436–443. https://doi.org/10.1177
/0956797618822969, PubMed: 30730789
Zhu, Q., & Bingham, G. P. (2011). Human readiness to throw: The
size–weight illusion is not an illusion when picking the best
objects to throw. Evolution and Human Behavior, 32(4), 288–293.
https://doi.org/10.1016/j.evolhumbehav.2010.11.005
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
e
d
u
o
p
m
i
/
l
a
r
t
i
c
e
-
p
d
f
/
d
o
i
/
i
/
/
.
1
0
1
1
6
2
o
p
m
_
a
_
0
0
0
4
0
1
9
2
8
3
1
9
o
p
m
_
a
_
0
0
0
4
0
p
d
/
.
i
f
b
y
g
u
e
s
t
t
o
n
0
7
S
e
p
e
m
b
e
r
2
0
2
3
OPEN MIND: Discoveries in Cognitive Science
41
