The Art of the Impossible
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Gendler-09 2/10/02 12:45 PM Page 337
The Art of the Impossible
S
Prize: One hundred dollars to the first person who identifies a picture of a
logical impossibility. I may be willing to pay more for the painting itself.
This finder’s fee is simply for pointing out the picture. Let me explain more
precisely what I seek.
1 Illegal Pictures
There is a genre of children’s picture puzzles that is marked by the question
‘What is wrong with this picture?’ Well, that goat does not belong in the
library.That clock is mirror-reversed. Ostriches do not fly. . . . The job of the
viewer is to spot the incongruities.
An impossible picture features a nomic incongruity—a violation of a law.
There are many pictures that depict scientifically impossible situations. René
Magritte’s Collective Invention features a reverse mermaid: woman from foot to
waist, fish from waist to gills.
An impossible situation need not involve an impossible object. Many of
Magritte’s paintings feature ordinary objects in impossible relationships. Zeno’s
Arrow simply shows a huge rock that fails to be gravitationally related to the
earth. Actually, all ‘impossible objects’ involve impossible relationships. For
An ancestor of this chapter was presented at the University of Saskatchewan. I thank Karl Pfeifer,
Walter Sinnott-Armstrong, and the editors of this volume,Tamar Gendler and John Hawthorne,
for comments and imaginative suggestions. I thank Milton Katz for permission to reprint one of
his figures, and István Orosz for permission to reprint his drawings.Gendler-09 2/10/02 12:45 PM Page 338
Roy Sorensen
instance, the impossibility of Magritte’s reverse mermaid involves an imposs-
ible relation among body parts.
Empirical background is needed to infer that Magritte’s reverse mermaid
cannot be actual.Maybe empirical knowledge sometimes suffices for the identi-
fication of a necessary falsehood. Perhaps reverse mermaids are ‘metaphysically
impossible’.The essentialist,Saul Kripke (),has argued that ‘unicorn’is a nec-
essarily empty term. He thinks that species terms work like names. Under
Kripke’s causal theory, only objects that bear the appropriate historical relation
with a name can be denoted by that name.So Kripke must deny that a picture of
a unicorn depicts an animal that could exist. He would not be claiming that the
impossibility of unicorns could be inferred from the picture alone. Knowledge
that unicorns do not exist is a posteriori, the result of scientific investigation.
I am interested in pictures that depict a priori impossibilities. Analyticity
is a traditional source of apriority.A statement is semantically analytic if its truth-
value is determined by the meanings of its words.W.V. Quine () excited
controversy about these statements that persists today.Although I am personally
content with the analytic/synthetic distinction, I confine the search to a picture
that avoids this controversy. Quine does not object to syntactically analytic state-
ments.These statements owe their truth-value just to their logical words.
Any logical truth is syntactically analytic.A logical truth is a theorem of a cor-
rect theory of what entails what. Standard logic (first-order predicate logic with
identity) forms the core of this theory.Thus the class of logical truths includes any
theorem found in logic textbooks.The negation of a logical truth is a logical
falsehood. So a perceptual depiction of a logical falsehood suffices for the prize.
Although ‘logical falsehood’ is clear enough,‘perceptually depicts a logical
falsehood’ is obscure. There are no plausible, precise theories of depiction.
Prize-seekers need not be discouraged.People make discoveries without being
able to define what they have discovered.
In a way, I am being strict. For I am not issuing the reward for a picture of a
mere conceptual impossibility.One reason,aside from the desire for a clear goal,
is that I am satisfied that a number of artists have composed scenes that violate
geometrical truths. For instance, the relative proximity relations of the columns
in István Orosz’s Cavalier (Fig. ) are inconsistent. I think most philosophers
should be receptive to the general possibility of depicting the impossible. For
most philosophers agree that it is possible to believe the impossible.And if it is
possible to believe the impossible, then what would stand in the way of graph-
ically representing the impossible?
Consider purely pictorial instructions.When frustrated by Ikea’s pamphlets
for assembling furniture (which are designed to rely on no knowledge of aGendler-09 2/10/02 12:45 PM Page 339
The Art of the Impossible
Fig. 1. Cavalier, by István Orosz. Reproduced by courtesy of the artist.
language), I have doubted the possibility of executing the instructions. I have
always been wrong. But have I been necessarily mistaken?
There are prominent philosophers who do not believe that one can believe
the impossible. Robert Stalnaker () maintains that the object of belief
must be a nonempty set of possible worlds.Ruth Marcus () claims that belief
relates to possibility as knowledge to truth.That is,belief has an external defeas-
ibility condition.When we learn that p is impossible, we retract our attribu-
tion of belief. Or so she argues. Others insist that we can believe only what
we can understand, and that anyone who understands a contradiction realizes
that it is not true—and so does not believe it.Causal theorists say that the object
of belief is the state of affairs that would cause that belief under optimal
conditions.There are no such conditions for impossibilities. Some devotees of
the principle of charity (which instructs us to interpret agents as rational) claim
that belief in impossibilities is unintelligible. Others say that the appearance of
contradiction should always trigger the postulation of an ambiguity.Gendler-09 2/10/02 12:45 PM Page 340
Roy Sorensen
All of these anti-contradiction strategies sound good in theory,but fall flat in
practice.I long believed that ‘The American Thanksgiving Holiday is on the last
Thursday of November which is the fourth Thursday in November’. Only in
November , which contains five Thursdays, did I realize that these two
definite descriptions only partially overlap. Of course, I long knew that
November has more than twenty-eight days and that there are only seven days
in a week and that the first day of the month cycles forward each year.But I did
not pull together all these analytical truths.
If you do not think that the believability of contradictions can be established
by the Method of Humiliating Confession, I also offer a Cartesian argument.
The essential idea is that belief that someone believes at least one contradiction
is infallible (Sorensen ).After all,if I mistakenly believe that it is impossible
to believe the impossible, then that very mistake would itself be a belief in an
impossibility.
In my opinion, the only theory that permits belief in the impossible is the
linguistic account of the object of belief.To believe is to believe something that
resembles a sentence—if not a sentence of a natural language, then a sentence
in the ‘language of thought’.
Are pictures sentences? John M. Kennedy (: ) speaks of a ‘language
of lines’,and supplies a vocabulary of concave corners,convex corners,occlud-
ing edges, and occluding bounds. His discussion of surface layouts can be
understood as an articulation of the syntax for constructing outline pictures.
The mere fact that there are computer programs for constructing illustrations
shows that important kinds of pictures are combinatoric. However, many
nonlinguistic phenomena are combinatoric: chemistry, checkers, building-
block toys. It is one thing to convey information in a modular fashion. It is
another to be the object of a propositional attitude.
2 Pseudo-Pictures
Those who regard pictures as sentences are often unclear about whether impossi-
ble pictures actually qualify as sentences. Linguists say that a language is a set
of sentences defined by a vocabulary and a grammar specifying how the words
can be combined into sentences.Therefore, ungrammatical sentences are not
part of the language.Thus, if one characterizes impossible pictures as ungram-
matical sentences of the picture language (Huffman ),then one should not
count them as pictures.This seems harsh. Kennedy attempts a compromise:
Combining incompatible words makes an ‘impossible’ sentence, a sentence that can
have no direct referent in reality.An example is ‘Colorless green ideas sleep furiously.’Gendler-09 2/10/02 12:45 PM Page 341
The Art of the Impossible
The sentence is grammatical—it is not nonsense like ‘furiously sleep ideas green
colorless.’A drawing,too,can show impossible things,things that cannot have a direct
equivalent in reality. (Kennedy : )
If ‘Colorless green ideas sleep furiously’ expressed an impossibility, then it
would have a negation that expresses a necessary truth. But Chomsky regards
‘It is not the case that colorless green ideas sleep furiously’ as equally meaning-
less. He takes his most famous utterance ‘Colorless green ideas sleep furiously’
to illustrate the fact that a meaningless sentence can conform to the grammar
of a language. He thinks that the sentence violates semantic rules. By contrast,
Chomsky (correctly) thinks that contradictory statements fully conform to all
rules of the language. They merely express propositions that are necessarily
false. A grammar that fails to generate contradictory English sentences is
an inadequate grammar. Grammaticality cannot be a necessary or sufficient
condition for possibility.
Many of those who reject the idea that pictures are sentences will still be
inclined to regard meaningless pictures as failed attempts at picturing. Happily,
prize-seekers need not take sides. Contradictions are meaningful. If there were
literally a language of outlines, contradictory pictures would be sentences
within that language.
3 Pictures have a Role within Propositions
I agree with most philosophers in denying that pictures are discursive. I also
conform to my colleagues’view that pictures cannot be believed on their own.
Photographs do not lie. Nor do they tell the truth.They can be evidence of the
truth by virtue of the optical information they carry.But bare photographs can
no more be believed than bare fingerprints.
I can believe that a picture of a flying saucer is undoctored. I can believe that
a town square in Holland remains as a sixteenth-century artist drew it. But I
cannot believe the picture itself. Nevertheless, pictorial representations (draw-
ings, maps, photographs) figure in what I believe.As David Kaplan (: )
observes:‘Many of our beliefs are of the form “The color of her hair is——,”
or “The song he was singing went——,”where the blanks are filled with images,
sensory impressions, or what have you, but certainly not words.’Although raw
images lack truth-values and so cannot figure as premises or conclusions, they
can be part of premises that do have truth-values:
() The color of her hair is——.
() The color of her sister’s hair is also——.
() At least two women have hair that is——.Gendler-09 2/10/02 12:45 PM Page 342
Roy Sorensen
The argument is sound, because I am thinking of two women who make the
premises true and because the argument is valid. Some of our beliefs are
demonstrative. Demonstratives cannot be reduced to qualitative descriptions.
Hence, pictures can play an essential role in forming the objects of belief.
Nevertheless, I am not interested in the contradiction ‘The color of her hair
is——and is not——’. Although the image plays a role in constituting this
demonstrative contradiction, the image is not doing any logical work.
I am not trying to raise the standard of representation to an impossible
height. If I thought a picture of a logical impossibility were impossible, then I
would feel safe in posting a large prize. In fact, I expect to pay the $ finder’s
fee. I may even wind up paying someone who does not actually believe that
it is possible to picture a logical impossibility. For all he needs to do is to
persuade me.This conditional proof can exploit my concession that conceptu-
ally impossible pictures are possible.The prize could be won simply by demon-
strating the following hypothetical: If there are pictures of conceptual
impossibilities, then this is a picture of a logical impossibility.
The issue for me is the step from conceptual impossibility to logical impossi-
bility. Prize-seekers will find it useful to see what standard of evidence I have
applied to the acceptance of conceptual impossibilities.
4 Historical Background
On the basis of introspection, the British empiricists believed that ideas have
pictorial properties. A speaker uses sentences to describe his mental images.
The pictorial mode of representation is epistemically prior to the discursive
mode.Nevertheless,the empiricists imposed an important,famous limit on the
expressive scope of pictures. David Hume writes:
Tis an establish’d maxim in metaphysics,That whatever the mind clearly conceives includes
the idea of possible existence, or in other words, that nothing we imagine is absolutely impossible.
We can form the idea of a golden mountain, and from thence conclude that such a
mountain may actually exist.We can form no idea of a mountain without a valley,and
therefore regard it as impossible. (–: )
A picture of a conceptual impossibility would generate counter-examples to
Hume’s principle that anything which is conceivable is possible. People would
look at the picture and thereby conceive an impossible scene.The artist would
have proved a philosophical proposition just as Clyde Tombaugh proved
the astronomical proposition that there is a ninth planet by photographing
Pluto in .Gendler-09 2/10/02 12:45 PM Page 343
The Art of the Impossible
Photographs can only be of actual objects. But drawings can prove the poss-
ibility of uninstantiated objects. A mathematician can convince an engineer
that a larger cube can pass through a smaller cube by drawing a smaller cube
with a diagonal tunnel. (A cube with a -meter face has a diagonal equal to the
square root of meters.) The proof works even if no one bothers to build the
perforated cube.
If drawing X demonstrates the possibility of X, then we appear to have a
quick proof that it is impossible to draw an impossible object. Drawing an
impossible object would show that it is possible for an impossible thing to exist.
Contradiction.Therefore, it is impossible to draw an impossible object.
This proof is sound.But only when read de re (as referring to a thing and then
reporting a feature of it). For instance, the de re report ‘The discoverer of the
largest prime number is being drawn as a winner of the Fields Medal’ entails
that the discoverer of the largest prime number exists. But some depiction is
de dicto (as concerning a representation). For instance, the de dicto report ‘In the
picture, the discoverer of the largest prime number is receiving the Fields
Medal’ does not entail that there is a discoverer of the largest prime number.
Nor does it entail that it is possible for someone to discover the largest prime
number.Any person who earns the $ finder’s fee will be giving me a de dicto
report. He will not be claiming to have discovered an impossibility that has
secured the attention of a faithful portraitist.
5 Requirements
Philosophical tradition and common sense converge on what counts as an
acceptable depiction of the logically impossible. None of the requirements
below are intended to indulge personal idiosyncrasies.
. Openness to Inspection
A description of an impossible situation should be detailed enough to convey
the nature of the impossibility. Ditto for depiction. Paul Tidman’s () joke
picture of a square circle (Fig. ) violates this requirement. Since Hume is not
Fig. 2. Square circle, side view.Gendler-09 2/10/02 12:45 PM Page 344
Roy Sorensen
present to balk Tidman’s evasiveness, I balk on Hume’s behalf. If evasive per-
spectives are permitted, anyone can ‘draw’ anything (see Fig. ). A genuine
●
Fig. 3. Any object as seen in the distance.
depiction must place no limit on potential detail. I do not insist on limitless
actual detail. I merely require that the specimen be open to view.
Well, let’s not be chauvinistic about vision. Any sense modality will do. A
depiction via smell or a less-known sense would be equally acceptable.
Roger Shepherd () devised a tone that seems to rise endlessly.
Jean-Claude Risset () has developed aesthetic possibilities of this and other
acoustic illusions (such as ever-accelerating beats) in his computer music. For
example, in Little Big Boy, there is a sound which goes down the scale but ends
up higher in pitch.The endless ‘nontransitive descent’ represents the dropping
of the atomic bomb on Hiroshima.Visual illusions are better known than
auditory illusions because artists have long been able to draw trick figures with
just a pencil.Auditory illusions generally require careful control by computer
synthesis. However, musicologists have discovered notated pitch circularity
dating back as far as (Braus : ).
The inconsistencies of paradoxical music are at the level of the medium of
representation, rather than at the level of the thing represented. It is like the
inconsistency inspired by the light–dark spectrum from nonblack to black.We
perceive the spectrum as devoid of transition points, but the spectrum as a
whole as embodying a complete transformation.
There is a strand of the empiricist tradition that favors touch over sight.The
young George Berkeley would have actually preferred a sculpture of a round
square—something he could put his hands on. In his New Theory of Vision,
Berkeley argued that touch is the primary sense modality; vision tells us about
reality only after we learn how to correlate what we see with what we feel.
. No Equivocation
In Taxicab geometry all squares are round squares (Krause ). In this form
of non-Euclidean geometry, distance is measured by how a taxi travels on aGendler-09 2/10/02 12:45 PM Page 345
The Art of the Impossible
coordinate plane.A circle is a figure whose perimeter is everywhere equidistant
from its center. Consequently, all circles are squares, and all squares are circles.
Thus a round square in Taxicab geometry is a tautologous figure rather than a
contradictory one. But this contrast is achieved by adopting a different mean-
ing for ‘distance’. When people say that round squares are impossible, they
should be read as making a claim in the framework of Euclidean geometry.
Similarly, I give no quarter (much less $) to any candidate who fiddles with
the meaning of ‘there is’,‘and’,‘etc.’, etc.
Drawings made in axonomic perspective or anamorphic perspective have
the superficial appearance of impossibility. But unfamiliarity should not be
confused with incongruity.Alternative systems of representation differ without
necessarily disagreeing. I want to see a genuine clash with logic.
Stick to the standard logical concept of ‘contradiction’. Soviet artists repres-
ented ‘historical contradictions’ in all relevant detail.Their usage echoes Georg
Hegel and Karl Marx.These philosophers used ‘contradiction’ broadly. (Daniel
Goldstick () points out that Hegel and Marx also used it in the narrower
sense more familiar to contemporary logicians.) Hegel and Marx included
phenomena analogous to gainsaying in a dialogue.Adorno elaborated this into
a dialectical metaphysics in which contradiction plays a central role.This is the
famous process which begins with a thesis.The thesis stimulates an anti-thesis.
The anti-thesis stimulates synthesis.The resulting synthesis between thesis and
anti-thesis is itself a more comprehensive thesis.Accordingly, this higher thesis
stimulates a higher anti-thesis and another round of synthesizing. Each contra-
diction is the effect of a limited vantage point. By building on the remains of
past positions, dialectical descendants command higher ground and a more
sweeping vista. Contradictions precipitate and sustain their own transcend-
ence. Soviet artists were instructed on how the history of thought, and indeed,
just plain history, is built on the backs of dead contradictions. These artists
brought new meaning to the theory of perspective.
Communism encourages an itchy trigger finger. Contradictions abound—
worldwide. Graham Priest (), uses ‘contradiction’ in a way that is intended
to encompass ordinary scenes such as Vladimir being in a doorway (because
Vladimir is both in and out of the room).
In addition to being broader than the logical sense of ‘contradiction’, the
dialectical conception of contradiction is also narrower:the dialectical concep-
tion implies that all contradictions are divisible into self-consistent conjuncts
that have the stereotypical P & ~P form. Many important logical contradic-
tions are not divisible in this way. Consider Hegel’s belief that the law of
identity is false.This logical falsehood, ~(x) (x x), is not divisible into self-
consistent conjuncts. Nor can we divide Bertrand Russell’s early belief thatGendler-09 2/10/02 12:45 PM Page 346
Roy Sorensen
there is a set for every property. Nor Ludwig Wittgenstein’s Tractarian belief
that there is a decision procedure for all logical truths (Fogelin : ch. ).
Each philosopher contradicted himself. But none were ‘of two minds’.
Those who believe that anything can be depicted also believe an indivisible
contradiction.To see why, first note that some pictures depict other pictures.
For instance,Watteau’s L’Enseigne de Gersaint features an art merchant selling his
merchandise. Here is a logical truth: there is no picture that depicts all and only
those pictures that do not depict themselves. If this picture depicts itself, then it
does not depict itself.But if it does not depict itself,then it must be amongst the
pictures it depicts. Contradiction. James F. Thomson (: ) discusses a
whole family of contradictions that have this logical form.
This logic exercise proves decisively that there are logically impossible
depictions.Artists are imaginative people. But imagination is not a resource for
evading logical limits. My $ fee can still be earned, because I want only a
picture that depicts a logical impossibility, not a picture that is itself logically
impossible.
For the sake of administrative ease, I will pay a $ bonus for an indivisible
contradiction.The assumption that all impossible figures are divisible into self-
consistent components is commonly made by philosophers—for instance,
Max Cresswell ().The assumption is made uniformly by psychologists.
In impossibles, each part is ecological, but the combination of the parts violates
nature.They could not exist, so they are imaginary, but the fact that they are imagin-
ary does not make them impossible. To make an imaginary object, parts are
combined in possible ways.The combination can be possible but be a combination
that does not exist. For example, there is nothing about surfaces and air spaces that
rules out a horse with a horn, like a unicorn. Nature has not seen fit to evolve
unicorns,but it could do so without contravening its own ways with surfaces and air.
The parts of a unicorn are ecological.The combination of parts breaks no laws of
solidity.In language,one may claim ‘I saw a unicorn,a horse with a horn.’In language,
as in pictures, to be imaginative is to combine familiar parts in possible but novel
ways,whereas to be impossible is to combine the parts in novel ways that violate rules
of nature. (Kennedy : –)
All mathematical analyses of impossible figures have conformed to the idea that
impossible figures are built from possible parts. For instance, Diego Uribe has
analyzed an infinite class of impossible figures as jig-saw puzzles of just thirty-
two equilateral triangles consisting of special bar elements (Ernst b: ).
This is the basis for software (available free over the Web) that enables you to
mechanically construct impossible objects by manipulating these triangles.But
if we take the analogy with language seriously, we should doubt that theseGendler-09 2/10/02 12:45 PM Page 347
The Art of the Impossible
analyses exhaust the stock of impossible figures. All natural languages can
express infinitely many indivisible contradictions. If it is possible to pictorially
represent a contradiction, then it should be possible to pictorially represent an
indivisible contradiction.
An indivisibly inconsistent picture would side-step the problem of distin-
guishing inconsistency from doubt. Consider ‘continuity’ errors in movies. For
instance,in the last ten minutes of Mission:Impossible ,secret agent Ethan Hunt
is riding a motorcycle in a chase scene.The last two digits of his license plate
shift from to .This production error does not make the movie inconsist-
ent about the license plate number. Instead, the conflicting depictions merely
create doubt whether the license plate ends with or .Whenever the con-
tradiction is divisible, there is the opportunity to interpret the scene in this
uninteresting way. Depiction of an indivisible contradiction would avoid
this hitch.
. The Depiction must be Perceptual
On a purely stipulational conception of ‘depict’, merely intending x to be
an F makes x an F.Thus, if a child scribbles on a page and says that the scrib-
ble is his mother, then the scribble is a depiction of his mother.Thinking so
makes it so.
Actually, I think that this subjectivist construal of stipulation is miscon-
ceived. Stipulation is more complicated and defeasible (Horowitz ).
Couples who simply declare themselves to be married do not thereby become
married. It might be pleasant to think of them as married. But they can only
marry with the help of the right sort of official conducting the proper sort
of ceremony. Artistic stipulation has a similar but fainter institutional infra-
structure. Like the preacher, the artist is participating in a practice that requires
knowledge of procedures and institutional backing. Nearly all of us are artists
in the capacious sense that we draw simple pictures. Outline drawings are
understood by toddlers without training (Hochberg and Brooks ).
Consequently, outline drawings are understood in all cultures (Kennedy :
ch. ). True, unfamiliar objects are misconstrued as more familiar objects.
But that kind of error only underscores a firm grasp of how drawings repre-
sent objects. Prehistoric cave paintings show that this ability has been around
for a very long time. Special training and the infrastructure of an art commu-
nity considerably amplify our stipulative capacity. The same applies to other
stipulative activities, such as the construction of thought experiments
(Sorensen ). Every healthy adult constructs experiments that edify byGendler-09 2/10/02 12:45 PM Page 348
Roy Sorensen
virtue of reflections on their design rather than by execution. But only those
who are inducted into special fields of science and philosophy magnify this
power.
In any case, if subjectivist ‘depiction’ sufficed, then my oldest son Maxwell
would deserve the $ reward.At age three, he loved the color green. In fact,
he loved it to the exclusion of all other colors.Maxwell became a green maxim-
izer. He drew a picture of a ‘green all over rainbow’ with a single green
crayon. Since a rainbow must be multicolored, no scene could match my son’s
description.
The problem with Maxwell’s picture is that it does not reveal what it would
be like to see a uniformly green rainbow. In artistic contexts,‘depict’ is used in
a way that allows failure.When students take art classes, they want to learn how
to render objects perceptually. Techniques such as drawing in perspective
capitalize on the running start we all have from folk optics.The students already
know how to depict objects discursively via pure stipulation (or stipulation plus
an ancillary stick figure).Suppose the art instructor says,‘Drawing is not as hard
as it looks.All you need to do is to decide what your marks on the canvas are
intended to represent.Then, presto, you are done.’The art students will rightly
demand a tuition refund.
One of René Magritte’s most famous pictures,TheTreason of the Pictures ( This
is not a pipe) consists of a picture of a pipe along with the caption ‘This is not a
pipe’.Peter Strawson might be tempted to say that this is not a depiction of any-
thing.According to Strawson (: –), a statement of the form P & not-P
says nothing because the ‘not-P’ merely cancels out the P. Others interpret the
picture as making the point that the picture of the pipe is not itself a pipe.This
illustrates a standard alternative to viewing a picture as depicting an impossibil-
ity: one attributes an ambiguity.To forestall this attribution of an ambiguity,
suppose the caption had instead been ‘This is not a picture of a pipe’.Would
Magritte then have pictured a contradiction?
Well, maybe. But it would not be the kind of picture I seek. I want the con-
tradiction to be within the picture, not between the picture and its caption.
I am not forbidding the kind of illocutionary variety that Wittgenstein alludes
to when he notes that a picture of a boxer can be used to report or instruct or
inquire. But I do forbid examples in which the content of the picture plays no
role.For instance,the picture plays no role in the pictorial conundrum (inspired
by Peter Geach ()) shown in Figure . Here is the enigma: Some pictures
are well-titled, in that they accurately describe the picture. Other pictures are
ill-titled, because they are descriptively inaccurate. But now consider Ill-titled.
If Ill-titled is ill-titled, then its title accurately describes the picture, and so
Ill-titled is well-titled. But if Ill-titled is well-titled, then the picture’s title fails toGendler-09 2/10/02 12:45 PM Page 349
The Art of the Impossible
Fig. 4. Ill-titled, inspired by Peter Geach.
describe itself, and so is ill-titled. Contradiction. Notice that the dilemma is
independent of anything hanging above the caption.
I also ban pragmatic paradoxes such as Fall of the Undepictable Domino (Fig.).
The very act of depicting the domino undermines its status as undepictable.
And let there be no crucial reliance on labels. Suppose there are two figures
in a picture, one labeled ‘Albanian tomato’ and the other labeled ‘Something
that cannot coexist with an Albanian tomato’. This picture is inconsistent,
but only discursively so. There are subtler ways to smuggle in discursive
elements.There are no words in Thought Clouds (Fig. ).Thought clouds are
the cartoonist’s iconographic symbols for thoughts. Embedding thought
clouds within thought clouds suggests a kind of cognitive impossibility.This
appearance of impossibility is embraced by some logicians.They try to solve the
liar paradox by insisting that all thoughts be ‘grounded’ (Burge ). It is not
clear that there is anyth-ing really impossible about an infinite regress of
embedded thoughts. The feeling that ungrounded thoughts are impossible
bears a suspicious resemblance to the feeling that an infinite past is impossible.
But my main reservation about Thought Clouds is its employment of thoseGendler-09 2/10/02 12:45 PM Page 350
Roy Sorensen
Fig. 5. Fall of the undepictable Domino.
Fig. 6. Thought clouds.
discursive-looking icons. I want an impossible picture, not an impossible
pictogram.
Fallacious geometrical proofs often rely on mislabeled diagrams.The same
applies to figures in rule books.The official rule book for Little League Baseball
mandates that home plate be an irregular pentagon (Fig. ). This figure is
impossible because it requires the existence of a (, , ) right triangleGendler-09 2/10/02 12:45 PM Page 351
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⬙
.⬙ .⬙
⬙ ⬙
Fig. 7. Home plate.
(Bradley ).According to the Pythagorean theorem,the squares of the sides
of a right triangle must add up to the square of the hypotenuse: ab c.
But . This example illustrates Wittgenstein’s
contention that many contradictions are inconveniences rather than disasters.
‘Home plate’ is used in definitions of ‘strike zone’, ‘out’, ‘run’, and so on.Yet
thousands of valid Little League baseball games have been played with home
plates that only approximate a regulation home plate.
Belief in a contradiction sometimes leads to a disaster. For instance, schedul-
ing inconsistencies have put trains on collision courses. But a priori errors
are no more likely to lead to disaster than a posteriori errors.We are more apt
to regret an a priori error because we had everything needed to detect the
mistake.But we only police our calendars and calculations with the same vigil-
ance as we check our empirical assumptions.This is good evidence that the
consequences of a priori error are only about as serious as the consequences of
a posteriori error.
These points about quality control generalize to the visual system.A priori
perceptual errors are bad, but may be acceptable given the right trade-off for
speed, generality, and ruggedness. Just as a busy street portraitist may resign
himself to some errors of perspective, the visual system may stray fromGendler-09 2/10/02 12:45 PM Page 352
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Euclidean geometry when representing a scene.These violations of the law do
not constitute dissidence.
. Adverbial Inconsistency is not Enough
There is a process/product ambiguity in ‘inconsistent description’. When
under police interrogation, most people inconsistently describe their past
activities.These inconsistent descriptions are not descriptions of an inconsist-
ent world in which the suspect is both present and not present at work on ,
January at : a.m. Instead, the testifier is describing a consistent state of
affairs in an inconsistent manner. He makes de re reference to his past and then
inadvertently assigns inconsistent properties to this sequence of events.
Most stories inherit a consistency constraint from the author’s belief that
inconsistencies are inaccuracies.The story purports to be accurate testimony.
Inconsistencies in this kind of fictional testimony must therefore be treated
in the way inconsistencies of factual narratives are treated. Standardly, the
story-teller is embarrassed by his inconsistencies and regards them as mistakes.
We should not interpret screen writers for Mission: Impossible as describing an
impossible world in which secret agent Ethan Hunt descends head first to
within one inch of a vault floor, yet then has room enough to prevent a bead of
sweat from hitting the floor by catching it with his outstretched hand.
A world is not the sort of thing that can be inconsistent.The only bearers of
inconsistency are representations. Consistency is just the absence of inconsist-
ency.Nonrepresentations are trivially consistent because they do not even have
an opportunity to be inconsistent.
Just as story-telling is parasitic on factual testimony, so depictions are para-
sitic on factual drawing.The artist purports to be presenting an accurate visual
record. Consider art students learning how to draw in perspective.They are
embarrassed by their inconsistent renderings of size and proximity relation-
ships. William Hogarth (–) lampooned these errors in his widely
reprinted drawing False Perspective.
Some errors in perspective are forced by competing aesthetic desiderata.
Artists deliberately sacrifice coherence for the sake of other aesthetic advantages.
For example, a fifteenth-century painting of the Archangel Gabriel telling
Mary about her future son recesses a middle pillar for the sake of an uninter-
rupted foreground. (The picture is reproduced in Ernst (a: ).) The
artist was probably aware of the inconsistency. But this does not mean he was
depicting a miraculous violation of geometry.
The opposite of forced inconsistency is gratuitous inconsistency. In Gary
Trudeau’s comic strip Doonesbury, there are often conflicting depictions ofGendler-09 2/10/02 12:45 PM Page 353
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background material.A cup will appear in one frame,disappear in the next,and
then reappear in a third frame.Trudeau introduces these inconsistencies in a
playful manner. He is not depicting strange appearances and reappearances of
household bric-à-brac. He is flippantly depicting ordinary scenes.
Good for him! When you need to say something vividly, say it with a
contradiction. Caricatures are easier to recognize than pictures with strict
adherence to geometrical fidelity (Ryan and Schwartz ; Dwyer ).
James Shellow, the defense attorney for Sandy Murphy, contended that her
husband (the Las Vegas millionaire Ted Binion) died in from the syner-
gistic effect of heroin,alcohol,and the prescription sedative Xanax.The attorney
explained that in this case . (Admittedly, the jury felt it did not
add up: Murphy was convicted along with her lover.)
. An Inconsistent Infrastructure is Not Enough
The art of inconsistency must be distinguished from art that merely rests on
inconsistent perceptual processes.Consider traditional engraving.The engraver
creates shades of gray by scratching sharp black lines into a white surface.Take
a good look at George Washington’s engraved picture on a one-dollar bill.
Washington’s face looks gray even after you notice that the picture is composed
solely of fine black lines.All engraved portraits exploit the ‘spreading effect’: at
a sufficiently fine scale,black and white are optically fused into gray.Varying the
density of the lines renders shadows and shades of gray. Many report that the
optical fusion does not wipe out the perception of black and white.The same
surface is seen simultaneously as gray all over and as black and white all over.
Unlike the Necker cube,there is no alternation between consistent interpreta-
tions.There is a single inconsistent interpretation.Yet the portrait of George
Washington is perfectly pedestrian.Inconsistent processes can yield a consistent
product.
The spreading effect can be explained in terms of competing homunculi
(Hurvich ). One feature detector analyzes the fine lines as just fine lines.
A rival feature detector averages the black lines with the white spaces to obtain
feature gray. These homunculi are not supervised, so neither is silenced or
muted. Consequently, the observer sees the same surface both ways.
A parallel explanation can be offered for the waterfall illusion. If you stare at
a waterfall and then look at neighboring rocks, the rocks appear to move while
remaining stationary. Staring at the waterfall adapts some position detectors,
but not others. When your eyes turn to the rocks, these adapted detectors
indicate that a movement in the opposite direction of the waterfall is taking
place. However, your unadapted detectors declare that the rocks are notGendler-09 2/10/02 12:45 PM Page 354
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moving. Absent the intervention of a censor, we see the rocks both ways at
once. Some psychologists have interpreted this as an example of seeing the
logically impossible:
although the after-effect gives a very clear illusion of movement, the apparently
moving features nevertheless seem to stay still! That is, we are still aware of features
remaining in their ‘proper’ locations even though they are seen as moving.What we
see is logically impossible! (Frisby : )
Tim Crane () thinks this shows that concepts cannot be part of perception.
One of the standard tests for ambiguity is the contradiction test. If a competent
speaker believes that x is F and x is not G,then F and G must have distinct mean-
ings M that is, express different concepts. However, in the waterfall illusion, the
speaker is inclined to believe that the rock is moving and not moving.The only
way to retain the contradiction test and deny ambiguity is to abandon the
assumption that concepts are involved in the speaker’s visual judgment.
D. H. Mellor boggles at how a judgment can be inconsistent if it does not
involve concepts. Just what could be the contradiction? A contradiction is a
proposition, so necessarily involves concepts. He goes on to deny that there is
any tendency to believe a contradiction.The waterfall illusion simply involves
two inclinations that cancel out:
We could,however,be inclined to believe that Fa,while also being inclined to believe
that ~Fa.And that, I submit, is what happens in the Waterfall Illusion.There isn’t sim-
ply, as Crane claims,‘a contradiction in the one content of one attitude’. Rather we
are conscious of seeing that a moves while also seeing that it doesn’t.One of these two
perceptual experiences gives us the corresponding belief, say that a doesn’t move,
which then suppresses the rival inclination to believe that it does. (: )
Mellor is proposing a divide-and-conquer solution.There is merely disagree-
ment between two self-consistent perceptual experiences.
I disagree with Crane and Mellor.The rocks are perceived inconsistently,but
it does not follow that the observer perceives a contradiction.The observer sees
ordinary rocks via an inconsistent homuncular process. Such inconsistent
processes are common.What is uncommon is our awareness of the inconsist-
ency. Only in atypically simple situations do we notice incoherences that are
systemic to experience.
. Ambiguity is Not Enough
The famous psychological reaction to ambiguous figures is ambivalent alterna-
tion between equally plausible, consistent interpretations. This instability is
exploited in István Orosz’s balcony scene. One cannot tell which corner of theGendler-09 2/10/02 12:45 PM Page 355
The Art of the Impossible
Fig. 8. Balcony, by István Orosz.
Reproduced by courtesy of the artist.
balcony is closer.The eye just vacillates between both interpretations.But visual
ambiguity can stimulate reactions other than ambivalence. Consider what
happens when the Necker cube (Fig. ) is stretched. At the stage of greatest
elongation, the dominant interpretation is inconsistent. As Barbara Gillam
notes, ‘Most observers report that for much of the time it appears to be an
impossible object with both ends pointing towards them at the same time’
(: ).They realize that this is possible only if the figure is bent. But the
figure is perceived as straight rather than bent.Gendler-09 2/10/02 12:45 PM Page 356
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Fig. 9. Necker sequence.
Our visual system’s ambivalent reaction to the Necker cube is often said to
illustrate the system’s insistence on consistency. However, the elongation
sequence suggests that consistency is negotiable. Moving top to bottom,
the inconsistent interpretation begins as a weak alternative to the consistent
alternatives. But as the elongation increases, the inconsistent interpreta-
tion becomes the dominant interpretation.Thus the inconsistent interpreta-
tion prevails even though the observer is being primed on consistent
interpretations.
A consistent interpretation is always logically available. Any ‘impossible
figure’ can be interpreted as a consistent drawing by treating the drawing as a
two-dimensional assembly of lines or as a conglomerate of distinct pictures.
With opposite deviousness, one can also interpret any possible figure as an
impossible figure. It is just a matter of connecting consistent dots in an incon-
sistent way (see Fig. ).
Logical availability does not imply psychological availability. Our visual
system is cognitively impenetrable. It cannot be modified to accommodate
the discovery of new possibilities. For instance, topologists have acquired an
excellent algebraic understanding of four-dimensional objects.They can even
calculate an impossible object that would be perceived by beings who can
perceive four-dimensional objects (Kim ). But they cannot visualize the
objects and so cannot grasp the depiction at first hand.Gendler-09 2/10/02 12:45 PM Page 357
The Art of the Impossible
y
x
–
Fig. 10. The quantum sine wave.
Psychological research on inconsistency in spatial representation suggests
that we routinely represent consistent states of affairs inconsistently. People
memorize local geography by employing heuristics (Moar and Bower ):
Turns are at right angles. Alternative paths are aligned perpendicularly. The
greater the number of turns, the longer the distance. Stylized subway maps are
pitched to these simplifications.We regularly fall into inconsistency when we
apply these heuristics to the street layout of our home towns. Compare this
inconsistency to the sort that mechanical calculators evince when they give
conflicting answers to / ? and / ? To save memory, the
calculator rounds off, and so treats divided by as a number slightly less
than a third. Rounding errors are common, but generally can be ignored.
Similarly, people make navigation manageable by rounding off geographical
irregularities.
Given the strong analogy between space and time, we may conjecture that
parallel heuristics lead to inconsistent mental diaries.Since we represent objects
and events in a system of space and time, I further conjecture that ordinary
experience is normally inconsistent.Most of the inconsistency goes unnoticed.
For instance, the truncated pyramid (Fig. ) is experienced without dis-
sonance.Are other animals more sensitive? The prey of such an animal would
have an opportunity to conceal itself as a nonexistent entity (see Fig. ).The
hyper-logical predator notes the inconsistent vertices. He moves on, leaving
the starfish unmolested. No organism implements this camouflage technique.
This suggests that all animals tolerate inconsistency.Gendler-09 2/10/02 12:45 PM Page 358
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Fig. 11. The truncated pyramid.
Fig. 12. The impossible starfish.
6 Why I am Optimistic
My picture of the impossible starfish is composed of five Penrose triangles.
Unlike the truncated pyramid, the Penrose triangle stimulates dissonance.
Unlike the elongated Necker cube,the dissonance tracks genuine incoherence.
There really is something awry in the picture.The vertices are each possible,
but not co-possible. One sees this without relying on labels or captions.The
inconsistency is within the picture itself.
Is inconsistency too abstract a relation to be ‘in the picture itself ’? The worry
becomes less pressing when one dwells on the range of properties to which
perceivers are sensitive.When the objects in view are fewer than four, we areGendler-09 2/10/02 12:45 PM Page 359
The Art of the Impossible
able to appreciate their exact number immediately.This enumerative process is
called ‘subitizing’. It is immediate, scale-insensitive, and virtually infallible.
Counting,the process used for objects that exceed three,is slow,scale-sensitive,
and error-prone.When one, two, or three objects are involved, we just see the
number of objects. Even if arithmetic is not reducible to logic, statements such
as have close logical counterparts. Illusions about the number of items
in a picture might be harnessed to form a picture of logical impossibility.
That is, if numerical properties are perceivable, then it seems likely that logical
properties are also perceivable.
Or consider the difference between an asymmetrical picture and its mirror
image. The two pictures have the same constituents. The internal relations
between the parts are the same.Yet they clearly differ perceptually.
7 Inconsistency does not Reduce to Ambiguity
Ambiguous figures such as the Jastrow duck-rabbit show that the numerically
same picture elements can be organized differently.When I see the figure as a
duck, the marks on the page are organized duck-wise.When I see the figure as
a rabbit,the marks are organized rabbit-wise.The difference between these two
perceptions is therefore not reducible to the marks on the page. Nor is it
reducible to the marks plus the low-level topographical relations that hold
between the marks.The relational difference is at a higher level: being a bill
rather than a pair of ears.I cannot convey the exact difference between the two
perceptions by drawing a duck and drawing a rabbit.The best I can do is to
distort the original picture to bias one interpretation over the other. The
additional contours and shadows will prime people to see the doctored figure
as a duck rather than a rabbit. But they see something topologically distinct
from the original.After all, the original picture and its doctored descendant are
not even identical with respect to the marks composing the picture.
‘Seeing as’ is compatible with there being a uniquely correct interpretation.
An astronomer may undergo Gestalt switches while viewing a lunar crater.
His visual system cannot settle on whether the feature is concave (a crater) or
convex (a mountain).Yet there is a fact of the matter. Or consider an ambigu-
ous photograph of a staircase (Fig. ). There is no way to tell whether the
picture is taken from the bottom of the staircase or the top.You can see the steps
as going up into the darkness or have a Gestalt switch in which they are going
down into the darkness. But there is an objectively correct answer.The origin
of the photograph settles the matter. (Confession: Actually both answers
are wrong. I took the photograph from beneath a stairwell.To duplicate theGendler-09 2/10/02 12:45 PM Page 360
Roy Sorensen
Fig. 13. Photograph of a staircase, by the author.
original orientation, hold the photograph above your head with the light side
closest to you.)
The artist’s intention can also ensure a fact of matter. But what if the artist
wants us to interpret his depicted staircase as going both up and down? Even if
we accept the artist’s authority on the matter, I would object that this is the
wrong kind of impossible picture. Just as the ambiguity of the Jastrow duck-
rabbit is internal,the inconsistency of a prize-winning picture must be internal.
The Penroses themselves seem unconcerned about the distinction between
a picture that looks as if it depicts an impossibility and a picture that really
depicts an impossibility. Lionel Penrose delighted in the construction of little
staircases and ramps that look impossible when photographed (or viewed with
one eye) from the appropriate angle. I like impossible construction projects
too—see ‘The impossible plumber’s son’ (Fig. ).The top pipe segments are
actually about a foot apart.The impossible boy darting through my carefully
staged scene is actually my two-year-old, Zachary Sorensen.Gendler-09 2/10/02 12:45 PM Page 361
The Art of the Impossible
Fig. 14. ‘The Impossible Plumber’s Son’,
photograph by the author.
Lionel Penrose’s son, Roger Penrose, analyzes the Penrose triangle as an
ambiguous figure. Mathematically, there can be no genuinely impossible
figures, so the analysans of Penrose’s topological investigations are consistent
figures that are perceived inconsistently.This is the main approach of mathemat-
ical psychologists. Accordingly, they regard ‘impossible object’ as a misnomer.
Some prefer to call the figures ‘improbable objects’.
Roger Penrose’s () ambiguity analysis cannot handle figures that are
ambiguous between interpretations that are themselves impossible figures (see
Fig. ). If all inconsistent interpretations are actually ambiguous, then Katz’s
figure would involve second-order ambiguity. But all higher-order ambiguity
collapses into first-order ambiguity (Sorensen ). Ambiguity is discretion
over meaning. If there is ambiguity about whether a term is ambiguous, then
the speaker has discretion over whether he has discretion about the term’s
meaning. But then he does have discretion about the term’s meaning. For he
can decide to exercise his discretion in favor of having discretion over whether
the term is ambiguous. Katz’s figure is not ambiguously ambiguous. Since it is
ambiguous between genuinely inconsistent interpretations, it is a counter-
example to the claim that all visual inconsistency is disguised ambiguity.Gendler-09 2/10/02 12:45 PM Page 362
Roy Sorensen
Fig. 15. Katz’s figure, from Katz 1984: 221.
Reproduced by courtesy of Milton Katz.
The linguistic analogue of Katz’s figure is ‘Some bats are not bats’. The
sentence has a reading that is a contradiction about a certain kind of winged
mammal and a reading that is a contradiction about a certain piece of baseball
equipment.Although the sentence is contradictory under all disambiguations,
the sentence is not itself a contradiction. Similarly, Katz’s figure is not itself
inconsistent. It is merely inconsistent under each of its disambiguations.
A Penrose triangle could be composed of smaller Penrose triangles.Thus an
impossible figure can be composed of other impossible figures.The linguistic
analogue of these figures is a conjunction such as ‘Some electrons are not
electrons and it is not the case that some electrons are not electrons’.This state-
ment is contradictory at two levels.At the base level it is contradictory by virtue
of having a contradictory conjunct. But it is also contradictory in virtue of its
overarching ‘P & ~P’ form.
The hierarchy could be continued upwards indefinitely. Could it extend
endlessly downward? A negative answer is implied by those who think that all
impossible figures can be ultimately decomposed into possible figures. But
perhaps there could be impossible ‘gunk’. Gunk is infinitely divisible matter.
A Penrose triangle might be composed of smaller and smaller Penrose triangles,
ad infinitum.
8 Pictorial Consistency is an Extrinsic Property
In M. C. Escher’s lithograph Ascending and Descending, the Penrose triangle is
used to construct a finite staircase that seems to rise endlessly. Escher also uses
the Penrose triangle to form a perpetual motion machine in Waterfall and an
incoherent alignment of pillars in Belvedere.Gendler-09 2/10/02 12:45 PM Page 363
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Shegeo Fukuda has pieced together physical models of Escher’s Waterfall
and Belvedere. Photographs of them (taken from exactly the right angle) look
like the corresponding Escher lithographs. (The photographs are reproduced
on pages – of Ernst (b).) However,these are not photographs of impos-
sible objects. They are photographs of miniature buildings that are bent and
broken to appear impossible. So if Escher depicts an impossible object, then it
cannot be simply in virtue of the graphical properties of the picture. Nor can
they be depicted simply in virtue of inconsistency within the perceptual
processes.Shegeo Fukuda’s photographs are perceptually equivalent to Escher’s
lithographs. Moral: the perceptual equivalent of a depiction of an impossible
object need not itself be a depiction of an impossible object.This undermines
attempts to define impossible depictions as those that stimulate inconsistent
perceptions.
Fukuda’s photograph of the Belvedere model also defeats attempts to define
impossible depictions as depictions which dispose people to form necessarily
false beliefs. If Graham Priest (: ) mistakenly believes that Fukuda’s
picture is of an impossible state of affairs, then his belief is necessarily false. For
if a state of affairs is possible, then it is necessarily possible. Thus there are
pictures of consistent states of affairs that promote beliefs in inconsistencies.
Shegeo Fukuda’s photograph further demonstrates that ‘depicts an imposs-
ible object’ involves an extrinsic property.The ‘narrow psychology’ of the artist
and his audience is not enough to ensure that a picture depicts an impossibility.
We must consider the origin of the picture.
The same point applies to a factual drawing of Fukuda’s Belvedere model.
Suppose a scientific draftsman draws the Belvedere model as seen through a
pinhole from the appropriate angle.The drawing will be perceptually equivalent
to Escher’s drawing, but will not be a depiction of an impossible object. For a
factual drawing represents in the same way as a photograph.A photograph rep-
resents an object just in case there is an appropriate causal connection between
the object and the image.The beliefs of the photographer are irrelevant.That is
why a photographer can be surprised by the content of his photograph.
9 Nonepistemic Conceiving
Fred Dretske () focused much attention on the distinction between epi-
stemic seeing (‘seeing that’),which entails belief,and nonepistemic seeing,which
is neutral with respect to belief. I can nonepistemically see a three-legged frog
even though I believe it is a hallucination. I nonepistemically see object O just
in case there is an appropriate causal connection between object O and myself.You can also read
