Phil 2511: Paradoxes
L12 Impossible Statements and Escher’s
Impossible Objects
[Note: This lecture derives from a paper of mine, `Reflexivity,
Contradiction, Paradox and M.C. Escher’, Leonardo 29/4 (1996). Copies of the article (which contains the Escher
pictures) are available on request.]
1.
In
this lecture, I hope to throw some light on the cassationists claim that there
are certain `impossible statements’ by looking at some pictures of `impossible
objects’. Representational pictures show
how things are or could be; Escher's pictures of impossible objects, though
having much else in common with ordinary representational pictures, contain
conflicts of visual clues artfully contrived by the artist, and represent
objects that could not exist in reality.
Paradoxical sentences have much in common with ordinary sentences ---
they are composed of familiar vocabulary and are in perfect grammatical order
--- yet they too embody (or so I shall argue) contradictory signals and, in a
similar way, are deprived of the power possessed by normal statements to
represent (truly or falsely) how things are.
2.
Before
the Renaissance, painters used various means for conveying distance and
depth. The four principal methods were
occlusion, size, shading, and what we might call `invocation of the viewer's
world knowledge'. Occlusion is the most
obvious: one object can be represented as being behind another if the image of
the first is partially blocked out, or occluded by the image of the second. Another way of indicating depth is through
the size of representation. If we have
reason to suppose that two objects are of roughly the same size, then we
picture one as being much further away than the other by making its
representation much smaller. Depth and
distance can also be represented using shading.
A foreground object illuminated by the sun from in front will cast a
shadow over part of the background. And
the deeper one goes, the darker it gets.
The use of occlusion, size and shading are clearly illustrated in
Escher's wood engraving `Depth'.
Finally, world knowledge of a variety of sorts can be invoked in a
variety of ways.
3.
It
should be clear that, with these representational techniques at his or her
disposal, the artist has ample resources for thoroughly confusing the viewer. For example, by the use of perspective, the
artist might suggest that X is behind Y, but through shading, might
simultaneously suggest that Y is behind X.
It is a fact about us that lines drawn in a certain way on a flat (2-dimensional) surface `look’
3-dimensional.
4.
An
impossible object that much influenced Escher was the Penrose tribar
Take three straight lines that meet at a point.
These are most naturally seen as co-planar. Add another two lines,
however, and something strange sometimes happens: we have the urge to see the
figure as three-dimensional, an urge which becomes irresistible as we add lines
to complete the `Necker Cube'.
5.
In
the Penrose tribar, the suggestion that the three bars are mutually orthogonal
is done not by using the geometrical convention for representing right angles,
but by the way the joints are drawn. If we look at the joints of the tribar,
our world knowledge about joints tells us that the three bars must be mutually
orthogonal, so that the ends of the depicted object cannot join up. Yet the picture shows them as joined. Three straight lines forming a closed figure
can, as we have seen, represent either a triangle of bars in a plane or three
bars in different planes. The detail of
the tribar picture --- in particular, the way the joints are drawn using
occlusion so as to show the plane in which pairs of joined members lie ---
encourages us to see both at once.
6.
There
are two of Escher's pictures which most readily spring to mind when thinking of
the reflexive paradoxes. These are `Print
Gallery' and `Drawing Hands'.
7.
In
order to avoid confusion in the discussion, let's call an object in the world a
W-object and the object in a picture a P-object --- so that typically
P-objects, whatever their appearance, are composed of ink, paint or some other
drawing medium. In the lithograph `Print
Gallery', we have what purports to be a representation of a W-boy looking at a
picture, which, of course, is composed of P-objects. But, as the picture curves around, we notice
that it includes that very boy. In other
words, the P-boy is, per impossibile a W-boy!
8.
Something
similar occurs in `Drawing Hands'. Focus
on one of the drawing hands. Were this
the whole picture, we should naturally see this as a representation of a W-hand
holding a W-pencil, and, if that pencil were executing a drawing of an apple,
then that drawing would have to be a representation of a P-object, because we
don't create W-apples with a pencil, but only P-apples. So, if the depicted W-hand is drawing a hand,
the latter must be a P-hand. Yet that
P-hand is shown doing a drawing, and that is crazy, because only
flesh-and-blood hands can draw; P-hands, made of carbon, can't. Notice too that the picture is
symmetrical. So we have no sufficient
reason for saying that one object represented in the picture is the W-hand, the
other a P-hand rather than vice-versa.
9.
Let's
compare the situation here to one in which two companies, say the Hopewell
Construction Company of Hong Kong and the French firm Dragages are tendering to build the Hong Kong to Beijing superhighway. Under normal circumstances, each company
would put in its sealed bid, naming a price for doing the job. But since each company knows that there is
only one other company tendering, each tries to outsmart the other in its
attempt to undercut the competitor. Hopewell's bid reads:
Hopewell offers to build the highway for $1000 less
than Dragages.
Now, had Dragages named their price for the job, we should know
what price Hopewell was offering. However,
when Dragage's letter of tender is opened, this is what it says:
Dragages offers to build the highway for $1000 less than Hopewell.
Had Dragages named a figure, the Hopewell bid would have been legitimate, but, under
the circumstances just described, it becomes empty, a non-bid. For its being an offer is contingent upon
Dragages' making an offer, but, as things turn out, whether Dragages has made
an offer is itself dependent upon whether Hopewell has.
10. The problem with paradoxically conflicting
statements is that we apparently cannot ascribe plausible truth-values to the
statements. But, as the example just
given demonstrates, there are speech-acts other than statements which give rise
to parallel problems. So the fundamental
issue is not specifically to do with truth and falsity (offers are neither true
nor false), but with the means by which any such conflicts arise.
11. Compare now the Hopewell/Dragages fiasco
with the following, which constitutes a paradox.
P1: P2 is false
P2: P1 is false
12. This is a paradox because, although P1 and
P2 can be consistently assigned opposite truth-values (i.e., if one is true,
the other is false), symmetry demands that they be assigned the same truth
value. We might now be inclined to say that, just as Hopewell and Dragages issued
sentences purporting to make offers, yet no offers materialized, so similarly
P1 and P2 are sentences which purport to make statements, yet fail to do so;
the sentences fail to yield any statements that can be evaluated as being
either true or false. The position we
have reached at this point is a sophisticated version of an approach to
paradoxes called cassatio, popular in the early mediaeval period. The sophistication lies in the distinction
between a sentence and the use of a sentence to perform a speech act, such as
the making of a statement or an offer.
13. Although the set of staircases in Escher’s
`Ascending and Descending' is, in a certain sense, reflexive --- the
steps form a closed loop, leading back to themselves, this is not necessary for
the production of illusion. Were the
stairs adjoining the `penthouse' omitted, the courses of the brickwork would
still indicate that ascending three flights of stairs does not get you to a
point higher than the one from which you started.
14. A similar point may be made about the
lithograph `Waterfall', in which a stream of water follows a closed
loop. If we look at just a fragment of
this lithograph, for example, the channel immediately above the waterwheel
leading away from the viewer, we can see that we are being given conflicting
visual signals --- the water appears to be running almost level, but one end of
the channel is raised much higher than the other by a set of pillars. Were the pillar-representations not to look
like pillars which raise the channel one end, then no illusion would result.
15. World knowledge of pillars also plays a
prominent role in our finding Escher's `Belvedere' perplexing, and to
recognize this is to account for that perplexity and therefore, in a certain
sense, to solve the picture. By a
`solution' to a paradoxical picture, I mean an understanding of how the
illusion is pulled off.
16. In this picture there is a tension between
the clues to the construction of the Belvedere as supplied by occlusion, size,
shading and perspective, and those supplied by world knowledge. For it is part of such knowledge that
classical, load-bearing stone pillars are vertical; the Leaning Tower of Pisa
is about at the limit beyond which collapse will occur. Look at the pair of pillars, second and third
from the left, supporting the top floor of the Belvedere. Were it physically possible, and usual, for
load-bearing pillars to criss-cross with a lean of about 30 degrees, then at
least this part of the structure would not appear to us disturbingly
paradoxical. But they can't; so it
does. By contrast, the impossible
gallows in Pieter Brueghel's `The Magpie on the Gallows' (1568) is not so
visually unsettling because we can easily interpret the gallows as being rather
rickety, the verticals badly bent and twisted, as is quite possible with old
timber.
17. Qua P-objects, pillar-representations are
just blobs of ink; qua components of a picture of something in a real or
imagined world, however, they represent pillars. Similarly, words, qua marks in a written
sentence or qua sounds in a spoken one have no meaning. The mark `suit' is common to both sentences
`Il suit un cochon' and `My suit is cotton'.
But, when words are used in discourse (for example, to make an
assertion, issue a command, ask a question) they have meaning --- `pilier', in
the mouth of someone speaking French is a device for referring to pillars. There is no mystery in this: speakers are
able to use words to refer to objects in the world and to mean what they say,
in virtue of the training they received as infants. When words are used, then conflicts of
meaning can occur. For example, the
sentence `The level channel has one end raised higher than the other by
pillars' produces an `Eh??!!??'-reaction, as does the sentence `The dead man
spoke with an Irish accent'. This
`Eh??!!??'-reaction is the counterpart of the visual bafflement we suffer when
confronted with the contradictory signals in an absurd picture.
18. Just as the third dimension jumps out at us
in a perspectival drawing --- we cannot help but see the lines on the flat
canvas as representing three-dimensional figures --- so words when used acquire
a life; a declarative sentence when used is no longer just a bundle of marks or
sounds but is (typically) an assertion.
The conventions of perspective, and the other techniques we have
considered ensure that drawings can produce illusions. Similarly, the conventions governing the use
of language can produce absurdities, as in the `Eh??!!??' cases above, and paradoxes. Take the simplest case of the latter. Someone who utters
L.
L is
false
purports to assert L, i.e., to claim L to be true, yet what he
asserts is that L is false. The speaker
seems to be asserting and denying the same thing in the same breath.
19. As we have seen, lines in a picture can have
multiple and mutually incompatible interpretations depending on different clues
supplied by the surroundings. And the
context in which a sentence occurs can determine contradictory interpretations. A sentence in one context may yield an
`impossible' statement, i.e. a non-statement; in another context may yield a
true description. What should we say
about such cases? `The Belvedere',
again, provides us with a useful starting point. The ladder in the Belvedere is inside the
lower floor, leans inwards, yet finishes outside the upper floor. The illusion can be understood, as Bruno
Ernst shows, in terms of an impossible cuboid.
20. What happens here is that, in creating the
picture, one sets up visual clues in such a way that a straight path between X
and Y lies behind an object O. But then
a straight, uninterrupted strip is drawn between X and Y and, where it
traverses the representation of O, one simply occludes the relevant part of
that representation, so that O now appears to be behind XY. Notice that, for such an illusion to work, a
`normal' visual situation has to be created by other elements in the picture,
for otherwise there are no visual expectations to be shattered. This is why, in Belvedere and other impossible
figures of Escher's, there is so much normal use of perspective and of
representations of familiar objects such as pillars and flights of stairs which
provide visually plausible transitions from one space to another.
21. Compare a picture of a possible cuboid
with the picture of an impossible one.
We can assign no consistent value to the length of the sides of the
impossible cuboid – we have a picture to which no object can correspond, just
as we can have a Liar sentence to which no statement corresponds.