Supposedly, early 20th century packaging for Quaker Oats depicted the eponymous Quaker holding a package of the oats, where the art on this package depicted the Quaker holding a package of the oats, which itself depicted the Quaker holding a package of the oats, ad infinitum. I have not been able to locate an photograph of the packaging, but more than one philosopher and mathematician has attributed an early interest in the nature of the infinite to childhood contemplation of this image. Here, however, I want to examine a different phenomenon: whether artwork that depicts itself in this way can lead to paradoxes.

Let’s begin with two well-known puzzles. The older of the two– the Liar paradox – was known to ancient Greek philosophers, and challenges the following platitudes about truth:

(T 1 ) A sentence is true if and only if what it says is the case.

(T 2 ) Every sentence is exactly one of true and false.

Consider the Liar sentence:

This sentence is false.

Is the Liar sentence true or false? If it is true, then what it says must be the case. It says it is false, so this means it is false. If it’s false, then, since it says it is false, what it says is indeed the case. But this would make it true. So the Liar sentence is true if and only if it is false, violating the platitudes.

The second puzzle is the Russell paradox, discovered by Bertrand Russell at the beginning of the 20th Century. This paradox involves collections, or sets, of objects, and two central theses:

(S 1 ) Given any property P, there will be a set of objects containing all and only the objects that have P.

(S 2 ) Sets are themselves objects, and can be contained in sets.

Given (S 2 ), we can divide objects into two types: Those that contain themselves (such as the set containing all sets whatsoever) and those that do not contain themselves (such as the set of all kittens). Thus, “is a set that does not contain itself” picks out a perfectly good property, and so by (S 1 ) there should be a set – let’s call it R – containing exactly those things that have this property. So:

A set is a member of R if and only if it is not a member of itself.

Now, is R a member of itself? Either it is or it isn’t. If R is a member of itself then R isn’t a member of itself. And if R isn’t a member of itself then R is a member of itself. Either way, R both is and isn’t a member of itself. Again, a contradiction.

There is another puzzle that seems intimately connected to these two paradoxes, however, that has not (as far as I know) been noticed or studied – the paradox of the impossible painting. This paradox stems from two principles governing the notion of depiction (or representation) rather than truth or set-theoretic membership.

First, it seems, at least at first glance, that we can paint anything that we can describe – if I tell you to paint a forest with exactly 28 trees, then you can produce a painting fitting that description. Thus:

(D 1 ) Given any description D, we can create a painting that depicts things exactly as described in D.

Second, there is nothing to prevent a painting from being depicted within another painting – for example, Diego Velázquez’s Las Meninas depicts the painter working on another painting. Thus:

(D 2 ) Paintings can be depicted in paintings.

If some paintings can depict other paintings, then it seems like we can divide paintings into two types: those that depict themselves (such as the artwork on old Quaker Oats packaging) and those that do not. Thus, “a scene depicting all and only the paintings that do not depict themselves” is a perfectly good description, and so by (D 1 ) it should be possible to produce a painting – let’s call it I – that depicts things as described. So:

A painting is depicted in I if and only if it does not depict itself.

Should I depict itself ? In other words, if you are creating this painting, should you include a depiction of I itself within the scene? If you include I in the painting, then I is a painting that depicts itself, so it should not be depicted in I after all. But if you don’t include I in the painting, then I is a painting that does not depict itself, so it should have been included. Either way, you can’t create a painting that depicts things exactly as described.

The paradox of the impossible painting is distinct from both the Liar paradox and the Russell paradox, since it involves depiction rather than truth or set-membership. But it has features in common with each. Most obviously, circularity plays a central role in all three paradoxes: the Liar paradox involves sentences that says something about themselves, the Russell paradox involves sets that are members of themselves, and the paradox of the impossible painting involves paintings that depict themselves.

Nevertheless, the paradox of the impossible painting has features not shared by the Liar paradox, and other features not shared by the Russell paradox. First, the Liar paradox involves a sentence that clearly exists (and is grammatical, etc.) that must be accounted for, while the Russell paradox can be seen in different terms, as a sort of proof that the Russell set R just doesn’t exist, and that we need to revise (S 1 ) accordingly. The proper response regarding the paradox of the impossible painting is more like the latter – we are not tempted to think that the paradoxical painting does or could exist, but instead conclude that there is something wrong with (D 1 ).

There is another sense, however, in which the paradox of the impossible painting is more like the Liar paradox than the Russell paradox. The Liar paradox arguably arises because of circularity of reference: the Liar sentence refers to, or ‘picks out’, itself. And the paradox of the impossible painting arises because of circularity of depiction – that is, paintings that depict, or ‘pick out’, themselves. Reference and depiction are different, but, insofar as they are both ways of ‘picking out’, while set-theoretic membership is not, suggests that, in this respect at least, the paradox of the impossible painting has more in common with the Liar paradox than with the Russell paradox.

Thus, the paradox of the impossible painting ‘lies between’, or is a sort of hybrid of, the Liar paradox and the Russell paradox, with some features in common with the former and others in common with the latter. As a result, studying this puzzle further seems likely to reward us with deeper insights into these two much older and more well-known conundrums. Who knew oats could be so deep?

Note: This puzzle first arose in conversation with Catharine Diehl (Humboldt – Berlin), Daniel Hoek (NYU), Nathan Kellen (UConn), Beau Mount (Oxford), and Jeffrey Schatz (UC-Irvine, LPS) while we were all attending the Foundations of Logic and Mathematics Summer School at the Northern Institute of Philosophy (NIP) at the University of St Andrews.

Featured Image Credit: ‘Drawing Hands’ (1948) by M.C Escher. Public domain via WikiArt.org