In 1917 a pivotal event occurred for art and philosophy: Marcel Duchamp unveiled his artwork Fountain in Alfred Stieglitz’s New York studio. This was simply a porcelain urinal, signed ‘R. Mutt’.

Fountain was notorious, even for avant-garde artists. It has become one of the most discussed works of art of the 20th century. The Society of Independent Artists rejected it, though every artist who paid the exhibition fee was supposed to have their work shown. For almost a century, it has remained a difficult artwork. The philosopher John Passmore summed up Fountain as: ‘a piece of mischief at the expense of the art world’, though many have taken it very seriously.

No doubt there was some tomfoolery involved – Duchamp did not choose a urinal randomly. Yet there is more to Fountain than nose-thumbing. What makes this artwork so striking is its philosophical contribution.

Commentators often highlight the influence of Fountain on conceptual art, and this most ‘aggressive’ readymade, as Robert Hughes put it, has certainly had an enduring legacy. In 2004, it was voted the most important 20th-century work by hundreds of art experts. From Andy Warhol to Joseph Beuys to Tracey Emin, this urinal inspired artists to reconsider the traditional artwork. Instead of paintings and sculptures, art was suddenly Brillo boxes, an unmade bed, or a light-bulb plugged into a lemon: ordinary objects, some readymade, removed from their original contexts and placed on display in art galleries. The art critic Roberta Smith sums it up this way: ‘[Duchamp] reduced the creative act to a stunningly rudimentary level: to the single, intellectual, largely random decision to name this or that object or activity “art”.’ As we will see, Duchamp’s choice was not random at all, but Smith’s description points to the broader shock that Duchamp’s work prompted: if this can be art, then anything can.

Since then, scholars have discussed Fountain to demonstrate a shift away from aesthetics to thought. As the philosopher Noël Carroll notes, it’s possible to enjoy thinking about Duchamp’s work without actually looking at it, which cannot be said for Henri Matisse’s vivid paintings or Barbara Hepworth’s dignified stone sculptures.

These traditional ideas, as we will see, are all important to Fountain. But they do not go far enough. They treat Fountain as art, but of a mocking sort: a kind of intellectual heckling that nudged artists to taunt and scoff more academically at their own field. Our explanation of the artwork’s power is much more controversial: we believe that Fountain is art only insofar as it is not art. It is what it is not – and this is why it is what it is. In other words, the artwork delivers a true contradiction, what’s called a dialetheia. Fountain did not simply usher in conceptual art – it afforded us an unusual and intriguing concept to consider: a work of art that isn’t really a work of art, an everyday object that is not just an everyday object.

How is this possible? Let’s begin with the obvious: Duchamp’s Fountain really was a urinal. Not a painting or sculpture of a urinoir – though the latter might raise interesting philosophical questions – but the real thing, a token of a particular type – there were many visually indistinguishable urinals that came off the same production line. And just as importantly, Duchamp had no involvement whatsoever in designing or making the urinal that was the raw material for his artwork. His contribution was to sign the urinal, and exhibit it as art.

In his paper ‘Art, Philosophy, and the Philosophy of Art’ (1983), the philosopher Arthur Danto gave a helpful account of what happened when Duchamp did this. The urinal became, as Danto puts it, ‘about something’. It was no longer primarily a useful object – it was primarily a meaningful object.

This is because the urinal was now part of what Danto called ‘the artworld’, in a 1964 article of the same name. The artworld is, simply put, a milieu in which objects can gain a new power: to express something beyond their ordinary utility. They are part of a new category, ‘art’, and gain a message that can be distinguished from their use or exchange value, and from the new category itself (a distinction we will come back to later). This is the famous aspect of art that the eye cannot, in Danto’s word, ‘descry’, the non-visual aspect of some visual art.

Warhol made his Brillo boxes from plywood, rather than taking boxes from a dry goods store; Duchamp didn’t make a urinal, he displayed it in a new contextThis expression is not representation, as often understood: mimesis, or copying the likeness of some object. Mark Rothko’s most popular paintings, for example, do not represent any particular objects, but they certainly have a message: of awe, or the sublime. So, within the artworld, objects express ideas and feelings – sometimes by resemblance, sometimes not.

The point is not that objects outside the artworld cannot have a message – the signs on a public toilet are straightforwardly ‘about’ something. The point is that, once they are within the artworld, objects can gain a new significance over and above their common use. They do not simply name an object or show its function: they make a statement of some sort.

Duchamp’s Fountain embodied a statement about art itself. This, as we will see, is why it is dialetheic. Danto gives the example of Warhol’s soap boxes (the artistic grandchildren of Duchamp’s urinal), arguing that they ‘made some kind of statement about art, and incorporated into their identity the question of what that identity is’. The same was true of Duchamp’s urinoir, though it is important to realise that Warhol made highly realistic copies of Brillo boxes from plywood, rather than exhibiting the boxes from a dry goods store; Duchamp, in contrast, didn’t make a urinal, he merely displayed it in a new context.

To see the specifics of Duchamp’s message, it helps to briefly detail the historical background. In the early 20th century, visual art was still chiefly associated with craft: the physical transformation of paint, clay and so on. But it was often seen to have more value than simple craft: some intimation, through beauty, of spiritual or philosophical truths. As the sociologist Pierre Bourdieu noted in Distinction (1979), the artist worked with or against matter to afford some moral, metaphysical or at least aesthetic vision, which was ‘higher’ than ordinary commerce and labour.

Duchamp’s Fountain was the antithesis of this. There was no obvious craft – and certainly no fine artistry. The urinal was designed and manufactured to some standard, but one of utility, not aesthetics. Duchamp was overt about this, mocking critics who later tried to find the beauty in the urinal. ‘I threw… the urinoir into their faces,’ he complained in 1961, ‘and now they come and admire it for its beauty.’

As this suggests, the urinal was also a ‘low’ object: something to be urinated on, not gazed at for its loveliness or spiritual truths. It was, in the words of E H Gombrich, an attempt to ‘cock a snook at the solemnity and pomposity of Art with a capital A’. In other words, the message of Fountain was one of mockery: of modern ideals of art. It mocked, not by parodying fine art, but by being its stark opposite: uncrafted by the author, ugly, utilitarian, vulgar, ubiquitous, and so on.

they call it art because it is the right kind of thing, in the right time and place – and exhibited by the right kind of artist

This specific message is important, as it provides an argument against the thought that art is simply anything we call by that name. As we saw, Duchamp’s urinal was not chosen arbitrarily by the artist. The object’s very specific qualities contributed to its message. And this was achieved at a very specific historical era. Too early, and Fountain would have been incomprehensible as art, even for the avant-garde. Too late, and it would have been passé. Just as importantly, Duchamp already had some sway in the artworld – what Bourdieu calls ‘capital’, within the field of art.

In this light, it cannot be argued that something is art simply because it is called so by members of the artworld. Rather, they call it so because it is the right kind of thing, in the right time and place – and exhibited by the right kind of artist.

So we might say that any object can potentially be art, but never actually: the artworld is always a specific milieu that authorises some artists, messages and objects and not others. So Fountain is not art simply because it is called ‘art’. It is art because Duchamp deemed it so with his signature and exhibition, and the message of this deeming was recognised and, over time, accepted by members of the artworld.

It is also important to deal with the opposite claim: that Fountain is simply not art at all. Now, it was, indeed, not art; but to say that it was simply not art is to miss the point. True, the urinal does indeed have very low aesthetic value, as defined by the philosopher Monroe Beardsley. It is unlikely to afford what Beardsley calls an ‘aesthetic experience’, because its perceptual qualities are unremarkable. But this does not mean that Fountain is simply not art. Duchamp’s work is art with a higher conceptual than aesthetic value. But this judgment is possible only because it is art; because it is part of the artworld, and able to be evaluated for, among other things, its aesthetic value.

What makes this evaluation philosophically interesting is that Fountain, as an artwork, has a message: that it is not art. This is a vital point, and often missed by traditional accounts of the work. Fountain is screaming: ‘I am not art’, but doing so from a plinth in an art show. It carries its message by rejecting implicitly all the traditional markers for the category of art: beauty, craftsmanship, uniqueness, artistic personality, along with the standard ideals of edification, expression or aesthetic pleasure. This message is one of so-called ‘anti-art’. What gives the work its power is that it is not art; but that, at the same time, it is art.

The urinal, as we have seen, was chosen by Duchamp precisely because it was antithetical to the basic ideas of art in the early 20th century. Put simply: it is because the urinal is not art that it has its particular artistic message; and it is only because it is art that it has a message at all. Of course, it can carry other messages as well, for example, that contemporary ideas of art are misguided, that craft is not essential to art, that beauty in art is optional. But it can carry such messages only because, in the artworld of its time, it was not art.

So Duchamp’s contribution to the history of modern art can be put like this: ‘This both is art and is not art.’ It is an obvious and bald contradiction. Yet for many in the artworld, this proposition was – implicitly or explicitly – true. That’s what made Fountain so immediately fascinating, and what has invited so many essays, artworks and gallery visitors. Duchamp’s urinal rightly draws attention because it grounds a dialetheia: a true contradiction, something that traditional logicians believe to be an impossibility.

perhaps Duchamp’s artwork is just wrong about itself? Perhaps it’s teasing us

Dialetheism is the view that some contradictions are true, and so disputes what philosophers call the principle of non-contradiction: roughly, the idea that the same statement can’t be both true and false at the same time. Though some have challenged this principle in the history of Western philosophy – the most notable dissident being, arguably, Hegel – it has been high orthodoxy in Western philosophy since Aristotle’s convoluted and dubious defence of it in his Metaphysics 4. In the past 30 years, we have seen new advocates of dialetheism appear. The advocacy has its home in modern formal logic (as we shall see in a moment), and is supported by all of its tools. Perhaps surprisingly, there is now a lively debate in the area, because the principle of non-contradiction seems so firmly based in common sense. If an animal is a cat, it can’t simultaneously not be a cat. It is either Thursday or not Thursday: it can’t be both Thursday and not Thursday on the same day, here and now. But beware, as Ludwig Wittgenstein said, of an inadequate diet of examples!

In this case, the dialetheism works like this. The object’s category is ‘art’. And as art, it has a message. In this case, its message is about its own category: it says to the viewer – truly – ‘this is not art’. The dialetheia arises because the message requires the very category it rejects, art; and because this rejection is its message within this category. The rejection of its status as an artwork is the very thing that makes it an artwork, which it rejects – and so on.

This invites the reply that it is simply false to say that Fountain isn’t a work of art. It just carries the message that it is not. Thus, a sign might display the message: ‘This is written in red’, while in fact it is written in black. In short: perhaps Duchamp’s artwork is just wrong about itself? Perhaps it’s teasing us, saying ‘I might not really be art’.

But Fountain can carry the message that it is not art only because it is not art – because its very entry into the artworld is defined by its rejection of art. Had it simply been art in an unproblematic sense – if, for example, Duchamp had chosen to paint an oil painting of a urinal – it could not have carried this message. This contrasts with the sign that is what it is because it has a message inscribed on it. So consider René Magritte’s 1928-9 painting of a pipe. This literally bears the message ‘Ceci n’est pas une pipe.’ The very words carry a message. By contrast, Fountain bears no explicit message. It conveys its message by being what it is. It is not art, and that is how it conveys its message. That is precisely why it is art. Put another way: contradiction is essential to Fountain as art. And if it didn’t embody a contradiction, it wouldn’t be half as interesting; we wouldn’t still be talking about it.

It might seem that the paradox of the urinal is a cultural oddity: something that could happen only in the strange world of contemporary art; but, actually, it fits a much larger pattern of something being the case because it is not the case: p because it is not the case that p.

Perhaps the most frequently discussed candidates for dialetheia are the logical paradoxes of self-reference, such as the famous liar paradox (concerning the sentence ‘This very sentence is not true’). These are apparently genuine arguments that end in contradictions, with this logical form: p and it is not the case that p. The paradoxical arguments that deliver these contradictions can be of different kinds, but one of these is of the form with which we are now concerned.

Consider König’s paradox. This concerns ordinals. Ordinals are numbers that extend the familiar counting numbers 0, 1, 2, … beyond the finite. After all the finite numbers, there is a next number, and then a next number plus one, and so on. Crucially, these numbers preserve the property of the counting numbers that any collection of them has a least member. How far, exactly, the ordinals go is a somewhat vexed question, both mathematically and philosophically, but it is not contentious that there are many more ordinals than can be represented by the phrases of a language with a finite vocabulary, such as English. This can be shown by a perfectly rigorous mathematical proof.

Now, if there are ordinals that cannot be referred to in this way, then, by the properties of the ordinals, there must be a least such ordinal. Consider the phrase ‘the least ordinal that cannot be referred to’. This obviously refers to the number in question. This number, then, both can and cannot be referred to. But note that it is precisely the fact that it cannot be referred to that allows us to refer to it. That is, it is referable because it is not referable: p because it is not the case that p. In a similar way, Fountain is art because it is not art.

The similarity between the Fountain paradox and some paradoxes of self-reference invites the question of whether self-reference is involved in the former. A little thought shows that it is. The art work includes the message ‘this is not a work of art’, and so refers to itself.

While not all dialetheias involve self-reference, self-reference is obviously a rich source of them. Traditional discussions of self-reference have concentrated on spoken or written language. What our discussion shows is that they can occur in other media, such as the visual, too. These, just as much as ordinary language, can convey information; the information can be self-referential; and the self-reference can engender paradox. It was Duchamp’s genius to have found a way of presenting an object that was simultaneously both art and non-art. It is high time that we recognised that Duchamp’s contribution was profoundly and intentionally paradoxical.