A while ago I read a piece in Aeon by Damon Young and Graham Priest, which argued that Marcel Duchamp's Fountain1 both is and isn't art. Before I read it, I had an idea where they might be going with it and was prepared to be unimpressed. I've got a lot of respect for Graham Priest (I hadn't heard of Damon Young) so I read it anyway, but I was expecting it to be a case of everything looking like a nail when all you've got is a hammer. (Or at least, when you have a hammer you're particularly pleased with.) I'm happy to report that I was wrong, and I suppose I owe them an apology. Sorry. In fact I found the thing so interesting that it's shifted the way I think about dialetheism a bit, or at least crystallized some of my thinking about it.
Three Truth Values In The Fountain
Priest is famously a dialetheist, in that he thinks that some statements are true even though their negations are also true. An example is "this statement is not true". Priest thinks this statement is both true and not true. I'm less confident than he is; I guess I'm probably about 60% sure that it's both true and not true2. Now, once you've decided that contradictions can be true, it gives you a new option for solving other philosophical problems. Often philosophical problems arise because there's a group of statements that all seem compelling, but they're jointly inconsistent. For most of us, this means we have to give one of them up. Dialetheists can accept them all. This is what I was expecting from Young and Priest's Aeon piece. I thought they were going to say that there were compelling reasons to think Fountain is art, and compelling reasons to think it isn't, and then encourage us to embrace the contradiction here as we already should be doing elsewhere.
That's not what they did, though. Instead, they said that its not being art is the very thing that makes it art. Here's one way they explain it:
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's just a urinal, so it's not art, but presenting a piece of non-art as Duchamp did lets it convey a message in such a way as to make it count as art. (Even though the message is that it isn't art.) The key thing is that the message only works because it's not art. So if it isn't art then it is art, but this can't prevent it from not being art because if it did then there wouldn't be anything making it art anymore. (You can disagree with their intrerpretation of Fountain itself, but admitting it's an available interpretation should be enough to make the general point about dialetheism.) This back-and-forth dynamic parallels what you get with the liar sentence. If it's not true then it is true, but that can't prevent it from not being true, because then it wouldn't have anything making it true, so it wouldn't be true anymore. And then it would be true, and so on.
This isn't what I was expecting. They haven't just said that there are arguments on both sides and as dialetheists they have the option of accepting both sides; they've given an analysis of Fountain's status that parallels the dynamic with the liar paradox, and it seems reasonable for them to resolve both issues the same way. I think we can all agree that this is much better, and so I was right to pleasantly surprised.
Contradictions, Contradictions Everywhere
Now, one thing that dialetheists sometimes worry about is what stops contradictions being true all over the place. If contradictions can be true, why aren't more of them true? If its being the case that I'm eating cornflakes doesn't prevent it from being the case that I'm not eating cornflakes, what does prevent it? I think this is a very difficult question, and often when I think about it I think that maybe I shouldn't be a 60%-committed dialetheist at all. But dialetheists do have things they say about the issue. In his book Spandrels of Truth, which I'm afraid I haven't read much of, JC Beall argues that dialetheias only show up when we're talking about truth. Dialetheias are a consequence of enriching our language with the kind of truth predicate Beall thinks we should use, and the rest of the language avoids them.3 This gives a nice answer to the question of why there aren't dialetheias everywhere. We'll call someone with a view like that a truth dialetheist.
Unfortunately, the kind of dialetheia Priest and Young talk about suggests the phenomenon is more widespread than that. Perhaps with some purported dialetheias you can argue that they're covertly about truth. I can see someone doing that with Russell's paradox (does the set of all sets that don't contain themselves contain itself?) but with the Fountain case it seems a bit of a stretch. There are other cases dialetheists sometimes argue for too that occasionally seem plausible to me, for example inconsistencies in law, or games, or perhaps ethics. The ones that seem most plausible to me are the ones where a rule applies because it doesn't apply. Young and Priest draw attention to this factor themselves:
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.
So if you're someone like me, and you quite like the idea of dialetheias that fit this pattern but don't like any others, you've got an option for limiting them besides truth dialetheism. Dialetheias arise when something is a certain way because it isn't that way. To me this seems more satisfying, because if this kind of dynamic doesn't have to generate dialetheias, why does it generate them in the case of semantic paradoxes? There are undoubtedly ways to respond to that challenge, but let's suppose we go for the idea that dialetheias arise whenever something is the case because it isn't, but otherwise they don't arise. We'll call that view explanation dialetheism. This doesn't limit dialetheias as much as truth dialetheism, but it's still supposed to limit them a lot. But can it really do that? Will anything be safe?
Persistent Macroscopic Physical Dialetheias
Here's something Young and Priest say about the law of non-contradiction before explaining how Fountain manages to break it:
[T]he 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!
Now, Priest at least is holding back a bit on the esoteric dope here, because he's said things elsewhere that suggest he may think things can simultaneously be cats and not cats, and that it can be both Thursday and not Thursday. He's expressed sympathy (Priest 2006: chs 11-12) with the idea of dialetheias obtaining at the instant of change, so maybe as the clock strikes midnight it's both Thursday and not, and maybe if things become or stop being cats, around the beginning or end of a cat's life, there are things that are both cats and non-cats. But these situations don't last for very long. You don't go to the pet store and come back with something that both is and isn't a cat, and you don't schedule a meeting on Thursday and then wonder whether or not to go on the grounds that it both is and isn't Thursday. (Although people do set deadlines at 23:59 instead of midnight, presumably to avoid this confusion.)
Aside from instants of change, another place to look for physical dialetheias is in quantum mechanics. When things get very small they also get very weird, and dialetheists would be missing a trick if they didn't at least check to see if embracing contradictions can help us interpret quantum mechanics. Priest himself (2006: 180-1) makes this connection, though he does say his speculation is 'perhaps rather fanciful'.
We do still have some common-sense attachment to the law of non-contradiction, though. Maybe we can accept that physical dialetheias obtain for a moment here and there, or at the quantum level, but we don't expect them to hang around for a long time among middle-sized objects. We don't expect persistent macroscopic physical dialetheias. So, if dialetheias are confined to situations where things are a certain way because they're not that way, should we expect PMPDs? I think the answer is that it depends. Let's think about two more paradoxes: the Pinocchio paradox and the grandfather paradox.
The Pinocchio paradox [Eldridge-Smith and Eldridge-Smith 2010] arises because of Pinocchio's nose's disposition to grow when and only when he lies. If he says "my nose will be growing", then his nose should grow iff it doesn't. This fits into the explanation model of dialetheias: the nose will grow because it doesn't, but its growing would prevent it from growing, and so on. This puts pressure on us to make the semantic world and the physical world play by the same rules, because when you're Pinocchio there's a law-like connection between them. I don't really think someone like Beall should be too worried by it, though. And indeed Beall himself wasn't , but his explanation seemed to me to muddy the waters a bit by talking about fiction, so I'll explain in my own way how I think dialetheists should respond to the Pinocchio paradox.
If physical dialetheias are impossible and semantic dialetheias are possible, then it just won't be possible for someone's nose to have the exceptionless disposition to grow when they say something false and not grow when they say something true. We don't worry that it's not possible for someone's nose to have an exceptionless disposition to grow when they're wearing a red sock and not grow when they're wearing a blue sock, on the grounds that there would have to be an exception when they were wearing one of each. The Pinocchio paradox is the same. At least, I think it's the same. Of course, if PMPDs are possible, then Pinocchio's nose could both grow and not grow, and the Pinocchio disposition need not be impossible after all. But the possibility of the disposition turns on the possibility of physical dialetheias, so to say that the Pinocchio paradox problematically commits semantic dialetheists to the possibility of physical dialetheias is question-begging. (It's also possible that you don't need the Pinocchio paradox itself; you could just ask whether Pinocchio's nose grows or not when he says 'this sentence is not true'.)
What about the grandfather paradox? I think that one's actually a lot harder. The grandfather paradox is the old chestnut where a time-traveller goes back and kills their grandfather. This means the time-traveller never gets born, so they never kill their grandfather, so their grandfather lives and they are born after all, and so on. This seems to fit the explanation model of what generates dialetheias perfectly. You're born because you weren't born; your grandfather lives because he doesn't live, and so on.
One resolution is to say that this isn't possible, and if time travel was possible then it would be possible, and so time travel isn't possible either. I don't find very satisfying, because it doesn't seem like a good enough reason for time travel to be impossible. I fear that if I tried articulating the feeling further I might say something foolish, but perhaps you feel the same way.
A second resolution is to say that time travel is possible, but killing your grandfather before the relevant parent is conceived is not possible. This response says that there are various ways the world might be, and some of them are like those really carefully written sci-fi stories where all the time travel fits together properly, but none of them involve contradictions.
A third resolution is to say that you can go back and kill your grandfather, and this will generate a PMPD that spreads out throughout the universe from the point at which you arrive in the past onwards. We've established that the laws of physics permit killing, and so if they also permit time travel then the laws of logic will just have to deal with it.
What should the dialetheist say here? I think it's a lot harder to deal with this than with the Pinocchio paradox, because the grandfather paradox only uses dispositions that - we're assuming for the sake of argument - we have independent reason to think are possible. Now, I can imagine that some people who think time travel is fine as long as it doesn't generate paradoxes will say that I'm confused. And it's true, I am a bit confused. But I don't think I'm confused about the thing I think they'll think I'm confused about. David Lewis  wrote what has kind of become the classic paper defending the "only non-paradoxical time travel" view, which I called the second resolution. That paper supplies the materials to accuse me of being confused about different contexts for evaluating possibility. When we say the time traveller can kill their grandfather we're holding fixed facts about their dispositions, and when we say they can't we're holding fixed historical facts about how they got there in the first place, and having those dispositions is compatible with having that history. What's incompatible with having that history is manifesting the disposition at that time.
I don't know. It's hard arguing with David Lewis, and I don't really expect to persuade anyone that he's wrong here. So instead I'll try shift the parameters of the debate a bit. They're already shifted somewhat by the fact we're viewing dialetheism as a live option, which Lewis never properly did4. But there's something else going on here. Lewis, in a way, thought that basic physical facts don't really have metaphysical explanations. For Lewis, the universes (he thought there were lots) are just patchworks of local facts. Some look law-governed the way ours seems to; most are a complete mess. But there's no deep metaphysical sense in which one billiard ball hitting another makes the second shoot off to the pocket. One thing happens, another thing happens, we notice patterns and we give explanations in terms of the patterns. But there's no metaphysical explanation there. The proposition that grass is green is true because grass is green. That's real explanation, not just pattern recognition. But causal explanation is just pattern recognition. If the explanation dialetheist adopts this metaphysically lightweight account of causal explanations, they don't have pressure on them to resolve the grandfather paradox the same way they resolve the liar paradox. On this view the world isn't a machine; it's a jigsaw5. There are only so many ways of fitting pieces together, and time-travellers preventing their own births is not one of those ways.
So that's one way for the explanation dialetheist to avoid PMPDs. They can say that basic physical propositions can't be explained by their not being true, on the grounds that they're not really explained at all. But what if you're not a Lewisian/Humean? What if the world is a machine? Well, I think that in that case you've got problems. Cards on the table, I find the machine view of the universe much more appealing than the jigsaw view. And since I also think dialetheism is an appealing solution to the liar paradox but I'm pretty wary of PMPDs, I guess I've got problems. It's an odd combination of views, though. You're probably fine.
 There's some evidence that Duchamp wasn't the artist behind Fountain, and that it was by one of his female friends. The leading contenders are apparently Louise Norton and Baroness Elsa von Freytag-Loringhoven. I'm not competent to assess the strength of the evidence, and I'll talk as if Duchamp was the artist because that's what Young and Priest do. But bear in mind that it may have actually been by someone else.
 60% is pretty impressionistic, and it isn't based on an explicit calculation. I do find dialethic solutions to the liar paradox the most satisfying, but I don't find them completely satisfying; for example they don't do too well with Curry's paradox. (I don't really think anybody does.) I also don't think the issues are well enough understood yet that anybody should be all that sure about the answer, although prior to knowing what our answers are one might expect him to be entitled to be more confident in his answer than I am in mine, because he's got more information than I have. One other worry I have is that I only prefer the dialethic solutions because I find Priest's writing easier going than Hartry Field's (Field  is probably the main one on this), and so I've read and understood more of it, but if you start worrying about that sort of thing you might end up never being 60% sure of anything.
 I'm not sure whether Beall thinks of the language itself as generating the dialetheias, or whether he thinks of the dialethic propositions as being out there already and adding the truth predicate just enables the language to express them. Or perhaps it's a third option I haven't considered. I don't think it matters a great deal for what we're talking about here, and in any case both options seem like possibilities you might want to consider, depending on whether you think of truth as primarily applying to sentences or to propositions.
 Lewis declined to write an article for a collection on the law of non-contradiction [Beall, Priest and Armour-Garb 2004], and two letters from him to the editors are included in the collection instead [Lewis 2004]. He basically made the same move GE Moore used to make about hairy metaphysical and sceptical theses: the law of non-contradiction is much more certain than anything you might try to base a refutation of it on.
 I owe the idea of jigsaw explanations to Alastair Wilson [2017: n.13], who mentioned the idea in the context of the many-worlds interpretation of quantum mechanics:
It is also to possible to interpret EQM so that Everett worlds are causally isolated from one another, regarding the dynamical connections between worlds as giving rise to non-causal explanations. (Perhaps phenomena in each Everett world are explained non-causally by phenomena in nearby worlds, in the same sort of way that the shape of a jigsaw piece is explained by the shapes of the surrounding pieces.)
I'm not sure whether what he has in mind is the same thing I've got in mind, but that's where I got the metaphor from anyway.
- Beall, JC 2009: Spandrels of Truth (Oxford University Press)
- Beall, JC. 2011: 'Dialetheists against Pinocchio', Analysis 71(4): 689-91
- Eldridge-Smith, P. and Eldridge Smith, V. 2010: 'The Pinocchio paradox', Analysis 70(2): 212-215
- Field, H. H. 2008: Saving Truth From Paradox (Oxford University Press)
- Lewis, D. 1976: 'The Paradoxes of Time Travel', American Philosophical Quarterly 13(2): 145-152
- Lewis, D. 2004: 'Letters to Priest and Beall', pp176-7 in Priest, Beall and Armour-Garb 
- Priest, G., 2006: In Contradiction: A Study of the Transconsistent, 2nd ed., (Oxford University Press)
- Priest, G., Beall, JC and Armour-Garb, B. 2004: The Law of Non-Contradiction: New Philosophical Eassays (Oxford University Press)
- Wilson, A. 2017: 'The Quantum Doomsday Argument', British Journal for the Philosophy of Science 68(2): 597-615
- Young, D. and Priest, G. 'It is and it isn't', Aeon, 22/9/16, https://aeon.co/essays/how-can-duchamp-s-fountain-be-both-art-and-not-art, accessed 31/5/18