The Easy Case for Dialetheism: Timmy the Square Circle and Divaltopian Law

Dialetheism is the view that some contradictions are true. Put another way, dialetheists claim that there are propositions that are both true and false at the same time and in the same respect.

For many people, this is plain crazy. Others find it extremely counterintuitive but will grant it because they’ve heard quantum mechanics proves it. Others still may suspect it is a desperate response to certain logical paradoxes, such as the Liar.

I wish to argue that all of this is quite beside the point. I don’t understand quantum mechanics (at all), but I would be surprised if there were really no way to account for experimental data without recourse to true contradictions. I’m (somewhat) better versed in debates about logic. I can tell you with confidence: the paradoxes have plenty of coherent solutions. Philosophers disagree primarily on the relative costs and benefits of these solutions. If dialetheism were truly incoherent and demonstrably impossible, we wouldn’t be backed into it: cheaper options than insanity are for sale.

There is a much simpler reason to be a dialetheist: despite initial appearances, it is intuitively compelling and even quite obviously true. We need no special training in physics or logic to see this.

Before getting on with the argument, a quick clarification about a misinterpretation of dialetheism that I encounter alarmingly often: dialetheism is the view that there is at least one true contradiction. It is not the view that all contradictions are true. That view is actually nuts. For example, that my name is William Nava is only true, it is not also false. Similarly, dialetheism is true and not also false. With that out of the way…

Divaltopian law

The following is adapted from an argument I’ve heard Graham Priest make once or twice.

One night in Divaltopia, Veronique sadly passes away. She leaves behind no will. Though grief-stricken, her sister Mila wonders who will inherit Veronique’s mansion and priceless art collection. Mila decides to check what Divaltopian law has to say about this scenario. Thorough as always, Mila checks the republic’s constitution first. After detailing the structure of government and legislative procedure, the constitution ends with the following First Law:

All subsequent Divaltopian law, provided it pass through the heretofore detailed procedures, shall be interpreted literally and as allowing for no exceptions unless they be explicitly stated in the law itself. Furthermore, the prescription of a state of affairs by any article of Divaltopian law shall be sufficient for that state of affairs to be mandated by Divaltopian law.

Having finished the constitution, Mila reads the Divaltopian legal code. When she finally gets to the section about hereditary law, she’s distressed to read Law A:

Unless otherwise stated in a will, all property of the deceased shall immediately become property of the deceased’s eldest offspring, unless the deceased have no living children.

Veronique did have one child: Marcus. Marcus is a criminal and troublemaker, most recently imprisoned for having burned down some of Veronique’s art. Mila can’t believe Veronique’s property must go to this ungrateful arsonist. Frustrated but still committed to thoroughness, Mila reads on. She stops upon reading Law B:

Under no circumstances is a convicted felon allowed to own property.

Mila is at first elated. Then confused. She decides to call a lawyer.

Claim 1: According to Divaltopian law, Veronique’s property now belongs to Marcus.

Is Claim 1 true? Well, of course it is. Law A clearly states it. But then again, Law B just as clearly states that Marcus can’t own property – this is inconsistent with Claim 1. Claim 1 must therefore be false.

Ways out?

There’s no way to wiggle out here. The First Law was probably unnecessary for my case, but I included it so there be no doubt. The existence of Law A satisfies sufficient conditions for Claim 1 to be true and the existence of Law B satisfies sufficient conditions for it to be false. We’re therefore justified in saying that Claim 1 is both true and false at the same time and in the same respect.

Objections to the details of the scenario are, of course, irrelevant. It’s clearly possible. Yeah, these laws weren’t written with the most foresight in the world. But surely, more idiotic laws have been passed by real states. What’s more, it’s likely that contradictions of this sort actually abound in real law – they’re just more subtle and subject to interpretive wiggling.

It also irrelevant how Divaltopians actually deal with this situation. They may pass new laws to rectify the problem. Mila’s incredibly talented lawyer may find a way to convince a judge to privilege one of the laws over the other. Claim 1 isn’t about what will happen or how the situation will be interpreted. It is about what is the case according to the law. (If you want to be really sticky about it, we can add a “…at the time Mila read the laws” to Claim 1 to account for changes made to the law after the fact.)

Philosophical issues about what really counts as the law are also quite beside the point. Maybe you subscribe to “natural law” theory and you think the law isn’t identical to what’s in law books. Specifically, you think an absurd and poorly written piece of legal code isn’t really “law”. Fine. Change “According to Divaltopian law” in Claim 1 to “According to the Divaltopian legal code”.

As a last ditch attempt, you may want to say that law doesn’t exist so Claim 1 doesn’t have a truth value. After all, the law is a creation of the human mind – there is no physical thing that is identical with it. While it’s true that law isn’t physical, this move involves a highly unusual use of the word “exist”. It implies that marriages don’t exist either, nor do obligations. Maybe you can swallow such a view of “exist”. But can you really take it so far as to claim that “According to the law, Paul and Patricia Churchland are married” has no truth value?

You may be unconvinced. Maybe this was too easy. Can I really just make something up out of thin air to show that some contradictions are true?

“Timmy the Square Circle”

I once wrote a short story that I’m very proud of. It’s titled “Timmy the Square Circle” (TSC). Here’s the full text:

There once was a boy named Timmy. Timmy was literally a square circle. The end.

Claim 2: Timmy, from William Nava’s short story “Timmy the Square Circle”, has four corners.

Since Timmy is square, he has four corners, so Claim 2 is true. Since Timmy is also a circle (and a boy), it follows that he has no corners. So Claim 2 is false.

TSC is a work of fiction. In a much more compelling way than with the law, Timmy doesn’t exist. Nonetheless, we can make truth claims about fictional characters. “Harry Potter is a woman” is false. Philosophers of language disagree about how to account for this. I’ve never really understood the difficulty: the truth of a proposition about a fictional situation is true relative to the relevant fictional world. There is an implicit “within the world of the story” tagged on to every claim about fictional characters. Within the world of the Harry Potter series, it’s false that Harry Potter is a woman. Within the world of TSC, it is both true and false that Timmy has four corners.

There’s an important difference between the Harry Potter series and TSC: the world of Harry Potter, while physically impossible, is logically possible. The world of TSC is not. Perhaps this accounts for why claims about characters in TSC may have no truth value while claims about Harry Potter characters do?

It’s hard to see why this must be so, other than to avoid contradictions. But even if we could motivate it, the move has higher costs than it may seem. Suppose a perfectly realistic novel places Tony at the Mayflower Pub in London during the first inauguration of George W. Bush. This happens on page 19 and is recounted by a perfectly reliable narrator. Later, on page 637, the same reliable narrator mentions that due to an evacuation, London was empty of people during the third week of 2001. In the world of this novel, it is both true and false that Tony was at the Mayflower Pub on January 20, 2001. Does it follow that all propositions about this story have no truth value? Of course not.

There are possible ways out here. We can doubt the narrator. But if the narrator has shown no other signs of being unreliable, this is disingenuous. Clearly, the novelist made an honest error. It is intentionally misreading the novel to interpret from this mistake some deep point about the narrator. It is similarly disingenuous to introduce science fiction explanations such as alternate dimensions. Finally, it’s plainly false to say that the novel is “incoherent”. The novel’s events are all perfectly understandable and clear; they simply include a contradiction.

At best, we can infer that the novelist should have been a bit more careful. But, much as with the Divaltopian case, this sort of thing actually happens all the time. It’s called a plot hole – most stories, even great ones, have at least one.

(For an example of an intentionally inconsistent story which is more, uh, substantial than TSC, check out Graham Priest’s “Sylvan’s Box: a Short Story and Ten Morals“.)

Logical worries

There is much more that could be said here. I’m trying to avoid a rabbit hole of technicalities because, while they’re available, I don’t think they’re ultimately relevant. As an obvious example, I haven’t made any distinction between positively asserting the negation of a claim (“The property doesn’t belong to Marcus”) and merely denying a claim (“It is not the case that the property belongs to Marcus”). Though intuitively, the two imply the same states of affairs, their meaning isn’t identical. If we reject the law of excluded middle, for example, a negation and a denial of the same claim wouldn’t always have to have the same truth value.

While interesting and possibly fruitful in its own right, this rather technical distinction just doesn’t seem like it changes anything. When there’s a problem, a technical diagnosis may be necessary. But here, there’s actually no problem. It makes intuitive sense that the law can contradict itself.

It’s possible, however, for logical analysis to demonstrate that unexpected consequences follow from a claim. For example, the logical rule of explosion (“ex falso quodlibet“) supposedly shows that from a contradiction everything can be proved to be true. So, the idea goes, a little logical analysis will show that my intuitive acceptance of some contradictions implies the absurdity that every proposition is true. I’d have to grant the truth of, for example, “Every weekday except Tuesdays, William Nava rides a tiny purple giraffe to work”.

This specific objection is of no real concern. Explosion isn’t handed down from above. It depends on the disjunctive syllogism, a type of argument that only makes any intuitive sense to consider valid if we presuppose that contradictions can’t be true. Paraconsistent logicians have laid out perfectly coherent logics without explosion and which allow for true contradictions. This isn’t the place to argue for paraconsistent logic, nor am I qualified to do so; what I can say is that there is surprisingly little cost to adopting such a logic. It is possible, for example, to adopt a paraconsistent logic and still validate classical inferences in those restricted contexts where we know contradictions won’t prop up.

All this is simply to say: if we accept dialetheism, we have recourse to logics that are coherent, intuitive, and don’t involve great expressive or inferential costs. While this isn’t an argument for dialetheism, it shows that the objection to it from formal logic is much weaker than it initially appears.

Truth, meaning, and imagination

Let’s take stock of where we are. Claims 1 and 2 are, intuitively, both true and false. This seems positively obvious. But how is it, exactly, that contradictions can be true? Why the strong intuition that they can’t be? And what does all this have to say about propositions that aren’t about things that are totally made up?

Consider this general rule: statement X is true if and only if there is some state of affairs Y that, by virtue of the meaning of X, satisfies sufficient conditions for X to be true. Put another way: any statement, if it is meaningful, comes with implicit conditions for its truth. If those conditions are satisfied by the facts of how things actually are, then the statement is true. This is a trivial account of what it means for a statement to be true.

All that’s happening in the examples above is that a statement’s truth conditions are met and its negation’s truth conditions are met. It is natural to say in such a case that the statement is both true and false.

While at first odd, this really shouldn’t be surprising. The facts that supply the truth conditions for Claims 1 and 2 are facts about things that are made up. Human beings are capable of making up anything that can be described. It should, therefore, come as no surprise that these kinds of true contradictions are possible. If we have a strong intuition to the contrary, it’s because we’re importing this intuition from our experience of physical reality. Physical reality doesn’t seem to allow for true contradictions. But conceptual inventions are not similarly constrained.

I should defend that last point since it is the source of another common objection. Close your eyes. Try to visualize a square circle. Can you do it? You probably can’t. I don’t know of anyone who can. I’ve heard this used to argue that human beings can’t make up inconsistent claims. But this visual imagining test is arbitrary and inadequate. Statements aren’t mental pictures. “All happy families are alike; each unhappy family is unhappy in its own way” is impossible to visually picture. Though we can maybe mentally picture something sort of related (an image of two families, one happy, one unhappy), we can’t conjure any one image that is identical with the sentence’s claim. Even so, it is a coherent statement invented by a human. It follows that mental visualizability is not required for being a coherent man-made statement.

Are man-made statements constrained at all? Sure. In order to make a meaningful statement, you need to use the known grammar and vocabulary of a language to predicate some meaningful property of some real object. “ldjs” is not a statement. “Dog” is not a statement because there is no predicate; it is therefore not the sort of thing that can be true and/or false. “The mouse was frumious”, though grammatical, is not a statement because “was frumious” is not a meaningful predicate. “The present king of France is bald” is a controversial case. Since there is no present king of France, the sentence arguably doesn’t have a real object, though grammatically it does.

“Timmy was literally a square circle” is not a controversial case. “Timmy” is perfectly real in the relevant fictional world (like Harry Potter is). And “was literally a square circle” is a meaningful predicate. It means to have literally had the property of being square and the property of being a circle. We know “being square” is meaningful and “being a circle” is meaningful. We know what it means for an object to have two properties at once. All semantic and syntactic components of the statement are accounted for. Conditions for being meaningful are thereby met. No picturing necessary; no possibility verification required. The class of the meaningful is larger than the class of the possible.

It is perhaps now tempting to draw a sharp line: the world of man-made ideas allows for true contradictions, reality doesn’t. However, this line is not so sharp. What the arguments above have to say about the possibility of true contradictions outside the realm of the man-made is complicated and subtle. I discuss it in a follow-up post. While interesting, though, this matter is irrelevant to whether dialetheism is true. Dialetheism is merely the claim that at least one contradiction, of any sort, is true. I have supplied two.

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