Accepting that Reality is Paradoxical, Research Time, 27/01/2014

Graham Priest is not only a logician and
professor, but an advanced martial artist.
Graham Priest is a logician. This is one important fact about him that you should know. Out of all the achievements of philosophers in the last few decades, I think I have the most respect for what Graham Priest managed in the field of logic. This regards the second fact everyone should know about Graham Priest. He’s probably done more than any other single philosopher to pioneer paraconsistent and dialetheic logical systems, systems of logic that can accept and understand inconsistencies and contradictions. Priest often says it himself: there are true contradictions.

Intuitively, I don’t think that makes sense to a lot of people. It just seems obvious without even having to think about it that one part of reality can’t contradict another part. ‘A and not-A’ is impossible. Today, someone working in logic can use a dialetheic system, a logical calculus in which true contradictions make sense, and if someone accuses them of peddling nonsense, they can simply refer to the precedent of Priest. I’ve done this in conversations with some very narrow-minded philosophers who accused me of being stupid when I just wanted to talk about dialetheic logic. 

This is why I’m so impressed with the success of his career. When Graham Priest was just a young researcher working on his first projects and some older, more established logician scoffed at his work as nonsensical, he couldn’t refer to Graham Priest to back himself up. He hadn’t even yet become The Graham Priest that everyone refers to when they want to get stodgy old traditional logicians off their backs. All he had was his own ingenuity.

The argument he gives in Beyond the Limits of Thought for why you can have true contradictions goes like this. Consider two possible states of affairs, that A is true, and that A is false.

He writes, “Given two states of affairs, there are, in general, four possibilities: one but not the other holds, vice versa, both, or neither.”
  1. “A is true” is true, and “A is false” is false. One but not the other.
  2. “A is true” is false, and “A is false” is true. The other but not the first.
  3. “A is true” is false, and “A is false” is false. Neither.
  4. “A is true” is true, and “A is false” is true. Both.
Most commonly, we only believe that (1) and (2) are sensible statements. But the common belief is only one way among many that you can define a negation operator.

The third part of Priest’s Beyond the Limits of Thought engages with the work of Georg Cantor, Frank Ramsay, and Bertrand Russell to remove the paradoxes of self-reference from set theory. These paradoxes all have the basic logical form: a chain of reasoning involving a statement that refers to itself or a set that includes itself will lead to inevitable contradiction. Since there’s nothing in set theory’s basic rules that prevent such self-referring sets from existing, the logical operations of this kind of mathematics are always open to contradiction.

My former colleague in McMaster’s doctoral program, J, returned this semester from her post-doc to take the department’s rotating post as the visiting Bertrand Russell scholar. She gave a talk at the department’s visiting speakers’ series where she discussed some of Russell’s historical work on Leibniz. One of the priorities Russell explicitly had was examining the disparate, uncollected writings of Leibniz’s corpus looking for the principles and interpretations that would make them all internally consistent. He wrote that consistency is the best guide to truth.

A central premise of Russell’s thought (along with Ramsey’s, Cantor’s, Gottlob Frege’s, and just about everyone in the Western philosophical tradition up to Graham Priest) was that a contradiction could not exist, because the truth must be internally consistent. The entire revolutionary period in the creation of contemporary logic from the 1890s to 1910s was defined by the drive to remove the contradictions from self-referring propositions, or at least modify logic and set theory so those self-referring propositions can’t exist. Logic, set theory, and mathematics must be made consistent.

But Priest’s point (and Kurt Gödel’s, if you go back in the history) is that self-referring propositions are inevitable, and can’t be gotten rid of. You need a dialetheic understanding of truth because the ability to form a proposition that refers to itself will always result in contradictions. Any discussion of a totality (such as the set of all sets) refers to itself, because quantifying over all propositions will include the proposition that quantifies over all propositions; otherwise, you’ll fail to quantify over all propositions. When a set includes itself as a member, it simultaneously belongs and does not belong to itself. To a logical perspective that considers contradictions to be nonsense, this would make set theory (and therefore pretty much all the logic developed in the Frege-Russell-Ramsey revolutionary period) nonsense.

Only a dialetheic perspective, that is the perspective Priest developed and endorses, can understand self-referring sets, propositions, statements, and sentences. Only a dialetheic perspective is not offended by the conclusion that the set of all sets both belongs and does not belong to itself. The contradictions of self-reference that arise from set theory do not render set theory nonsensical, but dialetheic. The true heir of the revolution in logic isn’t whoever manages to make the contradictions of self-reference disappear (since no one can do that anyway), but whoever manages to make sense of the contradictions of self-reference. To the best of my knowledge, that person is Graham Priest.

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