I’ve been fascinated by dialetheism ever since I first heard about it in 2005, from RM Sainsbury’s wonderful book Paradoxes. The pleasure I derive from the notion that contradictions can be true is, I admit, the same as the dizziness obtainable from the best sort of science fiction; but I don’t think that constitutes an argument against the idea.
If you’re going to countenance true contradictions, but you are not yet ready to embrace the view of truth espoused by Malaclypse the Younger in Principia Discordia, then you need a system of logic in which a single contradiction does not necessarily entail everything whatsoever. Such systems are studied under the banner of Paraconsistent Logic.
One of the early hopes of this line of enquiry was that it would be possible to rehabilitate naïve set theory by embedding it into a paraconsistent setting. For example, let be the Russell set
From we can conclude that , and vice versa, but in a paraconsistent setting that need not be a problem. All we’ve done is to locate a true contradiction, an occasion for excitement rather than despair!
I’m obviously not a specialist in the area, but from what I’ve read as a curious outsider I have the impression that the wheels have rather come off this initially-plausible endeavour. Not all the paradoxes of naïve set theory are so comfortably dispatched; in particular, paraconsistency is no defence against Curry’s paradox. Some have proposed to modify the logic of implication to forbid that reasoning, though doesn’t that smack a little of desperation?
Anyway, the always-interesting Greg Restall has just lobbed rather a fun grenade into the ring. He has concocted a version of the paradox that doesn’t even use implication, though it does require an “initial” proposition from which everything follows.
(Incidentally, I have no idea whether or why anything is gained by using the word “property” instead of “set”, and tweaking the notation to use pointy brackets and curly epsilons rather than the other way around.)
The funny thing from my point of view is that I’m more suspicious of logical “units” than of implication. (They cause no end of trouble in linear logic for little or no gain, which is what motivated the topic of my thesis.)
Because I happen to be suspicious of , I’m therefore mildly suspicious of arguments that depend on it — though I take Restall’s point that would do just as well, it increases the demands made on the ambient logic.
Anyway, I thought it would be interesting to see how one could modify the argument to remove the dependency on , without making other new demands on the ambient logic. (The lovely thing about Restall’s proof is that it depends hardly at all on the logic: as far as I can see it assumes only equality, contraction and .) My variation is here – it assumes a (primitive or definable) subset predicate that satisfies the obvious introduction and elimination rules, but of the logic demands nothing more than contraction.
This may seem naive to any philosopher who should happen across it. Perhaps there are already well-known objections to such an argument. In the context of this particular philosophical discourse it is presumably weaker than Greg Restall’s version — it invites the objection that the subset predicate is disguised implication, for example — but personally it’s enough to convince me that tinkering with the logic is not going to rescue naïve set theory without unacceptable collateral damage. And that’s a shame, because it was a cool idea.