## Bad news for naïve sets

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 $R$ be the Russell set

$\{x | x \notin x\}$

From $R\in R$ we can conclude that $R\notin R$, 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 $\bot$, I’m therefore mildly suspicious of arguments that depend on it — I take Restall’s point that $(\forall x)(\forall y)x\in y$ would do just as well, though 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 $\bot$, 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 $\bot$.) My variation is here – it assumes a (primitive or derived) subset predicate that satisfies the obvious introduction and elimination rules, but of the logic demands nothing more than contraction.
This may well seem very 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 argument — 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 na\”ive set theory without unacceptable collateral damage. And that’s a shame, because it was a cool idea.

Because I happen to be suspicious of $\bot$, I’m therefore mildly suspicious of arguments that depend on it — though I take Restall’s point that $(\forall x)(\forall y)x\in y$ 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 $\bot$, 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 $\bot$.) 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.

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### 7 Responses to Bad news for naïve sets

1. Greg Restall says:

Sweet. I like that a lot. My former self would dislike the contraction. But now I have a story to tell about that (involving assertion, denial and the way proofs constrain them).

I absolutely love your version, as it works for properties too: just read $\subseteq$ as "is no weaker than", (being red is no weaker than being coloured; etc. being a renate is no weaker than being a cordate) and there's no commitment to any extensionality at all in what you've done. You're not saying that if X is no weaker than Y and Y is no weaker than X then X and Y are the same properties. So this is very very nice.

• I’m glad it’s of some interest. I wasn’t really expecting that!

• Greg Restall says:

There’s bits in “Multiple Conclusions“. In short:

I like Gentzen’s two-sided sequent calculus for classical logic, and I defend the interpretation of it for which XY says that a position in which you assert each member of X and deny each member of Y is ruled out by the Meaning of The Act. And asserting A twice has the same effect as asserting it once. The same goes for denying it.

Of course, I’m also a fan of Logics Without Contraction (I have been for years), but as a pluralist, I’m allowed to go both ways here. It’s just that this inferentialist reading of classical logic seems to me to get something important right.

2. Greg Restall says:

Oh, on the point of properties and not classes: it’s that some folks like extensionality for classes/sets. That’s all. And my main target was Field, and he endorses a naive property theory and not a naive set/class theory. That’s all.

On units. Yeah, they’re a pain. But here, if we’ve got quantifiers, we’ve got them, or at least enough to cause the pain.

• Oh, I see. Yes, I suppose the word “set” has extensional connotations: thanks for the explanation.

I’m impressed you got here so quickly! I assumed – wrongly as it turns out – that it was the middle of the night in Melbourne.

3. Greg Restall says:

Ah, my first comment wasn’t posted… Let me reconstruct it.

I love this version of the paradox. The proof is shorter than mine (hey, subset is simpler than identity) and you’re not assuming extensionality anywhere here, so it works on people who don’t like naïve sets but do like naive properties. Very nicely done.

• Sorry, your first comment got caught in the spam filter. I have now fished it out.