## Luca Incurvati’s Conceptions of Set, 9

Naive set theory entails contradictions. Really bad news. Or so most of us think. But what if we are prepared to be more tolerant of contradictions, e.g. by adopting a dialethic and paraconsistent logic, which allows there to be contradictions which are true (as well as false) and where contradictions don’t entail everything? Could we rescue the naive conception of set, accommodate e.g. the idea of a Russell set, by departing from classical logic in this way? A desperate measure, most of us will think. Even if willing, once upon a time, to pause to be amused by varieties of dialethic logic, at this late stage in the game, I don’t have much patience left for the idea of going naive about sets by going far-too-clever-by-half about logic. But Luca is evidently a lot more patient that I am! He devotes Chapter 4 of his book to investigating various suggestions about how to save naive set theory by revising our logic. How does the story go?
Luca very helpfully divides his discussion into three main parts, corresponding to three dialethic strategies.
The first he labels the The Material Strategy — we adopt a non-classical logic which keeps the material conditional, so that is still simply equivalent to , while we reject the classically valid disjunctive syllogism, and hence material modus ponens.
Graham Priest initially thought that a ‘simple and natural choice’ here is his LP, the Logic of Paradox. But LP doesn’t validate the transitivity of the material conditional, and this hobbles the proof of various elementary theorems of set theory — even the usual proof of Cantor’s Theorem fails. And on the usual definition of set identity in terms of coextensionality, Leibniz’s Law fails too.
Can we tinker with LP to avoid these troubles? Luca mentions a few options: all of them have equally unattractive features — giving us a set theory that is too weak to be useful. So, Luca’s verdict seems right: the prospects of saving naive set theory as a formal. . .

News source: Logic Matters