Extensionality and Restriction in Naive Set Theory

The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Resta [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads to triviality.