Extensionality and Restriction in Naive Set Theory

Abstract

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 Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads to triviality.

Free of Detachment: Logic, Rationality, and Gluts

“…The dominant reply by glut theorists over the last four decades is to concede (1), for reasons briefly reviewed in § below, but, towards answering (2) and (3), proceed on a quest for (non‐material) detachable conditionals, generally following the lead of Anderson and Belnap's work and that of others in the ‘relevance' tradition . But as Dunn, Meyer and Routley noted early on , Curry's paradox riddles the quest for detachable conditionals with severe problems; and overcoming the problems makes for very complicated, philosophically awkward semantics , and indeed often engenders the need to find yet other detachable conditionals to serve other pressing needs (e.g., restricted quantification) .…”

Symmetric operators on modal pseudocomplemented De Morgan algebras

“…via Russells paradox. Arguing that contradictions are not necessarily devastating in a paraconsistent setting, these authors reinstate the axiom scheme of Comprehension as acceptable, allow the formation of the Russell set R, and conclude that both R ∈ R and R ∈ R are true, see [8,23,31,32,33,34]. Now, on PST of the second group the following papers can be mentioned [9,10].…”

Strong, universal and provably non-trivial set theory by means of adaptive logic

“…Most of the proposed set theories use a relevant logic that does not validate the rule of contraction to avoid Curry's paradox (cf. [5], [6], [19] and [20]). One can also use other logics without contraction (cf.…”