Extensionality and restriction in naive set theory (Q965901)

The author discusses the possibility of having a nontrivial (but possibly inconsistent) naive set theory (i.e. one with comprehension and extensionality) within a paraconsistent relevant logic. He shows that one such set theory, proposed by \textit{J. Beall, R. T. Brady, A. P. Hazen, G. Priest} and \textit{G. Restall} [J. Philos. Log. 35, No. 6, 587--598 (2006; Zbl 1111.03020)], which uses a restricted arrow given by \(B\mapsto C := B \wedge t \rightarrow C\), for some theorem \(t\), and a subset relation \(a\subseteq b := \forall x( x \in a \mapsto x \in b)\) with the rule \(a\subseteq b \wedge b\subseteq a \vdash A(a) \mapsto A(b)\), is trivial.