 It is unclear, however, that the conditional of Beall, et al. (2006) is the right connective to use, at least with regard to some weaker relevance logics. One virtue of some weaker systems is that they can be used to formalise naive set theory (see  section 6).  Zach Weber (2010) has formalised naive set theory, using this conditional to define the subset relation. The resulting system, unfortunately, is trivial in that every formula is provable in it.