Chris Rodriguez

We show that Wiredu’s result in [26] is not the doom for connexive set theories, not even for those based in logics similar to CC1, one of the original target logics. For this purpose, we present the necessary assumptions for Wiredu’s proof, making some precisions on the connexive requirements. Then we present a non-reflexive variant of CC1 in which Wiredu’s proof can be blocked. Finally, we discuss the prospects of a connexive set theory based on both the non-reflexive and non-transitive variants of CC1, which, even assuming non-triviality, are not very rosy.

According to common wisdom, the connective tonk defined by Prior trivializes any theory that contains it. However, it should not be forgotten that whether an argument holds or not depends to a large extent on the underlying notion of logical consequence. Logical consequence is usually assumed to be Tarskian, that is, reflexive, transitive and monotonic. However, Belnap had already conjectured that tonk might not be so problematic in a non-transitive logic, which Cook finally proved in 2005. In this paper we improve on Cook’s result in two ways: our hypothesis is simpler (namely, we use fewer interpretations than he did and we do not rely on a disjunctive consequence relation) and it is not ad hoc (namely, our working consequence relation is not designed merely to avoid triviality under tonk).