Quantifiers in connexive logic (in general and in particular)

Abstract Connexive logic has room for two pairs of universal and particular quantifiers: one pair, $\forall $ and $\exists $, are standard quantifiers; the other pair, $\mathbb{A}$ and $\mathbb{E}$, are unorthodox, but we argue, are well-motivated in the context of connexive logic. Both non-standard quantifiers have been introduced previously, but in the context of connexive logic they have a natural semantic and proof-theoretic place, and plausible natural language readings. The results are logics that are negation inconsistent but non-trivial.