Abstract
Abstract Connexive logic has room for two pairs of universal and particular quantifiers: one pair, ∀ and ∃, are standard quantifiers; the other pair, 𝔸 and 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.