 as strongly connexive, whereas if the conditions Unsat1 and Unsat2 are not both satisfied, the system is only called weakly connexive. Kapsner motivates the extra conditions by two intuitions, namely that it is not the case that a formula  A should imply or be implied by its own negation, and that if A implies B, then A does not imply not-B (and if A implies not-B, then A does not imply B). These intuitions may, however, also be seen to motivate ⊬ ~A → A and ⊬ A → ~ A, respectively (A →B) ⊬ (A → ~B) and (A → ~B) ⊬ (A → B) instead of Unsat1 and Unsat2. Moreover, imposing Unsat1 and Unsat2 precludes systems that satisfy the variable sharing property (i.e., broadly relevant or sociative logics in Routley’s (1989) terminology, for which it holds that if A → B is a theorem, then A and B share at least one propositional variable) and satisfy the deduction theorem from being connexive. So far only few strongly connexive logics satisfying the non-symmetry of implication condition are known, namely the system CC1, which, however, “is an awkward system in many ways” (McCall 2012, p. 429), see section 4.1, and the Boolean connexive relatedness logics from Jarmużek and Malinowski 2019a, see section 4.4. In Wansing and Unterhuber 2019, logics that satisfy AT and AT′ but validate Boethius’ theses only in rule form ((A → B) ⊢ ~(A → ~B) and (A → ~B) ⊢ ~(A → B)) are called weakly connexive. Weiss (2019) considers a language with classical negation, ¬, classical implication, ⊃, and another binary conditional, →, (notation adjusted). He calls a logic half-connexive if it validates the Weak Boethius’ Theses: