The last 50 years of research has taught us that conditionals are non-monotonic in the antecedent. That is, they invalidate Antecedent Strengthening. Many accounts have been developed for such conditionals, starting with Stalnaker and Lewis. These accounts converge roughly to Burgess’ conditional logic \({{\,\mathrm{\textsf{B}}\,}}\) or the non-monotonic reasoning system \(\textbf{P}\) . The latter two have Cautious Monotonicity as a weak replacement for Antecedent Strengthening. Lewis’ weakest conditional logic \({{\,\mathrm{\textsf{V}}\,}}\) or its non-monotonic reasoning counterpart system \(\textbf{R}\) are obtained by adding a stronger proxi for Antecedent Strengthening – the law of Rational Monotonicity. I argue that Rational Monotonicity is too much while Cautious Monotonicity is not enough monotonicity. I investigate two other monotonicity postulates, which are jointly weaker than Rational Monotonicity. One of them is related to Disjunctive Rationality, the other one is related to the idea of replacing the preservation law encoded in Rational Monotonicity by a pivoting principle – instead of conserving a conclusion when adding a compatible supposition, the principle recommends pivoting from a disjunction to another disjunction when the reason for the first is undermined by such an addition. This gives rise to three logics in between \({{\,\mathrm{\textsf{B}}\,}}\) and \({{\,\mathrm{\textsf{V}}\,}}\) and three other logics in between \({{\,\mathrm{\textsf{BN}}\,}}\) and \({{\,\mathrm{\textsf{VN}}\,}}\) obtained by strengthening \({{\,\mathrm{\textsf{B}}\,}}\) and \({{\,\mathrm{\textsf{V}}\,}}\) by a consistency axiom. I prove soundness, completeness and decidability results for these six new logics, using set-selection semantics, its correspondence theory, and filtrations. In the second follow-up paper, I investigate five other semantics for the new logics: well founded strict order semantics, a new closeness semantics, several weakenings of ranking semantics, and new versions of Lewis’ and Burgess’ similarity semantics.
