The principles of Conditional Excluded Middle (CEM) and Simplification of Disjunctive Antecedents (SDA) have received substantial attention in isolation. Both principles are plausible generalizations about natural language conditionals. There is however little discussion of their interaction. This paper aims to remedy this gap and explore the significance of having both principles constrain the logic of the conditional. Our negative finding is that, together with elementary logical assumptions, CEM and SDA yield a variety of implausible consequences. Despite these incompatibility results, we open up a narrow space to satisfy both. We show that, by simultaneously appealing to the alternative-introducing analysis of disjunction and to the theory of homogeneity presuppositions, we can satisfy both. Furthermore, the theory that validates both principles resembles a recent semantics that is defended by Santorio on independent grounds. The cost of this approach is that it must give up the transitivity of entailment: we suggest that this is a feature, not a bug, and connect it with recent developments of intransitive notions of entailment.