This paper presents the first possible world semantics for concessive conditionals (i.e., even if A, C conditionals) constructed in a compositional way. First, the meaning of if is formalized through a semantics that builds on the proposal given by Stalnaker [1968]. A major difference from Stalnaker’s approach is that irrelevant conditionals (i.e., conditionals where the antecedent and the consequent have no connection) are false in this new setting. Second, the meaning of even is analyzed through a formal semantics based on the notion of scale. This analysis overcomes the problems arising in standard approaches, in which even is analyzed with the help of pragmatic presuppositions. Finally, the two particles are combined in order to provide a formal analysis of even if. This theory predicts the major phenomena concerning the behavior of concessive conditionals and without any call to pragmatic explanations. More generally, this approach creates the possibility of a compositional analysis of other conditionals such as if then or only if forms.