Broad focus (or informational integration or nonautonomy) is lexically and contextually constrained, but these constraints are not well understood. On a standard theory of focus interpretation, the presupposition of a broad focus is verified whenever those of two narrow foci are. I argue that to account for cases where two narrow foci are preferred, it is necessary to assume that broad focus competes with two narrow foci and implicates the opposite of what they presuppose. Central constraints on thetic statements are (...) thus accounted for in an Optimality Theory (OT) enriched Alternative Semantics. (shrink)