  As for weakly contraction-free responses, the failure of CP has sometimes been motivated using “worlds” semantics of the sort that involve a distinction between logically possible and impossible worlds (e.g., Beall 2009; Nolan 2016). To refute CP we need the truth of \(\alpha \vdash_\mathcal{T} \beta\) and the falsity of \(\vdash_\mathcal{T} \alpha{\rightarrow}\beta\). On the target “worlds” approaches \(\vdash_\mathcal{T}\) is defined as truth preservation over a proper subset of worlds (in a model), namely, the “possible worlds” of the model. Hence, for \(\alpha \vdash_\mathcal{T} \beta\) to be true is for there to be no possible world (in any model) at which \(\alpha\) is true and \(\beta\) untrue. In turn, to refute \(\vdash_\mathcal{T}\alpha{\rightarrow}\beta\) we need a possible world at which \(\alpha{\rightarrow}\beta\) is untrue. How does that happen? Because connectives are defined in a way that takes account of all (types of) worlds in the model (possible and, if there be any, impossible) there’s an option for \(\alpha{\rightarrow}\beta\) to be untrue at a possible world in virtue of \(\alpha\) being true and \(\beta\) being untrue at an impossible world. And that’s just what happens on the target approaches. (Exactly how one defines the truth-at-a-world and falsity-at-a-world conditions for the arrow depends on the exact “worlds” approach at issue.)