  What makes this instance of the Curry-Paradox Lemma particularly troublesome is that it poses an obstacle to one common response to Curry’s paradox, namely the weakly  contraction-free response discussed in  section 4.2.1.  That response depended on  rejecting the rule CP of single-premise conditional proof, one direction of the single-premise “deduction theorem”. But this is a rule that has seemed difficult to resist for a consequence connective (Shapiro 2011; Weber 2014; Zardini 2013). If \(\beta\) is a consequence of \(\alpha\) according to the consequence relation of theory \(\mathcal{T}\), where this theory has \({\Rightarrow}\) as its own consequence connective, then \(\mathcal{T}\) must surely contain the consequence claim \(\alpha {\Rightarrow}\beta\). Likewise, this variety of Curry paradox poses an obstacle for detachment-free responses, which require rejecting the rule MP. If a theory with its own consequence connective contains both \(\alpha\) and the consequence conditional \(\alpha {\Rightarrow}\beta\), then it must surely contain \(\beta\) as well. Or so, at least, it has seemed. Admittedly, the proponent of a weakly detachment-free response will argue that MP for  \({\Rightarrow}\) illicitly builds in transitivity  (see section 4.2.2).  Still, what seems  inescapable is the converse of CP, the rule CCP that is the other direction of the single-premise deduction theorem. If a theory contains the consequence conditional \(\alpha {\Rightarrow}\beta\), then surely \(\beta\) follows from \(\alpha\) according to the theory. That would still rule out a strongly detachment-free response.