 Suppose that we have used one of the above methods to show, for some theory of truth, sets, or properties, that the theory is Curry-complete (in virtue, say, of containing a Curry sentence for each sentence of the language, or for an explosive sentence). To conclude that the theory in question is trivial, it now suffices to give an argument for the Troubling Claim. This is the claim that for every theory \(\mathcal{T}\), if there is a Curry sentence for \(\pi\) and \(\mathcal{T}\), then \(\vdash_{\mathcal{T}} \pi\). Such an argument will make use of assumptions about the logical behavior of the conditional \({\rightarrow}\) mentioned in  Definition 1.  Assuming the Troubling Claim must be resisted, this accordingly places constraints on the behavior of this conditional.