In the Unexpected Hanging Paradox, the prisoner tries to narrow down their date of execution using seemingly sound logical reasoning. They instead arrive at a contradiction. When the paradox is conveyed to many people, some people claim that the prisoner was justified in ruling out Friday as their date of execution, but no other day. No one I've spoken to has suggested a formal logic that incorporates such a restriction. I suspect that Girard's Linear Logic is such a logic.
The fact that the date of execution is supposed to be a surprise can get "spent" once the cut rule is applied. This should be sufficient to rule out Friday, but no other day - hence resolving the paradox.
Has linear logic been applied in such a way? Can substructural logics be applied to such things?