We shed light on an old problem by showing that the logic LP cannot define a binary connective obeying detachment in the sense that every valuation satisfying and also satisfies , except trivially. We derive this as a corollary of a more general result concerning variable sharing.
"A Note on Freedom from Detachment in the Logic of Paradox." Notre Dame J. Formal Logic 54 (1) 15 - 20, 2013. https://doi.org/10.1215/00294527-1731353