Naïve non-substructural solutions to the Validity Paradox
摘要
We present naïve non-substructural solutions to the Validity Paradox. The main idea behind these solutions is to break the link between a valuation satisfying an inference and a valuation not being a counterexample to that inference. A philosophical consequence of this technical move is that, in some of these theories, Validity Detachment will be valid while at the same time invalid, while in others Validity Detachment won’t be invalid, though it will not be valid either. No matter what path is chosen, it will not be possible to have the problematic instances of Validity Detachment as axioms of the derivations that lead to the paradoxical result. The first cannot have them as axioms because the problematic instances are not valid, and the second cannot have them either because they are invalid. Given the details of these proposals, new forms of both (i) naïve validity, and (ii) non-substructurality arise.