Quantum states as virtual singletons: converting duality into symmetry

Abstract: In a predicative framework from basic logic, defined for a model of quantum parallelism by sequents, we characterize a class of first order domains, termed {\em virtual singletons}, which allows a generalization of the notion of duality, termed {\em symmetry}. Although consistent with the classical notion of duality, symmetry creates an environment where negation has fixed points, for which the direction of logical consequence is irrelevant. Symmetry can model Bell's states. So, despite its nonsense in a traditional logical setting, symmetry can hide the peculiar advantage for the treatment of information, that is proper of quantum mechanics.