Extending classical logic for reasoning about quantum systems

A logic extending classical reasoning and supporting quantum reasoning is presented. The quantum logic is obtained by applying the exogenous semantics approach to propositional logic. The design is guided by the postulates of quantum mechanics and inspired by applications in quantum computation and information. The models of the quantum logic are superpositions of classical valuations. In order to achieve a recursive axiomatization, the superpositions are taken in inner product spaces over algebraic closures of arbitrary real closed fields.