Embedding classical logic into basic orthologic with a primitive modality

In the present paper we give the first proof-theoretical example of an embedding of classical logic into a quantum-like logic. This is performed in the framework of basic logic, where a proof-theoretical approach to quantum logic is convenient. We consider basic orthologic, that corresponds to a sequential formulation of paraconsistent quantum logic, and which is given by basic orthologic added with weakening and contraction, in a language with Girard's negation. In the paper we first consider a convenient cut-free calculus for classical logic, in the same language; then, in a language enriched with a new kind of literals, we introduce basic orthologic with a primitive modality, where classical logic is embeddable. Similarly to Girard's negation, our modality is not a connective but conceivable as an operator defined inductively on the set of formulae. This allows us to obtain a calculus which enjoys cut-elimination.