Fractional semantics for classical logic

This article presents a new (multivalued) semantics for classical propositional logic. We begin by maximally extending the space of sequent proofs so as to admit proofs for any logical formula; then, we extract the new semantics by focusing on the axiomatic structure of proofs. In particular, the interpretation of a formula is given by the ratio between the number of identity axioms out of the total number of axioms occurring in any of its proofs. The outcome is an informational refinement of traditional Boolean semantics, obtained by breaking the symmetry between tautologies and contradictions