A non-classical refinement of the interpolation property for classical propositional logic

We refine the interpolation property of the {∧,∨,¬}-fragment of classical propositional logic, showing that if 2 ¬φ, 2 ψ and φ ψ then there is an interpolant χ, constructed using at most atomic formulas occurring in both φ and ψ and negation, conjunction and disjunction, such that (i) φ entails χ in Kleene’s strong three-valued logic and (ii) χ entails ψ in Priest’s Logic of Paradox.