Interpolation and the projective Beth property in well-composed logics

We study the interpolation and Beth definability problems in propositional extensions of minimal logic J. Previously, all J-logics with the weak interpolation property (WIP) were described, and it was proved that WIP is decidable over J. In this paper, we deal with so-called well-composed J-logics, i.e., J-logics satisfying an axiom (⊥ → A) ∨ (A → ⊥). Representation theorems are proved for well-composed logics possessing Craig’s interpolation property (CIP) and the restricted interpolation property (IPR). As a consequence, we show that only finitely many well-composed logics share these properties and that IPR is equivalent to the projective Beth property (PBP) on the class of well-composed J-logics.