The projective Beth property in well-composed logics

The interpolation and Beth definability problems are proved decidable in well-composed logics, i.e., in extensions of Johansson's minimal logic $\mathrm J$ satisfying an axiom $(\perp\to A)\vee(A\to\perp)$. In previous studies, all $\mathrm J$-logics with the weak interpolation property (WIP) were described and WIP was proved decidable over $\mathrm J$. Also it was shown that only finitely many wellcomposed logics possess Craig's interpolation property (CIP) and the restricted interpolation property (IPR), and moreover, IPR is equivalent to the projective Beth property (PBP) on the class of logics in question. These results are applied to prove decidability of IPR and PBP in well-composed logics. The decidability of CIP in such logics was stated earlier. Thus all basic versions of the interpolation and Beth properties are decidable on the class of wellcomposed logics.