The modal completeness proofs of Guaspari and Solovay (1979) for their systems R and R – are improved and the relationship between R and R – is clarified.

To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical completeness theorem with respect to PA is obtained for LR.

The modal completeness proofs of Guaspari and Solovay for their systems R and R⁻ are improved and the relationship between R and R⁻ is clarified.