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This paper presents Rasiowa–Sikorski deduction systems (R–S systems) for logics \(\mathsf {CPL}\), \(\mathsf {CLuN}\), \(\mathsf {CLuNs}\) and \(\mathsf {mbC}\). For each of the logics two systems are developed: an R–S system that can be supplemented with admissible cut rule, and a \(\mathbf {KE}\)-version of R–S system in which the non-admissible rule of cut is the only branching rule. The systems are presented in a Smullyan-like uniform notation, extended and adjusted to the aims of this paper. Completeness is proved by the use of abstract refutability properties which are dual to consistency properties used by Fitting. Also the notion of admissibility of a rule in an R–S-system is analysed.

Rasiowa–Sikorski Deduction Systems with the Rule of Cut: A Case Study