Abstract
Hilbert systems L ⊳ and sequential calculi [L ⊳] for the versions of logics L= T,S4,B,S5, and Grz stated in a language with the single modal noncontingency operator ⊳ A=□A∨□¬ A are constructed. It is proved that cut is not eliminable in the calculi [L ⊳], but we can restrict ourselves to analytic cut preserving the subformula property. Thus the calculi [T ⊳], [S4 ⊳], [S5 ⊳] ([Grz ⊳], respectively) satisfy the (weak, respectively) subformula property; for [B 2 ⊳], this question remains open. For the noncontingency logics in question, the Craig interpolation property is established.
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Zolin, E.E. Sequential Reflexive Logics with Noncontingency Operator. Mathematical Notes 72, 784–798 (2002). https://doi.org/10.1023/A:1021485712270
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DOI: https://doi.org/10.1023/A:1021485712270