The B5-Modalized Propositional Logic

A B
₅
-modalized propositional logic will be proposed in this paper which there are five modalities [t], [T], [⊥], [a], [f] to represent the five values t, T, ⊥, a, f in B
₅
, respectively, and a Gentzen-typed deduction system will be given so that the system is sound and complete with the B
₅
-valued semantics of the logic.