On Natural Eight-Valued Reasoning

It is known that many-valued paraconsistent logics are useful for expressing uncertain and inconsistency-tolerant reasoning in a wide range of Computer Science. Some four-valued and sixteen-valued paraconsistent logics have especially been well-studied. Some four-valued logics are not so fine-grained, and some sixteen-valued logics are enough fine-grained, but rather complex. In this paper, a natural eight-valued paraconsistent logic in between four-valued and sixteen-valued logics is introduced as a Gentzen-type sequent calculus. A triplet valuation semantics is introduced for this logic, and the completeness theorem for this semantics is proved. The cut-elimination theorem for this logic is proved, and this logic is shown to be decidable.