Paraconsistency and Consistency Understood as the Absence of the Negation of any Implicative Theorem
Paraconsistency and Consistency Understood as the Absence of the Negation of any Implicative Theorem

Summary/Abstract: As is stated in its title, in this paper consistency is understood as the absence of the negation of any implicative theorem. Then, a series of logics adequate to this concept of consistency is defined within the context of the ternary relational semantics with a set of designated points, negation being modelled with the Routley operator. Soundness and completeness theorems are provided for each one of these logics. In some cases, strong (i.e., in respect of deducibility) soundness and completeness theorems are also proven. All logics in this paper are included in Lewis’ S4. They are all paraconsistent, but none of them is relevant.

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