Exactly true and non-falsity logics meeting infectious ones

In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding ‘exactly true’ and ‘non-falsity’ versions of four-valued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135; Shramko (2019). Dual-Belnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). First-degree entailment and its relatives. Studia Logica, 105(6), 1291–1317] to the case of infectious logics, namely to the case of Deutsch's logic introduced in Deutsch [Relevant analytic entailment. The Relevance Logic Newsletter, 2(1), 26–44; The completeness of S. Studia Logica, 38(2), 137–147]. The particular systems obtained in this way are and . We present them in the form of sequent calculi and prove corresponding soundness and completeness theorems. We illuminate the connection between , and two well-known systems, strong Kleene three-valued logic and Priest's Logic of Paradox . This connection allows us to investigate the characterisation of the entailment relations associated with and as well as to introduce the notion of ‘infectious analogue’ of a certain logic. We also study implicative extensions of and and prove soundness and completeness theorems for them as well.