A Characterisation of Some $$mathbf {}$$-Like Logics

In Béziau (Log Log Philos 15:99–111, 2006) a logic (mathbf {Z}) was defined with the help of the modal logic (mathbf {S5}). In it, the negation operator is understood as meaning ‘it is not necessary that’. The strong soundness–completeness result for (mathbf {Z}) with respect to a version of Kripke semantics was also given there. Following the formulation of (mathbf {Z}) we can talk about (mathbf {Z})-like logics or Beziau-style logics if we consider other modal logics instead of (mathbf {S5})—such a possibility has been mentioned in [1]. The correspondence result between modal logics and respective Beziau-style logics has been generalised for the case of normal logics naturally leading to soundness–completeness results [see Marcos (Log Anal 48(189–192):279–300, 2005) and Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 34(4):229–248, 2005)]. In Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 37(3–4):185–196, 2008), (Bull Sect Log 38(3–4):189–203, 2009) some partial results for non-normal cases are given. In the present paper we try to give similar but more general correspondence results for the non-normal-worlds case. To achieve this aim we have to enrich original Beziau’s language with an additional negation operator understood as ‘it is necessary that not’.