A Characterisation of Some \(\mathbf {Z}\)-Like Logics

Abstract

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’.

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Correspondence to Marek Nasieniewski.

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This work was completed with the partial support of Polish National Science Centre (NCN), Grant No. 2016/23/B/HS1/00344.

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Mruczek-Nasieniewska, K., Nasieniewski, M. A Characterisation of Some \(\mathbf {Z}\)-Like Logics. Log. Univers. 12, 207–219 (2018). https://doi.org/10.1007/s11787-018-0184-9

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Keywords

  • Beziau’s logic \(\mathbf {Z}\)
  • Kripke semantics
  • modal logics
  • non-normal worlds
  • completeness
  • modalized negations

Mathematics Subject Classification

  • Primary 03B53
  • Secondary 03B45
  • 03B22