Logica Universalis

, Volume 12, Issue 1–2, pp 207–219 | Cite as

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

  • Krystyna Mruczek-Nasieniewska
  • Marek NasieniewskiEmail author
Open Access


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’.


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

Mathematics Subject Classification

Primary 03B53 Secondary 03B45 03B22 


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© The Author(s) 2018

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Authors and Affiliations

  • Krystyna Mruczek-Nasieniewska
    • 1
  • Marek Nasieniewski
    • 1
    Email author
  1. 1.Department of LogicNicolaus Copernicus University in ToruńToruńPoland

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