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On Modal Logics Defining Jaśkowski’s D2-Consequence

Chapter
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 26)

Abstract

Jaśkowski’s logic D 2 (as a set of formulae) was formulated with the help of the modal logic S5 (see Jaśkowski, Stud Soc Sci Torun I(5):57–77, 1948; Stud Soc Sci Torun I(8):171–172, 1949). In Furmanowski (Stud Log 34:39–43, 1975), Perzanowski (Rep Math Log 5:63–72, 1975), Nasieniewski and Pietruszczak (Bull Sect Logic 37(3–4):197–210, 2008) it was shown that to define D 2 one can use normal and regular logics weaker than S5. In his paper Jaśkowski used a deducibility relation which we will denote by⊢ D 2 and which fulfilled the following condition: A 1,,A n ⊢; D 2 B iff \(\ulcorner \lozenge {A}_{1}^{\bullet }\rightarrow (\ldots \rightarrow (\lozenge {A}_{n}^{\bullet }\rightarrow \lozenge {B}^{\bullet })\ldots \,)\urcorner \in \mathbf{S5}\), where (−) is a translation of discussive formulae into the modal language. We indicate the weakest normal and the weakest regular modal logic which define D 2 -consequence.

Keywords

Regular Logic Weak Modal Logics Pietruszczak Discussive Logic Smallest Normal Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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Copyright information

© Springer Science+Business Media Dordrecht. 2013

Authors and Affiliations

  1. 1.Department of LogicNicolaus Copernicus UniversityToruńPoland

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