A Kotas-Style Characterisation of Minimal Discussive Logic

• Main Authors: Krystyna Mruczek-Nasieniewska, Marek Nasieniewski
• Format: Article
• Language: English
• Published: MDPI AG 2019-10-01
• Series: Axioms
• Subjects: discussive logics; the smallest discussive logic; discussive operators; seriality; accessibility relation; kotas’ method; modal logic
• Online Access: https://www.mdpi.com/2075-1680/8/4/108

In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski&#8217;s original model of discussion. The obtained system can be treated as the minimal discussive logic. It is determined by frames with serial accessibility relation. As the smallest one, this logic can be treated as a basis which could be extended to richer discussive logics that are obtained by varying accessibility relation and resulting in a lattice of discussive logics. One has to remember that while formulating discussive logics there is no one-to-one determination of discussive logics by modal logics. For example, it is proved that Jaśkowski&#8217;s logic <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="sans-serif">D</mi> <mn mathvariant="sans-serif">2</mn> </msub> </semantics> </math> </inline-formula> can be expressed by other than <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mn>5</mn> </mrow> </semantics> </math> </inline-formula> modal logics. In this paper we consider a deductive system for the sketchily described minimal logic. While formulating the deductive system, we apply a method of Kotas that was used to axiomatize <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="sans-serif">D</mi> <mn mathvariant="sans-serif">2</mn> </msub> </semantics> </math> </inline-formula>. The obtained system determines a logic <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="sans-serif">D</mi> <mn mathvariant="sans-serif">0</mn> </msub> </semantics> </math> </inline-formula> as a set of theses that is contained in <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="sans-serif">D</mi> <mn mathvariant="sans-serif">2</mn> </msub> </semantics> </math> </inline-formula>. Moreover, any discussive logic that would be expressed by means of the provided model of discussion would contain <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="sans-serif">D</mi> <mn mathvariant="sans-serif">0</mn> </msub> </semantics> </math> </inline-formula>, so it is the smallest discussive logic.