Abstract
In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski’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’s logic D 2 can be expressed by other than S 5 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 D 2 . The obtained system determines a logic D 0 as a set of theses that is contained in D 2 . Moreover, any discussive logic that would be expressed by means of the provided model of discussion would contain D 0 , so it is the smallest discussive logic.
Highlights
Stanisław Jaśkowski’s aim was to propose a calculus that would allow for explication of inconsistent theories by means of some consistent framework
Any discussive logic that would be expressed by means of the provided model of discussion would contain D0, so it is the smallest discussive logic
From the formal point of view, the underlying modal logic that will be used for the formulation of the proposed variant of discussive logic, will be the deontic normal logic D
Summary
Stanisław Jaśkowski’s aim was to propose a calculus that would allow for explication of inconsistent theories by means of some consistent framework. As a result Jaśkowski developed a logic denoted as D2 that was meant to be a basis for calculus that would not lead in general to overfull set of conclusions when applied to inconsistent set of premisses. He used the scenario of a discussion as a model case. In Jaśkowski’s intuitive model, operators take only auxiliary role and modal operators are not present in the object language of the discussive logic Such a variant seems to be natural and has been considered in [1]. Our aim is to indicate the weakest logic that arises from a natural variant of Jaśkowski model of discussion and give an adequate deductive system for such a logic
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