On the Mutual Definability of the Notions of Entailment, Rejection, and Inconsistency

Abstract

Abstract:In this paper, two axiomatic theories T´and T 1 are constructed, which are dual to Tarski's theory T + (1930) of deductive systems based on classical propositional calculus. While in Tarski's theory T + the primitive notion is the classical consequence function (entailment) Cn + , in the dual theory T´it is replaced by the notion of Słupecki's rejection consequence Cn´and in the dual theory T 1 it is replaced by the notion of the family Incons of inconsistent sets. The author has proved that the theories T + , T´, and T 1 are equivalent.

Refutation Systems: An Overview and Some Applications to Philosophical Logics

“…During the 1970s, Słupecki together with his students Bryll and Wybraniec-Skardowska developed a general systematic theory of rejected propositions, in the style of Tarski's theories of consequence relations and deductive systems, see in [Słupecki et al, 1971] and [Słupecki et al, 1972]. Follow up and related works along that line include [Staszek, 1971], [Staszek, 1972], and the more recent [Wybraniec-Skardowska, 2016], where two notions of axiomatic refutation systems, dual to Tarski's concept of deductive system for PL were constructed and proved to be equivalent.…”