Symmetric consequence relations

Abstract

We have developed investigations on a quite universal concept of logic in context purely set theoretic. In this article, we highlight the symmetric consequence relations as a particular and special case of consequence relations with multiple conclusions. The motivation for this approach comes from sequent calculus of Gentzen, which associates a set of sentences to another set of sentences, the antecedent and the consequent. In a special way, about the antecedent set we must consider a conjunction of sentences, while the consequent is formed by a disjunction. For this reason, Gentzen worked with finite sequences. Since we deal with any sets, then we must recognize the validity of each member of the antecedent and symmetrically of some member of the consequent. We show this case is still a deductive system of Tarski.