The fragment of Classical Logic that respects the Variable-Sharing Principle

We examine the set of formula-to-formula valid inferences Classical Logic, where premise and conclusion share at least a propositional variable in common. review fact, already proved literature, that such system is identical to first-degree entailment fragment R. Epstein's Relatedness it non-transitive logic sort investigated by S. Frankowski others. Furthermore, we provide semantics calculus for this logic. The defined terms \(p\)-matrix built on top 5-valued extension 3-element weak Kleene algebra, whereas Gentzen-style sequent left right negation rules are subject linguistic constraints.