A Cut-free Gentzen Calculus with Subformula Property for First-degree Entailments in Lc

In this paper, we introduce and study a multiple-conclusion cut-free propositional Gentzen calculus with subformula property for first-degree entailments (in Dunn’s sense [4], [5]) in Dummett’s linear intermediate logic LC [3] that is obtained from Gentzen’s propositional calculus LK [6] by restricting the rules of introduction of negation to the right side of sequents by allowing their right side formulas to be negations of formulas alone. By a method that adopts some ideas underlying Schütte’s reduction method [12] (cf. [13]) for the Cut-free version of Gentzen’s predicate calculus LK [6], we prove that a sequent is derivable in the calculus under consideration iff it is valid in the three-element chain Stone algebra. As a consequence, we show that our calculus does axiomatize first-degree entailments in Dummett’s linear intermediate logic and that Cut is permissible in the calculus