The rule of procedure Re in \({\L}ukasiewicz's\) many-valued propositional calculi (Q1076008)

In this paper several complete axiomatizations of \({\L}ukasiewicz's\) \(\aleph_ 0\)-valued and m-valued propositional calculuses are defined in which C and N are the only primitive functors and the sole primitive rule of inference is the rule Re, that when P, Q, R are formulas and S is the result of replacing one occurrence of the subformula CPQ in R by Q, if P and R are correct formulas, then S is a correct formula. Modus ponens is a special case of this rule. For the infinite-valued systems independence of axioms is proved by means of matrices.