The Degrees of Maximality of the Intuitionistic Propositional Logic and of Some of Its Fragments

Professor Ryszard Wójcicki once asked whether the degree of maxi-mality of the consequence operation C determined by the theorems of the intuitionistic propositional logic and the detachment rule for the implication connective is equal to 2 2 ℵ 0 ? The aim of the paper is to give the affirmative answer to the question. More exactly, it is proved here that the degree of maximality of C Ψ-the Ψ-fragment of C, is equal to 2 2 ℵ 0 , for every Ψ ⊆ {→, ∧, ∨, ¬} such that →∈ Ψ. ¿From now an Ψ stands for a subset of {→, ∧, ∨, ¬} such that →∈ Ψ, F Ψ = (F Ψ , Ψ) for the absolutely free algebra determined by Ψ via infinite but denumerable set V of variables, and C Ψ denotes the consequence operation in F Ψ (see [4]) determined by the detachment rule for the implication connective together with all rules of the form r(∅, α) where α ranges over all theorems of the intuitionistic propositional logic which contain only connectives from the set Ψ. For Ψ = {→, ∧, ∨, ¬} instead of C Ψ we write C. Observe that the Waisberg separation theorem yields: