 Although LK and NK are both commonly labeled as “natural deduction” systems, it is the latter which better deserves the title due to its more natural, human-like, approach to proof construction. The rules of NK are typically presented as acting on standard logic formulas in an implicitly understood context, but they are also commonly given in the literature as acting more explicitly on “judgements”, that is, expressions of the form Γ ⊢ α where Γ is a set of formulas and α is a formula. This form is typically understood as making the metastatement that there is a proof of α from Γ (Kleene 1962). Following Gentzen 1935 and Prawitz 1965 here we take the former approach. The system NK has no logical axioms and provides two introduction-elimination rules for each logical connective: