Generalised elimination rules and harmony

Standard natural deduction rules for Int (intuitionistic predicate logic) in the style of Gentzen and Prawitz are presumed to be familiar. The theory of cut-elimination for sequent calculus rules is very clear; whether a derivation in a sequent calculus is cut-free or not is easily defined, according to the presence or absence of instances of the Cut rule. Normality of a (natural) deduction is less clear; there are many inequivalent definitions in the literature. For implicational logic it is easy; but rules such as the elimination rule for disjunction cause problems with the notion of ’maximal formula occurrence’, and more problems when minor premisses have vacuous discharge of assumptions. One proposed solution is the use of generalised elimination rules, i.e. GE-rules. These can also be motivated in terms of Prawitz’ inversion principle: “the conclusion obtained by an elimination does not state anything more than what must have already been obtained if the major premiss of the elimination was inferred by an introduction” [17]. See the invaluable survey [11] for a full discussion, including the idea’s antecedents in the work of Lorenzen. The standard elimination rule for disjunction is already a GE rule: