Natural deduction with general elimination rules.

*(English)*Zbl 1021.03050Summary: The structure of derivations in natural deduction is analyzed through isomorphism with a suitable sequent calculus, with twelve hidden convertibilities revealed in usual natural deduction. A general formulation of conjunction and implication elimination rules is given, analogous to disjunction elimination. Normalization through permutative conversions now applies in all cases. Derivations in normal form have all major premisses of elimination rules as assumptions. Conversion in any order terminates.

Through the condition that, in a cut-free derivation of the sequent \(\Gamma\Rightarrow C\), no inactive weakening or contraction formulas remain in \(\Gamma\), a correspondence with the formal derivability relation of natural deduction is obtained: All formulas of \(\Gamma\) become open assumptions in natural deduction, through an inductively defined translation. Weakenings are interpreted as vacuous discharges, and contractions as multiple discharges. In the other direction, non-normal derivations translate into derivations with cuts having the cut formula principal either in both premisses or in the right premiss only.

Through the condition that, in a cut-free derivation of the sequent \(\Gamma\Rightarrow C\), no inactive weakening or contraction formulas remain in \(\Gamma\), a correspondence with the formal derivability relation of natural deduction is obtained: All formulas of \(\Gamma\) become open assumptions in natural deduction, through an inductively defined translation. Weakenings are interpreted as vacuous discharges, and contractions as multiple discharges. In the other direction, non-normal derivations translate into derivations with cuts having the cut formula principal either in both premisses or in the right premiss only.

##### MSC:

03F05 | Cut-elimination and normal-form theorems |