This chapter gives, first, the calculus of natural deduction, together with its basic structural properties such as the normalization of derivations and the subformula property of normal derivations. Next, the calculus is extended by mathematical rules, and it is shown that normalization works also in such extensions. The theory of equality is treated in detail, as a first example. Finally, predicate logic with an equality relation is studied. It is presented as an extension of predicate logic without equality, and therefore normalization of derivations applies. The question of the derivability of an atomic formula from given atomic formulas, i.e., the word problem for predicate logic with equality, is solved by a proof-theoretical algorithm.
Natural deduction with general elimination rules
Gentzen's rules of natural deduction for intuitionistic logic have proved to be remarkably stable. There has been variation in the way the closing of assumptions is handled. In 1984, Peter Schroeder-Heister changed the rule of conjunction elimination so that it admitted an arbitrary consequence similarly to the disjunction elimination rule. We shall do the same for the rest of the elimination rules and prove normalization for natural deduction with general elimination rules.
Natural deduction is based on the idea that proving begins in practice with the making of assumptions from which consequences are then drawn. Thus, the first rule of natural deduction is that any formula A can be assumed. Formally, by writing
we construct the simplest possible derivation tree, that of the conclusion of A from the assumption A.