An emendation of the axiom system of Hilbert and Ackermann for the restricted calculus of predicates

Extract

The fundamental role of the restricted calculus of predicates in applications of symbolic logic, and particularly in Hubert's Beweistheorie as summed up by Hilbert and Bernays, makes it important that this logical calculus should be accurately defined. The first standard formulation of the calculus was that of Hilbert and Ackermann's Grundzüge der theoretischen Logik. This employed (in the first three editions) a finite set of axioms and rules of derivation, with rules of substitution included. A reaction by Hilbert and Ackermann's successors to persistent difficulty encountered with the rules of substitution has been to omit these rules, and instead enlarge the set of axioms and the other rules of derivation so as to encompass all possible substitutions. Such an enlargement seems to me to be undesirable. As an alternative, this note is designed to put the original approach of Hilbert and Ackermann for once and for all on a sound basis.