 The notion of truth-functional semantics for propositional logic, using the familiar truth tables, and the notion of completeness of an axiom system, was not developed until soon after the publication of PM by Bernays (1926). As a result there is no attempt to find a short list of axioms that will be complete, and so at later stages of the work there is no simple appeal to “tautological consequences” which might be easily justified by semantic considerations.