A new proof of completeness.
R. L. Goodstein · Notre Dame Journal of Formal Logic · 1972
We present a new proof of the completeness of the formalisation P of sentence logic based on the first four axioms of Russell's Principia, with substitution and modus ponens as rules of inference.For the sake of brevity we take for granted various elementary properties of P, for instance that conjunction and disjunction are commutative and associative and that each distributes over the other; that rvΊr is provable in P\ that from A -> P and B -* P we may infer (Av B) -^ P, and from P -» A, P -» B we may infer P -• (A & B).It follows that if T denotes the provable sentence r v Ίr, and F denotes ΊT then the equivalences p{pvF) & (Ip vT)