Logic in Russell's Principles of Mathematics (Q1374210)

The author gives a thorough discussion and reconstruction of the logical calculus proposed by Bertrand Russell in his ``Principles of Mathematics'' of 1903. The author starts with considerations on Russell's philosophy of logic. Like Frege before him, Russell advocated a ``syntactic approach'' according to which logic is a universal language with content in its own right. This is opposed to the ``semantic approach'' of Tarski and his precursors in the algebraic tradition regarding logic as the study of uninterpreted deductive calculi which can express a content, if an interpretation over an appropriate domain is provided. The author critically examines Russell's nonstandard view on the propositional calculus as a system which permits bindable predicate variables in subject and predicate positions. He compares it with Peano's universal quantification. After discussing Russell's definitions of logical primitives he proves the completeness of the propositional calculus with respect of the propositional analogs of sentential tautologies, and the semantical completeness with respect to nonpropositional analogs of sentential tautologies. In sum 29 theorems are proved.