Russellian Simple Type Theory

In this paper I advocate a reexamination of mathematical logic in the tradition of Principia Mathematica and the early writings of Russell ([PoM], [OD], [ML]), to ask in what form it should now be studied in order to preserve certain important contributions of Russell and in order to compare it with the somewhat different Fregean tradition. This should not be as ramified type theory with axioms of reducibility, as this involves excessive complication,l and Russell himself conceded the objections against the axioms of reducibility (see [Intr2nd]); and not ramified type theory without axioms of reducibility, as this is insufficient for classical mathematics even if one adds the axioms of extensionality which Russell, influenced by Wittgenstein, suggests in [Intr2nd]. The modified version of Russell's mathematical logic which is now usual is simple theory of types ([Chwistek 1, 2], [Ramsey], [Carnap 1], [G6del]), with reliance on Tarski's resolution ([Tarski 2], [Church 6]) of the semantical antinomies.2 This is often taken as an extensional theory with explicit addition of axioms of extensionality ([Carnap 1], [G6del], [Tarski 1], [Quine 1], [Henkin], [Church 5]). If one is concerned primarily, or exclusively, with extensional logic, this is indeed the best approach, but it loses some of the characteristically Russellian contributions to logic. Especially the motivation for Russell's contextual definitions of class abstracts and of descriptions largely disappears, and it may seem better just to introduce primitive notations for both of these and to adopt a theory of descriptions which is more like Frege's than Russell's.3 Having formulated extensional logic in this way, if we then