Generality of Logical Types

Abstract

My aim is to examine logical types in Principia Mathematica from two (partly independent) perspectives. The first one pertains to the ambiguity of the notion of logical type as introduced in the Introduction (to the first edition). I claim that a distinction has to be made between types as called for in the context of paradoxes, and types as logical prototypes. The second perspective bears on typical ambiguity as described in Russell and Whitehead’s "Prefatory Statement of Symbolic Conventions", inasmuch as it lends itself to a comparison with specific systems of modern typed lambda-calculus. In particular, a recent paper shows that the theory of logical types can be formalized in the way of a lambda-calculus. This opens the way to an interesting reconciliation between type theories in the Russellian sense of the word, and type theories in the modern sense. But typical ambiguity is not taken into account in that paper. I would like to take up that question of typical ambiguity, by extending the typed lambda-calculus to be used.