A Note on <em>Principia</em>'s *38 on Operations

Abstract

Principia Mathematica *38 introduces what it calls “Relations and Classes Derived from a Double Descriptive Function”. The notion of a relation-e (relation in extension) so derived is called an operation, and of course all dyadic relation-e theorems rely ultimately on the comprehension axiom schema for relations in intension given at *12.11. But in attempting to give a general pattern of definition, *38 uses the odd-looking “x?y ” which lends itself to the misconception that  is itself an operation sign. The informal summary makes matters worse, writing “E!(x?y)” which is ungrammatical. This paper argues that with P, R and S as relation-e variables and ?, ?, and ? as class variables, operations are comprehended by wffs such as “ P=x?y”, “?=???” and “P=R?S”. Relying on triadic relations-e, I explain how the sign ? can be entirely avoided using comprehension. Along the way, puzzling cases such as ? and ? are resolved. [Greek and other symbols do not repoduce here. See the first page of the paper for the correct abstract.--Ed.]