A Note on Principia’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 α, β and μ as relation-e variables and D, E, and P as class variables, operations are comprehended by wffs such as “P = x♀ y”, “μ = α♀β” 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 [inline-graphic 01i] and [inline-graphic 02i] are resolved.