Did Russell had something like the notion of domain in the sense defined now by mathematics textbooks?

Question

The expression ∀x(ϕx → ψx) is supposed to mean that, in Russell's parlance, ϕx → ψx is true "for all values of x". However, what are those values that Russell is referring to?

At some point in a now distant past, all mathematicians would have probably happily subscribed to the idea that if x is a number, then x² is positive or null. Russell himself would have said that "x is a number" implies "x ≥ 0" for all values of x. Yet, after mathematicians decided that it was acceptable to say that there is a number i such that i² = -1, the expression ∀x("x is a number" → x ≥ 0) is no longer true.

I suppose that this sort of problem is solved by the notion of domain, whereby one can specify in advance that x belongs to some restrictive set of values, for example, in this case, ℕ. If the domain is ℕ, then "x is a number" implies "x ≥ 0" for all values of x if only because for all values of x now means "for all values of x in the domain", which means in this case x ∈ ℕ.

Thank you to correct me if I am wrong there.

Still, I couldn't find where Russell would have introduced the notion of domain as it is now defined in all mathematics textbooks. So, my question is:

Did Russell had something like the notion of domain in the sense defined now by mathematics textbooks?

Thank you for any scholarly references.

Answer 1

We can see Rusell's Theory of Types sketched into Appendix B of Principles (1903) and fully described in Mathematical Logic as Based on the Theory of Types (1908).

See III. The Meaning and Range of Generalized Propositions, page 233:

"all men are mortal " will mean " if x is a man, x is mortal, where x may have any value within a certain range." "Hence the values of x must be somehow restricted within some legitimate totality. This seems to lead us to the traditional doctrine of a "universe of discourse" within which x must be supposed to lie. [...] it follows that the function "if x is a man, x is mortal" must have a certain range of significance".