[seanhunter]

The modern formulation of functions as sets doesn’t require type theory but is entirely congruent with Russell’s definition, just much less cumbersome. In this view, φ is a relation on the set (D X C) where D and C are the domain and codomain of the function (which he calls the “range of significance of x” and the “range of significance of φ(x)” respectively). So since he’s talking about propositional functions, here C is the set {true, false} and D is all the things that are like whatever x is ie the set {x’: x’ is of the same type as x}.

Now a relation is just a particular type of predicate (ie it too is a set) so here we have x ~ y if φ(x) = y for all (x,y) in (D X C).

Notice here both the propositional function and the type are sets.