The Principles of Mathematics (1903)

§ 85

It is to be observed that, according to the theory of propositional functions here advocated, the ϕ in ϕx is not a separate and distinguishable entity: it lives in the propositions of the form ϕx, and cannot survive analysis. I am highly doubtful whether such a view does not lead to a contradiction, but it appears to be forced upon us, and it has the merit of enabling us to avoid a contradiction arising from the opposite view. If ϕ were a distinguishable entity, there would be a proposition asserting ϕ of itself, which we may denote by ϕ(ϕ); there would also be a proposition not-ϕ(phi), denying ϕ(ϕ). In this proposition we may regard ϕ as a variable; we thus obtain a propositional function. The question arises: Can the assertion in this propositional function be asserted of itself? The assertion is non-assertability of self, hence if it can be asserted of itself, it cannot, and if it cannot, it can. This contradiction is avoided by the recognition that the functional part of a propositional function is not an independent entity. As the contradiction in question is closely analogous to the other, concerning predicates not predicates of themselves, we may hope that a similar solution will apply there also.(§ 85 ¶ 1)