4.061 If one does not notice that a proposition has a sense independent of the facts, then one can easily believe that true and false are relations, with the same rights, between signs and the signified.

One could then say, e.g., that “p” signifies in the true way what “~p” signifies in the false way, etc.

Isn’t what “one could say” here quite correct? Don’t p and ~p refer to the same fact or state of affairs, according to the TLP itself? On this, see below. The point now is that a proposition’s having a sense does not depend on any fact (cf. 2.0211). Logic is not metaphysics. Sense is independent of truth/reality. What then of the picture theory? What of 4.03? Perhaps there is a sense in which propositions are independent of facts and another in which they are not.

Marie McGinn (p. 44) on 4.061-4.063: “[Wittgenstein’s] aim is to show that insofar as Frege holds that true and false propositions designate distinct but equivalent entities, the True and the False, he fails to make the relation between sense and truth and falsity perspicuous. In treating the Bedeutung of true sentences as an equivalent and distinct object from the Bedeutung of false sentences, Wittgenstein believes that Frege fails to make it clear that each proposition with sense essentially has two poles—a true pole and a false pole—each of which excludes the other.”