The meaning of a statement S is the difference it makes in the world, should it be true. A meaningful statement makes some difference.
But a difference is a comparison between two different conditions. Specifically, it is the comparison between the way the world would be if S is true (call that Consequences(S)), against the way the world would be if S is false (call that Consequences(not-S)).
For S to be meaningful means that Consequences(S) is different from Consequences(not-S).
By substituting, for not-S to be meaningful means that Consequences(not-S) is different from Consequences(not-not-S). But by double negation elimination, that's the same as saying Consequences(not-S) is different from Consequences(S). And that's the same condition as for S to be meaningful.
So, the condition for S to be meaningful is the same condition as the condition for not-S to be meaningful. Therefore, if S is meaningful, so is not-S, and vice versa.
Incidentally, the resolution to the paradox of the original question goes as follows.
Suppose that the principle of verifiability is itself verifiable. Then it follows that the contradictory of the principle must itself be a meaningful statement. For a statement can be said to be verifiable only if it has a significant contradictory. But the contradictory of the principle of verifiability asserts that there is some statement which is meaningful and which possesses no observable consequences. Hence it follows that it is significant to say: some statements are meaningful, and they don't have verifiable consequences. We must then conclude that the principle of verifiability is false. (Feuer, L. S. (1951). The Paradox of Verifiability)
The second to last line obtains the proposition, "it is significant to say: some statements are meaningful, and they don't have verifiable consequences." Let the sentence, "some statements are meaningful, and they don't have verifiable consequences" be named P. In order to obtain a counterexample to the principle of verifiability, we only have half the picture; we have shown that P is meaningful, but we have yet to show that P lacks verifiable consequences. Does it? The "paradox" fails to establish this.
In fact, there is evidence that P may have verifiable consequences. Namely, an intellectual community that believes P will behave differently from an intellectual community that believes not-P; the second community will reject as meaningless certain propositions that the first community will not. Belief in P influences behavior. And perhaps one of the communities will perform better, or worse, than the other, in regards to certain practical measures, such as the quality of technology produced by the community. The performance of the community, based on belief in P or not-P, could be considered a verifiable consequence of P. If P is true, we might expect the P-believing community would perform better, and if P is false, we might expect the not-P-believing community would perform better.