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topynate comments on Unsolved Problems in Philosophy Part 1: The Liar's Paradox - Less Wrong
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If it's meaningless, it doesn't assert anything.
A sharper formulation of the paradox just came to my mind. Consider the statements X = "X is not true" and Y = "X isn't true". (The difference in spelling is intentional.) If X is meaningless, then X isn't true, therefore Y is true. But it's a very weird state of affairs if replacing "isn't" by "is not" can make a true sentence meaningless!
The apostrophe in this sentence isn't needed for comprehension.
When you state ~~A(S) iff ~S, you are formally substituting S for ~A(S), but the meaning of "~~A(S) iff ~S" is "the set of truth-conditions for ~~A(S) is the same as the set of truth-conditions for ~S". But this assumes that there exists a set of truth-conditions for ~S, which assumes that there exists a set of truth-conditions for S, i.e. that S is meaningful, by your definition.