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!
Good point. I take the claim that a sentence S is meaningless as equivalent to the claim that S has no truth-conditions. Let A be any schema for the conditions on which a sentence has truth-conditions, so that for each English sentence S, A(S) is true iff S is meaningful/has truth-conditions. Let S be the sentence ~A(S). Then S has truth-conditions iff A(S) iff ~~A(S) iff ~S. Contradiction. Nowhere was it assumed that the contradictory sentence was meaningful.
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.
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.
Good point. I take the claim that a sentence S is meaningless as equivalent to the claim that S has no truth-conditions. Let A be any schema for the conditions on which a sentence has truth-conditions, so that for each English sentence S, A(S) is true iff S is meaningful/has truth-conditions. Let S be the sentence ~A(S). Then S has truth-conditions iff A(S) iff ~~A(S) iff ~S. Contradiction. Nowhere was it assumed that the contradictory sentence was meaningful.
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.