The Card paradox and blackboard paradox are interesting in that if we declare the Liar paradox to be meaningless, these paradoxes are meaningless or meaningful depending on the state of the world.
So what? “The King of France is bald” is meaningful when France has a king and meaningless the rest of the time. And “The statement on the blackboard in Carslaw Room 201 is false” is meaningless when said blackboard is blank.
To me, “the king of France is bald” (when there is no KoF) and “the statement on the blackboard is false” (when there is nothing on the blackboard) are in the third category, which isn’t quite the same as either “false” or “meaningless” but is distinctly nearer “false” than “meaningless”.
I would say a statement is meaningless, or at least meaningless to me, in so far as I have (or I suspect anyone could have) no clear conception of how its truth value depends on the state of the world. In the case of, e.g., “the king of France is bald” it’s pretty straightforward even though for some worlds—e.g., those with no king of France—its truth value might be undefined, and for others—e.g., those where there is a king of France, and he has a little bit of hair left—it might be unclear. Contrast, say, “the Absolute enters into, but is itself incapable of, evolution and progress” (an example used by A J Ayer) where it’s hard to see how to do better than “it’s true if the Absolute enters into, etc.”; or “The Mungle pilgriffs far awoy / Religeorge too thee worled” (from John Lennon) where it seems unlikely that any proposition with truth values is intended at all.
“The King of France is bald” is meaningful when France has a king and meaningless the rest of the time.
If you express this claim straightforwardly in first-order predicate logic, it can be either true or false depending on the structure you choose:
∀x. KingOf(x, France) → Bald(x) — True, because there are no counterexamples
∃x. KingOf(x, France) ∧ Bald(x) — False, because there is no satisfying value of x
Bald(KingOfFrance) — Erroneous because the universe does not contain an element “KingOfFrance”
If in France it is customary for the king to have his head shaved, then the first formalization is always true, and furthermore the original sentence has an ordinary interpretation which is still true when there is no king (though it is arguably better written as “The Kings of France are bald”, to emphasize the scope of the claim, in that case).
The point I intend is that “meaningless the rest of the time” is not fundamental to all reasonable interpretations of the sentence, but a choice you made. (I’d also agree with gjm’s comment that “contains a false assumption” is different from “meaningless”. (And, yes, first-order predicate logic does not include that distinction.))
So what? “The King of France is bald” is meaningful when France has a king and meaningless the rest of the time. And “The statement on the blackboard in Carslaw Room 201 is false” is meaningless when said blackboard is blank.
I would distinguish between
false,
meaningless, and
relying on a false assumption.
To me, “the king of France is bald” (when there is no KoF) and “the statement on the blackboard is false” (when there is nothing on the blackboard) are in the third category, which isn’t quite the same as either “false” or “meaningless” but is distinctly nearer “false” than “meaningless”.
I would say a statement is meaningless, or at least meaningless to me, in so far as I have (or I suspect anyone could have) no clear conception of how its truth value depends on the state of the world. In the case of, e.g., “the king of France is bald” it’s pretty straightforward even though for some worlds—e.g., those with no king of France—its truth value might be undefined, and for others—e.g., those where there is a king of France, and he has a little bit of hair left—it might be unclear. Contrast, say, “the Absolute enters into, but is itself incapable of, evolution and progress” (an example used by A J Ayer) where it’s hard to see how to do better than “it’s true if the Absolute enters into, etc.”; or “The Mungle pilgriffs far awoy / Religeorge too thee worled” (from John Lennon) where it seems unlikely that any proposition with truth values is intended at all.
If you express this claim straightforwardly in first-order predicate logic, it can be either true or false depending on the structure you choose:
∀x. KingOf(x, France) → Bald(x) — True, because there are no counterexamples
∃x. KingOf(x, France) ∧ Bald(x) — False, because there is no satisfying value of x
Bald(KingOfFrance) — Erroneous because the universe does not contain an element “KingOfFrance”
If in France it is customary for the king to have his head shaved, then the first formalization is always true, and furthermore the original sentence has an ordinary interpretation which is still true when there is no king (though it is arguably better written as “The Kings of France are bald”, to emphasize the scope of the claim, in that case).
The point I intend is that “meaningless the rest of the time” is not fundamental to all reasonable interpretations of the sentence, but a choice you made. (I’d also agree with gjm’s comment that “contains a false assumption” is different from “meaningless”. (And, yes, first-order predicate logic does not include that distinction.))