(A) The sentence A is not true.
I've never come across an argument against the lack of propositional content position on (A). I've read references to (A)'s lack of content – though that was it! The subject is quickly passed over as if to say this:
The logic and the paradoxes apply even if the sentence has no propositional content.
However, what if the logic and the paradoxes don't apply if the sentence has no content? Or, to put that another way, perhaps the paradoxes only arise because it has no propositional content. (Indeed wouldn't this also apply to the Liar Paradox?)
I understand that all the self-referential paradoxes work and how they come about. However, that may only be because the lack of content is ignored (or dismissed). So, sure, it works fantastically well as logical puzzle if the lack of content is ignored. The problem is: Can that the lack of content be ignored?
I'm also having a problem tying my criticisms in with the philosophical and historical fact that these self-referential paradoxes have had a profound effect on logic, mathematics and philosophy.
So perhaps it's all down to syntax and not semantics. That is, it's about the form/syntax of the sentences (as well as the problems/puzzles/paradoxes): not the content. Though if that's true, isn't it a sleight of hand to use sentences which appear to have content?
If my quibbles about propositional content are irrelevant, I will still need to know why that's so. If it's all down to logic and syntax (or form) , then sentences like “Everything I say is a lie” or “(A) This sentence A is not true” are misleading.
And if it's all about p's, q's, entailment, etc., and not really about propositional content, then why use sentences like “What I am saying is a lie” or “(A) This sentence A is not true” at all? The pretence or appearance of having propositional content is either not needed at all or just a plain mistake. If it's not about propositional content, logicians and philosophers shouldn't use examples which appear to have propositional content.
However, despite not finding much in the philosophical literature, one person did write this: “The liar isn't about propositional content.”
Perhaps not. Though what about (A)? And why isn't the Liar Paradox also about content? Or, at the very least, why isn't content relevant? That's precisely the question.
Indeed the person who said that content is not required then went on to talk about the Liar Paradox. He said:
“It led to the collapse of logicism and indirectly to Gödel's incompleteness results (i.e., that in a formal system like Zermelo-Frankel set theory you can derive (G(F) = "This sentence cannot be proved in F".)”
Yet that's just history and context.
In any case, Gödel's “This sentence cannot be proved in F” doesn't seem like (A) or the Liar Paradox – at least it's not precisely the same.
Put it this way.
Sentence S in system X is true though it can't be proven to be true in X.
Propositional Content & Plain Content
It's not being argued here that sentences have to be truth-evaluable in order to be legitimate. Indeed a distinction should be made between propositional content and mere or plain content.
A sentence with propositional content is truth-evaluable (it can be true or false); whereas a sentence with only content need not be true or false (though it still has content).
For example, the sentence “Shut that door” has content; though not propositional content. It's a command. (In fact it has more content, as it were, than “I'm lying at this moment”.)
The bottom line is not that all these sentences or statements must have propositional content (or be capable of being true or false). The argument is that only (A) and the Liar Paradox must have propositional content in order to be true or false (at least in principle). If they aren't propositional in that sense, then they can't generate the paradoxes.
So this debate doesn't include sentences like “Shut that door” (neither true nor false) or “People shouldn't kill animals” (a normative/moral judgement which is neither true nor false).
Self-referential Sentences With Semantic Content
A sentence can be self-referential and also have semantic (or otherwise) content. It's just that (A) doesn't.
We could have:
“When I say that 'All cows have four legs is true', what I'm saying is false.”
Of course that doesn't really work. One, it's really more than one sentence. Two, it's a plain self-contradiction. On the other hand, there's nothing self-contradictory in:
(A) The sentence A is not true.
For example, someone gave this example:
“I am lying to you at this very moment.”
He then went on to say that "no one can object that that sentence has no content”.
Grammatically speaking, the sentence “I am lying to you at this very moment” is a super sentence. We all know what the individual words means and it - grammatically - seems to make sense. However, what is its propositional content?
It can have propositional content if the self-accusation of lying refers to other statements the speaker has made. (Those other sentences would then be false.) However, it's supposed to be a self-referential statement. What is X lying about? He can't be referring to his lying alone because in order to lie, you have to make a claim that is false and to know that it's false.
The fact is that he isn't lying or telling the truth. X is only making a grammatically- acceptable sentence which has no propositional content. Therefore he can't be lying.
(You can also ask this question: If S has no content as such, then why is it grammatically acceptable?)
“I am lying to you at this very moment.”
“I am singing to you at this very moment.”
The two aren't equivalent and not just because one is about lying and the other is about singing.
When someone says “I am singing to you at this very moment” he is either lying or telling the truth. (He could be singing those words.) That doesn't also work for “I am lying to you at this very moment”. It has the same grammatical form; though the sentences differ not just in content. The latter is neither true nor false. The former is either true or false. And even if they have the same grammatical form, one is has a truth-value and the other doesn't. (Indeed I would argue that because of this difference, it surely can't even be said that they have the same grammatical form.)
Again, because “I'm singing to you at this very moment” and “I'm lying to you at this very moment” have the same shape (or form), this creates the problems. They may well have the same grammatical shape. Though one could be true and the other is neither true nor false. That difference seems to be very clear.
The Barber Paradox
It's worth noting here that some of the other well-known paradoxes are of a different nature to (A) and the Liar Paradox.
For example, take the Barber Paradox.
The Barber Paradox isn't about a single sentence. It can only be established through a chain of arguments. It's self-referential in that it deals with the question of whether the barber does or doesn't shave himself; though it's not about a self-referential statement (or sentence) as in the Liar Paradox or (A).
Of course the Barber Paradox can be summed up in a single sentence, such as:
“I shave everyone who doesn’t shave themselves.”
However, it's still not a self-referential sentence like “I am lying at this very moment”. It's about a self-referential situation (as it were); though, unlike (A) and the Lair Paradox, it isn't about a single sentence referring to itself or a person referring to what he's currently saying. The sentence above sums up the situation. It's about a possible or impossible state of affairs; not about a self-referential statement.