This is a well-known paradox which I have a strong dislike for:
(A) The sentence A is not true.
Firstly, the reason why I dislike this statement, or paradox, is because statement A has no content. I believe it's not a real statement at all. However, since many logicians and philosophers don't take this view, I will take the statement as they take it – as being a genuine, if paradoxical, statement.
The first thing to say is that it's self-referential; just like the Liar Paradox: it's about itself.
We have the sentence “The sentence A is not true” which includes the symbol “A” and “A” stands for the sentence it is in and the words which surround it. That means that a symbol within a sentence references to the sentence which it is within! (Now is that acceptable or even meaningful?)
The Locution "(A) The sentence A is not true" Has No Content
So “A” = “The sentence A is not true.”
Now what, precisely, is true? Sentence A is true, apparently. What does sentence A say? It says that it is true – and that's it. It doesn't say its subject-term of phrase is true: it says that the whole sentence is true.
If we take out the “A”, what do we have left? This: “The sentence... is not true.” Since A only refers to the sentence, then why can't we take “A” out? But if we do, then what are we left with? I've already said that “The sentence A is not true” is without content: so it's even more the case that “The sentence... is not true” is without content.
We can boil this down even more. We have removed the “A” and now we can remove the “is true”. After all, the predicate “is true” is supposed to be applicable to something else. So what is the “is not true” in “The sentence A” applicable to? That's right, the “is not true” predicate is applicable to “The sentence”! So the two words, “The sentence”, are meant to be either true or false. Yet how can “The sentence” be either true or false when it says precisely nothing?
Nonetheless, forget all that!
What Logicians Think About "(A) The sentence A is not true"
Apparently, when we see
(A) The sentence A is not true.
it is meant to be the case that it couldn't be true. Like the Liar Paradox, it can't be true because if what it says is true, then it is false. But if it is false, it must be true because it is true that it is not true.
On the other hand, if we take the above to be false, then it must be true. After all, it it is saying that it isn't true. That must mean, then, that is true. So a statement which says it is false is true. Alternatively, a statement which says it isn't true, is false.
I have rejected this pseudo-statement and said it has no content. Another way in which you can more or less say the same thing is to say that it is self-referential. Or, more correctly, that truth cannot be applied self-referentially – especially when the statement has no content in the first place! Indeed it is self-referential precisely because it has no content.
What about a sentence which is both self-referential and which has content? Such as:
The sentence 'Snow is white' is not true.
Now that is not really a single sentence or statement at all. It is in fact two statements. We have “Snow is white” as well as “This sentence [S] is true”. Thus it isn't self-referential. The meta-language “The sentience S is not true” is being applied to the object-language “Snow is white”. The statement “Snow is white” clearly has content and the whole sentence “The sentence 'Snow is white' is not true” is not self-referential either because there is a meta-sentence and an object-sentence.
Despite all that, logicians defend the “The sentence A is not true” paradox for two reasons:
i) The phrase “is true” is an acceptable English predicate.
ii) The whole sentence is grammatically “unassailable”.
But is it grammatically unassailable? I don't think it is either logically or philosophically acceptable. And now I reject its grammar too.
Again, the argument is that we can grammatically assert the sentence and grammatically apply “is not true” to it. That depends on what is meant by “we can grammatically assert the sentence”. Can we? Grammar, unlike logic, is largely about what it is acceptable to say in order to make sense. Now “This sentence”, or “This sentence A is not true”, is not grammatically acceptable for precisely the reasons I have given. It is roughly equivalent to saying “I walk down” or “This is” - and no teacher of English grammar would accept either locution on their own.