The following is a well-known paradoxical statement:
(A) The sentence A is not true.
Isn't it the case that statement A has no semantic content? Is it not a real statement (or proposition) at all? Nonetheless, since many logicians and philosophers don't take this view, let's take it 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. (Here a list of self-referential paradoxes.)
We have the sentence “(A) The sentence A is not true” which includes the symbol A. A stands for the sentence it is in and the words which surround it. That means that a symbol within a sentence refers to the sentence which it is in. Now is that acceptable or even meaningful?
The Statement Has No Semantic Content
Now what, precisely, is true? Sentence A is true, apparently. What does sentence A say? It says that it is “not 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 from the statement, then 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've already removed the A and now we can remove the words “is not true”. After all, the predicate “is not 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 two words “the sentence” be either true or false when it says precisely nothing?
Forget all that!
What Logicians Think About A
Apparently, when we see
(A) The sentence A is not true.
it's meant to be the case that it couldn't be true. Like the Liar Paradox, it can't be true because if 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's 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 it's true. So a statement which says it's false, is true. Alternatively, a statement which says it isn't false, is true.
I've have rejected this pseudo-statement and said it has no content. Another way in which I can more or less say the same thing is to say that it's malignly self-referential. Or, more correctly, that truth can't be applied self-referentially – especially when the statement has no content in the first place. Indeed it's self-referential precisely because it has no content.
What about a sentence which is both self-referential and which has content? Such as:
(S) The sentence “Snow is white” is not true.
Now that's 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 metalanguage “The sentence S is not true” is being applied to the object-language's “Snow is white”. The statement “Snow is white” clearly has content and the whole sentence “The sentence 'Snow is white' is not true” isn't self-referential either because there's both a meta-sentence and an object-sentence.
Despite all that, logicians defend the “The sentence A is not true” paradox for two main 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's either logically or philosophically acceptable. And now I'll 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's meant by the words “we can grammatically assert the sentence”. Can we? Grammar, unlike logic, is largely about what is acceptable to say in order to make sense. Now “this sentence” (or “This sentence A is not true”) isn't grammatically acceptable for precisely the reasons given. It's roughly equivalent to saying “I walk down” (or “This is”) - and no teacher of English grammar would accept this locution on its own or without a sentential (or semantic) context.
Logicians can treat statements such as “Bricks have a sense of humour” or “The number 2 is blue” logically. That is, they can assign a truth-value to such statements and then treat them as syntactic strings from which they can derive further statements and conclusions. Similarly, we can programme the words “The number 2 is blue” into a computer and then that computer can grind out further propositions (or premises) and indeed a conclusion.
So we have a statement (“[A] The sentence A is not true”) which is perfectly grammatical and logical; but which, at the same time, has no semantic content. However, and to plow over old ground, it can also be said that we can't really call it a statement at all. Isn't it simply an arbitrary string of words which obey a logical or grammatical form and which then has had a truth-value assigned to it? (Actually, unlike “The number 2 is blue”, A has contradictory truth-values assigned to it!) At most, then, A is a logical string.
We can conclude, then, that this is why we have the philosophy of logic; which is over and above pure (or formal) logic.