Dialetheism holds that some statements are simultaneously true and false. Adherents tend to use the Liar’s statement “This sentence is false” as an example.
“This sentence is false” is both true and false (See my post on the Liar’s Paradox for why we might think this); but this is okay, since some sentences just are dialetheic, admitting of both truth values.
Unfortunately, even admitting dialetheism, a Liar Statement is not far away that would lead to the claim of its being dialetheic meaningless:
“This sentence is false and not dialetheic”
Is it true? If it’s true, it says that it’s false and not dialetheic (i.e. it’s not both true and false). Thus, it’s only false. But if it’s only false, this is exactly what it says it is. Therefore, it’s true (and, given what it says, not dialetheic). In each case, it’s true that it’s not dialetheic.
The claim is a conjunction and the truth determination of the conjunction may be illustrated as:
(Dialetheicly T and F) & T;
T/F & T
Note that what is true via the second conjunct is that the sentence is not dialetheic. Thus, if the sentence is to be dialetheic for the result of being true and false (about the first conjunct claim to be false, while the claim of the second conjunct to be not dialetheic is determined to be true in each case), then the result is that the claim is both true and false, and it’s not dialetheic. Thus, the claim is both true and false, and it’s not both true and false.
That is, “This sentence is false and not dialetheic” is both dialetheic and not dialetheic.