In your question you ask about a statement implying the "opposite" of itself, which isn't typically a well-defined wording, but in the comments you change to using "negation" which is a much more rigorously defined term, so I'll focus on that.
You actually aren't looking so much for a statement as a system of inference. "Mainstream" logic would state that what you are looking for can never occur. In fact, Aristotle called this the "law of non-contradiction."
It is impossible, then, that "being a man" should mean precisely not being a man, if "man" not only signifies something about one subject but also has one significance ... And it will not be possible to be and not to be the same thing, except in virtue of an ambiguity, just as if one whom we call "man", and others were to call "not-man"; but the point in question is not this, whether the same thing can at the same time be and not be a man in name, but whether it can be in fact.
In first order logic, this has been captured as a truism ¬(A∧¬A) for all A.
Now logic which asserts the law of non-contradiction is by far more common than logic which does not. Your first step to finding a statement that meets your needs would be to identify a system of logic within which this law does not hold. One example would be Stable Model Semantics which has two forms of negation, strong and weak. A weak negation of X (notated
not X) is true if it is not possible to prove X with the given rules of inference, while a strong negation of X (notated
~X) further requires you to be able to prove X is false. Obviously under these rules,
X∧~X is still a false statement, because you can never prove X to be both true and false.
X∧not X also can never be true because if you can prove
not X must be false. However,
(not X)∧(not ~X) can be true if the truth value of X cannot be proven.
That sort of solution may not be what you are looking for, but hopefully it's in the right direction. Also worth consider that, in hip-speak, "That's cool" and "That's hot" are synonymous, so there's some precedence there!