# Can a true statement also imply the opposite of itself?

It's unlikely that there could be a thesis that also is its own antithesis. Similarly, a formula usually isn't the "opposite" of itself if we use well-defined terminology.

Somehow I have a notion though that there could be statements or descriptions that also are their own opposites and maybe not even contradictory.

I don't have a very good example but maybe there was or I can make a good example later that is formal without raising question of definitions and also without self-referential statement and also not being a metatype.

The idea, if possible, is similar to words that are true about themselves compared to other words and the paradox that might infer.

• iep.utm.edu/par-liar Commented Jan 14, 2017 at 1:40
• Can you define or at least slightly clarify by example what you mean with "opposite of itself"? Do you mean a true statement which implies its negation? Because that doesn't exist. You don't seem to mean this because you say "not even contradictory" but I don't have an intuition of what else you could mean here?
– MM8
Commented Jan 14, 2017 at 17:49
• @Kurow The liar paradox is self-referential. I'm trying to get around that and making a statement that is not self-referential and very much like you mention, implies its negation. I know it should not exist. But logic and math have had some similar results earlier. I don't want "meta-level" or self-referential. I want more regular statement. I'm thinking both what kind of statement and formalization. Mathematics could be e.g. if there are unsolved problems that could imply that the problem is solved if it is unsolvable. It is still vague and I'm going to try and make clear example if I can. Commented Jan 14, 2017 at 18:04
• I think you are asking for something X, that if it were known that it exists would cause massive discomfort to logicians and mathematicians, who would then devise a system which does not imply this X. One such X in the past was the Russel Set, which I think you are familiar with given that you are trying to find something that is not self-referential in that way. There are some non-contradictory results which are "weird" in an intuitive sense, though. Are you familiar with Skolem's paradox? That does not produce an outright contradiction but at face-value makes a very odd claim.
– MM8
Commented Jan 14, 2017 at 18:37
• So you are looking for a sentence which does not refer back to itself, and which yet implies its negation. How did this question occur to you? Perhaps the answer would help the community focus on possible solutions. Commented Jan 15, 2017 at 5:34

If theses can be separated from their antitheses, then yes its 'unlikely'; but Hegel pointed out that this was an assumption that required justifying; and he held that no, it wasn't; theses, contained within themselves their counter-motion (ie negation); and hence to his theory of sublation; see Hegels Logic.

It's worth looking at the beginning of paragraph 39, from the Phenomenology:

The true and the false belong to those determinate thoughts that are regarded as motionless essences unto themselves, with one standing fixedly here and the other standing fixedly there, and each being isolated from the other and sharing no commonality.

to which he immediately counters:

Against that view, it must be be maintained that truth is not a stamped coin issued directly from the mint and ready for one's pocket. Nor is there "a" false, no more than there is "an" evil

and

...The false, for it is only the false which is being spoken of here, would be the other, the negative of substance which, as the content of knowledge, is the true.

• "This rock exists." How does that contain its negation? Commented Jan 15, 2017 at 2:06
• @stoddard: its not much of a thesis...but I'd start with what do you mean by existence; can you explain that? Or is it a matter, simply like Johnson, I can kick this rock and therefore it is exists? Commented Jan 15, 2017 at 2:25
• what do you mean by thesis? Provide a thesis that has its negation contained within it that isn't an obvious paradox like Yablo's paradox Commented Jan 15, 2017 at 2:25
• no, let's not use an example that has material implication or anything. Let's use a clear example of a thesis. Hegel sounds like hogwash Commented Jan 15, 2017 at 2:26
• @Lothrop Stoddard: I'm autistic because I don't want to go along with your social and political attitudes? That's a strange way of understanding autism. Commented Apr 28, 2018 at 18:53

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 `X`, then `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!