Supernatural, or unnatural?

There may not be a logical fallacy here, but there appears to be some confusion as to the meaning of "supernatural".

Here is the argument : in set theory, there is such a thing as "the entire set", eg {A} and {not A}.

In another example, {the natural} and {the unnatural}, this also comprises the entire set.

So when someone uses the word "supernatural", since it obviously does not mean "natural", it must mean "unnatural".

Are you in agreement with this, or not? If not, why not?

I think that is fair. "supernatural" and "unnatural" could mean the same. Of course, there are connotations with each. Christians don't want to call God "unnatural" because that sounds like they are accusing him of having gay sex.

When debating this idea, I prefer "natural" and "not natural."

I think we need to agree on the definitions we're choosing for "natural", "unnatural", and "supernatural".  Depending on the definitions, the assumption that there's a di chotomy (as opposed to a tri chotomy) should come into question.

If we assume a dichotomy, then "supernatural" must be either "natural" or "unnatural", and the usual understanding of the prefix "un" meaning "not" would support the "supernatural is unnatural" notion.  However, I'm not sure this conclusion necessarily matches most situations.

We'd have a different situation if we were to define terms along the lines of (a) what happens in most cases ("natural") .. alligators being vicious and attacking (I suspect the generally-held opinion), (b) what is unusual to experience ("unnatural") ... an alligator serving as an emotional support animal (a recent new item from Florida featured a man with a super-docile alligator with demonstrated emotional benefits to its owner and to some medical patients), and (c) results that are unexplainable using traditional science ("supernatural") ... like grandma being able to tell when her favourite grandchild is on the way to visit her – or about some unexplainable effect that a given alligator has on a certain collection of other animals.  In this case, if something wasn't "natural" (or normal), it wouldn't need to be unnatural, it could be supernatural ... or even a forth classification altogether.

I don't so much see a fallacy ... mostly I see some assumptions and definitions that need to be specified and either accepted or rejected.  The biggest flaw I see in the argument is the assumption that when we use words containing "natural", there has to be a dichotomy with every chosen prefix meaning "the opposite of".   I'm not sure it's valid to assume that there's not a 3rd (or 4th, or ...) group of "not normal" options, as well.