Are these two simple logic statements equivalent?

Question

I've got two statements:

1. Set X always contains 3.
2. Set X set never contains not 3.

My question: Are these two statements logically equivalent?

I ask this question because of an argument between my brother... Maybe language plays a role.

Answer 1

No, they are not equivalent. "Not 3" presumably means "any number other than 3".

The null set {} contains no "not 3" elements, but it doesn't contain 3 either. Therefore it is a counterexample to the claim that the two statements are equivalent.

Answer 2

"Set $X$ always contains $3$," and "Set X never not contains 3" are equivalent. It is the property of dual negation of the modal quantifier. $$\Box\,(3\in X) ~\iff~ \neg \Diamond\, (3\notin X)$$

However, "Set X never contains not 3" is not the same thing at all.$$\neg\Diamond\,\exists x~(x\neq 3~\wedge~ x\in X)$$

The position of the "not" in a sentence is crucial to its meaning.

Consider $X=\{3, 4\}$. This is plausible under each of the first two statements, but is implausible under the third.