You can get a clearer-if-still-imperfect sense from contrasting upvotes on parallel,

I’m fairly certain that P(disagrees with blargtroll | disagrees with your proposal) >> P(agrees with blargtroll | disagrees with your proposal), simply because blargtroll’s counterargument is weak and its followups reveal some anger management issues.

For example, I would downvote both your proposal and blargtroll’s counterargument if I could—and by the Typical Mind heuristic so would everyone else :)

That said, I think you’re right in that this would not have received sufficiently many downvotes to become invisible.

In a topological space, defining

X ∨ Y as X ∪ Y

X ∧ Y as X ∩ Y

X → Y as Int( X^c ∪ Y )

¬X as Int( X^c )

does yield a Heyting algebra. This means that the understanding (but not the explanation) of /u/cousin_it checks out: removing the border on each negation is the “right way”.

Notice that under this interpretation X is always a subset of ¬¬X.:

Int(X^c) is a subset of X^c; by definition of Int(-).

Int(X^c)^c is a superset of X^c^c = X; since taking complements reverses containment.

Int( Int(X^c)^c ) is a superset of Int(X) = X; since Int(-) preserves containment.

But Int( Int(X^c)^c ) is just ¬¬X. So X is always a subset of ¬¬X.

However, in many cases ¬¬X is not a subset of X. For example, take the Euclidean plane with the usual topology, and let X be the plane with one point removed. Then ¬X = Int( X^c ) = ∅ is empty, so ¬¬X is the whole plane. But the whole plane is obviously not a subset of the plane with one point removed.