[mrkeen]

It really isn't a long enough section to get lost in.

The 'not accurate' diagram says that orange-less-than-yellow implies yellow-not-less-than-orange. Hard to find fault with.

> NO. Antisymmetry doesn't exclude `x = y`. Ties are permitted in the equality case. Antisymmetry for a non-strict order says that if both directions hold, the two elements must in fact be the same element. The author is describing strict comparison or total comparability intuition, not antisymmetry.

I like the article's "imprecise prose" better:

  You have x ≤ y and y ≤ x only if x = y

[gobdovan]

My comment is not long enough either to get lost in.

The prose "It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me" is inaccurate for describing antisymmetry. In the same short section, you first state the correct condition:

You have x ≤ y and y ≤ x only if x = y

from which it doesn't follow that "It also means that no ties are permitted". The "no ties" idea belongs to a stronger notion such as a strict total order, not to antisymmetry.

[wasabi991011]

The prose is correct.

You (presumably) aren't your grandmother, so we have x=/=y. Therefore by the biimplication, (x ≤ y and y ≤ x) is false i.e. either x ≤ y (I am better than my grandmother) or y ≤ x (my grandmother is better than me). The "neither" case is excluded by the law of totality.

[gobdovan]

> The "neither" case is excluded by the law of totality.

We literally said the same thing. It doesn't follow from antisymmetry.

My point is precisely that:

(x <= y /\ y <= x) -> x = y

does not entail

x <= y \/ y <= x

The second statement is totality/comparability, not antisymmetry.