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.
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.
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.
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: