I am rather unsure of the off-the-cuff formulation I gave of a possible “axiom of equality” with regards to intelligence, namely that given an ordering of intelligences based on the relation <=, if a<=b and b<=a then a=b. What this conventionally means is that the order is antisymmetric, in other words that it never happens that a <= b and b <= a unless a is b - there are no discrete elements a and b such that both a <= b and b <= a.

The situation I wanted to describe is one in which a is not identical to b, and yet both a <= b and b <= a so that there is a kind of symmetry between them; but this symmetry is not equality as such. It may be that what I was really after was an equivalence relation.

This area is fiddly in all sorts of ways, which I’ll try to write about soon - it would be nice to be able to contribute, even if absently, to this event.