Irreflexivity
A relation that nothing can bear to itself. Nothing is taller than itself, and on Augustine's account nothing begets itself. Irreflexivity is the one property that forces a relation's places to be filled by different individuals — which is why it, and not the number of places, turns out to be load-bearing.
A relation is irreflexive when nothing can stand in it to itself. "Taller than" is irreflexive; "at least as tall as" is not, since everything is at least as tall as itself.
This unglamorous property is the hinge of the whole "why exactly three?" question, and it is worth seeing why. If a relation has three places, one thing might fill all three. But if the relation is one that nothing can bear to itself, then whatever fills the first place cannot be what fills the second — and now the places really do force separate occupants. Irreflexivity, not arity, is what converts structure into a headcount.
Both of the strong theological bridges are irreflexivity claims about one specific relation. Augustine: nothing generates itself. Richard of St Victor: nobody has charity toward himself. Neither of them is counting anything.
Where this comes up
Sources
- F. A. Muller & S. Saunders, “Discerning Fermions”, as quoted in Dieks & Versteegh, Foundations of Physics 38 (2008)
- Augustine, De Trinitate I.1