I chose those examples of triadic relations to be as simple as possible without being completely trivial but they already exemplify many features we need to keep in mind in all the more complex cases as we use relational models of realistic phenomena and objective domains.
The mathematical examples are typical of many in linguistic, logical, and mathematical contexts where we start out with compact, ready-made axioms, definitions, equations, expressions, formulas, predicates, or terms that denote the relations of interest.
For example, we might be discussing dyadic relative terms like “parent of —” or “square of —” and triadic relative terms like “giver of — to —” or “sum of — and —”.
If we spend the majority of our time in contexts like that we may form the impression that all the relational concepts we’ll ever need can be requisitioned off-the-shelf from pre-fab stock, no assembly required.
That’s a pretty picture of our mental equipment. It may even be true if we cook the data long enough and fudge the meaning of pre-fab down to the level of amino acids or quarks or some other bosons on the bus.
As a practical matter, however, research pursued in experimental veins tends to push the envelope of pre-fab concepts into surprisingly novel realms of ideas.
I’ll discuss the examples of sign relations as I get more time …