Having been through this same discussion on many previous occasions I will try to sum up the more persistent confusions that never cease to bedevil the subject. Most of these arise from a failure to observe a number of critical distinctions.
- There is above all the distinction between relations and tuples. When it is necessary to emphasize the distinction I will describe relations as “relations in general” or “relations proper” while referring to tuples as “elementary relations”.
- There is the corresponding distinction between sign relations and elementary sign relations or triples of the form
Relations are, generally speaking, much more complex structures than elementary relations, so classifying relations is a much more complex affair than classifying elementary relations.
The same goes for sign relations and elementary sign relations. Almost all the literature you see on “classifying sign relations” actually goes no further than the much simpler task of classifying elementary sign relations. Classifying sign relations, in the proper sense of the word, is a task for the future.
- There is the distinction between formal or informational determination and causal or temporal determination. The latter form of determination is a special case of the former. A simple example of formal determination is found in such venerable phrases as “two points determine a line”. Pairs of points do not cause lines or precede them in time. Formal determination is defined at a higher level of abstraction than cause and time.
There is the distinction between dyadic forms of determination and triadic forms of determination. Here we run into a verbal problem. There is something about the word “determination” — possibly the grammatical category of “to determine” as a transitive verb with a lone direct object — that almost inexorably drags the mind down into the ruts of dyadic thinking, so it helps to use the more general and less biased idea of constraint.
In this more general perspective, the family of concepts including correspondence, determination, law, relation, structure, and so on all fall under the notion of constraint. Constraint is present in a system to the extent that one set of choices is distinguished by some mark from a larger set of choices. That mark may distinguish the actual from the possible, the desired from the conceivable, or any number of other divisions depending on the subject in view.
Thus we have a form of determination wherever we have a form of constraint. One of the most general ways of expressing a constraint is in terms of the subset relation:
- A dyadic relation is defined by the constraint where and are the domains of the relation
- A triadic relation is defined by the constraint where are the domains of the relation
- A sign relation is defined by the constraint where are the domains of the sign relation