The general name which applies to a thing by virtue of its being the first correlate of a relation is called a relative. That fact which, being looked upon in a particular way and considered as belonging to one of the objects concerned, becomes a relation is in itself, apart from the ordinary ordering of the objects, called the fundamentum relationis, or relationship.
A relative […] may be defined as the equivalent of a word or phrase which, either as it is (when I term it a complete relative), or else when the verb “is” is attached to it (and if it wants such attachment, I term it a nominal relative), becomes a sentence with some number of proper names left blank. A relationship, or fundamentum relationis, is a fact relative to a number of objects, considered apart from those objects, as if, after the statement of the fact, the designations of those objects had been erased. A relation is a relationship considered as something that may be said to be true of one of the objects, the others being separated from the relationship yet kept in view. Thus, for each relationship there are as many relations as there are blanks.
A relation is a character of an object consisting in its forming with others an ordered set of objects belonging to a general class of such sets. I use our English word relationship to denote a system of relations among a collection of objects.
That in the reality which corresponds to a proposition with a relative predicate is called the fundamentum relationis. A relationship is a system of such fundamenta.
A dyadic relation is a character whose being consists in the logical possibility of a definite fact concerning an ordered pair, or dyad, of subjects; the first of these being termed the relate, the second the correlate; and the relation is said to subsist between the relate and correlate when the fact in whose possibility its being consists actually has place between these objects. The relation, by itself, is, therefore, an ens rationis and mere logical possibility; but its subsistence is of the nature of a fact. When the quality of the fact concerning two objects is considered, without reference to any distinction between these subjects other than that which this fact establishes, and therefore regardless of which of them is relate, which correlate, its possibility is termed by the author a relationship. (It is a useful distinction, but cannot be translated into every language.)
I have, since 1870, written much about the logic of relations. In those writings, I have usually restricted the terms “relations” and “relationships” to existential relations and relationships. By a relationship I understand the conception of a fact about a set of things abstracted from the representation of the things themselves or, in other words, a predicate which requires more than one subject to complete a proposition, or conception of a fact.