It may help to clarify the relationship between logical relatives and mathematical relations. The word relative as used in logic is short for relative term — as such it refers to an article of language that is used to denote a formal object. So what kind of object is that? The way things work in mathematics, we are free to make up a formal object that corresponds directly to the term, so long as we can form a consistent theory of it, but it’s probably easier and more practical in the long run to relate the relative term to the kinds of relations that are ordinarily treated in mathematics and universally applied in relational databases.
In these contexts a relation is just a set of ordered tuples and — if you are a fan of strong typing like I am — such a set is always set in a specific setting, namely, it’s a subset of a specified Cartesian product.
Peirce wrote -tuples in the form and he referred to them as elementary -adic relatives. He expressed a set of -tuples as a “logical aggregate” or “logical sum”, what we would call a logical disjunction of elementary relatives, and he frequently regarded them as being arranged in the form of -dimensional arrays.
Time for some concrete examples, which I will give in the next post.