Peirce’s 1870 “Logic of Relatives” • Comment 11.9

Among the variety of regularities affecting dyadic relations we pay special attention to the -regularity conditions where is equal to

Let be an arbitrary dyadic relation.  The following properties can be defined.

We previously examined dyadic relations exemplifying each of these regularity conditions.  Then we introduced a few bits of terminology and special-purpose notations for working with tubular relations.

We arrive by way of this winding stair at the special cases of dyadic relations variously described as -regular, total and tubular, or total prefunctions on specified domains, or or both, and which are more often celebrated as functions on those domains.

If is a pre-function that happens to be total at then is known as a function from to typically indicated as

To say that a relation is total and tubular at is to say that is -regular at   Thus, we may formalize the following definitions.

For example, let and let be the dyadic relation depicted in the bigraph below.

We observe that is a function at and we record this fact in either of the manners or

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
• Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

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