Peirce’s 1870 “Logic of Relatives” • Comment 11.4

The task before us is to clarify the relationships among relative terms, relations, and the special cases of relations given by equivalence relations, functions, and so on.

The first obstacle to get past is the order convention Peirce’s orientation to relative terms causes him to use for functions.  To focus on a concrete example of immediate use in this discussion, let’s take the “number of” function Peirce denotes by means of square brackets and re-formulate it as a dyadic relative term in the following way.

To set the dyadic relative term within a suitable context of interpretation, let’s suppose corresponds to a relation where is the set of real numbers and is a suitable syntactic domain, here described as a set of terms.  The dyadic relation is at first sight a function from to   It is, however, not always possible to assign a number to every term in whatever syntactic domain we happen to pick, so we may eventually be forced to treat the dyadic relation as a partial function from to   All things considered, then, let’s try the following budget of strategies and compromises.

First, let’s adapt the arrow notation for functions in such a way as to allow detaching the functional orientation from the order in which the names of domains are written on the page.  Second, let’s change the notation for partial functions, or pre-functions, to mark more clearly their distinction from functions.  This produces the following scheme.

Until it becomes necessary to stipulate otherwise, let’s assume is a function in of written amounting to a functional alias of the dyadic relation and associated with the dyadic relative term whose rèlate lies in the set of real numbers and whose correlate lies in the set of syntactic terms.

Note.  Please refer to the article on Relation Theory for the definitions of functions and pre‑functions used in the above discussion.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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