As Peirce observes, it is not possible to work with relations in general without eventually abandoning all the more usual algebraic principles, in due time the associative law and even the distributive law, just as we already gave up the commutative law. It cannot be helped, as we cannot reflect on a law except from a perspective outside it, in any case, virtually so.
This could be done in the framework of the combinator calculus, and there are places where Peirce verges on systems of a comparable character, but here we are making a deliberate effort to stay within the syntactic neighborhood of Peirce’s 1870 Logic of Relatives. Not too coincidentally, it is for the sake of making smoother transitions between narrower and wider realms of algebraic law that we have been developing the paradigm of Figures and Tables indicated above.
In the next several episodes, then, I’ll examine the cases Peirce uses to illustrate the next level of complexity in the multiplication of relative terms, as shown in the Figures below.