As Peirce observes, it is not possible to work with relations in general without eventually abandoning all of one’s algebraic principles, in due time the associative law and maybe even the distributive law, just as we already gave up the commutative law. It cannot be helped, as we cannot reflect on a law unless from a perspective outside it, in any case, virtually so.
This could be done from the standpoint of the combinator calculus, and there are places where Peirce verges on systems that are very similar, 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 regimes of algebraic law that we have been developing the paradigm of Figures and Tables indicated above.
For the next few episodes, then, I will examine the examples that Peirce gives at the next level of complication in the multiplication of relative terms, for example, the three that are repeated below.