Peirce’s 1870 “Logic of Relatives” • Comment 10.6
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.
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should not be considered as altering at all the meaning of 
but as only a subjacent sign, serving to alter the arrangement of the correlates.
is secretly in love with 
as the “giver of a horse to an owner of that horse” then we may assume the associative and distributive laws remain in force, allowing us to derive the following equation.
is equal to 
but that no other form of product yields a non-null result. Scanning the implied terms of the triple product tells us that only the case 
is non‑null.
of the universe 
is invoked up front to anchor an explicit line of identity for the leading rèlate of the composition, while the terminal argument 
is generalized to the whole universe 
Doing this amounts to an act of abstraction from the particular application to 
This form of universal bracketing isolates the serial composition of the relations 
and 
to form the composite 
and 
Table 9 and Figure 10 present two ways of picturing the constraints involved in constructing the relational composition 
on each of the squares marked 
and similarly for the squares marked 
and 
meanwhile leaving all the blank squares empty.
are required to denote elements in the relations 
respectively.
will automatically bear the tokens of a pair of elements in the composite relation 
generally denotes no logical term. But 
may be considered as denoting some two 
’s. It is natural to write
alone will denote some two things. But this multiplication is not in general commutative, and only becomes so when it affects a relative which imparts a relation such that a thing only bears it to one thing, and one thing alone bears it to a thing. For instance, the lovers of two women are not the same as two lovers of women, that is,
is given by the abstract condition that 
but elements like those are commonly encountered in more concrete circumstances, acting as operators or transformations on other sets or spaces, and in that action they will often be represented as matrices of coefficients.
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