Peirce next considers a pair of compound involutions, stating an equation between them that is analogous to a law of exponents in ordinary arithmetic, namely,
Then will denote whatever stands to every woman in the relation of servant of every lover of hers; and will denote whatever is a servant of everything that is lover of a woman. So that
(Peirce, CP 3.77)
Articulating the compound relative term in set-theoretic terms is fairly immediate:
On the other hand, translating the compound relative term into a set-theoretic equivalent is less immediate, the hang-up being that we have yet to define the case of logical involution that raises a dyadic relative term to the power of a dyadic relative term. As a result, it looks easier to proceed through the matrix representation, drawing once again on the inspection of a concrete example.
There is a “servant of every lover of” link between and if and only if But the vacuous inclusions, that is, the cases where have the effect of adding non-intuitive links to the mix.
The computational requirements are evidently met by the following formula:
In other words, if and only if there exists a such that and