Peirce next considers a pair of compound involutions, stating an equation between them analogous to a law of exponents from 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
Articulating the compound relative term in set-theoretic terms is fairly immediate.
On the other hand, translating the compound relative term into its set-theoretic equivalent is less immediate, the hang-up being we have yet to define the case of logical involution raising one dyadic relative term to the power of another. As a result, it looks easier to proceed through the matrix representation, drawing once again on the inspection of a concrete example.
Involution Example 2
Consider a universe of discourse subject to the following data.
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
- Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
- Logic Syllabus • Relational Concepts • Relation Theory • Relative Term