We now have two ways of computing a logical involution raising a dyadic relative term to the power of a monadic absolute term, for example, for “lover of every woman”.
The first method applies set-theoretic operations to the extensions of absolute and relative terms, expressing the denotation of the term as the intersection of a set of relational applications.
The second method operates in the matrix representation, expressing the value of the matrix at an argument as a product of coefficient powers.
Abstract formulas like these are more easily grasped with the aid of a concrete example and a picture of the relations involved.
Involution Example 1
Consider a universe of discourse subject to the following data.
Figure 55 shows the placement of within and the placement of within
To highlight the role of more clearly, the Figure represents the absolute term by means of the relative term which conveys the same information.
Computing the denotation of by way of the class intersection formula, we can show our work as follows.
With the above Figure in mind, we can visualize the computation of as follows.
- Pick a specific in the bottom row of the Figure.
- Pan across the elements in the middle row of the Figure.
- If links to then otherwise
- If in the middle row links to in the top row then otherwise
- Compute the value for each in the middle row.
- If any of the values is then the product is otherwise it is
As a general observation, we know the value of goes to just as soon as we find a such that and in other words, such that but If there is no such then
Running through the program for each the only case producing a non-zero result is That portion of the work can be sketched as follows.
- Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
- Logic Syllabus • Relational Concepts • Relation Theory • Relative Term