Peirce’s 1870 “Logic of Relatives” • Comment 12.3
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, 
The first method applies set-theoretic operations to the extensions of absolute and relative terms, expressing the denotation of the term
The second method operates in the matrix representation, expressing the value of the matrix 
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 
![\begin{array}{*{15}{c}} X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i & \} \\[6pt] W & = & \{ & d, & f & \} \\[6pt] L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!e, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B15%7D%7Bc%7D%7D++X+%26+%3D+%26+%5C%7B+%26+a%2C+%26+b%2C+%26+c%2C+%26+d%2C+%26+e%2C+%26+f%2C+%26+g%2C+%26+h%2C+%26+i+%26+%5C%7D++%5C%5C%5B6pt%5D++W+%26+%3D+%26+%5C%7B+%26+d%2C+%26+f+%26+%5C%7D++%5C%5C%5B6pt%5D++L+%26+%3D+%26+%5C%7B+%26+b%5C%21%3A%5C%21a%2C+%26+b%5C%21%3A%5C%21c%2C+%26+c%5C%21%3A%5C%21b%2C+%26+c%5C%21%3A%5C%21d%2C+%26+e%5C%21%3A%5C%21d%2C+%26+e%5C%21%3A%5C%21e%2C+%26+e%5C%21%3A%5C%21f%2C+%26+g%5C%21%3A%5C%21f%2C+%26+g%5C%21%3A%5C%21h%2C+%26+h%5C%21%3A%5C%21g%2C+%26+h%5C%21%3A%5C%21i+%26+%5C%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
Figure 55 shows the placement of 
To highlight the role of 
Computing the denotation of
With the above Figure in mind, we can visualize the computation of 
- Pick a specific
- Pan across the elements
in the middle row of the Figure.
- If
otherwise
- If
otherwise
- Compute the value
for each
- If any of the values
is
then the product
is
otherwise it is
As a general observation, we know the value of 
Running through the program for each 
Resources
- Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
- Logic Syllabus • Relational Concepts • Relation Theory • Relative Term
cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)
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