Let us make a few preliminary observations about the operation of logical involution which Peirce introduces in the following words.
I shall take involution in such a sense that will denote everything which is an for every individual of Thus will be a lover of every woman.
In ordinary arithmetic the involution or the exponentiation of to the power is the repeated application of the multiplier for as many times as there are ones making up the exponent
In analogous fashion, the logical involution is the repeated application of the term for as many times as there are individuals under the term On Peirce’s interpretive rules, the repeated applications of the base term are distributed across the individuals of the exponent term In particular, the base term is not applied successively in the manner that would give something on the order of “a lover of a lover of … a lover of a woman”.
By way of example, suppose a universe of discourse numbers among its elements just three women, In Peirce’s notation the fact may be written as follows.
In that case the following equation would hold.
The equation says a lover of every woman in the aggregate is a lover of that is a lover of that is a lover of In other words, a lover of every woman in the universe at hand is a lover of and a lover of and a lover of
The denotation of the term is a subset of which may be obtained by the following procedure. For each flag of the form with collect the subset of elements which appear as the first components of the pairs in and then take the intersection of all those subsets. Putting it all together, we have the following equation.
It is instructive to examine the matrix representation of at this point, not the least because it effectively dispels the mystery of the name involution. First, we make the following observation. To say is a lover of every woman is to say loves if is a woman. That can be rendered in symbols as follows.
Reading the formula as “ loves if is a woman” highlights the operation of converse implication inherent in it, and this in turn reveals the analogy between implication and involution which accounts for the aptness of the latter name.
The operations defined by the formulas in the boolean domain are tabulated as follows.
It is clear the two operations are isomorphic, being effectively the same operation of type All that remains is to see how operations like these on values in induce the corresponding operations on sets and terms.
The term determines a selection of individuals from the universe of discourse which may be computed via the corresponding operation on coefficient matrices. If the terms and are represented by the matrices and respectively, then the operation on terms which produces the term must be represented by a corresponding operation on matrices, which produces the matrix In short, the involution operation on matrices must be defined in such a way that the following equation holds.
The fact that denotes individuals in a subset of tells us its matrix representation is a 1‑dimensional array of coefficients in indexed by the elements of The value of the matrix at the index in is written and computed as follows.
- Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
- Logic Syllabus • Relational Concepts • Relation Theory • Relative Term