Let us make a few preliminary observations about the operation of logical involution, as Peirce introduces it here:
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.
(Peirce, CP 3.77)
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 According to 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 like “a lover of a lover of … a lover of a woman”.
For example, suppose that a universe of discourse numbers among its elements just three women, This could be expressed in Peirce’s notation by writing:
Under these circumstances the following equation would hold:
This says that a lover of every woman in the given universe of discourse is a lover of that is a lover of that is a lover of In other words, a lover of every woman in this context is a lover of and a lover of and a lover of
The denotation of the term is a subset of that can be obtained as follows: For each flag of the form with collect the elements that appear as the first components of these ordered pairs, and then take the intersection of all these subsets. Putting it all together:
It is very instructive to examine the matrix representation of at this point, not the least because it effectively dispels the mystery of the name involution. First, let us make the following observation. To say that is a lover of every woman is to say that loves if is a woman. This 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 that accounts for the aptness of the latter name.
The operations defined by the formulas and for in the boolean domain are tabulated as follows:
It is clear that these operations are isomorphic, amounting to the same operation of type All that remains is to see how this operation on coefficient values in induces the corresponding operations on sets and terms.
The term determines a selection of individuals from the universe of discourse that may be computed by means of the corresponding operation on coefficient matrices. If the terms and are represented by the matrices and respectively, then the operation on terms that produces the term must be represented by a corresponding operation on matrices, say, that produces the matrix In other words, the involution operation on matrices must be defined in such a way that the following equations hold:
The fact that denotes the elements of a subset of means that the matrix is a 1-dimensional array of coefficients in that is indexed by the elements of The value of the matrix at the index is written and computed as follows: