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:
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