## Peirce’s 1870 “Logic of Relatives” • Comment 12.2

### Peirce’s 1870 “Logic of Relatives” • Comment 12.2

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 $x^y$ will denote everything which is an $x$ for every individual of $y.$  Thus $\mathit{l}^\mathrm{w}$ will be a lover of every woman.

(Peirce, CP 3.77)

In ordinary arithmetic the involution $x^y,$ or the exponentiation of $x$ to the power $y,$ is the repeated application of the multiplier $x$ for as many times as there are ones making up the exponent $y.$

In analogous fashion, the logical involution $\mathit{l}^\mathrm{w}$ is the repeated application of the term $\mathit{l}$ for as many times as there are individuals under the term $\mathrm{w}.$  On Peirce’s interpretive rules, the repeated applications of the base term $\mathit{l}$ are distributed across the individuals of the exponent term $\mathrm{w}.$  In particular, the base term $\mathit{l}$ 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, $\mathrm{W}^{\prime}, \mathrm{W}^{\prime\prime}, \mathrm{W}^{\prime\prime\prime}.$  In Peirce’s notation the fact may be written as follows.

$\mathrm{w} ~=~ \mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime}$

In that case the following equation would hold.

$\mathit{l}^\mathrm{w} ~=~ \mathit{l}^{(\mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime})} ~=~ (\mathit{l}\mathrm{W}^{\prime}), (\mathit{l}\mathrm{W}^{\prime\prime}), (\mathit{l}\mathrm{W}^{\prime\prime\prime})$

The equation says a lover of every woman in the aggregate $\mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime}$ is a lover of $\mathrm{W}^{\prime}$ that is a lover of $\mathrm{W}^{\prime\prime}$ that is a lover of $\mathrm{W}^{\prime\prime\prime}.$  In other words, a lover of every woman in the universe at hand is a lover of $\mathrm{W}^{\prime}$ and a lover of $\mathrm{W}^{\prime\prime}$ and a lover of $\mathrm{W}^{\prime\prime\prime}.$

The denotation of the term $\mathit{l}^\mathrm{w}$ is a subset of $X$ which may be obtained by the following procedure.  For each flag of the form $L \star x$ with $x \in W$ collect the subset $\mathrm{proj}_1 (L \star x)$ of elements which appear as the first components of the pairs in $L \star x$ 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 $\mathit{l}^\mathrm{w}$ at this point, not the least because it effectively dispels the mystery of the name involution.  First, we make the following observation.  To say $j$ is a lover of every woman is to say $j$ loves $k$ if $k$ is a woman.  That can be rendered in symbols as follows.

$j ~\text{loves}~ k ~\Leftarrow~ k ~\text{is a woman}$

Reading the formula $\mathit{l}^\mathrm{w}$ as “$j$ loves $k$ if $k$ 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 $x^y = z ~\text{and}~ (x\!\Leftarrow\!y) = z ~\text{for}~ x, y, z$ in the boolean domain $\mathbb{B} = \{ 0, 1 \}$ are tabulated as follows.

It is clear the two operations are isomorphic, being effectively the same operation of type $\mathbb{B} \times \mathbb{B} \to \mathbb{B}.$  All that remains is to see how operations like these on values in $\mathbb{B}$ induce the corresponding operations on sets and terms.

The term $\mathit{l}^\mathrm{w}$ determines a selection of individuals from the universe of discourse $X$ which may be computed via the corresponding operation on coefficient matrices.  If the terms $\mathit{l}$ and $\mathrm{w}$ are represented by the matrices $\mathsf{L} = \mathrm{Mat}(\mathit{l})$ and $\mathsf{W} = \mathrm{Mat}(\mathrm{w}),$ respectively, then the operation on terms which produces the term $\mathit{l}^\mathrm{w}$ must be represented by a corresponding operation on matrices, $\mathsf{L}^\mathsf{W} = \mathrm{Mat}(\mathit{l})^{\mathrm{Mat}(\mathrm{w})},$ which produces the matrix $\mathrm{Mat}(\mathit{l}^\mathrm{w}).$  In short, the involution operation on matrices must be defined in such a way that the following equation holds.

The fact that $\mathit{l}^\mathrm{w}$ denotes individuals in a subset of $X$ tells us its matrix representation $\mathsf{L}^\mathsf{W}$ is a 1‑dimensional array of coefficients in $\mathbb{B}$ indexed by the elements of $X.$  The value of the matrix $\mathsf{L}^\mathsf{W}$ at the index $u$ in $X$ is written $(\mathsf{L}^\mathsf{W})_u$ and computed as follows.

### Resources

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