## Peirce’s 1870 “Logic Of Relatives” • Comment 12.3

We now have two ways of computing a logical involution that raises a dyadic relative term to the power of a monadic absolute term, for example, $\mathit{l}^\mathrm{w}$ for “lover of every woman”.

The first method operates in the medium of set theory, expressing the denotation of the term $\mathit{l}^\mathrm{w}$ as the intersection of a set of relational applications:

$\displaystyle \mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x$

The second method operates in the matrix representation, expressing the value of the matrix $\mathsf{L}^\mathsf{W}$ with respect to an argument $u$ as a product of coefficient powers:

$\displaystyle (\mathsf{L}^\mathsf{W})_u ~=~ \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v}$

Abstract formulas like these are more easily grasped with the aid of a concrete example and a picture of the relations involved.

### Example 6

Consider a universe of discourse $X$ that is subject to the following data:

$\begin{array}{*{15}{c}} X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i & \} \\[6pt] W & = & \{ & d, & f & \} \\[6pt] L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!e, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \} \end{array}$

Figure 55 shows the placement of $W$ within $X$ and the placement of $L$ within $X \times X.$

 (55)

To highlight the role of $W$ more clearly, the Figure represents the absolute term $\mathrm{w}"$ by means of the relative term $\mathrm{w},\!"$ that conveys the same information.

Computing the denotation of $\mathit{l}^\mathrm{w}$ by way of the set-theoretic formula, we can show our work as follows:

$\displaystyle \mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x ~=~ L \cdot d ~\cap~ L \cdot f ~=~ \{ c, e \} \cap \{ e, g \} ~=~ \{ e \}$

With the above Figure in mind, we can visualize the computation of $\textstyle (\mathsf{L}^\mathsf{W})_u = \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v}$ as follows:

1. Pick a specific $u$ in the bottom row of the Figure.
2. Pan across the elements $v$ in the middle row of the Figure.
3. If $u$ links to $v$ then $\mathsf{L}_{uv} = 1,$ otherwise $\mathsf{L}_{uv} = 0.$
4. If $v$ in the middle row links to $v$ in the top row then $\mathsf{W}_v = 1,$ otherwise $\mathsf{W}_v = 0.$
5. Compute the value $\mathsf{L}_{uv}^{\mathsf{W}_v} = (\mathsf{L}_{uv} \Leftarrow \mathsf{W}_v)$ for each $v$ in the middle row.
6. If any of the values $\mathsf{L}_{uv}^{\mathsf{W}_v}$ is $0$ then the product $\textstyle \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v}$ is $0,$ otherwise it is $1.$

As a general observation, we know that the value of $(\mathsf{L}^\mathsf{W})_u$ goes to $0$ just as soon as we find a $v \in X$ such that $\mathsf{L}_{uv} = 0$ and $\mathsf{W}_v = 1,$ in other words, such that $(u, v) \notin L$ but $v \in W.$  If there is no such $v$ then $(\mathsf{L}^\mathsf{W})_u = 1.$

Running through the program for each $u \in X,$ the only case that produces a non-zero result is $(\mathsf{L}^\mathsf{W})_e = 1.$  That portion of the work can be sketched as follows:

$\displaystyle (\mathsf{L}^\mathsf{W})_e ~=~ \prod_{v \in X} \mathsf{L}_{ev}^{\mathsf{W}_v} ~=~ 0^0 \cdot 0^0 \cdot 0^0 \cdot 1^1 \cdot 1^0 \cdot 1^1 \cdot 0^0 \cdot 0^0 \cdot 0^0 ~=~ 1$

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