Peirce’s 1870 “Logic Of Relatives” • Comment 12.1

To get a better sense of why the above formulas mean what they do, and to prepare the ground for understanding more complex relational expressions, it will help to assemble the following materials and definitions:

$X$ is a set singled out in a particular discussion as the universe of discourse.

$W \subseteq X$ is the monadic relation, or set, whose elements fall under the absolute term $\mathrm{w} = \text{woman}.$ The elements of $W$ are referred to as the denotation or the extension of the term $\mathrm{w}.$

$L \subseteq X \times X$ is the dyadic relation associated with the relative term $\mathit{l} = \text{lover of}\,\underline{~~~~}.$

$S \subseteq X \times X$ is the dyadic relation associated with the relative term $\mathit{s} = \text{servant of}\,\underline{~~~~}.$

$\mathsf{W} = (\mathsf{W}_x) = \mathrm{Mat}(W) = \mathrm{Mat}(\mathrm{w})$ is the 1-dimensional matrix representation of the set $W$ and the term $\mathrm{w}.$

$\mathsf{L} = (\mathsf{L}_{xy}) = \mathrm{Mat}(L) = \mathrm{Mat}(\mathit{l})$ is the 2-dimensional matrix representation of the relation $L$ and the relative term $\mathit{l}.$

$\mathsf{S} = (\mathsf{S}_{xy}) = \mathrm{Mat}(S) = \mathrm{Mat}(\mathit{s})$ is the 2-dimensional matrix representation of the relation $S$ and the relative term $\mathit{s}.$

Recalling a few definitions, the local flags of the relation $L$ are given as follows:

$\begin{array}{lll} u \star L & = & L_{u\,@\,1} \\[6pt] & = & \{ (u, x) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}. \\[9pt] L \star v & = & L_{v\,@\,2} \\[6pt] & = & \{ (x, v) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that have}~ v ~\text{in the 2nd place}. \end{array}$

The applications of the relation $L$ are defined as follows:

$\begin{array}{lll} u \cdot L & = & \mathrm{proj}_2 (u \star L) \\[6pt] & = & \{ x \in X : (u, x) \in L \} \\[6pt] & = & \text{loved by}~ u. \\[9pt] L \cdot v & = & \mathrm{proj}_1 (L \star v) \\[6pt] & = & \{ x \in X : (x, v) \in L \} \\[6pt] & = & \text{lover of}~ v. \end{array}$

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