## Peirce’s 1870 “Logic Of Relatives” • Comment 8.5

Since multiplication by a dyadic relative term is a logical analogue of matrix multiplication in linear algebra, all of the products that we computed above can be represented in terms of logical matrices, that is, arrays of boolean $\{ 0, 1 \}$ coordinate values. Absolute terms and dyadic relatives are represented as 1-dimensional and 2-dimensional arrays, respectively.

The equations defining the absolute terms are given again below, first as logical sums of individual terms and then as n-tuples of boolean coordinates.

$\begin{array}{ccr*{11}{c}l} \mathbf{1} & = & \mathrm{B} & +\!\!, & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[4pt] & = & (1 & , & 1 & , & 1 & , & 1 & , & 1 & , & 1 & , & 1) \\[20pt] \mathrm{b} & = & & & & & & & & & & & & & \mathrm{O} \\[4pt] & = & (0 & , & 0 & , & 0 & , & 0 & , & 0 & , & 0 & , & 1) \\[20pt] \mathrm{m} & = & & & \mathrm{C} & & & & & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[4pt] & = & (0 & , & 1 & , & 0 & , & 0 & , & 1 & , & 1 & , & 1) \\[20pt] \mathrm{w} & = & \mathrm{B} & & & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & & & & & & \\[4pt] & = & (1 & , & 0 & , & 1 & , & 1 & , & 0 & , & 0 & , & 0) \end{array}$

Since we are going to be regarding these tuples as column arrays, it is convenient to arrange them into a table of the following form:

$\begin{array}{c|cccc} & \mathbf{1} & \mathrm{b} & \mathrm{m} & \mathrm{w} \\ \hline \mathrm{B} & 1 & 0 & 0 & 1 \\ \mathrm{C} & 1 & 0 & 1 & 0 \\ \mathrm{D} & 1 & 0 & 0 & 1 \\ \mathrm{E} & 1 & 0 & 0 & 1 \\ \mathrm{I} & 1 & 0 & 1 & 0 \\ \mathrm{J} & 1 & 0 & 1 & 0 \\ \mathrm{O} & 1 & 1 & 1 & 0 \end{array}$

Here are the dyadic relative terms again, followed by their representation as coefficient matrices, in this case bordered by row and column labels to remind us what the coefficient values are meant to signify.

$\begin{array}{*{13}{c}} \mathit{l} & = & \mathrm{B\!:\!C} & +\!\!, & \mathrm{C\!:\!B} & +\!\!, & \mathrm{D\!:\!O} & +\!\!, & \mathrm{E\!:\!I} & +\!\!, & \mathrm{I\!:\!E} & +\!\!, & \mathrm{O\!:\!D} \end{array}$

$\begin{array}{c|*{7}{c}} \mathit{l} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{I} & \mathrm{J} & \mathrm{O} \\ \hline \mathrm{B} & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ \mathrm{C} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathrm{D} & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \mathrm{E} & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ \mathrm{I} & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \mathrm{J} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathrm{O} & 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{array}$

$\begin{array}{*{13}{c}} \mathit{s} & = & \mathrm{C\!:\!O} & +\!\!, & \mathrm{E\!:\!D} & +\!\!, & \mathrm{I\!:\!O} & +\!\!, & \mathrm{J\!:\!D} & +\!\!, & \mathrm{J\!:\!O} \end{array}$

$\begin{array}{c|*{7}{c}} \mathit{s} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{I} & \mathrm{J} & \mathrm{O} \\ \hline \mathrm{B} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathrm{C} & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \mathrm{D} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathrm{E} & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ \mathrm{I} & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \mathrm{J} & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ \mathrm{O} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}$

Here are the matrix representations of the products that we calculated before:

$\begin{matrix} \mathit{l}\mathbf{1} & = & \text{lover of anything} & = \end{matrix} \\[10pt] \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix}1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 0 \\ 1\end{bmatrix}$

$\begin{matrix} \mathit{l}\mathrm{b} & = & \text{lover of a black} & = \end{matrix} \\[10pt] \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix}0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1\end{bmatrix} = \begin{bmatrix}0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0\end{bmatrix}$

$\begin{matrix} \mathit{l}\mathrm{m} & = & \text{lover of a man} & = \end{matrix} \\[10pt] \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0\end{bmatrix}$

$\begin{matrix} \mathit{l}\mathrm{w} & = & \text{lover of a woman} & = \end{matrix} \\[10pt] \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix}1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0\end{bmatrix} = \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 0 \\ 1\end{bmatrix}$

$\begin{matrix} \mathit{s}\mathbf{1} & = & \text{servant of anything} & = \end{matrix} \\[10pt] \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix}1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}0 \\ 1 \\ 0 \\ 1 \\ 1 \\ 1 \\ 0\end{bmatrix}$

$\begin{matrix} \mathit{s}\mathrm{b} & = & \text{servant of a black} & = \end{matrix} \\[10pt] \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix}0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1\end{bmatrix} = \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0\end{bmatrix}$

$\begin{matrix} \mathit{s}\mathrm{m} & = & \text{servant of a man} & = \end{matrix} \\[10pt] \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0\end{bmatrix}$

$\begin{matrix} \mathit{s}\mathrm{w} & = & \text{servant of a woman} & = \end{matrix} \\[10pt] \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix}1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0\end{bmatrix} = \begin{bmatrix}0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 1 \\ 0\end{bmatrix}$

$\begin{matrix} \mathit{l}\mathit{s} & = & \text{lover of a servant of ---} & = \end{matrix} \\[10pt] \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$

$\begin{matrix} \mathit{s}\mathit{l} & = & \text{servant of a lover of ---} & = \end{matrix} \\[10pt] \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$