# Work χ

## Peirce’s 1870 “Logic of Relatives” • Preliminaries

### LaTeX

$\begin{array}{ll} \multicolumn{2}{l}{\text{Table 1.~~Absolute Terms (Monadic Relatives)}} \\[4pt] \mathrm{a}. & \text{animal} \\ \mathrm{b}. & \text{black} \\ \mathrm{f}. & \text{Frenchman} \\ \mathrm{h}. & \text{horse} \\ \mathrm{m}. & \text{man} \\ \mathrm{p}. & \text{President of the United States Senate} \\ \mathrm{r}. & \text{rich person} \\ \mathrm{u}. & \text{violinist} \\ \mathrm{v}. & \text{Vice-President of the United States} \\ \mathrm{w}. & \text{woman} \end{array}$

$\begin{array}{ll} \multicolumn{2}{l}{\text{Table 2.~~Simple Relative Terms (Dyadic Relatives)}} \\[4pt] \mathit{a}. & \text{enemy} \\ \mathit{b}. & \text{benefactor} \\ \mathit{c}. & \text{conqueror} \\ \mathit{e}. & \text{emperor} \\ \mathit{h}. & \text{husband} \\ \mathit{l}. & \text{lover} \\ \mathit{m}. & \text{mother} \\ \mathit{n}. & \text{not} \\ \mathit{o}. & \text{owner} \\ \mathit{s}. & \text{servant} \\ \mathit{w}. & \text{wife} \end{array}$

$\begin{array}{ll} \multicolumn{2}{l}{\text{Table 3.~~Conjugative Terms (Higher Adic Relatives)}} \\[4pt] \mathfrak{b}. & \text{betrayer to ------ of ------} \\ \mathfrak{g}. & \text{giver to ------ of ------} \\ \mathfrak{t}. & \text{transferrer from ------ to ------} \\ \mathfrak{w}. & \text{winner over of ------ to ------ from ------} \end{array}$

## Peirce’s 1870 “Logic of Relatives” • Selection 5

$\text{Figure 1. Equivalent Terms}~ \mathrm{v}" = \mathrm{p}"$

$\text{Figure 2. Equivalent Terms}~ \mathit{s}(\mathrm{m} ~+\!\!,~ \mathrm{w})" = \mathit{s}\mathrm{m} ~+\!\!,~ \mathit{s}\mathrm{w}"$

## Peirce’s 1870 “Logic of Relatives” • Selection 7

### Giver of a Horse to a Lover of a Woman

#### LaTeX (s=2)

$\mathfrak{g}_{\dagger\ddagger} {}^\dagger\mathit{l}_\parallel {}^\parallel\mathrm{w} {}^\ddagger\mathrm{h}.$

## Peirce’s 1870 “Logic of Relatives” • Proto-Graphical Syntax

$\text{Figure 3. Giver of a Horse to a Lover of a Woman}$

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

### LaTeX

#### Othello Universe 2.0

$\begin{array}{ccr*{11}{c}l} \mathbf{1} & = & \mathrm{B} & +\!\!, & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[2pt] & = & (1 & , & 1 & , & 1 & , & 1 & , & 1 & , & 1 & , & 1) \\[10pt] \mathrm{b} & = & & & & & & & & & & & & & \mathrm{O} \\[2pt] & = & (0 & , & 0 & , & 0 & , & 0 & , & 0 & , & 0 & , & 1) \\[10pt] \mathrm{m} & = & & & \mathrm{C} & & & & & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[2pt] & = & (0 & , & 1 & , & 0 & , & 0 & , & 1 & , & 1 & , & 1) \\[10pt] \mathrm{w} & = & \mathrm{B} & & & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & & & & & & \\[2pt] & = & (1 & , & 0 & , & 1 & , & 1 & , & 0 & , & 0 & , & 0) \end{array}$

#### Othello Universe 1.0

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

#### Othello Column Array

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

#### Logical Matrix L

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

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

#### Logical Matrix S

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

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

#### Logical Matrix Products

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

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

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

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

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

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

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

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

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

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