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Peirce’s 1870 “Logic of Relatives” • Preliminaries

PNG

Version 2.0

Absolute Terms (Monadic Relatives)

Simple Relative Terms (Dyadic Relatives)

Conjugative Terms (Higher Adic Relatives)

Version 1.0

Absolute Terms (Monadic Relatives)

Simple Relative Terms (Dyadic Relatives)

Conjugative Terms (Higher Adic Relatives)

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

Equivalent Terms v = p

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

Equivalent Terms s(m +, w) = sm +, sw

\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

PNG (Full)

Giver of a Horse to a Lover of a Woman

Giver of a Horse to a Lover of a Woman

PNG (108 px)

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

Giver of a Horse to a Lover of a Woman

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

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

PNG

Othello Universe

Othello Universe

Othello Universe • Large

Othello Universe

Othello Universe • Small

Othello Universe

Othello Column Array • Large

Othello Column Array

Othello Column Array • Small

Othello Column Array

Logical Matrix L

Logical Matrix L

Logical Matrix S

Logical Matrix S

Logical Matrix L1

Logical Matrix L1

Logical Matrix LO

Logical Matrix LO

Logical Matrix LM

Logical Matrix LM

Logical Matrix LW

Logical Matrix LW

Logical Matrix S1

Logical Matrix S1

Logical Matrix SO

Logical Matrix SO

Logical Matrix SM

Logical Matrix SM

Logical Matrix SW

Logical Matrix SW

Logical Matrix LS

Logical Matrix LS

Logical Matrix SL

Logical Matrix SL

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}