## Peirce’s 1870 “Logic of Relatives” • Comment 8.4

### Peirce’s 1870 “Logic of Relatives” • Comment 8.4

I continue with my commentary on CP 3.73, developing the Othello example as a way of illustrating Peirce’s formalism.

To familiarize ourselves with the forms of calculation available in Peirce’s notation, let us compute a few of the simplest products we find at hand in the Othello universe.

Here are the absolute terms:

$\begin{array}{*{15}{c}} \mathbf{1} & = & \mathrm{B} & +\!\!, & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{b} & = & \mathrm{O} \\[6pt] \mathrm{m} & = & \mathrm{C} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{w} & = & \mathrm{B} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} \end{array}$

Here are the dyadic relative terms:

$\begin{array}{*{13}{c}} \mathit{l} & = & \mathrm{B} \!:\! \mathrm{C} & +\!\!, & \mathrm{C} \!:\! \mathrm{B} & +\!\!, & \mathrm{D} \!:\! \mathrm{O} & +\!\!, & \mathrm{E} \!:\! \mathrm{I} & +\!\!, & \mathrm{I} \!:\! \mathrm{E} & +\!\!, & \mathrm{O} \!:\! \mathrm{D} \\[6pt] \mathit{s} & = & \mathrm{C} \!:\! \mathrm{O} & +\!\!, & \mathrm{E} \!:\! \mathrm{D} & +\!\!, & \mathrm{I} \!:\! \mathrm{O} & +\!\!, & \mathrm{J} \!:\! \mathrm{D} & +\!\!, & \mathrm{J} \!:\! \mathrm{O} \end{array}$

Here are a few of the simplest products among those terms:

$\begin{array}{lll} \mathit{l}\mathbf{1} & = & \text{lover of anything} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{O} \\[6pt] & = & \text{anything except} ~ \mathrm{J} \end{array}$

$\begin{array}{lll} \mathit{l}\mathrm{O} & = & \text{lover of Othello} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & \mathrm{O} \\[6pt] & = & \mathrm{D} \end{array}$

$\begin{array}{lll} \mathit{l}\mathrm{m} & = & \text{lover of a man} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} \end{array}$

$\begin{array}{lll} \mathit{l}\mathrm{w} & = & \text{lover of a woman} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{O} \end{array}$

$\begin{array}{lll} \mathit{s}\mathbf{1} & = & \text{servant of anything} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} \end{array}$

$\begin{array}{lll} \mathit{s}\mathrm{O} & = & \text{servant of Othello} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & \mathrm{O} \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} \end{array}$

$\begin{array}{lll} \mathit{s}\mathrm{m} & = & \text{servant of a man} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} \end{array}$

$\begin{array}{lll} \mathit{s}\mathrm{w} & = & \text{servant of a woman} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) \\[6pt] & = & \mathrm{E} ~+\!\!,~ \mathrm{J} \end{array}$

$\begin{array}{lll} \mathit{l}\mathit{s} & = & \text{lover of a servant of}\, \underline{~~~~} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\[6pt] & = & \mathrm{B} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{D} \end{array}$

$\begin{array}{lll} \mathit{s}\mathit{l} & = & \text{servant of a lover of}\, \underline{~~~~} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\[6pt] & = & \mathrm{C} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O} \end{array}$

Among other things, one observes that the relative terms $\mathit{l}$ and $\mathit{s}$ do not commute, that is, $\mathit{l}\mathit{s}$ is not equal to $\mathit{s}\mathit{l}.$

### Resources

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