Peirce’s 1870 “Logic Of Relatives” • Comment 8.4

To familiarize ourselves with the forms of calculation that are available in Peirce’s notation, let us compute a few of the simplest products that 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 these 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{b} & = & \text{lover of a black}  \\[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{b} & = & \text{servant of a black}  \\[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}.

This entry was posted in Logic, Logic of Relatives, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , . Bookmark the permalink.

One Response to Peirce’s 1870 “Logic Of Relatives” • Comment 8.4

  1. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

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