Peirce’s 1870 “Logic Of Relatives” • Comment 8.2

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

In the development of the story so far, we have a universe of discourse characterized by the following equations:

\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}

This much forms a basis for the collection of absolute terms to be used in this example. Let us now consider how we might represent an exemplary collection of relative terms.

Consider the genesis of relative terms, for example:

\begin{array}{l}  ^{\backprime\backprime}\, \text{lover of}\, \underline{~~~~}\, ^{\prime\prime}  \\[6pt]  ^{\backprime\backprime}\, \text{betrayer to}\, \underline{~~~~}\, \text{of}\, \underline{~~~~}\, ^{\prime\prime}  \\[6pt]  ^{\backprime\backprime}\, \text{winner over of}\, \underline{~~~~}\, \text{to}\, \underline{~~~~}\, \text{from}\, \underline{~~~~}\, ^{\prime\prime}  \end{array}

We may regard these fill-in-the-blank forms as being derived by a kind of rhematic abstraction from the corresponding instances of absolute terms.

The following examples illustrate the relationships that exist among absolute terms, relative terms, relations, and elementary relations.

  • The relative term ^{\backprime\backprime} \text{lover of}\, \underline{~~~~}\, ^{\prime\prime} can be derived from the absolute term ^{\backprime\backprime} \text{lover of Emilia} ^{\prime\prime} by removing the absolute term ^{\backprime\backprime} \text{Emilia} ^{\prime\prime}.

    Iago is a lover of Emilia, so the relate-correlate pair \mathrm{I} \!:\! \mathrm{E} is an element of the dyadic relation associated with the relative term ^{\backprime\backprime} \text{lover of}\, \underline{~~~~}\, ^{\prime\prime}.

  • The relative term ^{\backprime\backprime} \text{betrayer to}\, \underline{~~~~}\, \text{of}\, \underline{~~~~}\, ^{\prime\prime} can be derived from the absolute term ^{\backprime\backprime} \text{betrayer to Othello of Desdemona} ^{\prime\prime} by removing the absolute terms ^{\backprime\backprime} \text{Othello} ^{\prime\prime} and ^{\backprime\backprime} \text{Desdemona} ^{\prime\prime}.

    Iago is a betrayer to Othello of Desdemona, so the relate-correlate-correlate triple \mathrm{I} \!:\! \mathrm{O} \!:\! \mathrm{D} is an element of the triadic relation associated with the relative term ^{\backprime\backprime} \text{betrayer to}\, \underline{~~~~}\, \text{of}\, \underline{~~~~}\, ^{\prime\prime}.

  • The relative term ^{\backprime\backprime} \text{winner over of}\, \underline{~~~~}\, \text{to}\, \underline{~~~~}\, \text{from}\, \underline{~~~~}\, ^{\prime\prime} can be derived from the absolute term ^{\backprime\backprime} \text{winner over of Othello to Iago from Cassio} ^{\prime\prime} by removing the absolute terms ^{\backprime\backprime} \text{Othello} ^{\prime\prime}, ^{\backprime\backprime} \text{Iago} ^{\prime\prime}, and ^{\backprime\backprime} \text{Cassio} ^{\prime\prime}.

    Iago is a winner over of Othello to Iago from Cassio, so the elementary relative term \mathrm{I} \!:\! \mathrm{O} \!:\! \mathrm{I} \!:\! \mathrm{C} is an element of the tetradic relation associated with the relative term ^{\backprime\backprime} \text{winner over of}\, \underline{~~~~}\, \text{to}\, \underline{~~~~}\, \text{from}\, \underline{~~~~}\, ^{\prime\prime}.

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 8.2

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

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