## Peirce’s 1870 “Logic Of Relatives” • Comment 8.1

To my way of thinking, CP 3.73 is one of the most remarkable passages in the history of logic.  In this first pass over its deeper contents I won’t be able to accord it much more than a superficial dusting off.

Let us invent a concrete example to illustrate the use of Peirce’s notation.  Imagine a discourse whose universe $X$ will remind us of the cast of characters in Shakespeare’s Othello.

$X ~=~ \{ \mathrm{Bianca}, \mathrm{Cassio}, \mathrm{Clown}, \mathrm{Desdemona}, \mathrm{Emilia}, \mathrm{Iago}, \mathrm{Othello} \}$

The universe $X$ is “that class of individuals about which alone the whole discourse is understood to run” but its marking out for special recognition as a universe of discourse in no way rules out the possibility that “discourse may run upon something which is not a subjective part of the universe;  for instance, upon the qualities or collections of the individuals it contains” (CP 3.65).

In order to provide ourselves with the convenience of abbreviated terms while preserving Peirce’s conventions about capitalization, we may use the alternate terms ${}^{\backprime\backprime}\mathrm{u} {}^{\prime\prime}$ for the universe $X$ and ${}^{\backprime\backprime} \mathrm{Jeste} {}^{\prime\prime}$ for the character $\mathrm{Clown}.$  This permits the above description of the universe of discourse to be rewritten in the following fashion:

$\mathrm{u} ~=~ \{ \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{I}, \mathrm{J}, \mathrm{O} \}$

This specification of the universe of discourse could be summed up in Peirce’s notation by the following equation:

$\begin{array}{*{15}{c}} \mathbf{1} & = & \mathrm{B} & +\!\!, & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \end{array}$

Within this discussion, then, the individual terms are as follows:

$\begin{array}{*{7}{c}} ^{\backprime\backprime}\mathrm{B}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{C}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{D}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{E}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{I}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{J}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{O}^{\prime\prime} \end{array}$

Each of these terms denotes in a singular fashion the corresponding individual in $X.$

By way of general terms in this discussion, we may begin with the following set:

$\begin{array}{ccl} ^{\backprime\backprime}\mathrm{b}^{\prime\prime} & = & ^{\backprime\backprime}\mathrm{black}^{\prime\prime} \\[6pt] ^{\backprime\backprime}\mathrm{m}^{\prime\prime} & = & ^{\backprime\backprime}\mathrm{man}^{\prime\prime} \\[6pt] ^{\backprime\backprime}\mathrm{w}^{\prime\prime} & = & ^{\backprime\backprime}\mathrm{woman}^{\prime\prime} \end{array}$

The denotation of a general term may be given by means of an equation between terms:

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

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