## Peirce’s 1870 “Logic of Relatives” • Comment 8.3

### Peirce’s 1870 “Logic of Relatives” • Comment 8.3

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

It is critically important to distinguish a relation from a relative term.

• The relation is an object of thought which may be regarded in extension as a set of ordered tuples known as its elementary relations.
• The relative term is a sign which denotes certain objects, called its relates, as these are determined in relation to certain other objects, called its correlates.  Under most circumstances the relative term may be taken to denote the corresponding relation.

Returning to the Othello example, let us consider the dyadic relatives $^{\backprime\backprime} \text{lover of}\, \underline{~~~~}\, ^{\prime\prime}$ and $^{\backprime\backprime} \text{servant of}\, \underline{~~~~}\, ^{\prime\prime}.$

The relative term $\mathit{l}$ equivalent to the rhematic expression $^{\backprime\backprime} \text{lover of}\, \underline{~~~~}\, ^{\prime\prime}$ is given by the following equation.

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

In the interests of simplicity, let’s put aside all distinctions of rank and fealty, collapsing the motley crews of servant and subordinate under the heading of a single service, denoted by the relative term $\mathit{s}$ for $^{\backprime\backprime} \text{servant of}\, \underline{~~~~}\, ^{\prime\prime}.$  The terms of this unified service are given by the following equation.

$\begin{array}{*{11}{c}} \mathit{s} & = & \mathrm{C} \!:\! \mathrm{O} & +\!\!, & \mathrm{E} \!:\! \mathrm{D} & +\!\!, & \mathrm{I} \!:\! \mathrm{O} & +\!\!, & \mathrm{J} \!:\! \mathrm{D} & +\!\!, & \mathrm{J} \!:\! \mathrm{O} \end{array}$

The inclusion of $\mathrm{I} \!:\! \mathrm{C}$ under $\mathit{s}$ might be implied by the plot of the play but since it is so hotly arguable I will leave it out of the toll.

One thing more we need to watch out for:  There are different conventions in the field regarding the ordering of terms in their applications and different conventions are more convenient under different circumstances, so there’s little chance any one of them can be canonized once and for all.  In our current reading we apply relative terms from right to left and our conception of relative multiplication, or relational composition, needs to be adjusted accordingly.

### Resources

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