Peirce’s 1870 “Logic Of Relatives” • Comment 8.3

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

  • The relation is an object of thought that may be regarded in extension as a set of ordered tuples that are known as its elementary relations.
  • The relative term is a sign that 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 elementary relation \mathrm{I} \!:\! \mathrm{C} 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 that we need to be duly wary about: There are many different conventions in the field as to the ordering of terms in their applications and different conventions will be more convenient under different circumstances than others, so there does not appear to be much of a chance that any one of them can be canonized once and for all. In the current reading we are applying relative terms from right to left and so our conception of relative multiplication, or relational composition, will need to be adjusted accordingly.

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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 8.3

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

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