## Peirce’s 1870 “Logic of Relatives” • Selection 5

On to the next part of §3. Application of the Algebraic Signs to Logic.

### The Signs for Multiplication

I shall adopt for the conception of multiplication the application of a relation, in such a way that, for example, $\mathit{l}\mathrm{w}$ shall denote whatever is lover of a woman.  This notation is the same as that used by Mr. De Morgan, although he appears not to have had multiplication in his mind.

$\mathit{s}(\mathrm{m} ~+\!\!,~ \mathrm{w})$ will, then, denote whatever is servant of anything of the class composed of men and women taken together.  So that:

$\mathit{s}(\mathrm{m} ~+\!\!,~ \mathrm{w}) ~=~ \mathit{s}\mathrm{m} ~+\!\!,~ \mathit{s}\mathrm{w}.$

$(\mathit{l} ~+\!\!,~ \mathit{s})\mathrm{w}$ will denote whatever is lover or servant to a woman, and:

$(\mathit{l} ~+\!\!,~ \mathit{s})\mathrm{w} ~=~ \mathit{l}\mathrm{w} ~+\!\!,~ \mathit{s}\mathrm{w}.$

$(\mathit{s}\mathit{l})\mathrm{w}$ will denote whatever stands to a woman in the relation of servant of a lover, and:

$(\mathit{s}\mathit{l})\mathrm{w} ~=~ \mathit{s}(\mathit{l}\mathrm{w}).$

Thus all the absolute conditions of multiplication are satisfied.

The term “identical with ──” is a unity for this multiplication.  That is to say, if we denote “identical with ──” by $\mathit{1}$ we have:

$x \mathit{1} ~=~ x ~ ,$

whatever relative term $x$ may be.  For what is a lover of something identical with anything, is the same as a lover of that thing.

(Peirce, CP 3.68)

Peirce in 1870 is five years down the road from the Peirce of 1865–1866 who lectured extensively on the role of sign relations in the logic of scientific inquiry, articulating their involvement in the three types of inference, and inventing the concept of “information” to explain what it is that signs convey in the process.  By this time, then, the semiotic or sign relational approach to logic is so implicit in his way of working that he does not always take the trouble to point out its distinctive features at each and every turn.  So let’s take a moment to draw out a few of those characters.

Sign relations, like any brand of non-trivial triadic relations, can become overwhelming to think about once the cardinality of the object, sign, and interpretant domains or the complexity of the relation itself ascends beyond the simplest examples.  Furthermore, most of the strategies we’d normally use to control the complexity, like neglecting one of the domains, in effect, projecting the triadic sign relation onto one of its dyadic faces, or focusing on a single ordered triple $(o, s, i)$ at a time, can result in our receiving a distorted impression of the sign relation’s true nature and structure.

I find it helps me to draw, or at least to imagine drawing, diagrams of the following form, where I can keep tabs on what’s an object, what’s a sign, and what’s an interpretant sign, for a selected set of sign-relational triples.

Figure 1 shows how I would picture Peirce’s example of equivalent terms, $\mathrm{v} = \mathrm{p},$ where $\mathrm{v}"$ denotes the Vice-President of the United States, and $\mathrm{p}"$ denotes the President of the Senate of the United States.

$\text{Figure 1. Equivalent Terms}~ \mathrm{v}" = \mathrm{p}"$

Depending on whether we interpret the terms $\mathrm{v}"$ and $\mathrm{p}"$ as applying to persons who hold the offices at one particular time or as applying to all persons who have held the offices over an extended period of history, their denotations may be either singular of plural, respectively.

Terms referring to many objects are known as having general denotations or plural referents.  They may be represented in the above style of picture by drawing an ellipsis of three nodes like “o o o” at the object ends of sign relational triples.

For a more complicated example, Figure 2 shows how I would picture Peirce’s example of an equivalence between terms which comes about by applying the distributive law for relative multiplication over absolute summation.

$\text{Figure 2. Equivalent Terms}~ \mathit{s}(\mathrm{m} ~+\!\!,~ \mathrm{w})" = \mathit{s}\mathrm{m} ~+\!\!,~ \mathit{s}\mathrm{w}"$

### Resources

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