## Peirce’s 1870 “Logic Of Relatives” • Selection 6

We continue with §3. Application of the Algebraic Signs to Logic.

The application of a relation is one of the most basic operations in Peirce’s logic. Because relation applications are so pervasive and because Peirce treats them on the pattern of algebraic multiplication, the part of §3 concerned with “The Signs for Multiplication” will occupy our attention for many days to come.

### The Signs for Multiplication (cont.)

A conjugative term like giver naturally requires two correlates, one denoting the thing given, the other the recipient of the gift.

We must be able to distinguish, in our notation, the giver of $\mathrm{A}$ to $\mathrm{B}$ from the giver to $\mathrm{A}$ of $\mathrm{B},$ and, therefore, I suppose the signification of the letter equivalent to such a relative to distinguish the correlates as first, second, third, etc., so that “giver of ── to ──” and “giver to ── of ──” will be expressed by different letters.

Let $\mathfrak{g}$ denote the latter of these conjugative terms. Then, the correlates or multiplicands of this multiplier cannot all stand directly after it, as is usual in multiplication, but may be ranged after it in regular order, so that:

$\mathfrak{g}\mathit{x}\mathit{y}$

will denote a giver to $\mathit{x}$ of $\mathit{y}.$

But according to the notation, $\mathit{x}$ here multiplies $\mathit{y},$ so that if we put for $\mathit{x}$ owner ($\mathit{o}$), and for $\mathit{y}$ horse ($\mathrm{h}$),

$\mathfrak{g}\mathit{o}\mathrm{h}$

appears to denote the giver of a horse to an owner of a horse. But let the individual horses be $\mathrm{H}, \mathrm{H}^{\prime}, \mathrm{H}^{\prime\prime}, ~\text{etc.}$

Then:

$\mathrm{h} ~=~ \mathrm{H} ~+\!\!,~ \mathrm{H}^{\prime} ~+\!\!,~ \mathrm{H}^{\prime\prime} ~+\!\!, ~\text{etc.}$

$\mathfrak{g}\mathit{o}\mathrm{h} ~=~ \mathfrak{g}\mathit{o}(\mathrm{H} ~+\!\!,~ \mathrm{H}^{\prime} ~+\!\!,~ \mathrm{H}^{\prime\prime} ~+\!\!,~ \text{etc.}) ~=~ \mathfrak{g}\mathit{o}\mathrm{H} ~+\!\!,~ \mathfrak{g}\mathit{o}\mathrm{H}^{\prime} ~+\!\!,~ \mathfrak{g}\mathit{o}\mathrm{H}^{\prime\prime} ~+\!\!, ~\text{etc.}$

Now this last member must be interpreted as a giver of a horse to the owner of that horse, and this, therefore must be the interpretation of $\mathfrak{g}\mathit{o}\mathrm{h}.$ This is always very important. A term multiplied by two relatives shows that the same individual is in the two relations.

If we attempt to express the giver of a horse to a lover of a woman, and for that purpose write:

$\mathfrak{g}\mathit{l}\mathrm{w}\mathrm{h},$

we have written giver of a woman to a lover of her, and if we add brackets, thus,

$\mathfrak{g}(\mathit{l}\mathrm{w})\mathrm{h},$

we abandon the associative principle of multiplication.

A little reflection will show that the associative principle must in some form or other be abandoned at this point. But while this principle is sometimes falsified, it oftener holds, and a notation must be adopted which will show of itself when it holds. We already see that we cannot express multiplication by writing the multiplicand directly after the multiplier; let us then affix subjacent numbers after letters to show where their correlates are to be found. The first number shall denote how many factors must be counted from left to right to reach the first correlate, the second how many more must be counted to reach the second, and so on.

Then, the giver of a horse to a lover of a woman may be written:

$\mathfrak{g}_{12} \mathit{l}_1 \mathrm{w} \mathrm{h} ~=~ \mathfrak{g}_{11} \mathit{l}_2 \mathrm{h} \mathrm{w} ~=~ \mathfrak{g}_{2(-1)} \mathrm{h} \mathit{l}_1 \mathrm{w}.$

Of course a negative number indicates that the former correlate follows the latter by the corresponding positive number.

A subjacent zero makes the term itself the correlate.

Thus,

$\mathit{l}_0$

denotes the lover of that lover or the lover of himself, just as $\mathfrak{g}\mathit{o}\mathrm{h}$ denotes that the horse is given to the owner of itself, for to make a term doubly a correlate is, by the distributive principle, to make each individual doubly a correlate, so that:

$\mathit{l}_0 ~=~ \mathit{L}_0 ~+\!\!,~ \mathit{L}_0^{\prime} ~+\!\!,~ \mathit{L}_0^{\prime\prime} ~+\!\!,~ \text{etc.}$

A subjacent sign of infinity may indicate that the correlate is indeterminate, so that:

$\mathit{l}_\infty$

will denote a lover of something. We shall have some confirmation of this presently.

If the last subjacent number is a one it may be omitted. Thus we shall have:

$\mathit{l}_1 ~=~ \mathit{l},$

$\mathfrak{g}_{11} ~=~ \mathfrak{g}_1 ~=~ \mathfrak{g}.$

This enables us to retain our former expressions $\mathit{l}\mathrm{w}, \mathfrak{g}\mathit{o}\mathrm{h}, ~\text{etc.}$

(Peirce, CP 3.69–70)

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