## Peirce’s 1870 “Logic Of Relatives” • Selection 8

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

### The Signs for Multiplication (cont.)

Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.

Now the absolute term “man” is really exactly equivalent to the relative term “man that is ──”, and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term.

Then “man that is black” will be written:

$\mathrm{m},\!\mathrm{b}.$

But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one.

Then:

$\mathit{l},\!\mathit{s}\mathrm{w}$

will denote a lover of a woman that is a servant of that woman.

The comma here after $\mathit{l}$ should not be considered as altering at all the meaning of $\mathit{l}\,,$ but as only a subjacent sign, serving to alter the arrangement of the correlates.

In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates.

So:

$\mathrm{m},\!,\!\mathrm{b},\!\mathrm{r}$

interpreted like

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

means a man that is a rich individual and is a black that is that rich individual.

But this has no other meaning than:

$\mathrm{m},\!\mathrm{b},\!\mathrm{r}$

or a man that is a black that is rich.

Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number.

If, therefore, $\mathit{l},\!,\!\mathit{s}\mathrm{w}$ is not the same as $\mathit{l},\!\mathit{s}\mathrm{w}$ (as it plainly is not, because the latter means a lover and servant of a woman, and the former a lover of and servant of and same as a woman), this is simply because the writing of the comma alters the arrangement of the correlates.

And if we are to suppose that absolute terms are multipliers at all (as mathematical generality demands that we should}, we must regard every term as being a relative requiring an infinite number of correlates to its virtual infinite series “that is ── and is ── and is ── etc.”

Now a relative formed by a comma of course receives its subjacent numbers like any relative, but the question is, What are to be the implied subjacent numbers for these implied correlates?

Any term may be regarded as having an infinite number of factors, those at the end being ones, thus:

$\mathit{l},\!\mathit{s}\mathrm{w} ~=~ \mathit{l},\!\mathit{s}\mathrm{w},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1}, ~\text{etc.}$

A subjacent number may therefore be as great as we please.

But all these ones denote the same identical individual denoted by $\mathrm{w};$ what then can be the subjacent numbers to be applied to $\mathit{s}\,,$ for instance, on account of its infinite “that is” ’s? What numbers can separate it from being identical with $\mathrm{w}?$ There are only two. The first is zero, which plainly neutralizes a comma completely, since

$\mathit{s},_0\!\mathrm{w} ~=~ \mathit{s}\mathrm{w}$

and the other is infinity; for as $1^\infty$ is indeterminate in ordinary algebra, so it will be shown hereafter to be here, so that to remove the correlate by the product of an infinite series of ones is to leave it indeterminate.

Accordingly,

$\mathrm{m},_\infty$

should be regarded as expressing some man.

Any term, then, is properly to be regarded as having an infinite number of commas, all or some of which are neutralized by zeros.

“Something” may then be expressed by:

$\mathit{1}_\infty.$

I shall for brevity frequently express this by an antique figure one $(\mathfrak{1}).$

“Anything” by:

$\mathit{1}_0.$

I shall often also write a straight $1$ for anything.

(Peirce, CP 3.73)

### References

• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.  Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.