## Peirce’s 1870 “Logic of Relatives” • Comment 11.21

### Peirce’s 1870 “Logic of Relatives” • Comment 11.21

One more example and one more general observation and we’ll be caught up with our homework on Peirce’s “number of” function.

#### NOF 4.3

So if men are just as apt to be black as things in general,

$[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}],$

where the difference between $[\mathrm{m}]$ and $[\mathrm{m,}]$ must not be overlooked.

(Peirce, CP 3.76)

The protasis, men are just as apt to be black as things in general, is elliptic in structure and presents us with a potential ambiguity.  If we had no further clue to its meaning, it might be read as either one of the following statements.

1. Men are just as apt to be black as things in general are apt to be black.
2. Men are just as apt to be black as men are apt to be things in general.

The second interpretation, if grammatical, is pointless to state, since it equates a proper contingency with an absolute certainty.  So I think it is safe to assume the following paraphrase of what Peirce intends.

• Men are just as likely to be black as things in general are likely to be black.

Stated in terms of conditional probability, we have the following equation.

$\mathrm{P}(\mathrm{b}|\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}).$

From the definition of conditional probability:

$\mathrm{P}(\mathrm{b}|\mathrm{m}) ~=~ \displaystyle{\mathrm{P}(\mathrm{b}\mathrm{m}) \over \mathrm{P}(\mathrm{m})}.$

Equivalently:

$\mathrm{P}(\mathrm{b}\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}|\mathrm{m})\mathrm{P}(\mathrm{m}).$

Taking everything together, we have the following result.

$\mathrm{P}(\mathrm{b}\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}|\mathrm{m})\mathrm{P}(\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b})\mathrm{P}(\mathrm{m}).$

That, of course, is the definition of independent events, as applied to the event of being Black and the event of being a Man.  We may take that as the most likely reading of Peirce’s statement about frequencies:

$[\mathrm{m,}\mathrm{b}] ~=~ [\mathrm{m,}][\mathrm{b}].$

The terms of that equation can be normalized to produce the corresponding statement about probabilities.

$\mathrm{P}(\mathrm{m}\mathrm{b}) ~=~ \mathrm{P}(\mathrm{m})\mathrm{P}(\mathrm{b}).$

Let’s see if that reading checks out.

Let $N$ be the number of things in general.  Expressed in Peirce’s notation we have the equation $[\mathbf{1}] = N.$  On the assumption that $\mathrm{m}$ and $\mathrm{b}$ are associated with independent events, we obtain the following sequence of equations.

$\begin{array}{lll} [\mathrm{m,} \mathrm{b}] & = & \mathrm{P}(\mathrm{m}\mathrm{b}) N \\[6pt] & = & \mathrm{P}(\mathrm{m}) \mathrm{P}(\mathrm{b}) N \\[6pt] & = & \mathrm{P}(\mathrm{m}) [\mathrm{b}] \\[6pt] & = & [\mathrm{m,}] [\mathrm{b}]. \end{array}$

As a result, we have to interpret $[\mathrm{m,}]$ = “the average number of men per things in general” as $\mathrm{P}(\mathrm{m})$ = “the probability of a thing in general being a man”.  That seems to make sense.

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