ONT Re: Logic Of Relatives
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LOR. Note 63
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One more example and one more general observation, and then we will
be all caught up with our homework on Peirce's "number of" function.
| So if men are just as apt to be black as things in general:
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| [m,][b] = [m,b]
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| where the difference between [m] and [m,] must not be overlooked.
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| C.S. 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:
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 this paraphrase of what Peirce intends:
3. Men are just as likely to be black as things in general are likely to be black.
Stated in terms of the conditional probability:
4. P(b|m) = P(b)
From the definition of conditional probability:
5. P(b|m) = P(b m)/P(m)
Equivalently:
6. P(b m) = P(b|m)P(m)
Thus we may derive the equivalent statement:
7. P(b m) = P(b|m)P(m) = P(b)P(m)
And this, of course, is the definition of independent events, as
applied to the event of being Black and the event of being a Man.
It seems like a likely guess, then, that this is the content of Peirce's
statement about frequencies, [m,b] = [m,][b], in this case normalized to
produce the equivalent statement about probabilities, P(m b) = P(m)P(b).
Let's see if this checks out.
Let n be the number of things in general, in Peirce's lingo, n = [1].
On the assumption that m and b are associated with independent events,
we get [m,b] = P(m b)n = P(m)P(b)n = P(m)[b] = [m,][b], so we have to
interpret [m,] = "the average number of men per things in general" as
P(m) = the probability of a thing in general being a man. Seems okay.
Then again, it is too nigh unto midnight ...
Jon Awbrey
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