ONT Re: Logic Of Relatives
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LOR. Note 64
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Let's look at that last example from a different angle.
| 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
In different lights the formula [m,b] = [m,][b] presents itself
as an "aimed arrow", "fair sample", or "independence" condition.
The example apparently assumes a universe of "things in general",
encompassing among other things the denotations of the absolute
terms m = "man" and b = "black". That suggests to me that we
might well illustrate this case in relief, by returning to
our earlier staging of 'Othello' and seeing how well that
universe of dramatic discourse observes the premiss that
"men are just as apt to be black as things in general".
Here is the relevant data:
| 1 = B +, C +, D +, E +, I +, J +, O
|
| b = O
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| m = C +, I +, J +, O
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| 1, = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
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| b, = O:O
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| m, = C:C +, I:I +, J:J +, O:O
The "fair sampling" or "episkeptral arrow" condition is tantamount to this:
"Men are just as apt to be black as things in general are apt to be black".
In other words, men are a fair sample of things in general with respect to
the factor of being black.
Should this hold, the consequence would be:
[m,b] = [m,][b].
When [b] is not zero, we obtain the result:
[m,] = [m,b]/[b].
Once again, the absolute term b = "black" is most felicitously depicted
by way of its idempotent representation 'b' = b, = "black that is ---",
and thus it can be taken as a selective from the universe of discourse.
Here is the bigraph for the composition:
m,b = "man that is black",
here represented in the equivalent form:
m,b, = "man that is black that is ---".
B C D E I J O
o o o o o o o 1
| | | |
| | | | m,
| | | |
o o o o o o o 1
|
| b,
|
o o o o o o o 1
B C D E I J O
Thus we observe one of the more factitious facts
that hold in this universe of discourse, namely:
m,b = b.
Another way of saying that is:
b -< m.
That in itself is enough to puncture any notion
that b and m are statistically independent, but
let us continue to develop the plot a bit more.
Putting all of the general formulas and particular facts together,
we arrive at following summation of situation in the Othello case:
If the fair sampling condition holds:
[m,] = [m,b]/[b] = [b]/[b] = `1`,
In fact, however, it is the case that:
[m,] = [m,1]/[1] = [m]/[1] = 4/7.
In sum, it is not the case in the Othello example that
"men are just as apt to be black as things in general".
Expressed in terms of probabilities: P(m) = 4/7 and P(b) = 1/7.
If these were independent we'd have: P(mb) = 4/49.
On the contrary, P(mb) = P(b) = 1/7.
Again, all subject to the midnight judgment disclaimer ...
Jon Awbrey
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