Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.22
http://inquiryintoinquiry.com/2014/06/04/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-22/
Posted : June 4, 2014 at 3:00 pm
Author : Jon Awbrey
Peircers,
Let’s look at that last example from a different angle.
NOF 4.4
=======
<quote>
So if men are just as apt to be black as things in general,
• [m,][b] = [m,b],
where the difference between [m] and [m,] must not be overlooked.
</quote>(Peirce, CP 3.76)
In different lights the formula [m,b] = [m,][b] presents itself as
an ''aimed arrow'', ''fair sampling'', or ''stochastic 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 are the relevant data:
• b = O
• m = C +, I +, J +, O
• 1 = B +, C +, D +, E +, I +, J +, O
• b, = O:O
• m, = C:C +, I:I +, J:J +, O:O}
• 1, = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
The fair sampling 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].
As before, it is convenient to represent the absolute term b = black by means of the corresponding
idempotent term b, = black that is ____.
Consider the bigraph for the composition:
• m,b = man that is black.
This is represented below in the equivalent form:
• m,b, = man that is black that is ____.
Figure 53. Bigraph Representation of “Man that is Black”
☞http://inquiryintoinquiry.files.wordpress.com/2014/06/lor-1870-figure-53.jpg
Thus we observe one of the more factitious facts affecting this very special universe of discourse,
namely:
• m,b = b
This is equivalent to the implication b ⇒ m that Peirce would have written in the
form b ─< m.
That 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 the general formulas and particular facts
together, we arrive at following summation of the situation in the ''Othello'' case:
If the fair sampling condition were true, it would have the following
consequence:
• [m,] = [m,b] / [b] = [b] / [b] = 1.
On the contrary, we have the following fact:
• [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 terms we would have:
• P(m & b) = 4/49.
In point of fact, however, we have:
• P(m & b) = P(b) = 1/7.
Another way to see it is to observe that:
• P(b|m) = 1/4 while P(b) = 1/7.
Regards,
Jon
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