Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.23
http://inquiryintoinquiry.com/2014/06/05/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-23/
Posted : June 5, 2014 at 10:48 am
Author : Jon Awbrey
Peircers,
Peirce's description of logical conjunction and conditional probability via the logic of relatives
and the mathematics of relations is critical to understanding the relationship between logic and
measurement, in effect, the qualitative and quantitative aspects of inquiry. To ground this
connection firmly in mind, I will try to sum up as succinctly as possible, in more current notation,
the lesson we ought to take away from Peirce's last “number of” example, since I know the account I
have given so far may appear to have wandered widely.
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 ''statistical independence''
condition. The concept of independence was illustrated above by means
of a case where independence fails. The details of that counterexample
are summarized below.
Figure 54. Bigraph Representation of “Man that is Black”
☞http://inquiryintoinquiry.files.wordpress.com/2014/06/lor-1870-figure-53.jpg
The condition that “men are just as apt to be black as things in general” is expressed in terms of
conditional probabilities as P(b|m) = P(b), which means that the probability of the event b given
the event m is equal to the unconditional probability of the event b.
In the ''Othello'' example, it is enough to observe that P(b|m) = 1/4 while P(b) = 1/7 in order to
recognize the bias or dependency of the sampling map.
The reduction of a conditional probability to an absolute probability, as P(A|Z) = P(A), is one of
the ways we come to recognize the condition of independence, P(AZ) = P(A)P(Z), via the definition of
a conditional probability, P(A|Z) = P(AZ)/P(Z).
To recall the derivation, the definition of conditional probability plus the independence condition
yields P(A|Z) = P(AZ)/P(Z) = P(A)P(Z)/P(Z), in short, P(A|Z) = P(A).
Regards,
Jon
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