Changes

===Commentary Note 11.23===

===Commentary Note 11.23===

Let me try to sum up as succinctly as possible the lesson that we ought to take away from Peirce's last "number of" example, since I know that the account that I have given of it so far may appear to have wandered rather widely.

Let me try to sum up as succinctly as possible the lesson

 

that we ought to take away from Peirce's last "number of"

example, since I know that the account that I have given

of it so far may appear to have wandered rather widely.

 

 

<p>C.S. Peirce, CP 3.76</p>

| C.S. Peirce, CP 3.76

In different lights the formula [m,b] = [m,][b] presents itself

In different lights the formula [''m'',''b''] = [''m'',][''b''] presents itself as an "aimed arrow", "fair sample", or "independence" condition. I had taken the tack of illustrating this polymorphous theme in bas relief, that is, via detour through a universe of discourse where it fails.  Here's a brief reminder of the Othello example:

as an "aimed arrow", "fair sample", or "independence" condition.

I had taken the tack of illustrating this polymorphous theme in

bas relief, that is, via detour through a universe of discourse

where it fails.  Here's a brief reminder of the Othello example:

B  C  D  E  I  J  O

B  C  D  E  I  J  O

o  o  o  o  o  o  o  1

o  o  o  o  o  o  o  1

o  o  o  o  o  o  o  1

o  o  o  o  o  o  o  1

B  C  D  E  I  J  O

B  C  D  E  I  J  O

The condition, "men are just as apt to be black as things in general",

The condition, "men are just as apt to be black as things in general", is expressible in terms of conditional probabilities as P(''b''|''m'') = P(''b''), written out, the probability of the event Black given the event Male is exactly equal to the unconditional probability of the event Black.

is expressible in terms of conditional probabilities as P(b|m) = P(b),

written out, the probability of the event Black given the event Male

is exactly equal to the unconditional probability of the event Black.

Thus, for example, it is sufficient to observe in the Othello setting

Thus, for example, it is sufficient to observe in the Othello setting that P(''b''|''m'') = 1/4 while P(''b'') = 1/7 in order to cognize the dependency, and thereby to tell that the ostensible arrow is anaclinically biased.

that P(b|m) = 1/4 while P(b) = 1/7 in order to cognize the dependency,

and thereby to tell that the ostensible arrow is anaclinically biased.

This reduction of a conditional probability to an absolute probability,

This reduction of a conditional probability to an absolute probability, in the form P(''A''|''Z'') = P(''A''), is a familiar disguise, and yet in practice one of the ways that we most commonly come to recognize the condition of independence P(''AZ'') = P(''A'')P(''Z''), via the definition of a conditional probability according to the rule P(''A''|''Z'') = P(''AZ'')/P(''Z'').  To recall the familiar consequences, the definition of conditional probability plus the independence condition yields P(''A''|''Z'') = P(''AZ'')/P(''Z'') = P(''A'')P(''Z'')/P(''Z''), to wit, P(''A''|''Z'') = P(''A'').

in the form P(A|Z) = P(A), is a familiar disguise, and yet in practice

one of the ways that we most commonly come to recognize the condition

of independence P(AZ) = P(A)P(Z), via the definition of a conditional

probability according to the rule P(A|Z) = P(AZ)/P(Z).  To recall the

familiar consequences, the definition of conditional probability plus

the independence condition yields P(A|Z) = P(AZ)/P(Z) = P(A)P(Z)/P(Z),

to wit, P(A|Z) = P(A).

As Hamlet discovered, there's a lot to be learned from turning a crank.

As Hamlet discovered, there's a lot to be learned from turning a crank.

===Commentary Note 11.24===

===Commentary Note 11.24===