Changes

===Commentary Note 11.21===

===Commentary Note 11.21===

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.

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:

<p>So if men are just as apt to be black as things in general:</p>

: <p>[''m'',][''b''] = [''m'',''b'']</p>

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.

<p>where the difference between [''m''] and [''m'',] must not be overlooked.</p>

2. Men are just as apt to be black as men are apt to be things in general.

<p>(Peirce, CP 3.76).</p>

The second interpretation, if grammatical, is pointless to state,

since it equates a proper contingency with an absolute certainty.

 

 

 

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:

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.

: Men are just as likely to be black as things in general are likely to be black.

Stated in terms of the conditional probability:

Stated in terms of the conditional probability:

4.  P(b|m) = P(b)

: P(''b''|''m'') = P(''b'')

From the definition of conditional probability:

From the definition of conditional probability:

5.  P(b|m) = P(b m)/P(m)

: P(''b''|''m'') = P(''b'' & ''m'')/P(''m'')

Equivalently:

Equivalently:

6.  P(b m) = P(b|m)P(m)

: P(''b'' & ''m'') = P(''b''|''m'')P(''m'')

Thus we may derive the equivalent statement:

Thus we may derive the equivalent statement:

7.  P(b m) = P(b|m)P(m) = P(b)P(m)

: P(''b'' & ''m'') = P(''b''|''m'')P(''m'') = P(''b'')P(''m'')

And this, of course, is the definition of independent events, as

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.

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

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'').

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's see if this checks out.

Let n be the number of things in general, in Peirce's lingo, n = [1].

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.

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.

===Commentary Note 11.22===

===Commentary Note 11.22===