Peirce's 1870 “Logic Of Relatives ” • Comment 11.21

Post   : Peirce's 1870 =93Logic Of Relatives=94 =95 Comment 11.21
http://inquiryintoinquiry.com/2014/06/02/peirces-1870-logic-of-relatives-%e=
2%80%a2-comment-11-21/
Posted : June 2, 2014 at 3:00 pm
Author : Jon Awbrey

Peircers,

One more example and one more general observation and then we shall be all =
caught up with our =

homework on Peirce's =93number of=94 function.

NOF 4.4
=3D=3D=3D=3D=3D=3D=3D

<quote>

So if men are just as apt to be black as things in general,

=95 [m,][b]  =3D  [m,b],

where the difference between [m] and [m,] must not be overlooked.

</quote>(Peirce, CP 3.76)

The protasis, =93men are just as apt to be black as things in general=94, i=
s elliptic in structure, and =

presents us with a potential ambiguity.  If we had no further clue to its m=
eaning, it might be read =

as either of the following:

1.  Men are just as apt to be black as things in general are apt to be blac=
k.
2.  Men are just as apt to be black as men are apt to be things in general.

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 th=
e following paraphrase of =

what Peirce intends:

=95 Men are just as likely to be black as things in general are likely to b=
e black.

Stated in terms of the conditional probability:

=95 P(b|m)  =3D  P(b).

 From the definition of conditional probability:

=95 P(b|m)  =3D  P(b & m) / P(m).

Equivalently:

=95 P(b & m)  =3D  P(b|m)P(m).

Taking everything together, we obtain the following result:

=95 P(b & m)  =3D  P(b|m)P(m)  =3D  P(b)P(m).

This, of course, is the definition of independent events, as applied to the=
 event of being Black and =

the event of being a Man.  It seems to be the most likely guess that this i=
s the meaning of Peirce's =

statement about frequencies:

=95 [m,b]  =3D  [m,][b].

The terms of this equation can be normalized to produce the corresponding s=
tatement about probabilities:

=95 P(m & b)  =3D  P(m)P(b).

Let's see if this checks out.

Let N be the number of things in general.  In terms of Peirce=92s =93number=
 of=94 function, then, we have =

the equation [*1*] =3D N.  On the assumption that m and b are associated wi=
th independent events, we =

obtain the following sequence of equations:

=95 [m,b]  =3D  P(m & b)N
          =3D  P(m)P(b)N
          =3D  P(m)[b]
          =3D  [m,][b].

As a result, we have to interpret [m,] =3D =93the average number of men per=
 things in general=94
as P(m) =3D =93the probability of a thing in general being a man=94.  This =
seems to make sense.

Regards,

Jon

-- =


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