Peirce's 1870 “Logic Of Relatives ” • Selection 11

Post   : Peirce's 1870 =93Logic Of Relatives=94 =95 Selection 11
http://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-%e=
2%80%a2-selection-11/
Posted : April 29, 2014 at 12:30 am
Author : Jon Awbrey

Peircers,

Among other things, we may note in this section the
roots of a connection between logic and measurement.

We continue with =A73. Application of the Algebraic Signs to Logic.

<quote>

The Signs for Multiplication (concl.)
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D

The conception of multiplication we have adopted is that of the application=
 of one relation to =

another.  So, a quaternion being the relation of one vector to another, the=
 multiplication of =

quaternions is the application of one such relation to a second.

Even ordinary numerical multiplication involves the same idea, for 2 =D7 3 =
is a pair of triplets, and =

3 =D7 2 is a triplet of pairs, where =93triplet of=94 and =93pair of=94 are=
 evidently relatives.

If we have an equation of the form:

xy =3D z

and there are just as many x=92s per y as there are, ''per'' things, things=
 of the universe, then we =

have also the arithmetical equation:

[x][y] =3D [z].

For instance, if our universe is perfect men, and there are as many teeth t=
o a Frenchman (perfect =

understood) as there are to any one of the universe, then:

[t][f] =3D [tf]

holds arithmetically.

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

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

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

It is to be observed that:

[_1_] =3D *1*.

Boole was the first to show this connection between logic and probabilities=
.  He was restricted, =

however, to absolute terms.  I do not remember having seen any extension of=
 probability to =

relatives, except the ordinary theory of ''expectation''.

Our logical multiplication, then, satisfies the essential conditions of mul=
tiplication, has a unity, =

has a conception similar to that of admitted multiplications, and contains =
numerical multiplication =

as a case under it.

</quote>(Peirce, CP 3.76)

Regards,

Jon

-- =


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