Changes

==Selection 11==

==Selection 11==

<pre>

<blockquote>

| The Signs for Multiplication (concl.)

<p>'''The Signs for Multiplication''' (concl.)</p>

 

| The conception of multiplication we have adopted is that of

<p>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.</p>

| the application of one relation to another.  So, a quaternion

 

| being the relation of one vector to another, the multiplication

<p>Even ordinary numerical multiplication involves the same idea, for 2&nbsp;&times;&nbsp;3 is a pair of triplets, and 3&nbsp;&times;&nbsp;2 is a triplet of pairs, where "triplet of" and "pair of" are evidently relatives.</p>

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

 

<p>If we have an equation of the form:</p>

| Even ordinary numerical multiplication involves the same idea, for

 

| 2 x 3 is a pair of triplets, and 3 x 2 is a triplet of pairs, where

: <p>''xy'' = ''z''

| "triplet of" and "pair of" are evidently relatives.

 

<p>and there are just as many x's per y as there are, ''per'' things, things of the universe, then we have also the arithmetical equation:</p>

| If we have an equation of the form:

 

: <p>[''x''][''y''] = [''z''].</p>

| xy = z

 

<p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p>

| and there are just as many x's per y as there are,

 

| 'per' things, things of the universe, then we have

: <p>['t'][f] = ['t'f]</p>

| also the arithmetical equation:

 

<p>holds arithmetically.</p>

| [x][y] = [z].

 

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

| For instance, if our universe is perfect men, and there

 

| are as many teeth to a Frenchman (perfect understood)

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

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

 

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

| ['t'][f] = ['t'f]

 

<p>It is to be observed that:</p>

| holds arithmetically.

 

: <p>[!1!]  =  `1`.</p>

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

 

<p>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''.</p>

| [m,][b] = [m,b]

 

<p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it. (Peirce, CP 3.76).</p>

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

</blockquote>

| It is to be observed that:

| [!1!]  =  `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 multiplication, has a unity, has a conception similar to that of

| admitted multiplications, and contains numerical multiplication as

| a case under it.

| C.S. Peirce, CP 3.76

|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).

</pre>

===Commentary Note 11.1===

===Commentary Note 11.1===