Changes

<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>

<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>

<p>Even ordinary numerical multiplication involves the same idea, for <math>~2 \times 3~</math> is a pair of triplets, and <math>~3 \times 2~</math> is a triplet of pairs, where ''triplet of'' and ''pair of'' are evidently relatives.</p>

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

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

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

 

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

| align="center" | <math>xy ~=~ z</math>

 

<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>

 

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

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

 

| align="center" | <math>[x][y] ~=~ [z].</math>

<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>

<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>

 

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

| align="center" | <math>[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]</math>

 

<p>holds arithmetically.</p>

<p>holds arithmetically.</p>

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

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

 

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

| align="center" | <math>[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}]</math>

 

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

<p>where the difference between <math>[\mathrm{m}]</math> and <math>[\mathrm{m,}]</math> must not be overlooked.</p>

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

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

: <p>[!1!]  =  `1`.</p>

<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.</p>

 

<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>

<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>

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

|}

|}