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<p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p>

<p>The conception of multiplication we have adopted is that of the application of one relation to another. &hellip;</p>

<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 "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''</p>

| 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''] = [''tf'']</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>

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