We continue with §3. Application of the Algebraic Signs to Logic.
The Signs for Multiplication (concl.)
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 
If we have an equation of the form:
and there are just as many 
![[x][y] ~=~ [z].](https://s0.wp.com/latex.php?latex=%5Bx%5D%5By%5D+%7E%3D%7E+%5Bz%5D.&bg=ffffff&fg=000000&s=1&c=20201002 "[x][y] ~=~ [z].")
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:
![[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7Bt%7D%5D%5B%5Cmathrm%7Bf%7D%5D+%7E%3D%7E+%5B%5Cmathit%7Bt%7D%5Cmathrm%7Bf%7D%5D&bg=ffffff&fg=000000&s=1&c=20201002 "[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]")
holds arithmetically.
So if men are just as apt to be black as things in general:
![[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bm%2C%7D%5D%5B%5Cmathrm%7Bb%7D%5D+%7E%3D%7E+%5B%5Cmathrm%7Bm%2C%7D%5Cmathrm%7Bb%7D%5D&bg=ffffff&fg=000000&s=1&c=20201002 "[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}]")
where the difference between ![[\mathrm{m}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bm%7D%5D&bg=ffffff&fg=000000&s=1&c=20201002 "[\mathrm{m}]")
![[\mathrm{m,}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bm%2C%7D%5D&bg=ffffff&fg=000000&s=1&c=20201002 "[\mathrm{m,}]")
It is to be observed that:
![[\mathit{1}] ~=~ \mathfrak{1}.](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7B1%7D%5D+%7E%3D%7E+%5Cmathfrak%7B1%7D.&bg=ffffff&fg=000000&s=1&c=20201002 "[\mathit{1}] ~=~ \mathfrak{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.
(Peirce, CP 3.76)
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