[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 42

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| 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 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.
|
| If we have an equation of the form:
|
| xy  =  z
|
| 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:
|
| [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:
|
| ['t'][f]  =  ['t'f]
|
| holds arithmetically.
|
| So if men are just as apt to be black as things in general:
|
| [m,][b]  =  [m,b]
|
| where the difference between [m] and [m,] must not be overlooked.
|
| 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
|
| Charles Sanders Peirce,
|"Description of a Notation for the Logic of Relatives,
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).

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