[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 6

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| The Signs for Multiplication (cont.)
|
| A conjugative term like 'giver' naturally requires two correlates,
| one denoting the thing given, the other the recipient of the gift.
|
| We must be able to distinguish, in our notation, the
| giver of A to B from the giver to A of B, and, therefore,
| I suppose the signification of the letter equivalent to such
| a relative to distinguish the correlates as first, second, third,
| etc., so that "giver of --- to ---" and "giver to --- of ---" will
| be expressed by different letters.
|
| Let `g` denote the latter of these conjugative terms.  Then, the correlates
| or multiplicands of this multiplier cannot all stand directly after it, as is
| usual in multiplication, but may be ranged after it in regular order, so that:
|
| `g`xy
|
| will denote a giver to x of y.
|
| But according to the notation,
| x here multiplies y, so that
| if we put for x owner ('o'),
| and for y horse (h),
|
| `g`'o'h
|
| appears to denote the giver of a horse
| to an owner of a horse.  But let the
| individual horses be H, H', H", etc.
|
| Then:
|
| h  =  H +, H' +, H" +, etc.
|
| `g`'o'h  =  `g`'o'(H +, H' +, H" +, etc.)
|
|          =  `g`'o'H +, `g`'o'H' +, `g`'o'H" +, etc.
|
| Now this last member must be interpreted as a giver
| of a horse to the owner of 'that' horse, and this,
| therefore must be the interpretation of `g`'o'h.
|
| This is always very important.
|
| 'A term multiplied by two relatives shows that
|  the same individual is in the two relations.'
|
| If we attempt to express the giver of a horse to
| a lover of a woman, and for that purpose write:
|
| `g`'l'wh,
|
| we have written giver of a woman to a lover of her,
| and if we add brackets, thus,
|
| `g`('l'w)h,
|
| we abandon the associative principle of multiplication.
|
| A little reflection will show that the associative principle must
| in some form or other be abandoned at this point.  But while this
| principle is sometimes falsified, it oftener holds, and a notation
| must be adopted which will show of itself when it holds.  We already
| see that we cannot express multiplication by writing the multiplicand
| directly after the multiplier;  let us then affix subjacent numbers after
| letters to show where their correlates are to be found.  The first number
| shall denote how many factors must be counted from left to right to reach
| the first correlate, the second how many 'more' must be counted to reach
| the second, and so on.
|
| Then, the giver of a horse to a lover of a woman may be written:
|
| `g`_12 'l'_1 w h  =  `g`_11 'l'_2 h w  =  `g`_2(-1) h 'l'_1 w.
|
| Of course a negative number indicates that
| the former correlate follows the latter
| by the corresponding positive number.
|
| A subjacent 'zero' makes the term itself the correlate.
|
| Thus,
|
| 'l'_0
|
| denotes the lover of 'that' lover or the lover of himself, just as
| `g`'o'h denotes that the horse is given to the owner of itself, for
| to make a term doubly a correlate is, by the distributive principle,
| to make each individual doubly a correlate, so that:
|
| 'l'_0  =  L_0 +, L_0' +, L_0" +, etc.
|
| A subjacent sign of infinity may
| indicate that the correlate is
| indeterminate, so that:
|
| 'l'_oo
|
| will denote a lover of something.
| We shall have some confirmation
| of this presently.
|
| If the last subjacent number is a 'one'
| it may be omitted.  Thus we shall have:
|
| 'l'_1  =  'l',
|
| `g`_11  =  `g`_1  =  `g`.
|
| This enables us to retain our former expressions 'l'w, `g`'o'h, etc.
|
| C.S. Peirce, CP 3.69-70
|
| 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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