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==Selection 6==
 
==Selection 6==
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<pre>
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<blockquote>
| The Signs for Multiplication (cont.)
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<p>'''The Signs for Multiplication''' (cont.)</p>
|
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| A conjugative term like 'giver' naturally requires two correlates,
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<p>A conjugative term like 'giver' naturally requires two correlates, one denoting the thing given, the other the recipient of the gift.</p>
| one denoting the thing given, the other the recipient of the gift.
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|
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<p>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.</p>
| We must be able to distinguish, in our notation, the
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| giver of A to B from the giver to A of B, and, therefore,
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<p>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:</p>
| I suppose the signification of the letter equivalent to such
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| a relative to distinguish the correlates as first, second, third,
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<p>`g`xy</p>
| etc., so that "giver of --- to ---" and "giver to --- of ---" will
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| be expressed by different letters.
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<p>will denote a giver to ''x'' of ''y''.</p>
|
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| Let `g` denote the latter of these conjugative terms.  Then, the correlates
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<p>But according to the notation, ''x'' here multiplies ''y'', so that if we put for ''x'' owner ('o'), and for ''y'' horse (h),</p>
| or multiplicands of this multiplier cannot all stand directly after it, as is
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| usual in multiplication, but may be ranged after it in regular order, so that:
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<p>`g`'o'h</p>
|
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| `g`xy
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<p>appears to denote the giver of a horse to an owner of a horse.  But let the individual horses be ''H'', ''H''&prime;, ''H''&Prime;, etc.</p>
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| will denote a giver to x of y.
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<p>Then:</p>
|
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| But according to the notation,
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<p>''h'' = ''H'' +, ''H''&prime; +, ''H''&Prime; +, etc.</p>
| x here multiplies y, so that
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| if we put for x owner ('o'),
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<p>`g`'o'h = `g`'o'(''H'' +, ''H''&prime; +, ''H''&Prime; +, etc.)</p>
| and for y horse (h),
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|
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<p>= `g`'o' ''H'' +, `g`'o' ''H''&prime; +, `g`'o' ''H''&Prime; +, etc.</p>
| `g`'o'h
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|
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<p>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.</p>
| appears to denote the giver of a horse
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| to an owner of a horse.  But let the
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<p>This is always very important.</p>
| individual horses be H, H', H", etc.
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|
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<p>'A term multiplied by two relatives shows that the same individual is in the two relations.'</p>
| Then:
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|
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<p>If we attempt to express the giver of a horse to a lover of a woman, and for that purpose write:</p>
| h = H +, H' +, H" +, etc.
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|
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<p>`g`'l'wh,</p>
| `g`'o'h = `g`'o'(H +, H' +, H" +, etc.)
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|
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<p>we have written giver of a woman to a lover of her, and if we add brackets, thus,</p>
|          = `g`'o'H +, `g`'o'H' +, `g`'o'H" +, etc.
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|
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<p>`g`('l'w)h,</p>
| Now this last member must be interpreted as a giver
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| of a horse to the owner of 'that' horse, and this,
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<p>we abandon the associative principle of multiplication.</p>
| therefore must be the interpretation of `g`'o'h.
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|
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<p>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.</p>
| This is always very important.
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|
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<p>Then, the giver of a horse to a lover of a woman may be written:</p>
| 'A term multiplied by two relatives shows that
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the same individual is in the two relations.'
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<p>`g`<sub>12</sub>'l'<sub>1</sub>''wh'' = `g`<sub>11</sub>'l'<sub>2</sub>''hw'' = `g`<sub>2(–1)</sub>h'l'<sub>1</sub>''w''.</p>
|
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| If we attempt to express the giver of a horse to
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<p>Of course a negative number indicates that the former correlate follows the latter by the corresponding positive number.</p>
| a lover of a woman, and for that purpose write:
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|
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<p>A subjacent 'zero' makes the term itself the correlate.</p>
| `g`'l'wh,
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|
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<p>Thus,</p>
| we have written giver of a woman to a lover of her,
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| and if we add brackets, thus,
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<p>'l'<sub>0</sub></p>
|
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| `g`('l'w)h,
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<p>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:</p>
|
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| we abandon the associative principle of multiplication.
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<p>'l'<sub>0</sub> = L<sub>0</sub> +, L<sub>0</sub>&prime; +, L<sub>0</sub>&Prime; +, etc.</p>
|
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| A little reflection will show that the associative principle must
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<p>A subjacent sign of infinity may indicate that the correlate is indeterminate, so that:</p>
| in some form or other be abandoned at this point.  But while this
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| principle is sometimes falsified, it oftener holds, and a notation
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<p>'l'<sub>&infin;</sub></p>
| must be adopted which will show of itself when it holds.  We already
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| see that we cannot express multiplication by writing the multiplicand
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<p>will denote a lover of something. We shall have some confirmation of this presently.</p>
| directly after the multiplier;  let us then affix subjacent numbers after
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| letters to show where their correlates are to be found.  The first number
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<p>If the last subjacent number is a 'one' it may be omitted.  Thus we shall have:</p>
| shall denote how many factors must be counted from left to right to reach
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| the first correlate, the second how many 'more' must be counted to reach
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<p>'l'<sub>1</sub> = 'l',</p>
| the second, and so on.
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|
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<p>`g`<sub>11</sub> = `g`<sub>1</sub> = `g`.</p>
| Then, the giver of a horse to a lover of a woman may be written:
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|
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<p>This enables us to retain our former expressions 'l'w, `g`'o'h, etc.</p>
| `g`_12 'l'_1 w h  = `g`_11 'l'_2 h w  = `g`_2(-1) h 'l'_1 w.
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|
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<p>(Peirce, CP 3.69–70).</p>
| Of course a negative number indicates that
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</blockquote>
| the former correlate follows the latter
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| by the corresponding positive number.
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|
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| A subjacent 'zero' makes the term itself the correlate.
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|
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| Thus,
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|
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| 'l'_0
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|
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| denotes the lover of 'that' lover or the lover of himself, just as
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| `g`'o'h denotes that the horse is given to the owner of itself, for
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| to make a term doubly a correlate is, by the distributive principle,
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| to make each individual doubly a correlate, so that:
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|
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| 'l'_0  = L_0 +, L_0' +, L_0" +, etc.
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|
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| A subjacent sign of infinity may
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| indicate that the correlate is
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| indeterminate, so that:
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|
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| 'l'_oo
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|
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| will denote a lover of something.
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| We shall have some confirmation
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| of this presently.
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|
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| If the last subjacent number is a 'one'
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| it may be omitted.  Thus we shall have:
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|
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| 'l'_1  = 'l',
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|
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| `g`_11  = `g`_1  = `g`.
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| This enables us to retain our former expressions 'l'w, `g`'o'h, etc.
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|
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| C.S. Peirce, CP 3.69-70
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|
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| Charles Sanders Peirce,
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|"Description of a Notation for the Logic of Relatives,
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| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
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|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
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|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
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</pre>
      
===Commentary Note 6===
 
===Commentary Note 6===
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