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==Selection 8==
 
==Selection 8==
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<pre>
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<blockquote>
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| The Signs for Multiplication (cont.)
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<p>'''The Signs for Multiplication''' (cont.)</p>
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|
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| Thus far, we have considered the multiplication of relative terms only.
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<p>Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.</p>
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| Since our conception of multiplication is the application of a relation,
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| we can only multiply absolute terms by considering them as relatives.
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<p>Now the absolute term "man" is really exactly equivalent to the relative term "man that is ---", and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term.</p>
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|
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| Now the absolute term "man" is really exactly equivalent to
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<p>Then:</p>
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| the relative term "man that is ---", and so with any other.
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| I shall write a comma after any absolute term to show that
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: <p>"man that is black"</p>
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| it is so regarded as a relative term.
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|
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<p>will be written</p>
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| Then:
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|
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: <p>m,b.</p>
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| "man that is black"
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|
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<p>But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more.  It is convenient to take this additional correlate as the first one.</p>
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| will be written
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|
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<p>Then:</p>
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| m,b.
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|
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: <p>'l','s'w</p>
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| But not only may any absolute term be thus regarded as a relative term,
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| but any relative term may in the same way be regarded as a relative with
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<p>will denote a lover of a woman that is a servant of that woman.</p>
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| one correlate more.  It is convenient to take this additional correlate
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| as the first one.
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<p>The comma here after 'l' should not be considered as altering at all the meaning of 'l', but as only a subjacent sign, serving to alter the arrangement of the correlates.</p>
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|
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| Then:
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<p>In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates.</p>
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|
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| 'l','s'w
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<p>So:</p>
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|
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| will denote a lover of a woman that is a servant of that woman.
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: <p>m,,b,r</p>
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|
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| The comma here after 'l' should not be considered as altering at
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<p>interpreted like</p>
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| all the meaning of 'l', but as only a subjacent sign, serving to
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| alter the arrangement of the correlates.
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: <p>`g`'o'h</p>
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|
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| In point of fact, since a comma may be added in this way to any
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<p>means a man that is a rich individual and is a black that is that rich individual.</p>
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| relative term, it may be added to one of these very relatives
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| formed by a comma, and thus by the addition of two commas
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<p>But this has no other meaning than:</p>
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| an absolute term becomes a relative of two correlates.
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|
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: <p>m,b,r</p>
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| So:
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|
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<p>or a man that is a black that is rich.</p>
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| m,,b,r
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|
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<p>Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number.</p>
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| interpreted like
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|
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<p>If, therefore, 'l',,'s'w is not the same as 'l','s'w (as it plainly is not, because the latter means a lover and servant of a woman, and the former a lover of and servant of and same as a woman), this is simply because the writing of the comma alters the arrangement of the correlates.</p>
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| `g`'o'h
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|
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<p>And if we are to suppose that absolute terms are multipliers at all (as mathematical generality demands that we should}, we must regard every term as being a relative requiring an infinite number of correlates to its virtual infinite series "that is --- and is --- and is --- etc."</p>
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| means a man that is a rich individual and
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| is a black that is that rich individual.
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<p>Now a relative formed by a comma of course receives its subjacent numbers like any relative, but the question is, What are to be the implied subjacent numbers for these implied correlates?</p>
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|
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| But this has no other meaning than:
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<p>Any term may be regarded as having an infinite number of factors, those at the end being 'ones', thus:</p>
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|
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| m,b,r
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: <p>'l','s'w = 'l','s'w,!1!,!1!,!1!,!1!,!1!,!1!,!1!, etc.</p>
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|
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| or a man that is a black that is rich.
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<p>A subjacent number may therefore be as great as we please.</p>
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|
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| Thus we see that, after one comma is added, the
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<p>But all these 'ones' denote the same identical individual denoted by w;  what then can be the subjacent numbers to be applied to 's', for instance, on account of its infinite "that is"'s?  What numbers can separate it from being identical with w?  There are only two. The first is 'zero', which plainly neutralizes a comma completely, since</p>
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| addition of another does not change the meaning
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| at all, so that whatever has one comma after it
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: <p>'s',_0 w = 's'w</p>
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| must be regarded as having an infinite number.
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|
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<p>and the other is infinity;  for as 1<sup>&infin;</sup> is indeterminate in ordinary algbra, so it will be shown hereafter to be here, so that to remove the correlate by the product of an infinite series of 'ones' is to leave it indeterminate.</p>
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| If, therefore, 'l',,'s'w is not the same as 'l','s'w (as it plainly is not,
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| because the latter means a lover and servant of a woman, and the former a
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<p>Accordingly,</p>
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| lover of and servant of and same as a woman), this is simply because the
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| writing of the comma alters the arrangement of the correlates.
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: <p>m,<sub>&infin;</sub></p>
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|
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| And if we are to suppose that absolute terms are multipliers
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<p>should be regarded as expressing 'some' man.</p>
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| at all (as mathematical generality demands that we should},
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| we must regard every term as being a relative requiring
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<p>Any term, then, is properly to be regarded as having an infinite number of commas, all or some of which are neutralized by zeros.</p>
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| an infinite number of correlates to its virtual infinite
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| series "that is --- and is --- and is --- etc."
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<p>"Something" may then be expressed by:</p>
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|
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| Now a relative formed by a comma of course receives its
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: <p>!1!<sub>&infin;</sub>.</p>
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| subjacent numbers like any relative, but the question is,
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| What are to be the implied subjacent numbers for these
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<p>I shall for brevity frequently express this by an antique figure one (`1`).</p>
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| implied correlates?
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|
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<p>"Anything" by:</p>
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| Any term may be regarded as having an
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| infinite number of factors, those
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: <p>!1!<sub>0</sub>.</p>
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| at the end being 'ones', thus:
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|
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<p>I shall often also write a straight 1 for 'anything'.</p>
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| 'l','s'w = 'l','s'w,!1!,!1!,!1!,!1!,!1!,!1!,!1!, etc.
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|
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<p>(Peirce, CP 3.73).</p>
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| A subjacent number may therefore be as great as we please.
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|
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| But all these 'ones' denote the same identical individual denoted
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| by w;  what then can be the subjacent numbers to be applied to 's',
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| for instance, on account of its infinite "that is"'s?  What numbers
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| can separate it from being identical with w?  There are only two.
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| The first is 'zero', which plainly neutralizes a comma completely,
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| since
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|
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| 's',_0 w = 's'w
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|
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| and the other is infinity;  for as 1^oo is indeterminate
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| in ordinary algbra, so it will be shown hereafter to be
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| here, so that to remove the correlate by the product of
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| an infinite series of 'ones' is to leave it indeterminate.
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|
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| Accordingly,
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|
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| m,_oo
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|
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| should be regarded as expressing 'some' man.
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|
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| Any term, then, is properly to be regarded as having an infinite
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| number of commas, all or some of which are neutralized by zeros.
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|
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| "Something" may then be expressed by:
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|
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| !1!_oo.
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|
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| I shall for brevity frequently express this by an antique figure one (`1`).
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|
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| "Anything" by:
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|
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| !1!_0.
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|
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| I shall often also write a straight 1 for 'anything'.
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|
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| C.S. Peirce, CP 3.73
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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).
  −
</pre>
      
===Commentary Note 8.1===
 
===Commentary Note 8.1===
Jon Awbrey
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