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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 16:30, 16 June 2013
, 16:30, 16 June 2013→The Signs for Multiplication (cont.)
<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>
<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>
<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>
<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>
<p>Then "man that is black" will be written:</p>
<p>Then “man that is black” will be written:</p>
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| align="center" | <math>\mathrm{m},\!\mathrm{b}\!</math>
| align="center" | <math>\mathrm{m},\!\mathrm{b}\!</math>
<p>If, therefore, <math>\mathit{l},\!,\!\mathit{s}\mathrm{w}</math> is not the same as <math>\mathit{l},\!\mathit{s}\mathrm{w}</math> (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>
<p>If, therefore, <math>\mathit{l},\!,\!\mathit{s}\mathrm{w}</math> is not the same as <math>\mathit{l},\!\mathit{s}\mathrm{w}</math> (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>
<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>
<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>
<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>
<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>
<p>A subjacent number may therefore be as great as we please.</p>
<p>A subjacent number may therefore be as great as we please.</p>
<p>But all these ''ones'' denote the same identical individual denoted by <math>\mathrm{w}\!</math>; what then can be the subjacent numbers to be applied to <math>\mathit{s}\!</math>, for instance, on account of its infinite "''that is''"'s? What numbers can separate it from being identical with <math>\mathrm{w}\!</math>? There are only two. The first is ''zero'', which plainly neutralizes a comma completely, since</p>
<p>But all these ''ones'' denote the same identical individual denoted by <math>\mathrm{w}\!</math>; what then can be the subjacent numbers to be applied to <math>\mathit{s}\!</math>, for instance, on account of its infinite “''that is''”'s? What numbers can separate it from being identical with <math>\mathrm{w}\!</math>? There are only two. The first is ''zero'', which plainly neutralizes a comma completely, since</p>
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| align="center" | <math>\mathit{s},_0\!\mathrm{w} ~=~ \mathit{s}\mathrm{w}</math>
| align="center" | <math>\mathit{s},_0\!\mathrm{w} ~=~ \mathit{s}\mathrm{w}</math>
<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>
<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>
<p>"Something" may then be expressed by:</p>
<p>“Something” may then be expressed by:</p>
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| align="center" | <math>\mathit{1}_\infty\!</math>
| align="center" | <math>\mathit{1}_\infty\!</math>
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<p>I shall for brevity frequently express this by an antique figure one <math>(\mathfrak{1}).</math>
<p>I shall for brevity frequently express this by an antique figure one <math>(\mathfrak{1}).</math></p>
<p>"Anything" by:</p>
<p>“Anything” by:</p>
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| align="center" | <math>\mathit{1}_0\!</math>
| align="center" | <math>\mathit{1}_0\!</math>