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But not only do the significations of '=' and '<' here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations.  Equality is, in fact, nothing but the identity of two numbers;  numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.  So, to write 5 < 7 is to say that 5 is part of 7, just as to write ''f'' < ''m'' is to say that Frenchmen are part of men.  Indeed, if ''f'' < ''m'', then the number of Frenchmen is less than the number of men, and if ''v'' = ''p'', then the number of Vice-Presidents is equal to the number of Presidents of the Senate;  so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities. (Peirce, CP 3.66).

<p>But not only do the significations of <math>~=~</math> and <math>~<~</math> here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations.  Equality is, in fact, nothing but the identity of two numbers;  numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.  So, to write <math>~5 < 7~</math> is to say that <math>~5~</math> is part of <math>~7~</math>, just as to write <math>~\mathrm{f} < \mathrm{m}~</math> is to say that Frenchmen are part of men.  Indeed, if <math>~\mathrm{f} < \mathrm{m}~</math>, then the number of Frenchmen is less than the number of men, and if <math>~\mathrm{v} = \mathrm{p}~</math>, then the number of Vice-Presidents is equal to the number of Presidents of the Senate;  so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.</p>

 

<p>(Peirce, CP 3.66).</p>

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