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<p>But not only do the significations of &nbsp;<math>~=~</math>&nbsp; and &nbsp;<math>~<~</math>&nbsp; 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>But not only do the significations of &nbsp;<math>=\!</math>&nbsp; and &nbsp;<math><\!</math>&nbsp; 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>

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

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Peirce is here remarking on the principle that the measure ''v'' on terms "preserves" or "respects" the prevailing implication, inclusion, or subsumption relations that impose an ordering on those terms.

Peirce is here remarking on the principle that the measure <math>\mathit{v}\!</math> on terms ''preserves'' or ''respects'' the prevailing implication, inclusion, or subsumption relations that impose an ordering on those terms.  In these initiatory passages of the text, Peirce is using a single symbol &nbsp;<math><\!</math>&nbsp; to denote the usual linear ordering on numbers, but also what amounts to the implication ordering on logical terms and the inclusion ordering on classes.  Later, of course, he will introduce distinctive symbols for logical orders.  The links among terms, sets, and numbers can be pursued in all directions, and Peirce has already indicated in an earlier paper how he would construct the integers from sets, that is, from the aggregate denotations of terms.  I will try to get back to that another time.

In these initiatory passages of the text, Peirce is using a single symbol "<" to denote the usual linear ordering on numbers, but also what amounts to the implication ordering on logical terms and the inclusion ordering on classes.  Later, of course, he will introduce distinctive symbols for logical orders.

We have a statement of the following form:

Now, the links among terms, sets, and numbers can be pursued in all directions, and Peirce has already indicated in an earlier paper how he would "construct" the integers from sets, that is, from the aggregate denotations of terms.

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| If <math>\mathrm{f} < \mathrm{m},\!</math> then the number of Frenchmen is less than the number of men.

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| <math>\mathrm{f} < \mathrm{m} ~\Rightarrow~ [\mathrm{f}] < [\mathrm{m}].</math>

: "if ''f'' < ''m'', then the number of Frenchmen is less than the number of men"

 

 

: ''f'' < ''m'' &rArr; [''f''] < [''m'']

Here, the "<" on the left is a logical ordering on syntactic terms while the "<" on the right is an arithmetic ordering on real numbers.

In this setting the <math>^{\backprime\backprime}\!\!<\!^{\prime\prime}</math> on the left is a logical ordering on syntactic terms while the <math>^{\backprime\backprime}\!\!<\!^{\prime\prime}</math> on the right is an arithmetic ordering on real numbers.

The type of principle that comes up here is usually discussed under the question of whether a map between two ordered sets is "order-preserving" or not.  The general type of question may be formalized in the following way.

The type of principle that comes up here is usually discussed under the question of whether a map between two ordered sets is ''order-preserving'' or not.  The general type of question may be formalized in the following way.

: Let ''X''₁ be a set with an ordering denoted by "&lt;₁".

: Let ''X''₁ be a set with an ordering denoted by "&lt;₁".