I think the reader is beginning to get an inkling of the crucial importance of the “number of” function in Peirce’s way of looking at logic. Among other things it is one of the planks in the bridge from logic to the theories of probability, statistics, and information, in which setting logic forms but a limiting case at one scenic turnout on the expanding vista. It is, as a matter of necessity and a matter of fact, practically speaking at any rate, one way that Peirce forges a link between the eternal, logical, or rational realm and the secular, empirical, or real domain.
With that little bit of encouragement and exhortation, let us return to the nitty gritty details of the text.
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 is to say that is part of just as to write is to say that Frenchmen are part of men. Indeed, if then the number of Frenchmen is less than the number of men, and if 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)
Peirce is here remarking on the principle that the measure on terms preserves or respects the prevailing implication, inclusion, or subsumption relations that impose an ordering on those terms. In these introductory passages, 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. 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.
We have a statement of the following form:
If then the number of Frenchmen is less than the number of men.
This goes into symbolic form as follows:
In this setting the on the left is a logical ordering on syntactic terms while the on the right is an arithmetic ordering on real numbers.
The question that arises in this case is whether a map between two ordered sets is order-preserving. In order to formulate the question in more general terms, we may begin with the following set-up:
Let be a set with the ordering
Let be a set with the ordering
An order relation is typically defined by a set of axioms that determines its properties. Since we have frequent occasion to view the same set in the light of several different order relations, we often resort to explicit specifications like and so on to indicate a set with a given ordering.
A map is order-preserving if and only if a statement of a particular form holds for all and in namely, the following:
The “number of” map has just this character, as exemplified in the case at hand:
Here, the on the left is read as proper inclusion, in other words, subset of but not equal to, while the on the right is read as the usual less than relation.