Changes

===Commentary Note 11.17===

===Commentary Note 11.17===

I think that the reader is beginning to get an inkling of the crucial importance of the "number of" map in Peirce's way of looking at logic, for it's one of the plancks in the bridge from logic to the theories of probability, statistics, and information, in which 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.

I think that the reader is beginning to get an inkling of the crucial importance of

the "number of" map in Peirce's way of looking at logic, for it's one of the plancks

in the bridge from logic to the theories of probability, statistics, and information,

in which 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,

With that little bit of encouragement and exhortation, let us return to the nitty gritty details of the text.

let us return to the nitty gritty details of the text.

NOF 2.

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&nbsp;<&nbsp;7 is to say that 5 is part of 7, just as to write ''f''&nbsp;<&nbsp;''m'' is to say that Frenchmen are part of men.  Indeed, if ''f''&nbsp;<&nbsp;''m'', then the number of Frenchmen is less than the number of men, and if ''v''&nbsp;=&nbsp;''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).

| But not only do the significations of '=' and '<' here adopted fulfill all

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.

| 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

| for the terms themselves, in case no signs of operation occur in the

| C.S. Peirce, CP 3.66

Peirce is here remarking on the principle that the

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.

measure 'v' on terms "preserves" or "respects" the

prevailing implication, inclusion, or subsumption

relations that impose an ordering on those terms.

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.

 

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.

We will get back to that at another time.

We will get back to that at another time.

In the immediate example, we have this sort of statement:

In the immediate example, we have this sort of statement:

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

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

In symbolic form, this would be written:

In symbolic form, this would be written:

f < m =>  [f] < [m]

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

 

 

Here, the "<" on the left is a logical ordering on syntactic terms

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

while the "<" on the right is an arithmetic ordering on real numbers.

: Let ''X''₂ be a set with an ordering denoted by "&lt;₂".

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 "<_1".

What makes an ordering what it is will commonly be a set of axioms that defines the properties of the order relation in question.  Since one frequently has occasion to view the same set in the light of several different order relations, one will often resort to explicit forms like (''X'',&nbsp;&lt;₁), (''X'',&nbsp;&lt;₂), and so on, to invoke a set with a given ordering.

Let X_2 be a set with an ordering denoted by "<_2".

A map ''F'' : (''X''₁,&nbsp;&lt;₁) &rarr; (''X''₂,&nbsp;&lt;₂) is ''order-preserving'' if and only if a statement of a particular form holds for all ''x'' and ''y'' in (''X''₁,&nbsp;&lt;₁), specifically, this:

several different order relations, one will often

resort to explicit forms like (X, <_1), (X, <_2),

and so on, to invoke a set with a given ordering.

A map F : (X_1, <_1) -> (X_2, <_2) is "order-preserving"

: ''x'' &lt;₁ ''y'' &rArr; ''Fx'' &lt;₂ ''Fy''

for all x and y in (X_1, <_1), specifically, this:

x <_1 y =>  Fx <_2 Fy

The action of the "number of" map ''v'' : (''S'', &lt;₁) &rarr; ('''R''', &lt;₂) has just this character, as exemplified by its application to the case where ''x'' = ''f'' = "frenchman" and ''y'' = ''m'' = "man", like so:

The action of the "number of" map 'v' : (S, <_1) -> (R, <_2)

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

the case where x = f = "frenchman" and y = m = "man", like so:

| f < m =>  [f] < [m]

: ''f'' < ''m'' &rArr; ''vf'' < ''vm''

| f < m  =>  'v'f < 'v'm

Here, to be more exacting, we may interpret the "<" on the left

Here, to be more exacting, we may interpret the "<" on the left as "proper subsumption", that is, excluding the equality case, while we read the "<" on the right as the usual "less than".

as "proper subsumption", that is, excluding the equality case,

while we read the "<" on the right as the usual "less than".

===Commentary Note 11.18===

===Commentary Note 11.18===