ONT Re: Logic Of Relatives
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LOR. Note 59
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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,
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 < 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.
|
| C.S. Peirce, CP 3.66
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.
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.
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.
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"
In symbolic form, this would be written:
f < m => [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.
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 "<_1".
Let X_2 be a set with an ordering denoted by "<_2".
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, <_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"
if and only if a statement of a particular form holds
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, <_1) -> (R, <_2)
has just this character, as exemplified by its application to
the case where x = f = "frenchman" and y = m = "man", like so:
| f < m => [f] < [m]
|
| f < m => 'v'f < 'v'm
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".
Jon Awbrey
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