Peirce's 1870 “Logic Of Relatives ” • Selection 8

Post   : Peirce's 1870 “Logic Of Relatives” • Selection 8
http://inquiryintoinquiry.com/2014/02/17/peirces-1870-logic-of-relatives-%e2%80%a2-selection-8/
Posted : February 17, 2014 at 3:00 pm
Author : Jon Awbrey
Tags   : Peirce, Logic, Logic of Relatives, Mathematics, Relation Theory, Semiotics

Peircers,

Please see the blog post linked above for properly formatted
text as the orthographic complexities of Peirce's paper are
passing well beyond what I can convey in a plain text copy.

We continue with §3. Application of the Algebraic Signs to Logic.

<quote>

The Signs for Multiplication (cont.)
------------------------------------

Thus far, we have considered the multiplication of relative terms only.  Since our conception of
multiplication is the application of a relation, we can only multiply absolute terms by considering
them as relatives.

Now the absolute term “man” is really exactly equivalent to the relative term “man that is ──”, and
so with any other.  I shall write a comma after any absolute term to show that it is so regarded as
a relative term.

Then “man that is black” will be written:

m,b.

But not only may any absolute term be thus regarded as a relative term, but any relative term may in
the same way be regarded as a relative with one correlate more. It is convenient to take this
additional correlate as the first one.

Then:

l,sw

will denote a lover of a woman that is a servant of that woman.

The comma here after l should not be considered as altering at all the meaning of l , but as only a
subjacent sign, serving to alter the arrangement of the correlates.

In point of fact, since a comma may be added in this way to any relative term, it may be added to
one of these very relatives formed by a comma, and thus by the addition of two commas an absolute
term becomes a relative of two correlates.

So:

m,,b,r

interpreted like

goh

means a man that is a rich individual and is a black that is that rich individual.

But this has no other meaning than:

m,b,r

or a man that is a black that is rich.

Thus we see that, after one comma is added, the addition of another does not change the meaning at
all, so that whatever has one comma after it must be regarded as having an infinite number.

If, therefore, l,,sw is not the same as l,sw (as it plainly is not, because the latter means a lover
and servant of a woman, and the former a lover of and servant of and same as a woman), this is
simply because the writing of the comma alters the arrangement of the correlates.

And if we are to suppose that absolute terms are multipliers at all (as mathematical generality
demands that we should}, we must regard every term as being a relative requiring an infinite number
of correlates to its virtual infinite series “that is ── and is ── and is ── etc.”

Now a relative formed by a comma of course receives its subjacent numbers like any relative, but the
question is, What are to be the implied subjacent numbers for these implied correlates?

Any term may be regarded as having an infinite number of factors, those at the end being ones, thus:

l,sw  =  l,sw,1,1,1,1,1,1,1, etc.

A subjacent number may therefore be as great as we please.

But all these ones denote the same identical individual denoted by w; what then can be the subjacent
numbers to be applied to s , for instance, on account of its infinite “that is” ’s?  What numbers
can separate it from being identical with w?  There are only two.  The first is zero, which plainly
neutralizes a comma completely, since

s,_0 w  =  sw

and the other is infinity; for as 1^∞ is indeterminate in ordinary algebra, so it will be shown
hereafter to be here, so that to remove the correlate by the product of an infinite series of ones
is to leave it indeterminate.

Accordingly,

m,_∞

should be regarded as expressing some man.

Any term, then, is properly to be regarded as having an infinite number of commas, all or some of
which are neutralized by zeros.

“Something” may then be expressed by:

1_∞.

I shall for brevity frequently express this by an antique figure one (1).

“Anything” by:

1_0.

I shall often also write a straight 1 for anything.

(Peirce, CP 3.73)

</quote>

-- 

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