Peirce's 1870 “Logic Of Relatives ” • Comment 8.6

Post   : Peirce's 1870 “Logic Of Relatives” • Comment 8.6
http://inquiryintoinquiry.com/2014/02/23/peirces-1870-logic-of-relatives-%e2%80%a2-comment-8-6/
Posted : February 23, 2014 at 1:36 pm
Author : Jon Awbrey
Tags   : Peirce, Logic, Logic of Relatives, Mathematics, Relation Theory, Semiotics

Peircers,

The foregoing has hopefully filled in enough background that we
can begin to make sense of the more mysterious parts of CP 3.73.

<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.

</quote> (Peirce, CP 3.73)

In any system where elements are organized according to types, there tend to be any number of ways 
in which elements of one type are naturally associated with elements of another type.  If the 
association is anything like a logical equivalence, but with the first type being lower and the 
second type being higher in some sense, then one may speak of a semantic ascent from the lower to 
the higher type.

For example, it is common in mathematics to associate an element _a_ of a set A with the constant 
function f_a : X → A that has f_a (x) = a for all x in X, where X is an arbitrary set that is fixed 
in the context of discussion.  Indeed, the correspondence is so close that one often uses the same 
name “a” to denote both the element _a_ in A and the function a = f_a : X → A, relying on context or 
an explicit type indication to tell them apart.

For another example, we have the tacit extension of a k-place relation L ⊆ X_1 × … × X_k to a 
(k+1)-place relation L' ⊆ X_1 × … × X_{k+1} that we get by letting L' = L × X_{k+1}, that is, by 
maintaining the constraints of L on the first k variables and letting the last variable wander freely.

What we have here, if I understand Peirce correctly, is another such type of natural extension, 
sometimes called the diagonal extension.  This extension associates a k-adic relative or a k-adic 
relation, counting the absolute term and the set whose elements it denotes as the cases for k = 0, 
with a series of relatives and relations of higher adicities.

A few examples will suffice to anchor these ideas.

| NB. Please see the blog post for the rest of this content,
|     as it contains too much math formatting to render here.

☞http://inquiryintoinquiry.com/2014/02/23/peirces-1870-logic-of-relatives-%e2%80%a2-comment-8-6/

Regards,

Jon

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