[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 21

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

| 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.
|
| C.S. 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 frequently speaks
of a "semantic ascent" from the lower to the higher type.

For instance, it is very common in mathematics to associate an element m
of a set M with the constant function f_m : X -> M such that f_m (x) = m
for all x in X, where X is an arbitrary set.  Indeed, the correspondence
is so close that one often uses the same name "m" for the element m in M
and the function m = f_m : X -> M, relying on the context or an explicit
type indication to tell them apart.

For another instance, we have the "tacit extension" of a k-place relation
L c X_1 x ... x X_k to a (k+1)-place relation L' c X_1 x ... x X_k+1 that
we get by letting L' = L x 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 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.

Absolute terms:

m   =  "man"                =  C +, I +, J +, O

n   =  "noble"              =  C +, D +, O

w   =  "woman"              =  B +, D +, E

Diagonal extensions:

m,  =  "man that is ---"    =  C:C +, I:I +, J:J +, O:O

n,  =  "noble that is ---"  =  C:C +, D:D +, O:O

w,  =  "woman that is ---"  =  B:B +, D:D +, E:E

Sample products:

m,n  =  "man that is noble"  

     =  (C:C +, I:I +, J:J +, O:O)(C +, D +, O)

     =  C +, O

n,m  =  "noble that is man"

     =  (C:C +, D:D +, O:O)(C +, I +, J +, O)

     =  C +, O

n,w  =  "noble that is woman"

     =  (C:C +, D:D +, O:O)(B +, D +, E)

     =  D

w,n  =  "woman that is noble"

     =  (B:B +, D:D +, E:E)(C +, D +, O)

     =  D

Jon Awbrey

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