[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

LOR.  Note 27

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Peirce's comma operation, in its application to an absolute term,
is tantamount to the representation of that term's denotation as
an idempotent transformation, which is commonly represented as a
diagonal matrix.  This is why I call it the "diagonal extension".

An idempotent element x is given by the abstract condition that xx = x,
but we commonly encounter such elements in more concrete circumstances,
acting as operators or transformations on other sets or spaces, and in
that action they will often be represented as matrices of coefficients.

Let's see how all of this looks from the graphical and matrical perspectives.

Absolute terms:

1  =  "anybody"  =  B +, C +, D +, E +, I +, J +, O

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

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

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

Previously, we represented absolute terms as column vectors.
The above four terms are given by the columns of this table:

   | 1 m n w
---o---------
 B | 1 0 0 1
 C | 1 1 1 0
 D | 1 0 1 1
 E | 1 0 0 1
 I | 1 1 0 0
 J | 1 1 0 0
 O | 1 1 1 0

One way to represent sets in the bigraph picture
is simply to mark the nodes in some way, like so:

    B   C   D   E   I   J   O
1   +   +   +   +   +   +   +

    B   C   D   E   I   J   O
m   o   +   o   o   +   +   +

    B   C   D   E   I   J   O
n   o   +   +   o   o   o   +

    B   C   D   E   I   J   O
w   +   o   +   +   o   o   o

Diagonal extensions of the absolute terms:

1,  =  "anybody that is ---"  =  B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O

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

Naturally enough, the diagonal extensions are represented by diagonal matrices:

!1!| B C D E I J O
---o---------------
 B | 1 0 0 0 0 0 0
 C | 0 1 0 0 0 0 0
 D | 0 0 1 0 0 0 0
 E | 0 0 0 1 0 0 0
 I | 0 0 0 0 1 0 0
 J | 0 0 0 0 0 1 0
 O | 0 0 0 0 0 0 1

!m!| B C D E I J O
---o---------------
 B | 0 0 0 0 0 0 0
 C | 0 1 0 0 0 0 0
 D | 0 0 0 0 0 0 0
 E | 0 0 0 0 0 0 0
 I | 0 0 0 0 1 0 0
 J | 0 0 0 0 0 1 0
 O | 0 0 0 0 0 0 1

!n!| B C D E I J O
---o---------------
 B | 0 0 0 0 0 0 0
 C | 0 1 0 0 0 0 0
 D | 0 0 1 0 0 0 0
 E | 0 0 0 0 0 0 0
 I | 0 0 0 0 0 0 0
 J | 0 0 0 0 0 0 0
 O | 0 0 0 0 0 0 1

!w!| B C D E I J O
---o---------------
 B | 1 0 0 0 0 0 0
 C | 0 0 0 0 0 0 0
 D | 0 0 1 0 0 0 0
 E | 0 0 0 1 0 0 0
 I | 0 0 0 0 0 0 0
 J | 0 0 0 0 0 0 0
 O | 0 0 0 0 0 0 0

Cast into the bigraph picture of 2-adic relations,
the diagonal extension of an absolute term takes on
a very distinctive sort of "straight-laced" character:

    B   C   D   E   I   J   O
u   o   o   o   o   o   o   o
    |   |   |   |   |   |   |
1,  |   |   |   |   |   |   |
    |   |   |   |   |   |   |
u   o   o   o   o   o   o   o
    B   C   D   E   I   J   O

    B   C   D   E   I   J   O
u   o   o   o   o   o   o   o
        |           |   |   |
m,      |           |   |   |
        |           |   |   |
u   o   o   o   o   o   o   o
    B   C   D   E   I   J   O

    B   C   D   E   I   J   O
u   o   o   o   o   o   o   o
        |   |               |
n,      |   |               |
        |   |               |
u   o   o   o   o   o   o   o
    B   C   D   E   I   J   O

    B   C   D   E   I   J   O
u   o   o   o   o   o   o   o
    |       |   |
w,  |       |   |
    |       |   |
u   o   o   o   o   o   o   o
    B   C   D   E   I   J   O

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o