ONT Re: Differential Logic -- Series A
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DLOG. Note A11
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Let me return to Peirce's early papers on the algebra of relatives
to pick up the conventions that he used there, and then rewrite my
account of regular representations in a way that conforms to those.
Peirce expresses the action of an "elementary dual relative" like so:
| [Let] A:B be taken to denote
| the elementary relative which
| multiplied into B gives A.
|
| Peirce, 'Collected Papers', CP 3.123.
And though he is well aware that it is not at all necessary to arrange
elementary relatives into arrays, matrices, or tables, when he does so
he tends to prefer organizing dyadic relations in the following manner:
[ A:A A:B A:C |
| |
| B:A B:B B:C |
| |
| C:A C:B C:C ]
That conforms to the way that the last school of thought
I matriculated into stipulated that we tabulate material:
[ e_11 e_12 e_13 |
| |
| e_21 e_22 e_23 |
| |
| e_31 e_32 e_33 ]
So, for example, let us suppose that we have the small universe {A, B, C},
and the 2-adic relation m = "mover of" that is represented by this matrix:
m =
[ m_AA (A:A) m_AB (A:B) m_AC (A:C) |
| |
| m_BA (B:A) m_BB (B:B) m_BC (B:C) |
| |
| m_CA (C:A) m_CB (C:B) m_CC (C:C) ]
Also, let m be such that:
A is a mover of A and B,
B is a mover of B and C,
C is a mover of C and A.
In sum:
m =
[ 1 * (A:A) 1 * (A:B) 0 * (A:C) |
| |
| 0 * (B:A) 1 * (B:B) 1 * (B:C) |
| |
| 1 * (C:A) 0 * (C:B) 1 * (C:C) ]
For the sake of orientation and motivation,
compare with Peirce's notation in CP 3.329.
I think that will serve to fix notation
and set up the remainder of the account.
Jon Awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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