ONT Re: Differential Logic

[Jon Awbrey <jawbrey@oakland.edu>]

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Note 20

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| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| objects of your 'conception' to have.  Then,
| your 'conception' of those effects is the
| whole of your 'conception' of the object.
|
| Charles Sanders Peirce,
| "Maxim of Pragmaticism", CP 5.438.

By way of collecting a short-term pay-off for all the work --
not to mention the peirce-spiration -- that we sweated out
over the regular representations of the Klein 4-group V_4,
let us write out as quickly as possible in "relative form"
a minimal budget of representations of the symmetric group
on three letters, S_3 = Sym(3).  After doing the usual bit
of compare and contrast among these divers representations,
we will have enough concrete material beneath our abstract
belts to tackle a few of the presently obscur'd details of
Peirce's early "Algebra + Logic" papers.

Table 1.  Permutations or Substitutions in Sym_{A, B, C}
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
|         |         |         |         |         |         |
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
|         |         |         |         |         |         |
|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

Writing this table in relative form generates
the following "natural representation" of S_3.

    e  =  A:A + B:B + C:C

    f  =  A:C + B:A + C:B

    g  =  A:B + B:C + C:A

    h  =  A:A + B:C + C:B

    i  =  A:C + B:B + C:A

    j  =  A:B + B:A + C:C

I have without stopping to think about it written out this natural
representation of S_3 in the style that comes most naturally to me,
to wit, the "right" way, whereby an ordered pair configured as X:Y
constitutes the turning of X into Y.  It is possible that the next
time we check in with CSP that we will have to adjust our sense of
direction, but that will be an easy enough bridge to cross when we
come to it.

Jon Awbrey

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