[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 17

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So long as we're in the neighborhood, we might as well take in
some more of the sights, for instance, the smallest example of
a non-abelian (non-commutative) group.  This is a group of six
elements, say, G = {e, f, g, h, i, j}, with no relation to any
other employment of these six symbols being implied, of course,
and it can be most easily represented as the permutation group
on a set of three letters, say, X = {a, b, c}, usually notated
as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3.
Here are the permutation (= substitution) operations in Sym(X):

Table 17-a.  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

Here is the operation table for S_3, given in abstract fashion:

Table 17-b.  Symmetric Group S_3
o-------------------------------------------------o
|                                                 |
|                        o                        |
|                     e / \ e                     |
|                      /   \                      |
|                     /  e  \                     |
|                  f / \   / \ f                  |
|                   /   \ /   \                   |
|                  /  f  \  f  \                  |
|               g / \   / \   / \ g               |
|                /   \ /   \ /   \                |
|               /  g  \  g  \  g  \               |
|            h / \   / \   / \   / \ h            |
|             /   \ /   \ /   \ /   \             |
|            /  h  \  e  \  e  \  h  \            |
|         i / \   / \   / \   / \   / \ i         |
|          /   \ /   \ /   \ /   \ /   \          |
|         /  i  \  i  \  f  \  j  \  i  \         |
|      j / \   / \   / \   / \   / \   / \ j      |
|       /   \ /   \ /   \ /   \ /   \ /   \       |
|      o  j  \  j  \  j  \  i  \  h  \  j  o      |
|       \   / \   / \   / \   / \   / \   /       |
|        \ /   \ /   \ /   \ /   \ /   \ /        |
|         \  h  \  h  \  e  \  j  \  i  /         |
|          \   / \   / \   / \   / \   /          |
|           \ /   \ /   \ /   \ /   \ /           |
|            \  i  \  g  \  f  \  h  /            |
|             \   / \   / \   / \   /             |
|              \ /   \ /   \ /   \ /              |
|               \  f  \  e  \  g  /               |
|                \   / \   / \   /                |
|                 \ /   \ /   \ /                 |
|                  \  g  \  f  /                  |
|                   \   / \   /                   |
|                    \ /   \ /                    |
|                     \  e  /                     |
|                      \   /                      |
|                       \ /                       |
|                        o                        |
|                                                 |
o-------------------------------------------------o

I think that the NKS reader can guess how we might apply
this group to the space of propositions of type B^3 -> B.

By the way, we will meet with the symmetric group S_3 again
when we return to take up the study of Peirce's early paper
"On a Class of Multiple Algebras" (CP 3.324-327), and also
his late unpublished work "The Simplest Mathematics" (1902)
(CP 4.227-323), with particular reference to the section
that treats of "Trichotomic Mathematics" (CP 4.307-323).

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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