[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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

DAL.  Note 12

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

| 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.
|
| C.S. Peirce, "Maxim of Pragmaticism", 'Collected Papers', CP 5.438

One other subject that it would be opportune to mention at this point,
while we have an object example of a mathematical group fresh in mind,
is the relationship between the pragmatic maxim and what are commonly
known in mathematics as "representation principles".  As it turns out,
with regard to its formal characteristics, the pragmatic maxim unites
the aspects of a representation principle with the attributes of what
would ordinarily be known as a "closure principle".  We will consider
the form of closure that is invoked by the pragmatic maxim on another
occasion, focusing here and now on the topic of group representations.

Let us return to the example of the so-called "four-group" V_4.
We encountered this group in one of its concrete representations,
namely, as a "transformation group" that acts on a set of objects,
in this particular case a set of sixteen functions or propositions.
Forgetting about the set of objects that the group transforms among
themselves, we may take the abstract view of the group's operational
structure, say, in the form of the group operation table copied here:

o-------o-------o-------o-------o-------o
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
| ` * ` % ` e ` | ` f ` | ` g ` | ` h ` |
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
o=======o=======o=======o=======o=======o
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
| ` e ` % ` e ` | ` f ` | ` g ` | ` h ` |
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
o-------o-------o-------o-------o-------o
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
| ` f ` % ` f ` | ` e ` | ` h ` | ` g ` |
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
o-------o-------o-------o-------o-------o
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
| ` g ` % ` g ` | ` h ` | ` e ` | ` f ` |
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
o-------o-------o-------o-------o-------o
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
| ` h ` % ` h ` | ` g ` | ` f ` | ` e ` |
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
o-------o-------o-------o-------o-------o

This table is abstractly the same as, or isomorphic to, the versions
with the E_ij operators and the T_ij transformations that we took up
earlier.  That is to say, the story is the same, only the names have
been changed.  An abstract group can have a variety of significantly
and superficially different representations.  But even after we have
long forgotten the details of any particular representation there is
a type of concrete representations, called "regular representations",
that are always readily available, as they can be generated from the
mere data of the abstract operation table itself.

For example, select a group element from the top margin of the Table,
and "consider its effects" on each of the group elements as they are
listed along the left margin.  We may record these effects as Peirce
usually did, as a logical "aggregate" of elementary dyadic relatives,
that is to say, a disjunction or a logical sum whose terms represent
the ordered pairs of <input : output> transactions that are produced
by each group element in turn.  This yields what is usually known as
one of the "regular representations" of the group, specifically, the
"first", the "post-", or the "right" regular representation.  It has
long been conventional to organize the terms in the form of a matrix:

Reading "+" as a logical disjunction:

   G  =  e  +  f  +  g  + h,

And so, by expanding effects, we get:

   G  =

   e:e  +  f:f  +  g:g  +  h:h  +

   e:f  +  f:e  +  g:h  +  h:g  +

   e:g  +  f:h  +  g:e  +  h:f  +

   e:h  +  f:g  +  g:f  +  h:e

More on the pragmatic maxim as a representation principle later.

Jon Awbrey

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