[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 52

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Arisbe List, Cybernetics List,

Toward "Logic As Sign Transformation" (cont.)

To focus still more, let's return to that Splendid Theorem
noted by Leibniz, and let's look more carefully at the two
distinct ways of transforming its initial expression that
were used to arrive at an equivalent expression, each of
which, in its own way, made its tautologous character,
or its theorematic nature, as evident as it could be.

Just to remind you, here is the Splendid Theorem again:

o-----------------------------------------------------------o
| Praeclarum Theorema (PT)` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` a o ` o d ` ` o ad` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` O ` ` ` ` ` ` ` ` ` = ` ` ` ` ` ` ` ` ` O ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `((a(b))(d(c))((ad(bc)))) ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

The first way of transforming the expression that appears
on the left hand side of the equation can be described as
"proof-theoretic" in character.  This appeared in Note 35.

FOLG 35.  http://stderr.org/pipermail/inquiry/2005-November/003214.html

The other way of transforming the expression that appears
on the left hand side of the equation can be described as
"model-theoretic" in character.  This appeared in Note 50.

FOLG 50.  http://stderr.org/pipermail/inquiry/2005-November/003238.html

What we have here amounts to a couple of different styles
of "communicational conduct", or "conductive communication",
if you prefer, that is to say, two sequences of signs of the
form e_1, e_2, ..., e_n, each one beginning with a problematic
expression and eventually ending with a clear expression of the
appropriate "logical equivalence class" (LEC) to which each and
every sign or expression in the sequence belongs.

Ordinarily, any orbit through a locus of signs can be taken
to reflect an underlying sign-process, a case of "semiosis".
So what we have here are two very special cases of semiosis,
and what we might just find it useful to contemplate is how
to characterize them as two species of a very general class.

We are starting to delve into some fairly picayune details
of a particular sign system, non-trivial enough in its own
right but still rather simple compared to the types of our
ultimate interest, and though I believe that this exercise
will be worth the effort in prospect of understanding more
complicated sign systems, I feel that I ought to say a few
words about the larger reasons for going through this work.

My broader interest lies in the theory of inquiry as a special
application or a special case of the theory of signs.  Another
name for the theory of inquiry is "logic" and another name for
the theory of signs is "semiotics".  So I might as well have
said that I am interested in logic as a special application
or a special case of semiotics.  But what sort of a special
application?  What sort of a special case?  Well, I think
of logic as "formal semiotics" -- though, of course, I am
not the first to have said such a thing -- and by "formal"
we say, in our etymological way, that logic is concerned
with the "form", indeed, with the "animate beauty" and
the very "life force" of signs and sign actions.  Yes,
perhaps that is far too Latin a way of understanding
logic, but it's all I've got.

Now, if you think about these things just a little more,
I know that you will find them just a little suspicious,
for what besides logic would I use to do this theory of
signs that I would apply to this theory of inquiry that
I'm also calling "logic"?  But that is precisely one of
the things signified by the word "formal", for what I'd
be required to use would have to be some brand of logic,
that is, some sort of innate or inured skill at inquiry,
but a style of logic that is casual, catch-as-catch-can,
formative, incipient, inchoate, unformalized, a work in
progress, partially built into our natural language and
partially more primitive than our most artless language.
In so far as I use it more than mention it, mention it
more than describe it, and describe it more than fully
formalize it, then to that extent it must be consigned
to the realm of unformalized and unreflective logic,
where some say "there be oracles", but I don't know.

Still, one of the aims of formalizing what acts of reasoning
that we can is to draw them into an arena where we can examine
them more carefully, perhaps to get better at their performance
than we can unreflectively, and thus to live, to formalize again
another day.  Formalization is not the be-all end-all of human
life, not by a long shot, but it has its uses on that behalf.

Jon Awbrey

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