ONT Re: Propositional Equation Reasoning Systems
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PERS. Note 10
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Logic As Sign Transformation
We have been looking at various ways of transforming propositional expressions,
expressed in the formats of character strings and graphical structures, while
preserve certain aspects of their "meaning", just to risk using that vaguest
of all possible words, but only as a promissory note, hoefully to be cashed
out in a more meaningful species of currency as the discussion develops.
I cannot pretend acquaintance with or comprehension of all forms
of "intensions" (properties) that others might find of interest
in a given form of expression, nor can I speak for every form
of meaning that another might find of significance in a given
form of syntax. The best that I can do is to try to specify
what my object is in using these propositional expressions,
and, on a related note, what aspects of their syntax seem
to be conducive toward that object, and thus of interest
to preserve in some measure as I put these expressions
through the paces of their various transformations.
On behalf of this object I have been spinning in the form
of this thread a developing example base of propositional
expressions, both text-strings and data-structural forms,
along with many examples of step-wise transformations on
these expressions that preserve something of significant
logical import, something that I have from time to time
called their "logical equivalence class" (LEC), a thing
that I could as well call the "constraint information"
or the "denotative object" of the expression in view.
To focus still more, let us return to that Splendid Theorem noted by Leibniz,
and look at the two very distinct ways of transforming its initial expression
until we arrived, in either event, at an equivalent expression, one that 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
| |
| ((a(b))(d(c))((ad(bc)))) = |
| |
o-----------------------------------------------------------o
The first way of transforming the expression
on the left hand side of the equation can be
described as "proof-theoretic" in character.
Here is a link to the first proof of the PT:
http://suo.ieee.org/ontology/msg04526.html
The other way of transforming the expression
on the left hand side of the equation can be
described as "model-theoretic" in character.
Here's a link to the second proof of the PT:
http://suo.ieee.org/ontology/msg04530.html
What we have here are two different styles of "communicative action",
that is to say, two sequences of signs of the form s_1, s_2, ..., s_n,
each one beginning with a problematic expression and eventually ending
with a clear expression of the "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 about to delve into to 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". 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. And,
yes, perhaps that is too Latin a way of understanding it, but it's all I've got.
Now, if you think about these things just a bit, 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 hope to apply to this theory of inquiry that I also call "logic"?
But that is, among other things, part of the significance of the word "formal",
for what I use will be a kind of logic, an innate or inured skill at inquiry,
but a kind of logic that is casual, catch-as-catch-can, formative, inchoate,
informal, partly built into our natural language and partly more primitive
than language itself, and to the extent that I use it more than mention it,
mention it more than describe it, 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 do not 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 more
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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