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SUO: All Liar, No Paradox

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I was reminded of this old note of mine, initially
addressed to some denizens of the Peirce List, by
Jay Halcomb's remarks on the need to distinguish
"syntactic", "semantic", and dare I add "pragmatic"
brands of inconsistency affecting our formal systems.
Just by way of tossing yet another file into the ointment,
I thought that this humble example, aside from booting the
Liar beyond the reaches of our Architectronic Bootstraps,
might also serve to illustrate just how tricky it can be
to perform the required dissection.

By the way, I forgot to include the Figure,
which was critical to seeing the solution,
so here it is:

              S1
              •
              |
       S1     |
        •-----•
         \   /
          \ /
           •
           |
           |
           @

   (( S1 , ( S1 ) ))

Figure 0.  Statement 0

Cheers,

Jon Awbrey

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Re Jim's question:

> Perhaps the liar's paradox can not be reduced to diagrammatic form.
> Why does this paradox seem to resist deductive analysis?  Is there
> something in the purpose or context of the premises that is at the
> root of the problem?  Is there something about asserting ones own
> context, or denying one speaks from a context that is at the bottom
> of this paradox?

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Jim,

According to my understanding of it, the so-called Liar Paradox
is just the most simple-minded of fallacies, involving nothing
more mysterious than the acceptance of a false assumption,
from which anybody can prove anything at all.

Consider one of the forms in which it is commonly presented:

| Somebody writes down:
|
| > 1.  Statement 1 is false.
| 
| Then you are led to reason:
| If Statement 1 is false then
| by the principle that permits
| the substitution of equals in
| a true statement to obtain
| yet another true statement,
| one can derive the result:
|
| "Statement 1 is false" is false.
| Ergo, Statement 1 is true,
| and so on, and so on,
| ad nauseum infinitum.

Where did you go wrong?
Where were you misled?

As it happens, graphical reasoning does help
to clear this up -- at least, it did for me --
if only because the process of translating
the purported reasoning into another form
gave me a clue where the wool was pulled.

Just here, to wit, where it is writ:

> 1.  Statement 1 is false.

What is this really saying?
Well, it's the same as writing:

> Statement 1.  Statement 1 is false.

And what the heck does this dot.comment say?
It is inducing you to accept this identity:

> "Statement 1" = "Statement 1 is false".

That appears to be a purely syntactic indexing,
the sort of thing you are led to believe that
you can do arbitrarily, with logical impunity.
But you cannot, for syntactic identity implies
logical equivalence, and that is liable to find
itself constrained by iron bands of logical law.

And you cannot just assume what this result says:

> "Statement 1" = "Negation of Statement 1"

To write the last step in the form that I like:

> (( Statement_1 , ( Statement_1 ) ))

And this my friends, call it "Statement 0",
is purely and simply a false statement,
with no hint of paradox about it.

Statement 0 was slipped into your drink
before you were even starting to think.
A bit before you were led to substitute
you should have examined more carefully
the site proposed for the substitution!

For the principle that you rushed to use
does not permit you to substitute unequals
into a statement that is false to begin with,
not just in the first place, but even before,
in the zeroth place of argument, as it were,
and still expect to come up with a truth.

Now, let that be the end of that.

Jon

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