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ONT All Liar, No Paradox


ALNP.  Note 1


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.

Let us contemplate one of the shapes in which
the re*putative Liar Paradox is commonly cast:

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
of representation gave me a crucial clue
as to where the wool was being 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, not with logical impunity, assume the result
of this transmutation, which would be to say as much as this:

"Statement 1" = "Negation of Statement 1"

To write down 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.

Here is Statement 0 in cactus syntax:

|                             |
|                s_1          |
|                 o           |
|                 |           |
|          s_1    |           |
|           o-----o           |
|            \   /            |
|             \ /             |
|              o              |
|              |              |
|              |              |
|              @              |
|                             |
|       (( s_1, (s_1) ))      |
Figure 0.  Statement 0

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 Awbrey

inquiry e-lab: