Discussion:
Graham Cooper's 2012-05-16 reasoning seems similar to mine
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Pete Olcott
2017-06-17 03:22:37 UTC
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The pathological self-reference error of the Liar Paradox
is identical to the 1931 GIT.
No, it's not.
"yields a falsehood when preceded by its quotation"
yields a falsehood when preceded by its quotation.
----
By the way, how would you translate Quine's paradox
into the formal language of MTT?
IDK, you will be able to do that yourself soon enough.
I have the formal syntax of MTT almost complete in YACC BNF.

After I have the knowledge ontology / automatic proof generator complete, people will be able to test it with their own set of axioms.

Since MTT will implement sub atomic semantic compositionality anyone will be able to augment MTT in completely unlimited ways, even redefining all of the logical operators.

If will focus very little of my time on solving specific problems and focus almost all of my time on making MTT have unlimited extensibility.

The whole idea of MTT is to provide the complete mathematical foundation for the mathematics of meaning: sub atomic semantic compositionality.

It just occurred to me today that I may need to build an inference engine toolkit into MTT. I was initially considering simply providing inferencing functions. If I provide a toolkit, this may make MTT much more extensible. In any case knowledge ontology Tree-Walking must be available to MTT programmers.
What about Yablo's paradox, which has no self-reference?
How would you write that in MTT?
Yablo's paradox has an infinite chain of sentences S[k],
Probably some form of mathematical induction.
each of which claims that all of the following sentences
S[0] == " j > 0 -> ~S[j] "
S[1] == " j > 1 -> ~S[j] "
S[2] == " j > 2 -> ~S[j] "
...
S[k] == " j > k -> ~S[j] "
...
If S[0] is true, then (in particular) S[1] is false.
But S[0] also says S[2], S[3], ... are false, which
is what S[1] says. So, S[1] is true.
If S[0] is false, then there is some m where S[m]
S[m] is true; S[m+1] is false, but S[m+1] is true.
But no self-reference, so there would seem to be no
pathological self-reference, either.
----
And, maybe you could prove, for Olcott's G, and for
some formal system T
T |- G <-> !Provable-in-T([G])
Whenever you get around to it.
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
Peter Percival
2017-06-17 05:51:23 UTC
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Post by Pete Olcott
[...]
What about Yablo's paradox, which has no self-reference?
How would you write that in MTT?
Yablo's paradox has an infinite chain of sentences S[k],
Probably some form of mathematical induction.
I have a feeling that you don't know what mathematical induction is.
Post by Pete Olcott
each of which claims that all of the following sentences
S[0] == " j > 0 -> ~S[j] "
S[1] == " j > 1 -> ~S[j] "
S[2] == " j > 2 -> ~S[j] "
...
S[k] == " j > k -> ~S[j] "
...
If S[0] is true, then (in particular) S[1] is false.
But S[0] also says S[2], S[3], ... are false, which
is what S[1] says. So, S[1] is true.
If S[0] is false, then there is some m where S[m]
S[m] is true; S[m+1] is false, but S[m+1] is true.
But no self-reference, so there would seem to be no
pathological self-reference, either.
----
And, maybe you could prove, for Olcott's G, and for
some formal system T
T |- G <-> !Provable-in-T([G])
Whenever you get around to it.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Pete Olcott
2017-06-17 13:56:51 UTC
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<div class="moz-cite-prefix">On 6/14/2017 11:53 AM, George Greene
wrote:<br>
</div>
<blockquote
cite="mid:c6f6d186-23b9-41f4-8b81-***@googlegroups.com"
type="cite">
<pre wrap="">On Tuesday, June 13, 2017 at 5:17:44 PM UTC-4, Pete Olcott wrote:
</pre>
<blockquote type="cite">
<pre wrap="">G( ~Provable(G) ) is not a sentence in F because it is not a truth bearer.
</pre>
</blockquote>
<pre wrap="">
This is FALSE, you moron.

Just because the sentence isn't decidable from some set of axioms doesn't mean
it stops being OF TYPE "truth-bearer". The sentence DOES in fact bear a truth-value IN EVERY *MODEL* of the axioms because that's how "model"</pre>
</blockquote>
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Symbol&quot;;mso-fareast-font-family:
&quot;MS Mincho&quot;;mso-bidi-font-family:&quot;Segoe UI
Symbol&quot;;background:yellow">G @ ∀L ∈
Formal_Systems, ~∃Γ ⊂ L (Γ ⊢ G) <o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-bidi-font-family:&quot;Segoe UI
Symbol&quot;;background:white">"@" means
the LHS is assigned as an alias for the RHS<span
class="apple-converted-space"> .
</span><o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-bidi-font-family:&quot;Segoe UI
Symbol&quot;;background:white">There is no
referencing / dereferencing needed, G is one and the same
thing as the
expression that refers to G. G is not referring to its name, G
is referring to
itself.<span class="apple-converted-space"> </span></span></b><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;mso-bidi-font-family:
&quot;Segoe UI Symbol&quot;"><o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-fareast-font-family:&quot;MS
Mincho&quot;;mso-bidi-font-family:&quot;Segoe UI Symbol&quot;"><o:p> </o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-fareast-font-family:&quot;MS
Mincho&quot;;mso-bidi-font-family:&quot;Segoe UI Symbol&quot;">01
∀<span style="mso-spacerun:yes">      </span>(2)(5)<span
style="mso-spacerun:yes">  </span><span
style="background:yellow">// G is an
alias for this node</span><o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-fareast-font-family:&quot;MS
Mincho&quot;;mso-bidi-font-family:&quot;Segoe UI Symbol&quot;">02
</span></b><b><span style="font-size:11.0pt;font-family:&quot;MS
Mincho&quot;;mso-ascii-font-family:
&quot;Segoe UI Symbol&quot;;mso-hansi-font-family:&quot;Segoe
UI Symbol&quot;;mso-bidi-font-family:
&quot;Segoe UI Symbol&quot;" lang="HI">∈</span></b><b><span
style="font-size:11.0pt;font-family:
&quot;Segoe UI Symbol&quot;;mso-fareast-font-family:&quot;MS
Mincho&quot;;mso-bidi-font-family:
&quot;Segoe UI Symbol&quot;"><span style="mso-spacerun:yes">     
</span>(3)(4)<o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-fareast-font-family:&quot;MS
Mincho&quot;;mso-bidi-font-family:&quot;Segoe UI Symbol&quot;">03
L<o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-fareast-font-family:&quot;MS
Mincho&quot;;mso-bidi-font-family:&quot;Segoe UI Symbol&quot;">04
Formal Systems<o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-fareast-font-family:&quot;MS
Mincho&quot;;mso-bidi-font-family:&quot;Segoe UI Symbol&quot;">05
~<span style="mso-spacerun:yes">       </span>(6)<span
style="mso-spacerun:yes">      </span><o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-fareast-font-family:&quot;MS
Mincho&quot;;mso-bidi-font-family:&quot;Segoe UI Symbol&quot;">06
∃<span style="mso-spacerun:yes">       </span>(7)(9) <o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-fareast-font-family:&quot;MS
Mincho&quot;;mso-bidi-font-family:&quot;Segoe UI Symbol&quot;">07
⊂<span style="mso-spacerun:yes">       </span>(8)(3)<o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
mso-fareast-font-family:&quot;MS
Mincho&quot;;mso-bidi-font-family:&quot;Segoe UI Symbol&quot;">08
Γ<o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span
style="font-size:11.0pt;font-family:&quot;Segoe UI
Symbol&quot;;
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⊢<span style="mso-spacerun:yes">       </span>(8)(1)<span
style="mso-spacerun:yes">  </span><span
style="background:yellow">// cycle
indicates infinite evaluation loop error</span><o:p></o:p></span></b></p>
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<font size="+2"><b>The above specified G is proved to lack a truth
value in every model. </b></font><br>
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