Hypothesis: Paradox of self-reference such as the Halting Problem is an error of reasoning

[Peter Olcott]

Basis:
Lemma01: Meaning can only be correctly specified within an acyclic
directed graph:

a) Montague [meaning postulates] must be specified within acyclic
di-graphs. (connections between elements)

b) Connections between Montague [meaning postulates] (principle of
compositionality) must not produce cycles.

Lemma02: The [meaning postulate] of all self-reference paradoxes can
only be fully specified within a di-graph that contains cycles.

Lemma03: The Halting Problem and the Liar Paradox are both
self-reference paradoxes.

Conclusion:
The Halting Problem and the Liar Paradox are errors of
specification/reasoning because their complete [meaning postulates]
necessarily always contain cycles.

[Ben Bacarisse]

That's a fine collection of linguistic definitions, and I hope you find
the conclusion comforting. However, the halting theorem remains a
theorem of mathematics, no matter how you label it.

--
Ben.

[Ben Bacarisse]

This is a re-post because I messed up the headers...

[Peter Olcott]

The language of mathematics is insufficiently expressive to discern the
error of the fallacy of self-reference.

It can not be correctly concluded that an error does not exist on the
basis that this error can not be expressed within the limitations of any
specific mode of expression such as the language of mathematics.

It can only be correctly concluded that within this specific mode of
expression that this error can not be seen.

[Rupert]

On Tuesday, November 19, 2013 3:17:47 PM UTC+1, Peter Olcott wrote:
> It can not be correctly concluded that an error does not exist on the
> basis that this error can not be expressed within the limitations of any
> specific mode of expression such as the language of mathematics.
>
> It can only be correctly concluded that within this specific mode of
> expression that this error can not be seen.

If you derive the conclusion from accepted axioms using accepted rules of inference, then you can conclude that there's no error. (Unless of course you think we somehow got it wrong when we chose to accept these axioms and rules of inference, but you'd need to argue that point.)

You need to show a place in Turing's argument where he makes an inference that's not warranted by the accepted rules. Or makes an assertion that is not justified.

[Peter Percival]

Peter Olcott wrote:

> The language of mathematics is insufficiently expressive to discern the
> error of the fallacy of self-reference.

The language of mathematics is as expressive as any language. Why?
Because the language of mathematics includes the natural languages of
mathematicians.


--
Madam Life's a piece in bloom,
Death goes dogging everywhere:
She's the tenant of the room,
He's the ruffian on the stair.

[Ben Bacarisse]

Peter Olcott <OCR4Screen> writes:

> On 11/19/2013 8:00 AM, Ben Bacarisse wrote:
>> This is a re-post because I messed up the headers...
>>
>> Peter Olcott <OCR4Screen> writes:

<snip>

>>> Conclusion:
>>> The Halting Problem and the Liar Paradox are errors of
>>> specification/reasoning because their complete [meaning postulates]
>>> necessarily always contain cycles.
>>
>> That's a fine collection of linguistic definitions, and I hope you find
>> the conclusion comforting. However, the halting theorem remains a
>> theorem of mathematics, no matter how you label it.
>
> The language of mathematics is insufficiently expressive to discern
> the error of the fallacy of self-reference.

The same sets are decidable. The same sets are undecidable. Nothing
has changed since you first started this nearly a decade ago[1]. Had
you not been distracted by making your fortune from on-screen OCR[2] you
might have something more to show for the years of pondering these
questions.

For those not familiar with the history, Peter Olcott starts from a
theological position and does whatever is needed to resolve the ensuing
conflict:

Message-ID: <L7mdnZ0QUroncUbS...@giganews.com>
| If God can not solve the Halting Problem, then there is something
| wrong with the problem.

Of course, that does not mean he's wrong (he's wrong for very down to
earth reasons) but it does give some hints about how open to reason he
in on this matter. Given the flexibility in the notion of 'god' it
would surely have been simpler to decide that he *can* solve the halting
problem and more on from there. Since that, sadly, seems not be an
option, the solution is to find a way to label the question as invalid or
fallacious. Although this is misleading, it does, of course, alter the
question in any substantive way. The answer is still the same.

[1] It's been going on since at least 2004, but it may be even longer
than that.

[2] http://www.ocr4screen.com/Download.html A summer deadline has been
missed it seems. Oh well,
--
Ben.

[Peter Olcott]

On 11/19/2013 10:14 AM, Peter Percival wrote:
> Peter Olcott wrote:
>
>> The language of mathematics is insufficiently expressive to discern the
>> error of the fallacy of self-reference.
>
> The language of mathematics is as expressive as any language. Why?
> Because the language of mathematics includes the natural languages of
> mathematicians.
>
>

The language of humans (natural language) although sufficiently
expressive to show things such as the fallacy of self-reference has not
yet been made sufficiently precise such that this error can be discerned
by those accustomed to using conventional mathematical notation (or most
others).

The most typical response is along the lines of the lack of capability
to express this error using conventional mathematical notation indicates
that the error does not actually exist.

What is needed is a sufficiently expressive and completely precise
notational convention. Richard Montague provided the foundational basis
for such a notational system.

I propose one more detail to be added to the system proposed by Richard
Montague: Meanings can only be correctly connected together (principle
of compositionality) using an acyclic directed graph.

Fallacies of self-reference can not be completely specified within any
acyclic directed graph, this is what indicates that they are erroneous.

[Peter Olcott]

On 11/19/2013 11:03 AM, Ben Bacarisse wrote:
> [2]http://www.ocr4screen.com/Download.html A summer deadline has been
> missed it seems. Oh well,

http://pixels2words.com/Download.html

[wolfgang.m...@hs-augsburg.de]

Am Dienstag, 19. November 2013 17:14:24 UTC+1 schrieb Peter Percival:


>
> The language of mathematics is as expressive as any language. Why?
>
> Because the language of mathematics includes the natural languages of
>
> mathematicians.

And why does this language not contain definitions for all real numbers? Such that we have to seek those definitions in uncountably many different languages which are so foreign to the language of mathematics that their definitions cannot be translated into the language of mathematics?

Regards, WM

[Peter Olcott]

On 11/19/2013 9:39 AM, Rupert wrote:
> On Tuesday, November 19, 2013 3:17:47 PM UTC+1, Peter Olcott wrote:
>> It can not be correctly concluded that an error does not exist on the
>> basis that this error can not be expressed within the limitations of any
>> specific mode of expression such as the language of mathematics.
>>
>> It can only be correctly concluded that within this specific mode of
>> expression that this error can not be seen.
>
> If you derive the conclusion from accepted axioms using accepted rules of inference, then you can conclude that there's no error.

Unless this a very tentatively held conclusion it would tend to presume
omniscience upon the one holding the conclusion. It presumes that all
the rules of inference are both correct and complete. It think that the
former conclusion (that the current rules are correct) has a much more
substantial basis than the latter conclusion, that the set of rules is
complete.

I am proposing a basis for forming new rules of inference. This basis
that I am proposing is currently in its infancy. Very little is
currently known of the inherent structure of knowledge. A key element in
this new basis has not nearly reached any degree of significant
agreement, much less universal consensus is:

http://plato.stanford.edu/entries/compositionality/

I propose that the connections between elements of knowledge must derive
an acyclic directed graph.

I also propose that all paradoxes of self-reference can not be fully
specified within an acyclic di-graph, and this limitation shows their
error.

[Virgil]

In article <773d7c04-7999-41a4...@googlegroups.com>,

"As far as the laws of mathematics refer to
reality, they are not certain; and as
far as they are certain, they do not refer
to reality. It seems to me that complete
clearness as to this state of things first
became common property through that new departure
in mathematics which is known by the name of
mathematical logic or ŚAxiomatics.ą The progress
achieved by axiomatics consists in its having
neatly separated the logical-formal from its
objective or intuitive content; according to
axiomatics the logical-formal alone forms the
subject-matter of mathematics, which is not
concerned with the intuitive or other content
associated with the logical-formal. . . .
[On this view it is clear that] mathematics
as such cannot predicate anything about
perceptual objects or real objects. In
axiomatic geometry the words Śpoint,ą Śstraight
line,ą etc., stand only for empty
conceptual schemata."

Albert Einstein
--

[graham...@gmail.com]

On Tuesday, November 19, 2013 3:25:27 AM UTC-8, Peter Olcott wrote:
> Basis:
>
> Lemma01: Meaning can only be correctly specified within an acyclic
>
> directed graph:

This is essentially a 2nd Order Logic.

Your Hypothesis should be:

2nd Order Programming Languages have a larger set of High Order Functions
than 1st Order Programming Languages.



f = sin(cos(x))
e.g. SIZE( f ) = 11 characters


Not Turing Machine can do this using a simple unary interpretation of the machines output tape.

>
>
>
> a) Montague [meaning postulates] must be specified within acyclic
>
> di-graphs. (connections between elements)
>
>
>
> b) Connections between Montague [meaning postulates] (principle of
>
> compositionality) must not produce cycles.
>
>
>
> Lemma02: The [meaning postulate] of all self-reference paradoxes can
>
> only be fully specified within a di-graph that contains cycles.
>
>
>
> Lemma03: The Halting Problem and the Liar Paradox are both
>
> self-reference paradoxes.
>
>
>
> Conclusion:
>
> The Halting Problem and the Liar Paradox are errors of
>
> specification/reasoning because their complete [meaning postulates]
>
> necessarily always contain cycles.

Right! It's a very simple error.

H0: no turing machine can input the 'godel number' of all other turing machines
and output 1=halt or 0=not-halt


Hypothesis 0 (H0) is actually CORRECT!


H1: a subset of Turing Machines may be computationally complete.



T' C T


It is T' where the halt function lies.


Herc
--
www.PrologDatabase.com

[Rupert]

On Tuesday, November 19, 2013 10:01:25 PM UTC+1, Peter Olcott wrote:
> On 11/19/2013 9:39 AM, Rupert wrote:
> > On Tuesday, November 19, 2013 3:17:47 PM UTC+1, Peter Olcott wrote:
> >> It can not be correctly concluded that an error does not exist on the
> >> basis that this error can not be expressed within the limitations of any
> >> specific mode of expression such as the language of mathematics.
>
> >> It can only be correctly concluded that within this specific mode of
> >> expression that this error can not be seen.
>
> > If you derive the conclusion from accepted axioms using accepted rules of inference, then you can conclude that there's no error.
>
> Unless this a very tentatively held conclusion it would tend to presume
> omniscience upon the one holding the conclusion. It presumes that all
> the rules of inference are both correct and complete. It think that the
> former conclusion (that the current rules are correct) has a much more
> substantial basis than the latter conclusion, that the set of rules is
> complete.
>

What sense of completeness do you have in mind here? We know that first-order logic is semantically complete, but on the other hand theories such as PA or ZFC are deductively incomplete. But that's neither here nor there, there was no reason why I had to make any presumption of completeness in any sense. You just need the truth of the axioms and the soundness of the rules of inference. You'd probably want to acknowledge that your belief in this is fallible, but I don't see why you need to be especially tentative about it. The insolubility of the halting problem can be proved in PA, and my belief in the soundness of PA is not a tentative one. I also don't think you've offered any reason to doubt that Turing's proof can be converted into a formally correct proof in PA.

> I am proposing a basis for forming new rules of inference. This basis
> that I am proposing is currently in its infancy. Very little is
> currently known of the inherent structure of knowledge. A key element in
> this new basis has not nearly reached any degree of significant
> agreement, much less universal consensus is:
>
> http://plato.stanford.edu/entries/compositionality/
>
> I propose that the connections between elements of knowledge must derive
> an acyclic directed graph.
>

What connections, and where do the directions of the edges of the graph come from?

> I also propose that all paradoxes of self-reference can not be fully
> specified within an acyclic di-graph, and this limitation shows their
> error.

Well, there's not a lot that I can do with that. Which aspect of Turing's proof cannot be fully specified within an acyclic di-graph, what's the meaning of such a claim?

[Peter Olcott]

On 11/20/2013 9:28 AM, Rupert wrote:
> On Tuesday, November 19, 2013 10:01:25 PM UTC+1, Peter Olcott wrote:
>> On 11/19/2013 9:39 AM, Rupert wrote:
>>> On Tuesday, November 19, 2013 3:17:47 PM UTC+1, Peter Olcott wrote:
>>>> It can not be correctly concluded that an error does not exist on the
>>>> basis that this error can not be expressed within the limitations of any
>>>> specific mode of expression such as the language of mathematics.
>>
>>>> It can only be correctly concluded that within this specific mode of
>>>> expression that this error can not be seen.
>>
>>> If you derive the conclusion from accepted axioms using accepted rules of inference, then you can conclude that there's no error.
>>
>> Unless this a very tentatively held conclusion it would tend to presume
>> omniscience upon the one holding the conclusion. It presumes that all
>> the rules of inference are both correct and complete. It think that the
>> former conclusion (that the current rules are correct) has a much more
>> substantial basis than the latter conclusion, that the set of rules is
>> complete.
>>
>
> What sense of completeness do you have in mind here?

Omniscience

>
>> I am proposing a basis for forming new rules of inference. This basis
>> that I am proposing is currently in its infancy. Very little is
>> currently known of the inherent structure of knowledge. A key element in
>> this new basis has not nearly reached any degree of significant
>> agreement, much less universal consensus is:
>>
>> http://plato.stanford.edu/entries/compositionality/
>>
>> I propose that the connections between elements of knowledge must derive
>> an acyclic directed graph.
>>
>
> What connections, and where do the directions of the edges of the graph come from?

Connections from larger concepts to their constituent parts, recursively
down to their respective atomic elements of meaning.

>
>> I also propose that all paradoxes of self-reference can not be fully
>> specified within an acyclic di-graph, and this limitation shows their
>> error.
>
> Well, there's not a lot that I can do with that. Which aspect of Turing's proof cannot be fully specified within an acyclic di-graph, what's the meaning of such a claim?
>

http://plato.stanford.edu/entries/montague-semantics/

Since the science of formal semantics as applied to natural language has
mostly not been invented yet it is more feasible to start with the most
elemental form of self-reference and later build up this proof to
include increasingly more complex forms of self-reference.

The simplest form of the Liar Paradox is the simplest possible form of a
self-reference paradox:

[This proposition is false]

Within every possible world the above proposition can not be made coherent.

[Rupert]

On Wednesday, November 20, 2013 5:48:42 PM UTC+1, Peter Olcott wrote:
> On 11/20/2013 9:28 AM, Rupert wrote:
> > On Tuesday, November 19, 2013 10:01:25 PM UTC+1, Peter Olcott wrote:
> >> On 11/19/2013 9:39 AM, Rupert wrote:
> >>> On Tuesday, November 19, 2013 3:17:47 PM UTC+1, Peter Olcott wrote:
> >>>> It can not be correctly concluded that an error does not exist on the
> >>>> basis that this error can not be expressed within the limitations of any
> >>>> specific mode of expression such as the language of mathematics.
>
> >>>> It can only be correctly concluded that within this specific mode of
> >>>> expression that this error can not be seen.
>
> >>> If you derive the conclusion from accepted axioms using accepted rules of inference, then you can conclude that there's no error.
>
> >> Unless this a very tentatively held conclusion it would tend to presume
> >> omniscience upon the one holding the conclusion. It presumes that all
> >> the rules of inference are both correct and complete. It think that the
> >> former conclusion (that the current rules are correct) has a much more
> >> substantial basis than the latter conclusion, that the set of rules is
> >> complete.
>
> > What sense of completeness do you have in mind here?
>
> Omniscience
>

Doesn't mean anything to talk about a set of rules being omniscient.

> >> I am proposing a basis for forming new rules of inference. This basis
> >> that I am proposing is currently in its infancy. Very little is
> >> currently known of the inherent structure of knowledge. A key element in
> >> this new basis has not nearly reached any degree of significant
> >> agreement, much less universal consensus is:
>
> >> http://plato.stanford.edu/entries/compositionality/
>
> >> I propose that the connections between elements of knowledge must derive
> >> an acyclic directed graph.
>
> > What connections, and where do the directions of the edges of the graph come from?
>
> Connections from larger concepts to their constituent parts, recursively
> down to their respective atomic elements of meaning.
>
> >> I also propose that all paradoxes of self-reference can not be fully
> >> specified within an acyclic di-graph, and this limitation shows their
> >> error.
>
> > Well, there's not a lot that I can do with that. Which aspect of Turing's proof cannot be fully specified within an acyclic di-graph, what's the meaning of such a claim?
>
> http://plato.stanford.edu/entries/montague-semantics/
>
> Since the science of formal semantics as applied to natural language has
> mostly not been invented yet it is more feasible to start with the most
> elemental form of self-reference and later build up this proof to
> include increasingly more complex forms of self-reference.
>
> The simplest form of the Liar Paradox is the simplest possible form of a
> self-reference paradox:
>
> [This proposition is false]
>
> Within every possible world the above proposition can not be made coherent.

In the title of this thread you talk about the halting problem. I want to criticise your claim that Turing's proof of the insolubility of the halting problem contains in error. If you want to talk about the liar paradox instead, that's a different kettle of fish. I'd say you're right that it's not a coherent proposition.

[fom]

Not so. I read a wonderful analysis somewhere
that simply responds to every such statement
with the question "To which proposition does
"this" refer?"

In other words:

"blah... blah... blah.... 'Turtles have
two heads'; this proposition is false. It
is not... blah... blah... blah..."

would be considered an admissible use.

If one simply denies that the paradoxical
interpretation of the demonstrative is an
admissible pragmatic use, then the statement
is a discourse deixis requiring a proper
referent.

http://en.wikipedia.org/wiki/Deixis#Discourse

The wikipedia link would admit your interpretation,
but the analysis I read would not.

[Peter Olcott]

On 11/20/2013 10:59 AM, Rupert wrote:
> On Wednesday, November 20, 2013 5:48:42 PM UTC+1, Peter Olcott wrote:
>> On 11/20/2013 9:28 AM, Rupert wrote:
>>> On Tuesday, November 19, 2013 10:01:25 PM UTC+1, Peter Olcott wrote:
>>>> On 11/19/2013 9:39 AM, Rupert wrote:
>>>>> On Tuesday, November 19, 2013 3:17:47 PM UTC+1, Peter Olcott wrote:
>>>>>> It can not be correctly concluded that an error does not exist on the
>>>>>> basis that this error can not be expressed within the limitations of any
>>>>>> specific mode of expression such as the language of mathematics.
>>
>>>>>> It can only be correctly concluded that within this specific mode of
>>>>>> expression that this error can not be seen.
>>
>>>>> If you derive the conclusion from accepted axioms using accepted rules of inference, then you can conclude that there's no error.
>>
>>>> Unless this a very tentatively held conclusion it would tend to presume
>>>> omniscience upon the one holding the conclusion. It presumes that all
>>>> the rules of inference are both correct and complete. It think that the
>>>> former conclusion (that the current rules are correct) has a much more
>>>> substantial basis than the latter conclusion, that the set of rules is
>>>> complete.
>>
>>> What sense of completeness do you have in mind here?
>>
>> Omniscience
>>
>
> Doesn't mean anything to talk about a set of rules being omniscient.

I am not saying that the rules themselves are omniscient. I am saying
the the set of rules is known to be complete in the sense that no
additional rules can every possibly be discovered that are relevant to
the problem at hand.

> insolubility of thehalting problem contains in error. If you want
> to talk about the liar paradox instead,that's a different kettle

> of fish. I'd say you're right that it's not a coherent proposition.
>

I said: Paradox of self-reference (meaning all of them) such as...

I am saying that the error of the Halting Problem can be shown to be
analogous to the error of the Liar Paradox, thus equally an error.

The Halting Problem suffers from exactly the same incoherence as the
Liar Paradox, the difference is that the HP incoherence is far too
difficult to directly see using current notational conventions.

[George Greene]

On Tuesday, November 19, 2013 6:25:27 AM UTC-5, Peter Olcott wrote:
> Basis:
>
> Lemma01:
>

> a) Montague [meaning postulates] must be specified within acyclic
>
> di-graphs. (connections between elements)

Sez who? Seriously, why do you think you can defend this?
Appeal to authority is a fallacy.

> b) Connections between Montague [meaning postulates] (principle of
>
> compositionality) must not produce cycles.

That sentence is NOT grammatical. Connections between Montague AND WHAT??

> Lemma02: The [meaning postulate] of all self-reference paradoxes

Nothing nor anybody NEEDS TO HAVE a "meaning postulate".

> Lemma03:
>
Why are you calling these lemmas when you cannot prove them?

[George Greene]

On Tuesday, November 19, 2013 6:25:27 AM UTC-5, Peter Olcott wrote:

> Basis:
>
> Lemma01: Meaning can only be correctly specified within an acyclic
>
> directed graph:

THAT is NOT the RELEVANT basis!
THE BASIS is a PURELY SYNTACTIC paradigm.
It says that when you put certain strings in as inputs, you get
certain other strings out. Some of these strings can be output from NO
input. These strings are EASY to characterize. NOBODY GIVES A SHIT
whether YOU (or Montague) THINKS THEY *MEAN* anything OR NOT!!

[graham...@gmail.com]

So given

--------------
| 0 | 0 | 0 |
--------------
^


the end result from some TM is


--------------
| 0 | 1 | 1 |
--------------
^


this computes WHAT exactly George?


Herc

[Rupert]

Right, well my claim didn't rest on any claim of completeness in that sense.

Well, if you wish to put forward such a claim, then you need actually to show it.

> The Halting Problem suffers from exactly the same incoherence as the
> Liar Paradox, the difference is that the HP incoherence is far too
> difficult to directly see using current notational conventions.

Well, your claim remains unsupported.

[graham...@gmail.com]

On Wednesday, November 20, 2013 9:32:21 AM UTC-8, Peter Olcott wrote:
> The Halting Problem suffers from exactly the same incoherence as the
>
> Liar Paradox, the difference is that the HP incoherence is far too
>
> difficult to directly see using current notational conventions.

Yes and no. The Halting CONCLUSION relies on the Church Turing Thesis
which is purely an assumption.

CTT: All (full) computation models are equal.



Anyone can deny the Halting Conclusion and be entirely correct!

"Oh thats only for Turing Machines! Prove it holds for all languages!"








Do you agree / disagree with H0 and H1 ?

H0: no turing machine can input the 'godel number' of all other turing machines
and output 1=halt or 0=not-halt

H1: a subset of Turing Machines may be computationally complete.

Computationally Complete = Equivalent to Turing Machine Set of functions.




Herc
--
www.PrologDatabase.com

[Peter Olcott]

On 11/21/2013 2:49 AM, Franz Gnaedinger wrote:

> Last time I gave you the advice to look out for
> a more modest but realistic application of your
> knowledge-condensing machine, but you said No,
> it must be the machine of All Knowledge that goes
> against Goedel's proved theorems.

I do not think that is what I said, your paraphrase seems incorrect.
What I said would have been something along the lines that we must first
find the inherent essential structure of the universal set of all
knowledge before we can correctly begin correctly composing any subset
of this knowledge.

> Meanwhile someone
> else did what you refused to consider, a teenager,
> by then seventeen years old: he wrote an app that
> condenses online articles into summaries of four
> hundred words - you can read the summary, and if you
> like it you can download the entire article, saves
> you a lot of reading time. He sold his app to a big
> company, and made a lot of money. While you go on
> relying on the magic of words and dreaming of your
> God machine that will undo Goedel.
>

http://en.wikipedia.org/wiki/Cyc
http://www.cyc.com/

I am searching for the most efficient and effective strategy to complete
the CYC project such that it can by itself compose the complete meaning
postulates for any and all knowledge not yet contained within its [boot
strap] knowledge base.

[Peter Olcott]

I am lying about, lying about, lying ...
without ever getting to the actual object of the proposition.

Valid propositions must have objects that the proposition is about.

A statement of fact must have a possible world where it can be true, or
it is not even as much as a false statement.

If an utterance is assessed to see if it is a valid proposition and this
utterance lacks a possible world where it can be true, then this
utterance is not a valid proposition.

[Peter Olcott]

On 11/21/2013 1:16 AM, Rupert wrote:
> On Wednesday, November 20, 2013 6:32:21 PM UTC+1, Peter Olcott wrote:
>> On 11/20/2013 10:59 AM, Rupert wrote:
>>> On Wednesday, November 20, 2013 5:48:42 PM UTC+1, Peter Olcott wrote:
>>>> On 11/20/2013 9:28 AM, Rupert wrote:
>>>>> On Tuesday, November 19, 2013 10:01:25 PM UTC+1, Peter Olcott wrote:
>>>>>> On 11/19/2013 9:39 AM, Rupert wrote:
>>>>>>> On Tuesday, November 19, 2013 3:17:47 PM UTC+1, Peter Olcott wrote:
>>>>>>>> It can not be correctly concluded that an error does not exist on the
>>>>>>>> basis that this error can not be expressed within the limitations of any
>>>>>>>> specific mode of expression such as the language of mathematics.
>>
>>>>>>>> It can only be correctly concluded that within this specific mode of
>>>>>>>> expression that this error can not be seen.
>>
>>>>>>> If you derive the conclusion from accepted axioms using accepted rules of inference, then you can conclude that there's no error.
>>
>>>>>> Unless this a very tentatively held conclusion it would tend to presume
>>>>>> omniscience upon the one holding the conclusion. It presumes that all
>>>>>> the rules of inference are both correct and complete. It think that the
>>>>>> former conclusion (that the current rules are correct) has a much more
>>>>>> substantial basis than the latter conclusion, that the set of rules is
>>>>>> complete.
>>
>>>>> What sense of completeness do you have in mind here?
>>
>>>> Omniscience
>>
>>> Doesn't mean anything to talk about a set of rules being omniscient.
>>
>> I am not saying that the rules themselves are omniscient. I am saying

>> that the set of rules is known to be complete in the sense that no
>> additional rules can ever possibly be discovered that are relevant to

>> the problem at hand.
>>
>
> Right, well my claim didn't rest on any claim of completeness in that sense.

The only reason why things such as the Halting Problem and the
Incompleteness Theorem remain insufficiently refuted is that the current
degree of knowledge of the inherent structure of knowledge itself is not
yet sufficiently complete.

How about we say mostly unsupported or insufficiently supported, to say
that it is unsupported generally logically entails that not the
slightest trace of any degree of support has been provided what-so-ever.
It is extremely subtle nuances of meaning such as this that hides the
truth that I am attempting to show.

The support that I did provide (paraphrase) was that:

1) All paradoxes of self-reference inherently have cycles within the
directed graph of their meaning postulates.

2) Meaning postulates that have cycles within their directed graph are
incorrect.

3) Meaning postulates must only be constructed using acyclic di-graphs
as their basis.

[Rupert]

Actually, the reason that they're insufficiently refuted is that you haven't identified any problems with the arguments at all. The proofs are perfectly sound, and the theorems are correct.

You mean Turing's proof of the result that the halting problem is insoluble? It contains some kind of incoherence? Well, show me. Why is it incoherent? What's wrong with the argument?

> > Well, your claim remains unsupported.
>
> How about we say mostly unsupported or insufficiently supported, to say
> that it is unsupported generally logically entails that not the
> slightest trace of any degree of support has been provided what-so-ever.
>

Which is indeed the case.

> It is extremely subtle nuances of meaning such as this that hides the
> truth that I am attempting to show.
>

By all means attempt to show it. I am patiently waiting for the demonstration.

> The support that I did provide (paraphrase) was that:
>
> 1) All paradoxes of self-reference inherently have cycles within the
> directed graph of their meaning postulates.
>

Please give an example of this. Preferably actually showing us the graph.

> 2) Meaning postulates that have cycles within their directed graph are
> incorrect.
>

What's a meaning postulate?

> 3) Meaning postulates must only be constructed using acyclic di-graphs
> as their basis.

So why does this have anything to do with Turing's proof?

[Peter Olcott]

Whoops, let me correct that:
If no possible worlds exist where a proposition can be either true or
false, then the proposition is erroneous.

[Peter Olcott]

Statements such as this presume omniscience as their basis, and are thus
insufficiently supported when this omniscience is lacking.

The most that one can correctly say about the proofs and the theorems is
that they are very widely accepted as true and sound. This is not at all
the same thing as actually being *perfectly* sound and true.

In order for the correct conclusion to be made that they are indeed
*perfectly* sound and true, one would have to exhaustively show that no
refutation could ever possibly exist given an infinite amount of time to
form such a refutation. This would either take an infinite amount of
time or would require actual omniscience.

+-------+
| V
+---[This proposition is false]

[Five is greater than three]--->{numerical relation between(5, 3)}

The first graph points to itself as the object of the proposition.

The second graph does not point to itself as the object of its proposition.

>
>> 2) Meaning postulates that have cycles within their directed graph are
>> incorrect.
>>
>
> What's a meaning postulate?

http://www.glottopedia.org/index.php/Meaning_postulate

>
>> 3) Meaning postulates must only be constructed using acyclic di-graphs
>> as their basis.
>
> So why does this have anything to do with Turing's proof?
>

The self-reference aspect of the Halting Problem forms an error.

[fom]

That would seem to be the point.

And, thus you agree that you have misrepresented

[This proposition is false]

as a proposition.

> A statement of fact must have a possible world where it can be true, or
> it is not even as much as a false statement.
>

Facticity need not be conjoined with a
semantic theory. One may have a set of
facts relative to the story in a fictional
piece of literature. More formally, one
may have a logic and a deductive closure
with respect to that logic whose terms
are empty names.

The notion of "fact" is in that nether world
requiring some definition before it is used
quite so loosely. Moreover, your usage above
cannot tie it to actuality since you are using
it in a modal context.

Perhaps it is the word "true" which you are
using as an empty reference.

In like fashion, circularity is not necessarily
an error of reasoning. Early commentators on
this fact include Carnap and Ramsey. More recently,
I have read Feferman who observes that paradox
arises precisely where self-similarity is combined
with negation. Self-similarity alone is not necessarily
fatal. Church's lambda calculus originates with the
idea of functions that can be applied to themselves.

You really need to get your "facts" straight.

> If an utterance is assessed to see if it is a valid proposition and this
> utterance lacks a possible world where it can be true, then this
> utterance is not a valid proposition.
>

Well, if that is your reasoning, then you are a
little late.

The entire idea behind a formalized language is
to exclude expressions which are not propositions.

That would include the syntax above (self-purport
of being a proposition does not make the statement
propositional).

If you find that standard strategy objectionable,
then you may turn to natural language constructs
as you claim to be doing. However, then you
should stop using meaningless phrases like "valid
propositions". The pragmatics of natural language
is not based upon ideal language semantics.

[fom]

Well, instead of impressing us with assertions
of subtle nuance, list a few meaning postulates
and a few propositions and demonstrate the graphs.

[fom]

Well, not quite.

The assertion that the expression is a proposition
is erroneous.

Or, it is not that the proposition is erroneous. Rather,
is that it is unstable:

http://plato.stanford.edu/entries/truth-revision/

[Peter Olcott]

It was much more clear to refer to my reasoning about propositions using
the tri-state construct {True, False, Incorrect} because of the lack of
a universally agreed upon term to otherwise refer to this non-proposition.

[Doc O'Leary]

In article <HKSdnc8L-5O0bRDP...@giganews.com>,

Peter Olcott <OCR4Screen> wrote:

> I do not think that is what I said, your paraphrase seems incorrect.
> What I said would have been something along the lines that we must first
> find the inherent essential structure of the universal set of all
> knowledge before we can correctly begin correctly composing any subset
> of this knowledge.

Or, put another way, we need to define what intelligence is before we
can seriously attempt to construct it artificially. I've covered this
many times in past discussions. My current thinking remains that we
function as Expectation Engines.

> http://en.wikipedia.org/wiki/Cyc
> http://www.cyc.com/
>
> I am searching for the most efficient and effective strategy to complete
> the CYC project such that it can by itself compose the complete meaning
> postulates for any and all knowledge not yet contained within its [boot
> strap] knowledge base.

Give up on CYC. Start from scratch.

--
iPhone apps that matter: http://appstore.subsume.com/
My personal UDP list: 127.0.0.1, localhost, googlegroups.com, theremailer.net,
and probably your server, too.

[fom]

Three value logics often use the expression "neither".

But, again, if you are not using a bivalent logic,
then you need to be clear about such matters.

There are analyses of these problems in many-valued
logics already. Perhaps you could give us a survey
of what you have encountered and why those approaches
are unsatisfactory.

[Peter Olcott]

On 11/21/2013 12:28 PM, Doc O'Leary wrote:
> In article <HKSdnc8L-5O0bRDP...@giganews.com>,
> Peter Olcott <OCR4Screen> wrote:
>
>> I do not think that is what I said, your paraphrase seems incorrect.
>> What I said would have been something along the lines that we must first
>> find the inherent essential structure of the universal set of all
>> knowledge before we can correctly begin correctly composing any subset
>> of this knowledge.
>
> Or, put another way, we need to define what intelligence is before we
> can seriously attempt to construct it artificially.

I don't think that this is entirely true. We only need the capability to
reproduce the functional end results of intelligence, this may not
require any understanding at all of the underlying implementation details.

The key aspect that I was addressing in my prior response is that any
entirely adequate model of knowledge must make sure that this model
works with every element in the set of all knowledge. This will not
require omniscience it will only require fully understanding the
inherent structure of this set of all knowledge.

[Peter Olcott]

That would not sufficiently correspond to problems such as this:
1) What time is it (Yes or No) ?

2) Is the statement: "What time is it?" true or false?

In this case {neither} is explicitly excluded from the possible solution
set.

The fallacies of self-reference (like the above examples) correspond to
my original tri-state values:
1) true
2) false
3) erroneous

[Ben Bacarisse]

Peter Olcott <OCR4Screen> writes:
<snip>

> Whoops, let me correct that:
> If no possible worlds exist where a proposition can be either true or
> false, then the proposition is erroneous.

Same old stuff from 18 months ago. Every proposition of the form
"Turing machine M halts on input I" is either true or false in this
world. Similarly for the proposition "there is a TM that can decide
halting" (it's false in this world). None of these propositions is
erroneous by that criterion.

<snip>
--
Ben.

[Peter Olcott]

On 11/21/2013 6:20 PM, Ben Bacarisse wrote:
> Peter Olcott <OCR4Screen> writes:
> <snip>
>> Whoops, let me correct that:
>> If no possible worlds exist where a proposition can be either true or
>> false, then the proposition is erroneous.
> Same old stuff from 18 months ago. Every proposition of the form
> "Turing machine M halts on input I" is either true or false in this
> world.

No Turing Machines ever halt or fail to halt in any possible world
because they are defined as fictional.

[fom]

chuckle

Perhaps you should demonstrate that you have actually
investigated these matters. I do not fault you
for having a different opinion. I just do not believe
you have made the effort to learn about what you think
about.

[George Greene]

This is A STUPID
question. ALL TMs compute EXACTLY WHAT they compute.
And you can't allege that anything has been computed
by 1 input and 1 output.

[Rupert]

This is false. The statement is based on the fact that I have read the proof and confirmed that it is correct. No claim to omniscience is necessary.

> and are thus
> insufficiently supported when this omniscience is lacking.
>

No, the statement is perfectly adequately supported. The support for the statement is provided by the proofs of the theorems. People who are competent to understand the arguments are able to read them and confirm that they are correct. You haven't identified any flaw in the arguments. The belief that the arguments are correct is not infallible, but it's certainly quite adequately supported, very strongly supported indeed.

> The most that one can correctly say about the proofs and the theorems is
> that they are very widely accepted as true and sound. This is not at all
> the same thing as actually being *perfectly* sound and true.
>

I have read the arguments and confirmed that they are correct, and just about every other mathematician who has examined them in the last 80 years agrees with me. This does indeed provide very strong reason to think that the arguments are perfectly sound and true. The belief is not infallible. You might be able to identify a flaw in the arguments which somehow everyone missed for a period of 80 years. But you haven't made any attempt to do that. So there is excellent reason to think that the arguments are indeed perfectly sound and true.

> In order for the correct conclusion to be made that they are indeed
> *perfectly* sound and true, one would have to exhaustively show that no
> refutation could ever possibly exist given an infinite amount of time to
> form such a refutation. This would either take an infinite amount of
> time or would require actual omniscience.
>

No. It doesn't require omniscience. It just requires an ability to read and understand mathematical arguments and check that they are correct.

I can't see a directed graph here. The first thing seems to be a connected graph with two labelled vertices, but no direction for the edge. You need to do more by way of explaining its significance.

> >> 2) Meaning postulates that have cycles within their directed graph are
> >> incorrect.
>
> > What's a meaning postulate?
>
> http://www.glottopedia.org/index.php/Meaning_postulate
>

What's the directed graph of a meaning postulate?

> >> 3) Meaning postulates must only be constructed using acyclic di-graphs
> >> as their basis.
>
> > So why does this have anything to do with Turing's proof?
>
> The self-reference aspect of the Halting Problem forms an error.

Please be more specific about what you mean by "the self-reference aspect".

[Antti Valmari]

A much more readable and competent attempt to deny undecidability of
halting was written by Eric C.R. Hehner. He is an established computer
scientist with a decent publication record. To me the most rewarding
part (rewarding in a perhaps weird sense) was when I temporarily ignored
the flaws in his reasoning and thought about the consequences of his
conclusion: what would change if he were right. Here:

www.cs.toronto.edu/~hehner/PHP.pdf

The discussions in comp.theory eventually led me to do a literature
survey and research of my own on the proportion of hard instances among
all instances, given an incomplete halting tester. I found surprisingly
few papers; excluding one, they were surprisingly recent; they were
surprisingly diverse and unaware of each other; and the old one had been
cited surprisingly few times. After submitting the camera-ready version
but before the conference I realized that the part of my Theorem 9 about
type-B testers is wrong. The truth is formulated in the conference
slides and basically the opposite to what I claimed in the paper. Here
is an (uncorrected) extended version of the paper:

http://arxiv.org/abs/1307.7066

The slides of the conference talk are here:

http://www.cs.tut.fi/%7eava/ill_k.pdf

If anybody reading this is fluent in 40 year old versions of recursive
function theory and willing to help me, I would like to discuss the
second result of Nancy Lynch mentioned in my survey. If it can be
carried over to programming languages, it would be a very strong result.

--- Antti Valmari ---

[Peter Olcott]

To have actual complete certainty that it is correct without the
slightest trace of any presumption would require knowing with certainty
that no errors in any of the reasoning of this proof could ever be found
even if given an infinite amount of time to find such errors.

>> and are thus
>> insufficiently supported when this omniscience is lacking.
>>
> No, the statement is perfectly adequately supported.

Which is it [perfectly] or [adequately] ? There is a huge difference
between the two.

> The support for the statement is provided by the proofs of the theorems. People who are competent to understand the arguments are able to read them and confirm that they are correct. You haven't identified any flaw in the arguments. The belief that the arguments are correct is not infallible, but it's certainly quite adequately supported, very strongly supported indeed.

Okay, I would agree with that.

>> The most that one can correctly say about the proofs and the theorems is
>> that they are very widely accepted as true and sound. This is not at all
>> the same thing as actually being *perfectly* sound and true.
>>
> I have read the arguments and confirmed that they are correct, and just about every other mathematician who has examined them in the last 80 years agrees with me. This does indeed provide very strong reason to think that the arguments are perfectly sound and true.

Not perfectly, that would presume omniscience.

> The belief is not infallible. You might be able to identify a flaw in the arguments which somehow everyone missed for a period of 80 years. But you haven't made any attempt to do that. So there is excellent reason to think that the arguments are indeed perfectly sound and true.

Or in other words the sufficient basis for concluding that they are not
sound and true has not yet been established. In this I would agree.

>> In order for the correct conclusion to be made that they are indeed
>> *perfectly* sound and true, one would have to exhaustively show that no
>> refutation could ever possibly exist given an infinite amount of time to
>> form such a refutation. This would either take an infinite amount of
>> time or would require actual omniscience.
>>
> No. It doesn't require omniscience. It just requires an ability to read and understand mathematical arguments and check that they are correct.

*perfection* of knowledge (absolute complete and total) can rarely if
ever occur lacking omniscience. People tend to use the term [perfect]
when they actually mean {very excellent}. People also often say [all]
when they mean very many, and [never] when they mean {very few}, on and
on habitual sloppiness of precision that seems to be hardwired into most
human brains.

The ASCII text arrows: "V" and ">" show the direction.
I have to pull the elaborations from their wordless intuitions within my
mind.

Can you see how the infinitely recursive structure of the Liar Paradox
causes it to not have an object to which a truth value can be attached?

>>>> 2) Meaning postulates that have cycles within their directed graph are
>>>> incorrect.
>>> What's a meaning postulate?
>> http://www.glottopedia.org/index.php/Meaning_postulate
>>
> What's the directed graph of a meaning postulate?

Meaning Postulates are formed through connections between elements of
meaning including AtomicUnitsOfMeaning such that all of these
connections form an acyclic directed graph.

[Rupert]

We're not making a claim of infallibility here. We're claiming that it's as solidly justified a belief as any and you haven't offered the least grounds for doubt.

> >> and are thus
> >> insufficiently supported when this omniscience is lacking.
>
> > No, the statement is perfectly adequately supported.
>
> Which is it [perfectly] or [adequately] ? There is a huge difference
> between the two.
>

"Perfectly" modifies the "adequately".

One could paraphrase it as follows: the statement is supported in a perfectly adequate manner. Here "adequate" is the adjective and "perfectly" is the adverb. Does that make things clearer?

> > The support for the statement is provided by the proofs of the theorems. People who are competent to understand the arguments are able to read them and confirm that they are correct. You haven't identified any flaw in the arguments. The belief that the arguments are correct is not infallible, but it's certainly quite adequately supported, very strongly supported indeed.
>
> Okay, I would agree with that.
>
> >> The most that one can correctly say about the proofs and the theorems is
> >> that they are very widely accepted as true and sound. This is not at all
> >> the same thing as actually being *perfectly* sound and true.
>
> > I have read the arguments and confirmed that they are correct, and just about every other mathematician who has examined them in the last 80 years agrees with me. This does indeed provide very strong reason to think that the arguments are perfectly sound and true.
>
> Not perfectly, that would presume omniscience.
>

No. It doesn't. It is much more likely than not that the arguments are indeed perfectly sound and true in the sense of not having any errors at all. Our belief that this is the case is not infallible, but perfectly well-founded.

> > The belief is not infallible. You might be able to identify a flaw in the arguments which somehow everyone missed for a period of 80 years. But you haven't made any attempt to do that. So there is excellent reason to think that the arguments are indeed perfectly sound and true.
>
> Or in other words the sufficient basis for concluding that they are not
> sound and true has not yet been established. In this I would agree.
>

Good. So we're very well justified in thinking the theorems true, and you haven't offered the least reason to think otherwise. So are we done?

> >> In order for the correct conclusion to be made that they are indeed
> >> *perfectly* sound and true, one would have to exhaustively show that no
> >> refutation could ever possibly exist given an infinite amount of time to
> >> form such a refutation. This would either take an infinite amount of
> >> time or would require actual omniscience.
>
> > No. It doesn't require omniscience. It just requires an ability to read and understand mathematical arguments and check that they are correct.
>
> *perfection* of knowledge (absolute complete and total) can rarely if
> ever occur lacking omniscience. People tend to use the term [perfect]
> when they actually mean {very excellent}. People also often say [all]
> when they mean very many, and [never] when they mean {very few}, on and
> on habitual sloppiness of precision that seems to be hardwired into most
> human brains.
>

If a proof is correct, it is perfectly correct. That's the way that "perfect" is being applied in this context.

Right, so it looks as though the first graph has four edges and only one of them is directed, is that correct?

> I have to pull the elaborations from their wordless intuitions within my
> mind.
>

How do you obtain a directed graph of a meaning postulate?

> Can you see how the infinitely recursive structure of the Liar Paradox
> causes it to not have an object to which a truth value can be attached?
>

No.

> >>>> 2) Meaning postulates that have cycles within their directed graph are
> >>>> incorrect.
>
> >>> What's a meaning postulate?
>
> >> http://www.glottopedia.org/index.php/Meaning_postulate
>
> > What's the directed graph of a meaning postulate?
>
> Meaning Postulates are formed through connections between elements of
> meaning including AtomicUnitsOfMeaning such that all of these
> connections form an acyclic directed graph.
>

How do you obtain the directed graph? Give an example of a meaning postulate.

[Doc O'Leary]

In article <ceidnRFFcYvp-RPP...@giganews.com>,

Peter Olcott <OCR4Screen> wrote:

> On 11/21/2013 12:28 PM, Doc O'Leary wrote:
> > In article <HKSdnc8L-5O0bRDP...@giganews.com>,
> > Peter Olcott <OCR4Screen> wrote:
> >
> >> I do not think that is what I said, your paraphrase seems incorrect.
> >> What I said would have been something along the lines that we must first
> >> find the inherent essential structure of the universal set of all
> >> knowledge before we can correctly begin correctly composing any subset
> >> of this knowledge.
> >
> > Or, put another way, we need to define what intelligence is before we
> > can seriously attempt to construct it artificially.
>
> I don't think that this is entirely true. We only need the capability to
> reproduce the functional end results of intelligence, this may not
> require any understanding at all of the underlying implementation details.

You are wrong. Your way of thinking is precisely why very little
progress has been made in AI in decades. Or, sure, we may soon have
some "functional end results" like self-driving cars, but they have
*nothing* to do with artificial intelligence. Watson was just another
such cheat. The Turing Test itself is a cop-out.

> The key aspect that I was addressing in my prior response is that any
> entirely adequate model of knowledge must make sure that this model
> works with every element in the set of all knowledge. This will not
> require omniscience it will only require fully understanding the
> inherent structure of this set of all knowledge.

Yes, exactly what I referred to. The "inherent structure" of knowledge
with respect to intelligence is a core issue that seldom gets discussed.
I have often called out the proponents who hand wave "learning" as the
AI panacea, because they never seem to be able to explain *what* is
fundamentally being "learned". And, something I maintain is just as
important, what happens when when an intelligent system determines it is
*wrong* about something.

[Peter Olcott]

On 11/22/2013 12:21 PM, Doc O'Leary wrote:
> In article <ceidnRFFcYvp-RPP...@giganews.com>,
> Peter Olcott <OCR4Screen> wrote:
>
>> On 11/21/2013 12:28 PM, Doc O'Leary wrote:
>>> In article <HKSdnc8L-5O0bRDP...@giganews.com>,
>>> Peter Olcott <OCR4Screen> wrote:
>>>
>>>> I do not think that is what I said, your paraphrase seems incorrect.
>>>> What I said would have been something along the lines that we must first
>>>> find the inherent essential structure of the universal set of all
>>>> knowledge before we can correctly begin correctly composing any subset
>>>> of this knowledge.
>>> Or, put another way, we need to define what intelligence is before we
>>> can seriously attempt to construct it artificially.
>> I don't think that this is entirely true. We only need the capability to
>> reproduce the functional end results of intelligence, this may not
>> require any understanding at all of the underlying implementation details.
> You are wrong. Your way of thinking is precisely why very little
> progress has been made in AI in decades. Or, sure, we may soon have
> some "functional end results" like self-driving cars, but they have
> *nothing* to do with artificial intelligence. Watson was just another
> such cheat. The Turing Test itself is a cop-out.

I would tend to agree with your assessment of Watson and the Turing test.
What I am proposing is that once the structure of thought is fully
understood that human like reasoning will be (at least mostly) entailed
by the details of this structure.

>
>> The key aspect that I was addressing in my prior response is that any
>> entirely adequate model of knowledge must make sure that this model
>> works with every element in the set of all knowledge. This will not
>> require omniscience it will only require fully understanding the
>> inherent structure of this set of all knowledge.
> Yes, exactly what I referred to. The "inherent structure" of knowledge
> with respect to intelligence is a core issue that seldom gets discussed.

It is good to have agreement on this crucial point.

> I have often called out the proponents who hand wave "learning" as the
> AI panacea, because they never seem to be able to explain *what* is
> fundamentally being "learned".

Exactly !
Everyone seems to be skipping this fundamental prerequisite.

> And, something I maintain is just as
> important, what happens when when an intelligent system determines it is
> *wrong* about something.

Yes, learning from its mistakes.

[Peter Percival]

sci.lang removed

Peter Olcott wrote:

> Lemma03: The Halting Problem and the Liar Paradox are both
> self-reference paradoxes.

Why do you call the halting problem a self-reference paradox?


--
Madam Life's a piece in bloom,
Death goes dogging everywhere:
She's the tenant of the room,
He's the ruffian on the stair.

[Peter Olcott]

On 11/22/2013 1:28 PM, Peter Percival wrote:
> sci.lang removed
>
> Peter Olcott wrote:
>
>> Lemma03: The Halting Problem and the Liar Paradox are both
>> self-reference paradoxes.
>
> Why do you call the halting problem a self-reference paradox?
>
>

http://plato.stanford.edu/entries/self-reference/#ConConProCom

[Newberry]

On Wednesday, November 20, 2013 8:48:42 AM UTC-8, Peter Olcott wrote:
> On 11/20/2013 9:28 AM, Rupert wrote:
>
> > On Tuesday, November 19, 2013 10:01:25 PM UTC+1, Peter Olcott wrote:
>
> >> On 11/19/2013 9:39 AM, Rupert wrote:
>
> >>> On Tuesday, November 19, 2013 3:17:47 PM UTC+1, Peter Olcott wrote:
>
> >>>> It can not be correctly concluded that an error does not exist on the
>
> >>>> basis that this error can not be expressed within the limitations of any
>
> >>>> specific mode of expression such as the language of mathematics.
>
> >>
>
> >>>> It can only be correctly concluded that within this specific mode of
>
> >>>> expression that this error can not be seen.
>
> >>
>
> >>> If you derive the conclusion from accepted axioms using accepted rules of inference, then you can conclude that there's no error.
>
> >>
>
> >> Unless this a very tentatively held conclusion it would tend to presume
>
> >> omniscience upon the one holding the conclusion. It presumes that all
>
> >> the rules of inference are both correct and complete. It think that the
>
> >> former conclusion (that the current rules are correct) has a much more
>
> >> substantial basis than the latter conclusion, that the set of rules is
>
> >> complete.
>
> >>
>
> >
>
> > What sense of completeness do you have in mind here?
>
>
>
> Omniscience
>
>
>
>
>
> >
>

> >> I am proposing a basis for forming new rules of inference. This basis
>
> >> that I am proposing is currently in its infancy. Very little is
>
> >> currently known of the inherent structure of knowledge. A key element in
>
> >> this new basis has not nearly reached any degree of significant
>
> >> agreement, much less universal consensus is:
>
> >>
>
> >> http://plato.stanford.edu/entries/compositionality/
>
> >>
>
> >> I propose that the connections between elements of knowledge must derive
>
> >> an acyclic directed graph.
>
> >>
>
> >
>
> > What connections, and where do the directions of the edges of the graph come from?
>
>
>
> Connections from larger concepts to their constituent parts, recursively
>
> down to their respective atomic elements of meaning.
>
>
>
> >
>
> >> I also propose that all paradoxes of self-reference can not be fully
>
> >> specified within an acyclic di-graph, and this limitation shows their
>
> >> error.
>
> >
>
> > Well, there's not a lot that I can do with that. Which aspect of Turing's proof cannot be fully specified within an acyclic di-graph, what's the meaning of such a claim?
>
> >
>
>
>
> http://plato.stanford.edu/entries/montague-semantics/
>
>
>
> Since the science of formal semantics as applied to natural language has
>
> mostly not been invented yet it is more feasible to start with the most
>
> elemental form of self-reference and later build up this proof to
>
> include increasingly more complex forms of self-reference.
>
>
>
> The simplest form of the Liar Paradox is the simplest possible form of a
>
> self-reference paradox:
>
>
>
> [This proposition is false]
>
>
>
> Within every possible world the above proposition can not be made coherent.

Are we living in an impossible world then?

[Newberry]

On Wednesday, November 20, 2013 9:32:21 AM UTC-8, Peter Olcott wrote:
> On 11/20/2013 10:59 AM, Rupert wrote:
>

> > On Wednesday, November 20, 2013 5:48:42 PM UTC+1, Peter Olcott wrote:
>
> >> On 11/20/2013 9:28 AM, Rupert wrote:
>
> >>> On Tuesday, November 19, 2013 10:01:25 PM UTC+1, Peter Olcott wrote:
>
> >>>> On 11/19/2013 9:39 AM, Rupert wrote:
>
> >>>>> On Tuesday, November 19, 2013 3:17:47 PM UTC+1, Peter Olcott wrote:
>
> >>>>>> It can not be correctly concluded that an error does not exist on the
>
> >>>>>> basis that this error can not be expressed within the limitations of any
>
> >>>>>> specific mode of expression such as the language of mathematics.
>
> >>
>
> >>>>>> It can only be correctly concluded that within this specific mode of
>
> >>>>>> expression that this error can not be seen.
>
> >>
>
> >>>>> If you derive the conclusion from accepted axioms using accepted rules of inference, then you can conclude that there's no error.
>
> >>
>
> >>>> Unless this a very tentatively held conclusion it would tend to presume
>
> >>>> omniscience upon the one holding the conclusion. It presumes that all
>
> >>>> the rules of inference are both correct and complete. It think that the
>
> >>>> former conclusion (that the current rules are correct) has a much more
>
> >>>> substantial basis than the latter conclusion, that the set of rules is
>
> >>>> complete.
>
> >>
>
> >>> What sense of completeness do you have in mind here?
>
> >>
>
> >> Omniscience
>
> >>
>
> >
>

> > Doesn't mean anything to talk about a set of rules being omniscient.
>
>
>
> I am not saying that the rules themselves are omniscient. I am saying
>

> the the set of rules is known to be complete in the sense that no
>
> additional rules can every possibly be discovered that are relevant to
>
> the problem at hand.
>
>
>
> >
>

> >>>> I am proposing a basis for forming new rules of inference. This basis
>
> >>>> that I am proposing is currently in its infancy. Very little is
>
> >>>> currently known of the inherent structure of knowledge. A key element in
>
> >>>> this new basis has not nearly reached any degree of significant
>
> >>>> agreement, much less universal consensus is:
>
> >>
>
> >>>> http://plato.stanford.edu/entries/compositionality/
>
> >>
>
> >>>> I propose that the connections between elements of knowledge must derive
>
> >>>> an acyclic directed graph.
>
> >>
>
> >>> What connections, and where do the directions of the edges of the graph come from?
>
> >>
>
> >> Connections from larger concepts to their constituent parts, recursively
>
> >> down to their respective atomic elements of meaning.
>
> >>
>
> >>>> I also propose that all paradoxes of self-reference can not be fully
>
> >>>> specified within an acyclic di-graph, and this limitation shows their
>
> >>>> error.
>
> >>
>
> >>> Well, there's not a lot that I can do with that. Which aspect of Turing's proof cannot be fully specified within an acyclic di-graph, what's the meaning of such a claim?
>
> >>
>
> >> http://plato.stanford.edu/entries/montague-semantics/
>
> >>
>
> >> Since the science of formal semantics as applied to natural language has
>
> >> mostly not been invented yet it is more feasible to start with the most
>
> >> elemental form of self-reference and later build up this proof to
>
> >> include increasingly more complex forms of self-reference.
>
> >>
>
> >> The simplest form of the Liar Paradox is the simplest possible form of a
>
> >> self-reference paradox:
>
> >>
>
> >> [This proposition is false]
>
> >>
>
> >> Within every possible world the above proposition can not be made coherent.
>
> >
>

> > In the title of this thread you talk about the halting problem.
>
>
>
> > I want to criticise your claim that Turing's proof of the
>
> > insolubility of thehalting problem contains in error. If you want
>
> > to talk about the liar paradox instead,that's a different kettle
>
> > of fish. I'd say you're right that it's not a coherent proposition.
>
> >
>
>
>
> I said: Paradox of self-reference (meaning all of them) such as...
>
>
>
> I am saying that the error of the Halting Problem can be shown to be
>
> analogous to the error of the Liar Paradox, thus equally an error.
>
>
>

> The Halting Problem suffers from exactly the same incoherence as the
>
> Liar Paradox, the difference is that the HP incoherence is far too
>
> difficult to directly see using current notational conventions.

I tend to agree with this assessment.

[Newberry]

On Thursday, November 21, 2013 4:20:08 PM UTC-8, Ben Bacarisse wrote:
> Peter Olcott <OCR4Screen> writes:
>
> <snip>
>
> > Whoops, let me correct that:
>
> > If no possible worlds exist where a proposition can be either true or
>
> > false, then the proposition is erroneous.
>
>
>
> Same old stuff from 18 months ago. Every proposition of the form
>
> "Turing machine M halts on input I" is either true or false in this
>
> world.

There are syntactically correct pseudo-programs, to which the halting question does not apply.

[Newberry]

1) P
2) ~P
3) ~T(P)

[Newberry]

On Thursday, November 21, 2013 7:07:52 AM UTC-8, Rupert wrote:
> On Thursday, November 21, 2013 4:00:59 PM UTC+1, Peter Olcott wrote:
>
> > On 11/21/2013 1:16 AM, Rupert wrote:

>
> > > On Wednesday, November 20, 2013 6:32:21 PM UTC+1, Peter Olcott wrote:
>
> > >> On 11/20/2013 10:59 AM, Rupert wrote:
>

> > >>> On Wednesday, November 20, 2013 5:48:42 PM UTC+1, Peter Olcott wrote:
>
> > >>>> On 11/20/2013 9:28 AM, Rupert wrote:
>
> > >>>>> On Tuesday, November 19, 2013 10:01:25 PM UTC+1, Peter Olcott wrote:
>
> > >>>>>> On 11/19/2013 9:39 AM, Rupert wrote:
>
> > >>>>>>> On Tuesday, November 19, 2013 3:17:47 PM UTC+1, Peter Olcott wrote:
>
> > >>>>>>>> It can not be correctly concluded that an error does not exist on the
>
> > >>>>>>>> basis that this error can not be expressed within the limitations of any
>
> > >>>>>>>> specific mode of expression such as the language of mathematics.
>
> >
>
> > >>>>>>>> It can only be correctly concluded that within this specific mode of
>
> > >>>>>>>> expression that this error can not be seen.
>
> >
>
> > >>>>>>> If you derive the conclusion from accepted axioms using accepted rules of inference, then you can conclude that there's no error.
>
> >
>
> > >>>>>> Unless this a very tentatively held conclusion it would tend to presume
>
> > >>>>>> omniscience upon the one holding the conclusion. It presumes that all
>
> > >>>>>> the rules of inference are both correct and complete. It think that the
>
> > >>>>>> former conclusion (that the current rules are correct) has a much more
>
> > >>>>>> substantial basis than the latter conclusion, that the set of rules is
>
> > >>>>>> complete.
>
> >
>
> > >>>>> What sense of completeness do you have in mind here?
>
> >
>
> > >>>> Omniscience
>
> >
>

> > >>> Doesn't mean anything to talk about a set of rules being omniscient.
>
> >
>
> > >> I am not saying that the rules themselves are omniscient. I am saying
>

> > >> that the set of rules is known to be complete in the sense that no
>
> > >> additional rules can ever possibly be discovered that are relevant to
>
> > >> the problem at hand.
>
> >
>

> > > Right, well my claim didn't rest on any claim of completeness in that sense.
>
> >
>
> > The only reason why things such as the Halting Problem and the
>
> > Incompleteness Theorem remain insufficiently refuted is that the current
>
> > degree of knowledge of the inherent structure of knowledge itself is not
>
> > yet sufficiently complete.
>
> >
>
>
>
> Actually, the reason that they're insufficiently refuted is that you haven't identified any problems with the arguments at all. The proofs are perfectly sound, and the theorems are correct.
>
>
>

> > >>>>>> I am proposing a basis for forming new rules of inference. This basis
>
> > >>>>>> that I am proposing is currently in its infancy. Very little is
>
> > >>>>>> currently known of the inherent structure of knowledge. A key element in
>
> > >>>>>> this new basis has not nearly reached any degree of significant
>
> > >>>>>> agreement, much less universal consensus is:
>
> >
>
> > >>>>>> http://plato.stanford.edu/entries/compositionality/
>
> >
>
> > >>>>>> I propose that the connections between elements of knowledge must derive
>
> > >>>>>> an acyclic directed graph.
>
> >
>
> > >>>>> What connections, and where do the directions of the edges of the graph come from?
>
> >
>
> > >>>> Connections from larger concepts to their constituent parts, recursively
>
> > >>>> down to their respective atomic elements of meaning.
>
> >
>
> > >>>>>> I also propose that all paradoxes of self-reference can not be fully
>
> > >>>>>> specified within an acyclic di-graph, and this limitation shows their
>
> > >>>>>> error.
>
> >
>
> > >>>>> Well, there's not a lot that I can do with that. Which aspect of Turing's proof cannot be fully specified within an acyclic di-graph, what's the meaning of such a claim?
>
> >
>
> > >>>> http://plato.stanford.edu/entries/montague-semantics/
>
> >
>
> > >>>> Since the science of formal semantics as applied to natural language has
>
> > >>>> mostly not been invented yet it is more feasible to start with the most
>
> > >>>> elemental form of self-reference and later build up this proof to
>
> > >>>> include increasingly more complex forms of self-reference.
>
> >
>
> > >>>> The simplest form of the Liar Paradox is the simplest possible form of a
>
> > >>>> self-reference paradox:
>
> >
>
> > >>>> [This proposition is false]
>
> >
>
> > >>>> Within every possible world the above proposition can not be made coherent.
>
> >
>

> > >>> In the title of this thread you talk about the halting problem.
>
> >
>
> > >>> I want to criticise your claim that Turing's proof of the
>
> > >>> insolubility of thehalting problem contains in error. If you want
>
> > >>> to talk about the liar paradox instead,that's a different kettle
>
> > >>> of fish. I'd say you're right that it's not a coherent proposition.
>
> >
>
> > >> I said: Paradox of self-reference (meaning all of them) such as...
>
> >
>
> > >> I am saying that the error of the Halting Problem can be shown to be
>
> > >> analogous to the error of the Liar Paradox, thus equally an error.
>
> >
>

> > > Well, if you wish to put forward such a claim, then you need actually to show it.
>
> >
>

> > >> The Halting Problem suffers from exactly the same incoherence as the
>
> > >> Liar Paradox, the difference is that the HP incoherence is far too
>
> > >> difficult to directly see using current notational conventions.
>
> >
>
>
>

> You mean Turing's proof of the result that the halting problem is insoluble? It contains some kind of incoherence? Well, show me. Why is it incoherent? What's wrong with the argument?

If the machine halted then there would be a contradiction. Therefore the machine will not halt. There! We have just computed that it will not halt.

>
> > > Well, your claim remains unsupported.
>
> >
>
> > How about we say mostly unsupported or insufficiently supported, to say
>
> > that it is unsupported generally logically entails that not the
>
> > slightest trace of any degree of support has been provided what-so-ever.
>
> >
>
>
>
> Which is indeed the case.
>
>
>
> > It is extremely subtle nuances of meaning such as this that hides the
>
> > truth that I am attempting to show.
>
> >
>
>
>
> By all means attempt to show it. I am patiently waiting for the demonstration.
>
>
>
> > The support that I did provide (paraphrase) was that:
>
> >
>
> > 1) All paradoxes of self-reference inherently have cycles within the
>
> > directed graph of their meaning postulates.
>
> >
>
>
>
> Please give an example of this. Preferably actually showing us the graph.
>
>
>

> > 2) Meaning postulates that have cycles within their directed graph are
>
> > incorrect.
>
> >
>
>
>
> What's a meaning postulate?
>
>
>

[fom]

On 11/22/2013 12:14 PM, Rupert wrote:

<snip>

>>
>>>>> What's a meaning postulate?
>>
>>>> http://www.glottopedia.org/index.php/Meaning_postulate
>>
>>> What's the directed graph of a meaning postulate?
>>
>> Meaning Postulates are formed through connections between elements of
>> meaning including AtomicUnitsOfMeaning such that all of these
>> connections form an acyclic directed graph.
>>
>
> How do you obtain the directed graph? Give an example of a meaning postulate.
>

"[...]

"To capture this restriction formally,
Montague turned to the device of so-called
meaning postulates. This device, first
used by Carnap in 1947 ["Meaning and Necessity -
A Study in Semantics and Modal Logic"], is
best thought of as a kind of constraint on
possible models. Carnap introduced it to
deal with analytically true sentences that
cannot be analyzed as being logically true
(true as a consequence of their syntactic
form), such as 'All bachelors are unmarried'.
If B is the predicate 'is a bachelor' and
M is 'is married', then Carnap's example of
a meaning postulate is

Ax( B(x) -> ~M(x) )

The intent of this postulate is that in considering
possible models for our language, we are to restrict
ourselves to models in which this formula is
true. This means, in effect, that in constructing
a possible model we may still choose the extensions
of non-logical constants of M and B in any way
we wish, except that every individual in the
extension of B must be excluded from the extension
of M in any 'admissible' model. Otherwise, no
constraints are placed on the extensions of these
predicates."

"Introduction to Montague Semantics"
Dowty, Wall, and Peters


"A method of semantical meaning analysis is developed
in this chapter. It is applied to those expressions
of a semantical system S which we call designators; they
include (declarative) sentences, individual expressions
(i.e., individual constants or individual descriptions)
and predicators (i.e., predicate constants or compound
predicate expressions, including abstract expressions).
We start with the semantical concepts of *truth* and
*L-truth* (logical truth) of sentences. It is seen from
the definition of L-truth that it holds for a sentence
if its truth follows from the semantical rules alone without
reference to (extra-linguistic) facts. Two sentences are
called (materially) equivalent if both are true or both
are not true. The use of this concept of equivalence
is then extended to designators other than sentences. Two
individual expressions are equivalent if they stand for
the same individual. Two predicators (of degree one) are
equivalent if they hold for the same individuals. *L-equivalence*
(logical equivalence) is defined for both sentences and
other designators in such a manner that it holds for two
designators if and only if their equivalence follows from
the semantical rules alone. The concepts of equivalence
and L-equivalence in their extended use are fundamental
to our method.

"If two designators are equivalent, we say also that they
have the same extension. If they are, moreover, L-equivalent,
we say that they have also the same intension. Then we look
around for entities which might be taken as extensions or
as intensions for the various kinds of designators. We find
that the following choices are in accord with the two identity
conditions just stated. We take as the extension of a predicator
the class of those individuals to which it applies and, as its
intension, the property which it expresses; this is in accord
with customary conceptions. As the extension of a sentence we
take its truth-value (truth or falsity); as its intension, the
property expressed by it. Finally, the extension of an individual
expression is the individual to which it refers; its intension
is a concept of a new kind expressed by it, which we call an
individual concept. These conceptions of extensions and intensions
are justified by their fruitfulness; further definitions and
theorems apply equally to extensions of all types or to intensions
of all types.

"A sentence is said to be extensional with respect to a
designator occurring in it if the extension of the sentence
is a function of the extension of the designator, that is
to say, if the replacement of the designator by an equivalent
one transforms the whole sentence into an equivalent one. A
sentence is said to be intensional with respect to a designator
occurring in it if it is not extensional and if its intension
is a function of the intension of the designator, that is to
say, if the replacement of this designator by an L-equivalent
one transforms the whole sentence into an L-equivalent one. A
modal sentence (for example, 'it is necessary that...') is
intensional with respect to its subsentence. A psychological
sentence like 'John believes that it is raining now' is neither
extensional nor intensional with respect to its subsentence."


"Meaning and Necessity - A Study in Semantics and Modal Logic"
Rudolf Carnap




You will laugh when you figure out what a "meaning
postulate" is. Continuuing, ...


"A complete construction of the semantical system S_1,
which cannot be given here, would consist in laying down
the following kinds of rules:

1.
rules of formation, determining the admitted
forms of sentences;

2.
rules of designation for the descriptive constants;

3.
rules of truth, which we will explain now;

4.
rules of ranges, to be explained in the next section.

Of the rules of truth, we shall give here
only three examples [...]

"[...] The rules of truth together constitute a
recursive definition for 'true in S_1", because
they determine, in common with the rules of designation,
for every sentence in S_1 a sufficient and necessary
condition of its truth. Thereby, they give an interpretation
for every sentence."


"Meaning and Necessity - A Study in Semantics and Modal Logic"
Rudolf Carnap


What is important, for the moment are the "rules
of designation". Here is what Carnap wrote to
explain them:

"S_1 contains descriptive constants (that is, non-logical
constants) of individual and predicate types. The number
of predicates in S_1 is supposed to be finite, that of
individual constants may be infinite. For some of these
constants, which we shall use in examples, we state here
their meanings by semantical rules which translate them
into English.

"1-1. Rules of designation for individual constants

's' is a symbolic translation of 'Walter Scott'
'w' is a symbolic translation of '(the book) Waverley'

"1-2. Rules of designation for predicates

'Hx' is a symbolic translation of 'x is human (a human being)'
'RAx' is a symbolic translation of 'x is a rational animal'
'Fx' is a symbolic translation of 'x is (naturally) featherless'
'Bx' is a symbolic translation of 'x is a biped'
'Axy' is a symbolic translation of 'x is an author of y'

"The English words here used are supposed to be understood
in such a way that 'human being' and 'rational animal' mean
the same."


"Meaning and Necessity - A Study in Semantics and Modal Logic"
Rudolf Carnap


It is important to understand exactly why the statement
following the rules of designation is being made. Recall
from above,

"Carnap introduced it to deal with analytically
true sentences that cannot be analyzed as being
logically true (true as a consequence of their
syntactic form), such as 'All bachelors are
unmarried'."

"Introduction to Montague Semantics"
Dowty, Wall, and Peters


There are a number of participants in these newsgroups
who do not grasp that the analytical truth of "All bachelors
are unmarried" is not a notion of logical truth. Carnap
introduces these rules of designation specifically to
handle analytical truth separately, as intensions.

Carnap, however, does not distinguish analytical truth
and logical truth in the sense of the authors above.
Rather, Carnap distinguishes between *truth* and *L-truth*.
Since the latter is an abbreviation for "logical truth"
Carnap is using the phrase "logical truth" in the sense
which Frege attributed to intensional logicians. Recently
I had been branded a Nazi for trying to keep these notions
distinctly separated.

Carnap's statements for defining L-truth proceed through
notions from Leibniz and Wittgenstein:

"We shall introduce L-concepts with the help of the
concepts of state-description and range. Some ideas
of Wittgenstein were the starting point for the
development of this method.

"A class of sentences in S_1 which contains for every
atomic sentence either this sentence or its negation,
but not both, and no other sentences, is called a
*state-description* in S_1, because it obviously gives
a complete description of a possible state of the
universe of individuals with respect to all properties
and relations expressed by predicates of the system.
Thus the state-descriptions represent Leibniz' possible
worlds or Wittgenstein's possible states of affairs.

"It is easily possible to lay down semantical rules which
determine for every sentence in S_1 whether or not it
holds in a given state description. That a sentence
holds in a state-description means, in nontechnical
terms, that it would be true if the state-description
(that is, all sentences belonging to it) were true.
[...]

"The class of all those state-descriptions in which a
given sentence s_1 holds is called the range of s_1.
All the rules together, of which we have just given
five examples [standard model-theoretic statements, fom],
determine the range of any sentence in S_1; therefore,
they are called rules of ranges. By determining the
ranges, they give, together with the rules of designation
for the predicates and the individual constants, an
interpretation for all sentences in S_1, since to know
the meaning of a sentence is to know in which of the
possible cases it would be true and in which not, as
Wittgenstein has pointed out.

"The connection between these concepts and that of
truth is as follows: There is one and only one
state-description which describes the actual state
of the universe; it is that which contains all true
atomic sentences and the negations of those which
are false. Hence, it contains only true sentences;
therefore, we call it the true state-description. A
sentence of any form is true if and only if it holds
in the true state-description. These are only
incidental remarks for explanatory purposes; the
definition of L-truth will not make use of the
concept of truth"

"[...]

"2-1. Convention. A sentence s_1 is L-true in a
semantical system S if and only if s_1 is true in
S in such a way that its truth can be established
on the basis of the semantical rules of the system
S alone, without any reference to (extra-linguistic)
facts.

"This is not yet a definition of L-truth. It is an
informal formulation of a condition which any proposed
definition of L-truth must fulfill in order to be
adequate as an explication of our explicandum. Thus,
this convention has merely an explanatory and heuristic
function.

"How shall we define L-truth so as to fulfill the
requirement 2-1? A way is suggested by Leibniz'
conception that a necessary truth must hold in all
possible worlds. Since our state descriptions
represent the possible worlds, this means that a
sentence is logically true if it holds in all
state-descriptions. This leads to the following
definition:

"2-2. Definition. A sentence s_1 is *L-true* (in
S_1) <=df=> holds in every state-description (in S_1)."

"Meaning and Necessity - A Study in Semantics and Modal Logic"
Rudolf Carnap



Well, I have transcribed a lot here so that
the development of the notion can be appreciated.
It has to do with expressions whose analysis
led to Bolzano's conclusion that one could not
define simple substance (individuals) using term
logic. This is one of the paths by which the
foundations of mathematics came to endorse the
notion of "undefined language primitives" in its
dogma.

To give you a slightly more up to date sense of
what is really involved, here is an excerpt from
"Foundations of Intensional Semantics" by Fox and
Lappin,

"Now, consider the following theorem from
Johnstone ("Stone Spaces", 1982, p. 14) and
its corollary:

"Theorem 2(Johnstone): For any a, b in a distributive
lattice L such that ~( a <= b ), there is a homomorphism
h: L -> 2 in which h(a)=1 and h(b)=0.

"Corollary 1: For any a, b in a distributive lattice
L, if every homomorphism h: L -> 2 is such that
h(a)=h(b), then a=b

"Proof: [...]

"Corollary 1 specifies a direct connection between
provability and the identity relation between propositions
in a lattice. It has the consequence that if the
entailment relation among propositions can be modeled
as a partial order in a distributive lattice, identity
of propositions reduces to logical equivalence (mutual
entailment). If we formulate a semantic theory in
algebraic terms, we can use Corollary 1 as an important
measure of its success in avoiding this reduction.

"This result implies that there are only two basic
strategies that can be used to sustain a distinction
between logical equivalence and intensional identity.
On the first, impossible worlds are added to the model
theory to permit propositions that are equivalent across
the set of possible worlds to receive different truth-values.
These impossible worlds violate either the meaning
postulates governing the interpretations of non-logical
constants in the language or the interpretations of
logical constants fixed for the set of possible worlds.

"The second strategy involves weakening the partial
order relation that models entailment to a non-antisymmetric
preorder in which mutual entailment does not force
identity. This turns the algebraic structure of the
semantic theory into a bounded distributive prelattice
in which equivalent propositions are distinguished."


Hope this helps.

Mr. Olcott appears to be another fine person that
has failed to do his homework.

[graham...@gmail.com]

On Thursday, November 21, 2013 7:07:52 AM UTC-8, Rupert wrote:
> Actually, the reason that they're insufficiently refuted is that you haven't identified any problems with the arguments at all. The proofs are perfectly sound, and the theorems are correct.
>

By whose standards?

>
>
> You mean Turing's proof of the result that the halting problem is insoluble? It contains some kind of incoherence? Well, show me. Why is it incoherent? What's wrong with the argument?
>
>

It relies on the Church Turing thesis.

Without that, it is just a trivial fact like

~E(S) xeS<->~xeS


Nothing about un-computable functions can be determined from the actual result.



Herc

[graham...@gmail.com]

It wouldn't work anyway.

Try finding an intentional identity (why can't you just call it UNIFY(f1,f2) ?)
for the NON existence of Russell's set.

It's not a Partial Order (why can't you just call it transitive inference)
it's a O(n^n) pairwise negative comparison on V.

A 'history of sentences' (state description) to determine semantic truth
is a bit primitive isn't it? Do you know how many permutations there are
of an infinite string?


Herc
--
www.ProlotDatabase.com

[graham...@gmail.com]

> On Thursday, November 21, 2013 4:20:08 PM UTC-8, Ben Bacarisse wrote:
>
> > Same old stuff from 18 months ago. Every proposition of the form
>
> > "Turing machine M halts on input I" is either true or false in this
>
> > world.
>

OK! Every thing is either a MEMBER of a particular set
or it is NOT A MEMBER of a particular set.


Does that mean you cannot make a poor-specification of a set?


NO!


So why do you argue you cannot make a poor specification of a function?



Herc

[George Greene]

On Thursday, November 21, 2013 10:00:59 AM UTC-5, Peter Olcott wrote:
> The only reason why things such as the Halting Problem and the
>
> Incompleteness Theorem remain insufficiently refuted is that the current
>
> degree of knowledge of the inherent structure of knowledge itself is not
>
> yet sufficiently complete.

YOU ARE *LYING*.
These results have NOTHING WHATSOEVER TO DO with "the structure of knowledge".
Or of meaning, either, for that matter.
One aspect of "the structure of knowledge" of which YOU are IGNORANT
is that in the context of proofs from theorems from axioms,
NOTHING *MEANS* anything.
It DOES NOT MATTER how you ascribe "meaning" TO ANY of the parts of
the strings! The theorems come up proven COMPLETELY IRRESPECTIVE OF
their meaning or interpretation! They are true UNDER *ALL*POSSIBLE*
interpretation of their "meaning" -- that IS WHY their meaning is
simply irrelevant!
You (for example) canNOT HAVE an r that R's all&only those things that
don't R themselves, *REGARDLESS* of what r is OR of what R means!!

[fom]

On 11/22/2013 10:09 PM, graham...@gmail.com wrote:
>
>
> It wouldn't work anyway.
>
> Try finding an intentional identity (why can't you just call it UNIFY(f1,f2) ?)
> for the NON existence of Russell's set.
>

I am not certain I understand the question.

I imagine that Prolog has facilities for the
kind of manipulations involved with these logics.

Montague's system is typed.

> It's not a Partial Order (why can't you just call it transitive inference)
> it's a O(n^n) pairwise negative comparison on V.
>

Well, I might call it that if the texts
documenting the meaning of the words came
from a source that used those phrases.

> A 'history of sentences' (state description) to determine semantic truth
> is a bit primitive isn't it?

No. It will be exactly how people
will think about such matters when
the overuse of technology brings
mayhem to the world.

:-)

It is the future.

> Do you know how many permutations there are
> of an infinite string?
>

Infinitely many?

Truthfully, none.

[graham...@gmail.com]

INPUT



> > --------------
> > | 0 | 0 | 0 |
> > --^-----------


OUTPUT


> > --------------
> > | 0 | 1 | 1 |
> > ----------^---

>
> > this computes WHAT exactly George?
>
>
> This is A STUPID
>
> question. ALL TMs compute EXACTLY WHAT they compute.
>
> And you can't allege that anything has been computed
>
> by 1 input and 1 output.

What was the individual computation George?

Was it HALT( gcd() , (4,6) ) ?


Was it f(0) = 2 ?



For something you claim is 100% explicit

you have trouble working it out!




Herc

[graham...@gmail.com]

On Friday, November 22, 2013 8:57:39 PM UTC-8, fom wrote:
> On 11/22/2013 10:09 PM, graham...@gmail.com wrote:
>
>
>
> > A 'history of sentences' (state description) to determine semantic truth
>
> > is a bit primitive isn't it?
>
>
>
> No. It will be exactly how people
>
> will think about such matters when
>
> the overuse of technology brings
>
> mayhem to the world.
>
>
>
> :-)

In a world of possibilities you would never be able to make a retraction.

I guess it makes sense though... e.g.


ONCE YOU ACCEPTED
-----------------

1 not e CantorSet IF 1 e your 1st set
2 not e CantorSet IF 2 e your 2nd set
...


PROVED
------

SET OF ALL SUBSETS OF N > INFINITY!




Once you fall for a lure... you would be forever stuck,
unable to address your error in foundations and twisting
your perceptions and rationale at everyone to suit your
initial true sentences states.

Kind of like everyone in sci.logic!

Herc

[Julio Di Egidio]

"fom" <fom...@nyms.net> wrote in message
news:V9SdnWgRE5Qcgg3P...@giganews.com...

> There are a number of participants in these newsgroups
> who do not grasp that the analytical truth of "All bachelors
> are unmarried" is not a notion of logical truth.

Rather you don't grasp that that is a distinction all internal to
*mathematical* logic already. As for logic proper, i.e. the analysis of
arguments and reasoning via natural language, there is no such thing:
logically true statements are so by virtue of their *meaning*, and "syntax"
does not enter the picture at all.

> Recently
> I had been branded a Nazi for trying to keep these notions
> distinctly separated.

Rather a statement of yours was branded "nazi speech": so long to your
ability to make any distinctions at all, indeed so long to your reasoning
capabilities.

Julio

[Julio Di Egidio]

"Antti Valmari" <Antti....@c.s.t.u.t.f.i.invalid> wrote in message
news:l6nhi4$cue$1...@news.cc.tut.fi...

>
> A much more readable and competent attempt to deny undecidability of
> halting was written by Eric C.R. Hehner. He is an established computer
> scientist with a decent publication record. To me the most rewarding
> part (rewarding in a perhaps weird sense) was when I temporarily ignored
> the flaws in his reasoning

I find that line of reasoning quite compelling, although at an informal
level. Could you maybe tell which (at least in essence) are those flaws?

Julio

> and thought about the consequences of his
> conclusion: what would change if he were right. Here:
>
> www.cs.toronto.edu/~hehner/PHP.pdf

<snip>

[graham...@gmail.com]

He just did explain the flaw... most work-arounds to
'calculate' a halt value have the consequence of a
halt function being computable and the same paradox ensues.


But really you guys are all totally incompetent!

------------------------------------------------


PROCEDURE S()
BEGIN
IF HALT('S') GOSUB S
END


IF HALT('S)<->TRUE
IT IS THE WRONG VALUE

IF HALT('S')<->FALSE
IT IS THE WRONG VALUE



THIS *CLEARLY* DOES NOT PROVE

THE HALT FUNCTION TO DETERMINE WHETHER A T.M.
HALTS OR NOT IS AN UN-COMPUTABLE FUNCTION.



IT'S A JUMP IN REASONING BASED ON A POOR SPECIFICATION.

HALT() MUST BE A GLOBALLY APPLIED FUNCTION (A TEST HARNESS)
NOT A GENERAL PURPOSE PURE FUNCTION.


-------------

YOU CAN HAVE THIS:

HALT(TM1)
HALT(TM2)
HALT(TM3)
...




YOU CAN HAVE THIS:

TM1(HALT(..))
TM2(HALT(..))
TM3(HALT(..))




BUT YOU CAN'T COMBINE THE 2 WITH A NAIVE DEFN OF HALT().



There is a complexity to the Church Turing Thesis
that renders Chaitan's Omega as an erroneous defn - there is CLEARLY
no such specified real number based on uncomputable halt values.

A TM is not DESIGNED for program/function inspection.

Yes you CAN DO IT... by encoding the Tape Parameterisation
but then you are using a DIFFERENT TM ARCHITECTURE to
the one you started with with specification for HALT().


Herc

[Julio Di Egidio]

<graham...@gmail.com> wrote in message
news:8f30b8b4-4232-4001...@googlegroups.com...

> On Saturday, November 23, 2013 1:09:14 AM UTC-8, Julio Di Egidio wrote:
>> "Antti Valmari" <Antti....@c.s.t.u.t.f.i.invalid> wrote in message
>> news:l6nhi4$cue$1...@news.cc.tut.fi...
>>
>> > A much more readable and competent attempt to deny undecidability of
>> > halting was written by Eric C.R. Hehner. He is an established computer
>> > scientist with a decent publication record. To me the most rewarding
>> > part (rewarding in a perhaps weird sense) was when I temporarily
>> > ignored
>> > the flaws in his reasoning
>>
>> I find that line of reasoning quite compelling, although at an informal
>> level. Could you maybe tell which (at least in essence) are those flaws?
>>

>> > and thought about the consequences of his
>> > conclusion: what would change if he were right. Here:
>
> He just did explain the flaw...

What are you babbling about?

> most work-arounds to
> 'calculate' a halt value have the consequence of a
> halt function being computable and the same paradox ensues.

Here is the link that you have just snipped:
<http://www.cs.toronto.edu/~hehner/PHP.pdf>
Try and read it first.

> But really you guys are all totally incompetent!

And you are still in my spam filter.

Julio

[graham...@gmail.com]

Abstract
Either we leave the definition of the halting function incomplete, not
saying its result when applied to its own program, or we suffer inconsistency. If we
choose incompleteness, we cannot require a halting program to apply to programs
that invoke the halting program, and we cannot conclude that it is incomputable. If
we choose inconsistency, then it makes no sense to propose a halting program.
Either way, the incomputability argument is lost.
Keywords halting problem, computability





Exactly what I described above!

[fom]

Yup.

[Julio Di Egidio]

"fom" <fom...@nyms.net> wrote in message

news:SPCdnXmd9tk1UA3P...@giganews.com...

> On 11/23/2013 9:38 AM, graham...@gmail.com wrote:

>> Exactly what I described above!

What are you babbling about? That is the abstract of the article: I had
asked Mr Valmari why he thinks that the article is flawed...

> Yup.

Elective affinities.

Julio

[George Greene]

On Thursday, November 21, 2013 4:43:41 AM UTC-5, graham...@gmail.com wrote:
> Yes and no. The Halting CONCLUSION relies on the Church
> Turing Thesis which is purely an assumption.

NO, IT *DOESN'T*.
GOOD *GRIEF*!!
Why do people who will NOT READ BASIC TEXTBOOKS
continue to insist on LYING about what is in them?!??
Church's Thesis is about TMs and A NATURAL-LANGUAGE WORD
like "computable" or somesuch. Both the Halting Problem AND
its conclusion/solution ARE NOT about that!! The Halting
Problem DOES NOT GIVE A SHIT what is or isn't "really
truly computABLE" by humans or by whatever.
The Halting Problem Conclusion concludes (instead) that
NO TURING MACHINE solves the Halting Problem FOR TURING Machines!!
THAT IS ALL!!

[wugi]

Julio Di Egidio schreef op 23/11/2013 10:09:

> "Antti Valmari" <Antti....@c.s.t.u.t.f.i.invalid> wrote in message
> news:l6nhi4$cue$1...@news.cc.tut.fi...
>>
>> A much more readable and competent attempt to deny undecidability of
>> halting was written by Eric C.R. Hehner. He is an established computer
>> scientist with a decent publication record. To me the most rewarding
>> part (rewarding in a perhaps weird sense) was when I temporarily ignored
>> the flaws in his reasoning
>
> I find that line of reasoning quite compelling, although at an informal
> level. Could you maybe tell which (at least in essence) are those flaws?

Since decades I've been stuck with this layman's question about halting
decidability:
Is a simple line like
"IF RND < 0.0000000000001 THEN END"
allowed to be part of the program under examination?
If not, why not?
If yes, then how would one but conceive the idea of deciding upon its
halting or not?

guido google:wugi

[Julio Di Egidio]

"wugi" <br...@brol.be> wrote in message
news:l6r91d$cu1$1...@speranza.aioe.org...

> Since decades I've been stuck with this layman's question about halting
> decidability:
> Is a simple line like
> "IF RND < 0.0000000000001 THEN END"
> allowed to be part of the program under examination?
> If not, why not?
> If yes, then how would one but conceive the idea of deciding upon its
> halting or not?

Rather consider this:

while (!(rnd() < 1e-13)) {}

where we assume rnd() in [0, 1) reasonably distributed, and 0 < 1e-13 ==
true.

Unless I am missing something, of course it halts, eventually...

Julio

[Peter Percival]

wugi wrote:
> Julio Di Egidio schreef op 23/11/2013 10:09:
>> "Antti Valmari" <Antti....@c.s.t.u.t.f.i.invalid> wrote in message
>> news:l6nhi4$cue$1...@news.cc.tut.fi...
>>>
>>> A much more readable and competent attempt to deny undecidability of
>>> halting was written by Eric C.R. Hehner. He is an established computer
>>> scientist with a decent publication record. To me the most rewarding
>>> part (rewarding in a perhaps weird sense) was when I temporarily ignored
>>> the flaws in his reasoning
>>
>> I find that line of reasoning quite compelling, although at an informal
>> level. Could you maybe tell which (at least in essence) are those flaws?
>
> Since decades I've been stuck with this layman's question about halting
> decidability:
> Is a simple line like
> "IF RND < 0.0000000000001 THEN END"
> allowed to be part of the program under examination?

Yes, so long as RND is a Turing computable function. Note that the RNDs
found in program libraries are deterministic and only seemingly random.
The numbers they return are often called pseudo-random for that reason.

> If not, why not?
> If yes, then how would one but conceive the idea of deciding upon its
> halting or not?

No difference.

> guido google:wugi

[graham...@gmail.com]

So by un-computable you mean 'uncomputable by Turing Machines only' ?


You really are THICK George!


You SPECIFY halt() to have this property:


halt(TM1(x)) = yes/no
halt(TM2(x)) = yes/no
halt(TM3(x)) = yes/no



WHICH HAS ONLY 2 VALUES.



THEN, you USE Halt like this:

TM77(halt(TM33,X))




which is a 3 valued usage.



Herc









Herc

[George Greene]

On Saturday, November 23, 2013 7:27:13 PM UTC-5, graham...@gmail.com wrote:

> So by un-computable you mean 'uncomputable by Turing Machines only' ?

I don't mean ANYthing by "uncomputable" AND NEITHER DO YOU.
YOU CANNOT *SAY*WHAT*YOU* mean by "computable". So that is just
NOT relevant. We can ALL, however, say what we mean by TM-computable.
So those of us who are talking about what we CAN talk about will keep
talking. You, on the other hand, will just be an idiot.

>
>

> You SPECIFY halt() to have this property:

Specifying that a function has a property DOES NOT
guarantee that ANY such function exists.
In particular, you cannot specify an algorithm
that halts on and only on those algorithms that
don't halt ON THEMSELVES, if there is a way to make
one algorithm be an input for another algorithm --
AND THERE NECESSARILY ALWAYS IS.

[George Greene]

On Thursday, November 21, 2013 10:00:59 AM UTC-5, Peter Olcott wrote:

> The only reason why things such as the Halting Problem and the
>
> Incompleteness Theorem remain insufficiently refuted

Idiot, NO instance of Russell's paradox is "insufficiently"
refuted. They ARE ALL COMPLETELY refuted -- they are OBVIOUSLY ALL
contradictory. There cannot exist a unary predicate true of and only
of those predicates that are not true OF THEMSELVES, if you can encode
a predicate as input to another predicate -- AND YOU CAN.
There cannot exist a TM that halts on and only on all TMs that don't
halt ON THEMSELVES, if you can encode a TM as input to another TM --
AND YOU CAN.

The allegation that such a thing COULD exist *is* COMPLETELY AND PERFECTLY
refuted by noticing that IT LOGICALLY ENTAILS A CONTRADICTION in the
case of whether this thing, IF it existed, would or wouldn't be true
OF ITSELF, or would or wouldn't halt ON ITSELF.

The proof that Russell's paradox really is paradoxical IS TRIVIAL and
it DOES NOT REALLY EVEN INVOLVE *self*-"reference" -- it merely involves
diagonalization. It merely involves using 1 thing in both argument-places
of a binary predicate. There is no reason to think of two copies of
the same string as involving some NON-string "referring to" itself.
That is PURELY metaphorical and WAS INTENDED TO HELP you understand.
Tragic that it is having the opposite effect.
THIS is what is happening:
~(ErAx[rRx <-> ~xRx]).
THAT IS ALL.
The thing inside the [ ]'s is COMPLETELY PROVABLY false when you
instantiate x to r, so the Ax[ . . . ] is provably false.
THAT
*IS*
*ALL*.
There IS NOT
*more*
than this, going on, here!
All this introduction of Montague and specification and
other paradigms IS IRrelevant.

[George Greene]

On Tuesday, November 19, 2013 9:17:47 AM UTC-5, Peter Olcott wrote:

> The language of mathematics is insufficiently expressive to discern the
> error of the fallacy of self-reference.

Self-reference ISN'T fallacious, IDIOT.
There are PLENTY of self-references that ARE NOT problematic.
Even the ones that are contradictory (like Russell's Paradox
and the instances of it being debated here, such as Godel's
Theorem and the Halting Problem) are simply FALSE, NOT fallacious!

More to the point, in
Ax[rRx <-> ~xRx], there truly really is NOT ANY "self-reference"
OCCURRING IN THE FIRST PLACE.
Applying R to some arguments does NOT mean that R or r OR x is
"referring to" ANYthing.
Strings really are JUST strings.
It is a fact about both proofs and programs that
they CAN be ENCODED as strings, and that encoding
causes "reference", yes. But there is simply NOTHING
fallacious about that. There is nothing you can do about
the fact that programming languages exist.

[George Greene]

On Tuesday, November 19, 2013 9:17:47 AM UTC-5, Peter Olcott wrote:

> It can not be correctly concluded that an error does not exist on the
>
> basis that this error can not be expressed within the limitations of any
>
> specific mode of expression such as the language of mathematics.

Oh, horse-shit. The statements in question ARE ONLY IN AND ABOUT "the
language of mathematics", so it cannot even be insinuated (despite
the fact that you are doing it -- you are just a living example of the
fact that it cannot be LEGITIMATELY insinuated) THAT ANY other criterion
is relevant.

More to the point, we don't EVEN NEED mathematical language to see this --
it is JUST AS TRUE IN NATURAL language that you cannot have an r that
R's all and only those x's that don't R themselves.
EVERY NATURAL-language instance of that prohibition IS JUST AS MUCH A PROVEN *LOGICAL* theorem, AS all the mathematical ones!

[George Greene]

On Tuesday, November 19, 2013 8:58:02 AM UTC-5, Ben Bacarisse wrote:
> However, the halting theorem remains a
> theorem of mathematics, no matter how you label it.

No, really, it doesn't. The theorem about the non-existence of a TM
properly implementing what graham likes to call "The Halt specification"
FAR&LONG *PRECEDES* math.
It is a PURELY NATURAL-language OBVIOUSNESS that you cannot have an r
that R's all&only those x's that don't R themselves -- would said r
R itself? Or not? It is a contradiction for such an r to exist, SO
IT DOESN'T EXIST. NATURAL language ALREADY had to be logical BEFORE
there was math. BEFORE math, it was the case that it was either raining
or it wasn't raining. Before math, it was ALREADY the case that it couldn't
be both hailing AND NOT hailing (in one place at one time).
These are SIMPLE BASIC LAWS OF THOUGHT, NOT
complicated logical inferences or theorems depending on a whole lot of axioms.

You cannot have a TM that halts on all&only those TMs that don't halt on
themselves, for THE SAME reason that you cannot have a barber who shaves
all&only those barbers who don't shave themselves.
This IS *SIMPLE*,
which is why you have to fight efforts to complicate it.

The one piece of math involved in the TM case involves
the correspondence between TMs and their inputs -- you have
to prove that any TM can be accurately referred
to by a TM-input-string -- but even that almost IS NOT math -- that
is purely a labeling thing.
Is it "math" when I say that the keys on the piano are labeled A through G
with sharps&flats? NO, NOT really.

[George Greene]

On Tuesday, November 19, 2013 9:17:47 AM UTC-5, Peter Olcott wrote:

> >> The Halting Problem and the Liar Paradox are errors of
> >> specification/reasoning because their complete [meaning postulates]
> >> necessarily always contain cycles.

The Halting Problem ISN'T like the Liar Paradox, IDIOT.
It is like RUSSELL'S Paradox, and that DOES NOT HAVE ANY
"meaning postulates", IDIOT -- it's a LOGICAL result!
It is INDEPENDENT of the meaning of EVERYthing except "or" and "not"!!

[George Greene]

On Wednesday, November 20, 2013 2:11:09 AM UTC-5, graham...@gmail.com wrote:

> This is essentially a 2nd Order Logic.

The whole family of questions is from second order logic.
Russell's paradox is
Ax[rRx <-> ~xRx].
It's a paradox because it's a contradiction -- this is GUARANTEED to be false.
All you have to do to see that is instantiate x to r -- the thing inside
the [ ]'s canNOT POSSIBLY be true for ALL x because it in particular cannot
be true for the x that is r -- in that case, the thing inside the brackets
becomes rRr <-> ~rRr WHICH IS ALWAYS GUARANTEED TO BE FALSE,
IRrespective of ANY MEANINGS of r OR R !!
R, in particular, as a first-order binary relation, is not constrained
in any way in this argument and can therefore be universally generalized over --
this is false FOR ALL R, which makes it 2nd order.
ForAll R [ ~ Er[ Ax[xRr <-> ~xRx] ] ]
is the 2nd-order validity saying that these kinds of r's canNOT exist.
THIS IS a 2nd-order sentence that IS NECESSARILY obviously true.
It is COMPLETELY EASILY seen to be necessary.
Again, the fact that rRr <-> ~rRr IS NECESSARILY FALSE, REGARDLESS of the
"meaning" of R, is COMPLETELY clear, just as clearly as
P/\~P is necessarily false, regardless of the meaning of P.
That's a 0th-order contradiction but it is at the kernel of this
2nd-order validity (which would turn back into a contradiction if
took out the ~ right before the Er[ -- thinking that such an r
CAN exist WILL lead you into inconsistency.

[graham...@gmail.com]

On Saturday, November 23, 2013 7:43:57 PM UTC-8, George Greene wrote:
>
> The one piece of math involved in the TM case involves
>
> the correspondence between TMs and their inputs -- you have
>
> to prove that any TM can be accurately referred
>
> to by a TM-input-string -- but even that almost IS NOT math -- that
>
> is purely a labeling thing.
>
> Is it "math" when I say that the keys on the piano are labeled A through G
>
> with sharps&flats? NO, NOT really.

Fair enough perspective, its is the 'establishment' FOR mathematics to be carried out on.

But it gives you a arbitrary line in the sand to always jump to higher ground...
this
discussion is not about semantics its about HARD FACTS about whether:

CHAITAINS OMEGA is an infinite digit string (real number)
that lies outside the scope of all deterministic computers output.

It is ALSO a HARD FACT that mid-proof you CHANGE the interpretation
of the input and output of the turing machines from:

1 parameter input
to
2 parameter input (parsed as 1 parameter)


THIS is a DIFFERENT VIRTUAL MACHINE where your erroneous conclusion is made.


Herc

[Newberry]

On Saturday, November 23, 2013 7:29:09 PM UTC-8, George Greene wrote:
> On Thursday, November 21, 2013 10:00:59 AM UTC-5, Peter Olcott wrote:
>
>
>
> > The only reason why things such as the Halting Problem and the
>
> >
>
> > Incompleteness Theorem remain insufficiently refuted
>
>
>
> Idiot, NO instance of Russell's paradox is "insufficiently"
>
> refuted. They ARE ALL COMPLETELY refuted -- they are OBVIOUSLY ALL
>
> contradictory. There cannot exist a unary predicate true of and only
>
> of those predicates that are not true OF THEMSELVES,

There surely can. "This sentence is not true" is not true. Where is the contradiction in this?

[Peter Olcott]

On 11/24/2013 11:39 AM, Newberry wrote:
> On Saturday, November 23, 2013 7:29:09 PM UTC-8, George Greene wrote:
>> On Thursday, November 21, 2013 10:00:59 AM UTC-5, Peter Olcott wrote:
>>
>>
>>
>>> The only reason why things such as the Halting Problem and the
>>> Incompleteness Theorem remain insufficiently refuted
>>
>>
>> Idiot, NO instance of Russell's paradox is "insufficiently"
>>
>> refuted. They ARE ALL COMPLETELY refuted -- they are OBVIOUSLY ALL
>>
>> contradictory. There cannot exist a unary predicate true of and only
>>
>> of those predicates that are not true OF THEMSELVES,
> There surely can. "This sentence is not true" is not true. Where is the contradiction in this?

If "This sentence is not true" is not true.
then Not("This sentence is not true")<>"This sentence is true"

[George Greene]

On Wednesday, November 20, 2013 12:20:33 PM UTC-5, fom wrote:
> If one simply denies that the paradoxical
> interpretation of the demonstrative is an
> admissible pragmatic use, then

Then one IS A DAMN FOOL, because demonstratives are
BY FAR THE EASIEST and MOST strongly-conventionally-reinforced
*words(or pronouns)*-TO-*interpret* IN THE WHOLE language!

Seriously: a noun is, to oversimplify, " a person, place, or thing ". Since
pronouns stand for nouns, they, too, refer to persons, places, or things.
If I use ANY such referring noun or pronoun OTHER than a demostrative,
IT IS *HARDER* TO INTERPRET *THAN* the demonstrative.
Seriously, any referring noun-clause referring to a person (a who), a place
(a where), or a thing (a what), IS HARDER TO EVALUATE *THAN*ANY* simple use
of "this" is, to evaluate!

Indeed, trying to disallow "the paradoxical interpretation of" the demonstrative
is most LOUDLY proven to be horse-shit by the fact that UNLESS THE UTTERANCE
IS PRAGMATICALLY *AMBIGUOUS*, i.e., unless it does not REALLY have One Clear
Meaning AT ALL, there IS ONLY ONE pragmatically CORRECT interpretation of the
demonstrative (AND OF EVERY OTHER part of the sentence), SO OBVIOUSLY ONE *MAY*NOT* disallow THAT interpretation!

The rules for the demonstratives are very clear. They are the clearEST
simplest rules. They are evaluated in terms of things forming the very
FRAME, often the PHYSICAL frame, of the conversation/situation. Most of the
time they do NOT EVEN ADMIT ONE alternative interpretation. Most of the time
they have ONE MAXIMALLY clear interpretation. The fact that it happens to be
paradoxical in some cases is NOT the DEMONSTRATIVE's problem and THOU SHALT NOT claim to be resolving the paradox BY FALSELY alleging that there is some sort of multiplicity in available interpretations for the demonstrative (within the pragmatic context) -- There ALMOST NEVER is such!

[George Greene]

On Friday, November 22, 2013 8:13:45 PM UTC-5, Newberry wrote:
> There are syntactically correct pseudo-programs,
> to which the halting question does not apply.

If you are only going to present the program by itself, then, of course.
The point being that YOU NEVER GET to do that. Even though some TMs
ignore their input, all TMs BY DEFINITION have an input tape and the
question therefore always has to be asked of a TM,input-string PAIR.

[George Greene]

On Thursday, November 21, 2013 9:34:14 PM UTC-5, Peter Olcott wrote:
> No Turing Machines ever halt or fail to halt in any possible world
> because they are defined as fictional.

Stop lying.

[Newberry]

On Sunday, November 24, 2013 5:23:00 PM UTC-8, Peter Olcott wrote:
> On 11/24/2013 11:39 AM, Newberry wrote:
>
> > On Saturday, November 23, 2013 7:29:09 PM UTC-8, George Greene wrote:
>
> >> On Thursday, November 21, 2013 10:00:59 AM UTC-5, Peter Olcott wrote:
>
> >>
>
> >>
>
> >>
>
> >>> The only reason why things such as the Halting Problem and the
>
> >>> Incompleteness Theorem remain insufficiently refuted
>
> >>
>
> >>
>
> >> Idiot, NO instance of Russell's paradox is "insufficiently"
>
> >>
>
> >> refuted. They ARE ALL COMPLETELY refuted -- they are OBVIOUSLY ALL
>
> >>
>
> >> contradictory. There cannot exist a unary predicate true of and only
>
> >>
>
> >> of those predicates that are not true OF THEMSELVES,
>
> > There surely can. "This sentence is not true" is not true. Where is the contradiction in this?
>
>
>
> If "This sentence is not true" is not true.
>
> then Not("This sentence is not true")<>"This sentence is true"

I got lost in your notation. 'Not("This sentence is not true" ' is equivalent to ' "This sentence is not true" is false', but NOT equivalent to ' "This sentence is not true" is not true'. Not sure where you are going with this.

[George Greene]

> On 11/22/2013 1:28 PM, Peter Percival wrote:
> > Why do you call the halting problem a self-reference paradox?
>
> >
>
> >

On Friday, November 22, 2013 3:30:00 PM UTC-5, Peter Olcott wrote:
> http://plato.stanford.edu/entries/self-reference/#ConConProCom

That article is lame.
It doesn't even notice most relevant commonalities of structure of the paradoxes. It is also lame for your personal purpose as well since it explicitly gives an example of a paradox that refers withOUT SELF-referring.

The paradoxes (including the Halting Paradox and Godel's unprovability "paradox") ARE ALL RUSSELLIAN in structure and all arise because no thing
can bear ANY relation to all and only those things that don't bear the
relation to THEMselves. In the Godelian case, it is the predicate's relation
TO ITSELF, NOT to some Godel-number encoding it, that is problematic, and
in the Halting case, it is the TM's relation TO ITSELF, AND NOT to some
string encoding it, that is relevant. The point is that the encodings (and
therefore the relations) MUST exist. There is actually a better Stanford Encyclopedia of Philosophy article about the paradoxes. MAYBE YOU SHOULD HUNT IT DOWN.

[George Greene]

On Tuesday, November 19, 2013 6:25:27 AM UTC-5, Peter Olcott wrote:

> Conclusion:

>
> The Halting Problem and the Liar Paradox are errors of
> specification/reasoning because their complete [meaning postulates]
> necessarily always contain cycles.

Cycles aren't always bad and non-cycles aren't always safe.
You cited this article from the Stanford Encyclopedia of Philosophy
WITHOUT EVEN NOTICING that it addresses the question of just what kinds
of graphs might lead to paradox:
http://plato.stanford.edu/entries/self-reference/#ParWitSelRef
In particular, a conclusion reads:
"Given the insight that not only cyclic structures of reference can lead to paradox, but also certain types of non-wellfounded structures, it becomes interesting to study further these structures of reference and their potential in characterising the necessary and sufficient conditions for paradoxicality. This line of work was initiated by Gaifman (1988, 1992, 2000), and later pursued by Cook (2004), Walicki (2009) and Rabern, et al. (forthcoming). A complete graph-theoretical classification of which structures of reference admit paradox is still to be found."

Maybe you'd better get busy!

[graham...@gmail.com]

On Sunday, November 24, 2013 7:34:55 PM UTC-8, George Greene wrote:
> http://plato.stanford.edu/entries/self-reference/#ParWitSelRef
>
> In particular, a conclusion reads:
>
> "Given the insight that not only cyclic structures of reference can lead to paradox, but also certain types of non-wellfounded structures, it becomes interesting to study further these structures of reference and their potential in characterising the necessary and sufficient conditions for paradoxicality. This line of work was initiated by Gaifman (1988, 1992, 2000), and later pursued by Cook (2004), Walicki (2009) and Rabern, et al. (forthcoming). A complete graph-theoretical classification of which structures of reference admit paradox is still to be found."
>
>
>
> Maybe you'd better get busy!

QUESTION: Can a Turing Machine Computable Function
return 'FALSE' IFF the function that calls it
is called by at least one other function?



e.g. call the function naked()

f(g(naked())) <-> f(g(false)) 'g is called by atleast one other function'
g(naked) <-> g(true)


Herc

[pauljk]

"Julio Di Egidio" <ju...@diegidio.name> wrote in message news:l6re5j$32l$1...@dont-email.me...

It'll eventually halt only if the mantissa of the hardware floating point
delivered by rnd() has at least 13 digits. Internally the rnd() function
will have to be using arithmetic with significantly longer mantissas.

pjk

[Peter Olcott]

On 11/24/2013 8:16 PM, Newberry wrote:
> On Sunday, November 24, 2013 5:23:00 PM UTC-8, Peter Olcott wrote:
>> On 11/24/2013 11:39 AM, Newberry wrote:
>>
>>> On Saturday, November 23, 2013 7:29:09 PM UTC-8, George Greene wrote:
>>
>>>> On Thursday, November 21, 2013 10:00:59 AM UTC-5, Peter Olcott wrote:
>>
>>>>
>>
>>>>
>>
>>>>
>>
>>>>> The only reason why things such as the Halting Problem and the
>>
>>>>> Incompleteness Theorem remain insufficiently refuted
>>
>>>>
>>
>>>>
>>
>>>> Idiot, NO instance of Russell's paradox is "insufficiently"
>>
>>>>
>>
>>>> refuted. They ARE ALL COMPLETELY refuted -- they are OBVIOUSLY ALL
>>
>>>>
>>
>>>> contradictory. There cannot exist a unary predicate true of and only
>>
>>>>
>>
>>>> of those predicates that are not true OF THEMSELVES,
>>
>>> There surely can. "This sentence is not true" is not true. Where is the contradiction in this?
>>
>>
>>
>> If "This sentence is not true" is not true.
>>
>> then Not("This sentence is not true")<>"This sentence is true"
>
> I got lost in your notation. 'Not("This sentence is not true" ' is equivalent to ' "This sentence is not true" is false', but NOT equivalent to ' "This sentence is not true" is not true'. Not sure where you are going with this.

I will re-state my position after more thought:
1) "This sentence is not true" is not true.
2) "This sentence is not true" is not false.
3) Therefore "This sentence is not true" in not a valid proposition.

Objects of thought are divided into types: (Atoms of Meaning)
a) Individuals
b) Properties of individuals
c) Relations between individuals
d) Properties of such relations

The property of TruthValue is only associated with the Proposition
Type. It is not associated with any other type of utterance such as the
Question type, or an Exclamation Type.

The utterance: "This sentence is not true" is not a Proposition because
it lacks a TruthValue Property.

Thus to request the TruthValue property of the utterance:
"This sentence is not true" would form an incorrect question.

A TruthValue Property must have an object that it refers to. If it lacks
an object that it refers to then it is not a TruthValue Property.

If I say I am going to the store.
The store is the destination object of my travel.

If I say I am going to the...
There is a piece missing, so the Declaration(Propsosition) is malformed
because it is missing a required element.

[Julio Di Egidio]

"pauljk" <paul....@xtra.co.nz> wrote in message
news:l6vcgl$6gv$1...@dont-email.me...

> "Julio Di Egidio" <ju...@diegidio.name> wrote in message
> news:l6re5j$32l$1...@dont-email.me...
>> "wugi" <br...@brol.be> wrote in message
>> news:l6r91d$cu1$1...@speranza.aioe.org...
>>
>>> Since decades I've been stuck with this layman's question about halting
>>> decidability:
>>> Is a simple line like
>>> "IF RND < 0.0000000000001 THEN END"
>>> allowed to be part of the program under examination?
>>> If not, why not?
>>> If yes, then how would one but conceive the idea of deciding upon its
>>> halting or not?
>>
>> Rather consider this:
>>
>> while (!(rnd() < 1e-13)) {}
>>
>> where we assume rnd() in [0, 1) reasonably distributed, and 0 < 1e-13 ==
>> true.
>>
>> Unless I am missing something, of course it halts, eventually...
>

> It'll eventually halt only if the mantissa of the hardware floating point
> delivered by rnd() has at least 13 digits. Internally the rnd() function
> will have to be using arithmetic with significantly longer mantissas.

No, that's wrong: I have given the conditions, in particular you need 0 <
1e-13, regardless of the precision of the output of rnd().

Julio

[Julio Di Egidio]

"Julio Di Egidio" <ju...@diegidio.name> wrote in message

news:l6vfvq$p13$1...@dont-email.me...

In fact, the lower the precision, more quickly it will halt (on the
average). Just think the case where rnd() returns either 0.0 or 0.5...

Julio

[Peter Olcott]

On 11/25/2013 1:55 AM, Franz Gnaedinger wrote:
> On Sunday, November 24, 2013 1:24:27 PM UTC+1, Peter Olcott wrote:
>>
>> The reference did not merely use the term [Atomic Unit of Meaning] it
>> explained what this is.
>> Did you bother to look at the reference?
>
> No, I did not waste my time that way. I asked you
> for one example of a word with an atomic unit of
> meaning. You didn't bother giving me an example.

Simple Theory of Types (Atoms of Meaning)
http://en.wikipedia.org/wiki/History_of_type_theory#G.C3.B6del_1944

Atoms of meaning are below the level of meaning contained within words.
Every word is comprised of its constituent (Atoms of Meaning).

Objects of thought are divided into types: (Atoms of Meaning)
a) Individuals
b) Properties of individuals
c) Relations between individuals
d) Properties of such relations

The Concept {Is Larger Than} forms a type of relation between a pair of
other types. This concept also has its own constituent parts.
Conventionally this would be referred to as a two place predicate.

A two place predicate would itself be a type of relation.
It would be a relation between a Predicate type and its two
ObjectOfPredicate types.

> And you don't get the difference between a model
> and a theorem. The atomic unit of meaning is a model,
> a working model, just like the atom in early quantum
> theory was seen as a tiny tiny solar system. One
> formulates a model and looks how far one can get.
> The model of the mini-solar system was helpful
> for a while, and led quite far, but then it had to be
> abandoned for a more complex model of probability waves.
> You could get somewhere with an atomic unit of meaning,
> but it is not a theorem, you can't announce a machine
> that will condense all knowledge - a machine of all
> knowledge, a God machine - on the basis of a very limited
> working model. So again, give me an example of a word
> that has an atomic unit of meaning, and I'll show you
> how limited that working model is.
>

[Antti Valmari]

On 11/23/13 11:09, Julio Di Egidio wrote:
> "Antti Valmari" <Antti....@c.s.t.u.t.f.i.invalid> wrote in message
> news:l6nhi4$cue$1...@news.cc.tut.fi...
>>
>> A much more readable and competent attempt to deny undecidability of

>> halting was written by Eric C.R. Hehner. ...
>> the flaws in his reasoning ...

>
> I find that line of reasoning quite compelling, although at an informal
> level. Could you maybe tell which (at least in essence) are those flaws?
>

> Julio

He presents a number of examples of ill-defined concepts. Then he
presents the halting problem contradiction and says that it is
analogous, so couldn't it be that the halting function is ill-defined
instead of well-defined but uncomputable. All his examples work out also
under the interpretation that a function may be well-defined but
uncomputable. When reading, just be careful to distinguish program code
from definitions.

When deriving a contradiction from an ill-defined concept, the reason of
the contradiction indeed is that the definition is bad. If we assume the
existence of the smallest natural number that is different from itself,
then indeed we get weird consequences.

In the halting problem contradiction, the starting point is a
hypothetical piece of code that computes the halting function. In that
kind of a situation, there are two possible sources of the
contradiction: (A) the function that the code should compute does not
exist, or (B) the function does exist but the code does not. How could
we know which one is the right one?

Before continuing, let me tell that in this research field, only
deterministic programs are considered. So each time the same program is
started with the same input, it behaves the same. This assumption is
usually not said out loud, but it is there. I point this out because
some other posts wondered about this.

(A) requires accepting that the halting function is ill-defined. The
definition of the halting function is intuitively among the simplest
possible definitions, and mathematically it is similar to many others.
Intuitively, when some program is executed on some input, it either is
or is not the case that the execution eventually stops. Almost all
computer scientists and mathematicians find nothing wrong with this. If
something were wrong here, then we would have a nasty problem: what does
distinguish this definition from numerous others, so that we could
reject this one as ill-defined, without rejecting everything?

Some opponents of uncomputability confuse this by saying that the result
of the halting function is self-contradictory or undefined, when a
certain piece of code based on the halting tester is given as the input.
So, at least for this input, both the reply "eventually stops" and the
reply "does not eventually stop" is wrong, they say. This they call the
self-reference paradox. What they fail to notice or refuse to accept is
that this input is non-existent. The input is based on a halting tester,
but because the halting tester does not exist, also the input does not
exist. The contradiction does not arise from assuming the correct answer
for each program and instance. The contradiction arises from assuming
that there is a piece of code who finds the correct answer.

On the other hand, (B) requires accepting the existence of uncomputable
functions. It means abandoning the idea that for each well-defined
function, there is a program that computes it. But what evidence is
there that for each well-defined function, there is a program that
computes it? None. There only is some peoples' hope that it should be so.


>> and thought about the consequences of his
>> conclusion: what would change if he were right.

The established science partitions the attempts to define functions into
three classes:

(A) Bad definitions
(B) Good definitions yielding uncomputable functions
(C) Good definitions yielding computable functions

It seems to me that accepting Hehner's view would mean fusing the
classes (A) and (B) and giving new names:

(A union B) Bad definitions
(C) Good definitions

The fact that there is no halting tester would remain, it would just be
given a new explanation. The same holds for all undecidable problems. We
would lose a huge amount of established non-computer-science
mathematics. We would have a very difficult obligation of stating
criteria for well-definedness, so that we could avoid bad definitions in
the future.


--- Antti Valmari ---

[Antti Valmari]

On 11/24/13 00:08, wugi wrote:
> Since decades I've been stuck with this layman's question about halting
> decidability:
> Is a simple line like
> "IF RND < 0.0000000000001 THEN END"
> allowed to be part of the program under examination?

I assume that RND yields a random value.

Usually, for simplicity, it is assumed that the programs under
discussion are deterministic, so this line would be ruled out. After you
understand the standard theory, you can let randomness or nondeterminism
enter the game and analyse the consequences.

Randomness brings in new notions that may sound weird. Assume tossing a
fair coin until getting heads. Non-termination is possible but has
probability zero. So probability zero is not the same thing as
impossibility.


--- Antti Valmari ---

[fom]

Your conclusion here is similar to a remark in the
link Mr. Greene provided elsewhere. If one looks at
the paragraph just above the position given in the link,

http://plato.stanford.edu/entries/self-reference/#ExtAltKriTheTru

one will find that Kripke's approach to the liar paradox using
many-valued Kleene truth is different from others, but does not
eliminate the problem which introduces hierarchies.

In fact, the remark expresses this fact in another
way related to hierarchies. Just like there can be
truths of second-order logic which are not provable
in first-order logic, the truth that Kripke's object
language cannot express that the liar paradox is expressible
in the meta-language.

[graham...@gmail.com]

RE: "This sentence is not true"

On Monday, November 25, 2013 4:06:37 AM UTC-8, Peter Olcott wrote:
>
> I will re-state my position after more thought:
>
> 1) "This sentence is not true" is not true.

that is EXACTLY what the sentence says!

SEMANTICALLY it is true.

It is a weaker / external truth.. that the sentence is true ABOUT_ITSELF.



It has the same SEMANTIC TRUTH as a GODEL STATEMENT..

That ALL# LOGIC THEORIES are INCOMPLETE by not including the proof of Godels Statement... is SEMANTICS!



LOGIC DOES NOT NEED SEMANTIC TRUTHS!



THIS IS NOT A MEMBER OF THE SET OF ALL TRUE SENTENCES

is semantically_true.




It's also trivial to equate with 'X<->~X' and discard!

>
> 2) "This sentence is not true" is not false.
>
> 3) Therefore "This sentence is not true" in not a valid proposition.
>

Herc