'When Are Relations Neither True Nor False? [Logic]

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From: Nam Nguyen on 27 Mar 2010 14:26

Daryl McCullough wrote:
> Nam Nguyen says...
>> Daryl McCullough wrote:
>>> Nam Nguyen says...
>>>> Daryl McCullough wrote:
>>>>> Nam Nguyen says...
>>>>>> Daryl McCullough wrote:
>>>>>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation
>>>>>>> of the language of PA.
>>>>>> So you've agreed "G(PA) can be arithmetically false"?
>>>>> It is false in nonstandard models of PA.
>>>> Why don't we make it more precise.
>>> What I said was already perfectly precise.
>> If you ask me whether or not Pythagoras is provable in some T
>> and I answer you "2+2=4" is true, then what I answer might be
>> precise in certain context but is completely _irrelevant_
>> in the underlying discussion.
>
> Well, it seemed perfectly relevant (and precise) to me.

Well, you know that's not a basis to further discussion about foundation
of FOL reasoning. In any rate, do you think if there's any _valid_ context
that "G(PA) can be arithmetically false" is true? (Hope that you'd agree
this is a Yes/No question.)

From: Alan Smaill on 27 Mar 2010 14:52

stevendaryl3016(a)yahoo.com (Daryl McCullough) writes:

> Alan Smaill says...
>
>>FWIW you can look at Bourbaki's account of Goedel's incompleteness
>>theorem to note that they studiously avoid saying that the goedel
>>sentence is true (arithaally, absolutely, or in any other way).
>
> Is that because they feel that there is something different about
> the Godel sentence than other sentences of arithmetic of similar
> complexity, or because they would equally well refrain from saying
> that other arithmetic statements are true?

I don't think it's exactly either of these --
they have a general tendency towards formalisation as first and foremost
about proving theorems, so that truth only matters insofar as it's
a problem if the axioms are inconsistent (which even then can be
expressed in terms of provability). So Goedel shows a limitation
on our ability to provide complete consistent axiomatisations --
no need to mention truth, & better to avoid foundational squabbles.

> --
> Daryl McCullough
> Ithaca, NY
>

--
Alan Smaill email: A.Smaill at ed.ac.uk
School of Informatics tel: 44-131-650-2710
University of Edinburgh

From: Nam Nguyen on 27 Mar 2010 14:55

Nam Nguyen wrote:
> Daryl McCullough wrote:
>> Nam Nguyen says...
>>> Daryl McCullough wrote:
>>>> Nam Nguyen says...
>>>>> Daryl McCullough wrote:
>>>>>> Nam Nguyen says...
>>>>>>> Daryl McCullough wrote:
>>>>>>>> [G(PA)] is a *relative* truth. It's true in the standard
>>>>>>>> interpretation
>>>>>>>> of the language of PA.
>>>>>>> So you've agreed "G(PA) can be arithmetically false"?
>>>>>> It is false in nonstandard models of PA.
>>>>> Why don't we make it more precise.
>>>> What I said was already perfectly precise.
>>> If you ask me whether or not Pythagoras is provable in some T
>>> and I answer you "2+2=4" is true, then what I answer might be
>>> precise in certain context but is completely _irrelevant_
>>> in the underlying discussion.
>>
>> Well, it seemed perfectly relevant (and precise) to me.
>
> Well, you know that's not a basis to further discussion about foundation
> of FOL reasoning. In any rate, do you think if there's any _valid_ context
> that "G(PA) can be arithmetically false" is true? (Hope that you'd agree
> this is a Yes/No question.)

There's reason why the word "cranks" has a different meaning than
"standard mathematicians, logicians" and I believe the difference
is genuine.

It's just that the later somehow believe that they're aways invincible
in their methods of reasoning and they'd would slam the door shut on
a slightest hint their methods could be wrong.

They know the 1-many problem and yet somehow they could convince themselves
they'd fully understand the infinite complexity of the natural numbers.

Aren't there any conservative, objective, and rational mathematicians/logicians
left in these forums to further discussions about the current state of FOL
reasoning?

From: Nam Nguyen on 27 Mar 2010 15:32

Alan Smaill wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> Daryl McCullough wrote:
>>> Nam Nguyen says...
>>>> Daryl McCullough wrote:
>>>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation
>>>>> of the language of PA.
>>>> So you've agreed "G(PA) can be arithmetically false"?
>>> It is false in nonstandard models of PA.
>> Why don't we make it more precise. When we say F, a formula written
>> in the language of arithmetic, is true or false _by default_ we
>> mean it's being arithmetically true or false: i.e. true or false
>> in the natural numbers. So we're *not* talking about F is being
>> true/false in any general kind of models here.
>
> You haven't (and can't) give me an effective way to use
> this definition to decide truth or falsity in the natural numbers.

You and I have nothing to argue here, since I just repeated the
convention about what "being true" means that virtually all of us
would use.

>
>> The point is Alan said he wouldn't know what an absolute truth
>> of G(PA) would mean
>> and I've implicitly defined it for him, and
>> here is the explicit version:
>>
>> There's _no other_ context in which the meta statement
>> "G(PA) is arithmetically true in the natural number" would
>> be false.
>
> I'm not obliged to accept any particular definition of yours.

Sure. You could label/name the definition "Republican" or "Democrat"
if you'd like. But first of all _you_ are the one who asked for what
it might mean by G(PA) be an absolute truth; and I just gave you
an definition that I think is reasonable.

Secondly in spite of whatever definition might be, can you answer the
relevant question I asked before (in some form): whether or not in the
meta level there's a valid context in which the statement "G(PA) is
arithmetically false" is true?

(That's a sound technical question - independent of any definition of
"absolute truth" or of any naming of any such definition you might
protest.)

>
> FWIW you can look at Bourbaki's account of Goedel's incompleteness
> theorem to note that they studiously avoid saying that the goedel
> sentence is true (arithaally, absolutely, or in any other way).
>
>> The question I was hoping you'd answer one way or another is
>> whether or not there's a context in which the meta statement:
>>
>> "G(PA) is arithmetically true in the natural number"
>>
>> would be _false_ ?
>>
>> If your answer is "yes", then the [arithmetically-in-the-natural-
>> number] truth of G(PA) is a relative notion. Otherwise it's an
>> absolute notion.
>
> Why should I believe the only answers are no or yes?
> Your realist assumptions are showing.
>
>> Which answer would you have? And perhaps why?
>
> you need a bit more Zen.

You need to answer some direct technical question presented to you.

From: Alan Smaill on 27 Mar 2010 15:53

Nam Nguyen <namducnguyen(a)shaw.ca> writes:

> Alan Smaill wrote:
>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>
>>> Daryl McCullough wrote:
>>>> Nam Nguyen says...
>>>>> Daryl McCullough wrote:
>>>>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation
>>>>>> of the language of PA.
>>>>> So you've agreed "G(PA) can be arithmetically false"?
>>>> It is false in nonstandard models of PA.
>>> Why don't we make it more precise. When we say F, a formula written
>>> in the language of arithmetic, is true or false _by default_ we
>>> mean it's being arithmetically true or false: i.e. true or false
>>> in the natural numbers. So we're *not* talking about F is being
>>> true/false in any general kind of models here.
>>
>> You haven't (and can't) give me an effective way to use
>> this definition to decide truth or falsity in the natural numbers.
>
> You and I have nothing to argue here, since I just repeated the
> convention about what "being true" means that virtually all of us
> would use.

Well, I do disagree with you here. You will find some negative comments
from Torkel Franzen, for example, about the waving of the magic wand of
"standard model" as somehow settling the question of the meaning of all
arithemtic statements (that's my paraphrase, not his wording).

>>> The point is Alan said he wouldn't know what an absolute truth
>>> of G(PA) would mean
>>> and I've implicitly defined it for him, and
>>> here is the explicit version:
>>>
>>> There's _no other_ context in which the meta statement
>>> "G(PA) is arithmetically true in the natural number" would
>>> be false.
>>
>> I'm not obliged to accept any particular definition of yours.
>
> Sure. You could label/name the definition "Republican" or "Democrat"
> if you'd like. But first of all _you_ are the one who asked for what
> it might mean by G(PA) be an absolute truth; and I just gave you
> an definition that I think is reasonable.

Please, I did *not* ask for a definition!

> Secondly in spite of whatever definition might be, can you answer the
> relevant question I asked before (in some form): whether or not in the
> meta level there's a valid context in which the statement "G(PA) is
> arithmetically false" is true?
>
> (That's a sound technical question - independent of any definition of
> "absolute truth" or of any naming of any such definition you might
> protest.)

First, I addressed this in my previous response --
(see below) which you have ignored.

Second, if you want this to be a sound technical question, we need to
know what a "valid context" is, and what the metalogic in question is.

Finally I do not find it convincing that the answer to this question
corresponds to a notion of *absolute* truth, regardess of whether the answer
is yes or no.

>> FWIW you can look at Bourbaki's account of Goedel's incompleteness
>> theorem to note that they studiously avoid saying that the goedel
>> sentence is true (arithaally, absolutely, or in any other way).
>>
>>> The question I was hoping you'd answer one way or another is
>>> whether or not there's a context in which the meta statement:
>>>
>>> "G(PA) is arithmetically true in the natural number"
>>>
>>> would be _false_ ?
>>>
>>> If your answer is "yes", then the [arithmetically-in-the-natural-
>>> number] truth of G(PA) is a relative notion. Otherwise it's an
>>> absolute notion.
>>
>> Why should I believe the only answers are no or yes?
>> Your realist assumptions are showing.
>>
>>> Which answer would you have? And perhaps why?
>>
>> you need a bit more Zen.
>
> You need to answer some direct technical question presented to you.

I do not accept that your question has a yes/no answer,
as I already said. Why do you think it has?

--
Alan Smaill email: A.Smaill at ed.ac.uk