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

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From: Nam Nguyen on 10 Mar 2010 09:32

Daryl McCullough wrote:
> Nam Nguyen says...
>> MoeBlee wrote:
>>
>>> (1) Anyone who has studied mathematical logic already recognizes that
>>> a sentence that is neither logically true nor logically false has
>>> models in which it is true and models in which it is false.
>> Did you yourself quantifying-ly inspect uncountably many models
>> of, say, PA to know that?
>
> The beauty of mathematical proof is that you can be certain
> of the truth of a universal statement without checking every
> instance.

You meant as "certain" as the truth of GC or "There are infinitely many
counter examples of GC"?

From: Daryl McCullough on 10 Mar 2010 09:35

Nam Nguyen says...
>
>Daryl McCullough wrote:
>> Nam Nguyen says...
>>> MoeBlee wrote:
>>>
>>>> (1) Anyone who has studied mathematical logic already recognizes that
>>>> a sentence that is neither logically true nor logically false has
>>>> models in which it is true and models in which it is false.
>>> Did you yourself quantifying-ly inspect uncountably many models
>>> of, say, PA to know that?
>>
>> The beauty of mathematical proof is that you can be certain
>> of the truth of a universal statement without checking every
>> instance.
>
>You meant as "certain" as the truth of GC or "There are infinitely many
>counter examples of GC"?

Neither. I mean certain as the truth of "every consistent theory has a
countable model".

--
Daryl McCullough
Ithaca, NY

From: Nam Nguyen on 10 Mar 2010 09:47

Daryl McCullough wrote:
> Nam Nguyen says...
>
>> MoeBlee wrote:
>
>>> What I mean is: What mathematical theorems can you prove? Come on, if
>>> I don't have to prove any mathematical theorems, then I wouldn't need
>>> to propose any principles AT ALL!
>> You still could prove a lot of theorems under an edifice that complies
>> to these principles. In fact if you take the current FOL and strip off
>> anything that are (or _depends on_) the naturals or models from the
>> role of _inference_ then you'd obtain an edifice that would conform
>> to these 4 principles (at least from a first glance at the matter).
>
> Give an example of a nontrivial theorem in such a system. I don't
> think anyone would be interested in it, not even you.

How about ExAy[~(Sy=x)], in Q (in that edifice)? It's an arithmetic
theorem, got to be interesting, isn't it?

"Interesting" is subjective and *not* logical/reasoning. Proving is
logical/reasoning. If you're eager to claim some truths and in the
process sacrificing the rigidity of reasoning via syntactical proofs,
what's the point?

Iirc, in the French Evolution it was said something to the effect that
in the name of freedom, liberty, revolution, they did commit some
crimes. Well, I think in the name of interesting, induction, countably
infinite, we've harmed the rigor of reasoning by syntactical rules
of inference, in mathematical logic.

From: Nam Nguyen on 10 Mar 2010 09:49

Daryl McCullough wrote:
> Nam Nguyen says...
>> Daryl McCullough wrote:
>>> Nam Nguyen says...
>>>> MoeBlee wrote:
>>>>
>>>>> (1) Anyone who has studied mathematical logic already recognizes that
>>>>> a sentence that is neither logically true nor logically false has
>>>>> models in which it is true and models in which it is false.
>>>> Did you yourself quantifying-ly inspect uncountably many models
>>>> of, say, PA to know that?
>>> The beauty of mathematical proof is that you can be certain
>>> of the truth of a universal statement without checking every
>>> instance.
>> You meant as "certain" as the truth of GC or "There are infinitely many
>> counter examples of GC"?
>
> Neither. I mean certain as the truth of "every consistent theory has a
> countable model".

How certain is that while you don't know exactly what the naturals collectively
is?

From: Nam Nguyen on 10 Mar 2010 10:18

Nam Nguyen wrote:
> Daryl McCullough wrote:
>> Nam Nguyen says...
>>> Daryl McCullough wrote:
>>>> Nam Nguyen says...
>>>>> MoeBlee wrote:
>>>>>
>>>>>> (1) Anyone who has studied mathematical logic already recognizes that
>>>>>> a sentence that is neither logically true nor logically false has
>>>>>> models in which it is true and models in which it is false.
>>>>> Did you yourself quantifying-ly inspect uncountably many models
>>>>> of, say, PA to know that?
>>>> The beauty of mathematical proof is that you can be certain
>>>> of the truth of a universal statement without checking every
>>>> instance.
>>> You meant as "certain" as the truth of GC or "There are infinitely many
>>> counter examples of GC"?
>>
>> Neither. I mean certain as the truth of "every consistent theory has a
>> countable model".
>
> How certain is that while you don't know exactly what the naturals
> collectively
> is?

In any rate, do you (Daryl, MoeBlee) see anything wrong with the 4
principles? And if so why? [You both seem to have resisted their
"power". No?]