# Re: Re: The two types of truth

```Hi Bruno Marchal

Thanks for your patience. Beautiful stuff,
```

Roger Clough, rclo...@verizon.net
11/6/2012
"Forever is a long time, especially near the end." -Woody Allen

----- Receiving the following content -----
From: Bruno Marchal
Time: 2012-11-06, 07:43:06
Subject: Re: The two types of truth

On 05 Nov 2012, at 13:45, Roger Clough wrote:

> Hi Bruno Marchal
>
>
> Is sigma_6 truth truth with only a 6 sigma possibility of error ?

let P(x) be a decidable number property. Like being prime.

Note that if P(x) is decidable, then ~P(x) is decidable too. P(x), and
~P(x) are said sigma_0

Then, thanks to a theorem of Mostowski, you have a natural ladder of
degrees of insolubility:

sigma_1 the proposition with shape ExP(x) = "it exists a number
x such that it is the case that P(x)"

You can see that if a sigma_1 proposition is true, then, if you have
enough time you can know it in principle. Just test P(x) on 0, then on
1, then on 2, etc. If "ExP(x)" is true, you will find that x
eventially with that method.

Pi_1 the negation of of sigma_1 proposition. That is ~ExP(x),
which is equivalent with Ax~P(x). Do you see that. If is is false that
a number exists with the property P, it means that all numbers have
the property ~P.

Now Pi_1 are a priori more complex to prove that the sigma_1, as you
have to very that 0 has not p, and then 1 has not p, ad infinitum, in
case the proposition is true. Note that you can still refute such a
proposition in case it false, as you have just to verify the "ExP(x)"
to refute it.

Sigma_2 = Ay Ex P(x, y) Pi_2 Ey Ax P(x, y)
etc.

So a sigma_6 proposition would be AxEyAzErAtEkAnEmP(x,y,z,r,t,n,m)

Very complex proposition. For example with an oracle for the halting
problem, you can decide all sigma_1 truth or falsity, but you can't
decide a sigma_2 proposition.

Note this: sigma_1 completeness (the ability to decide the true, (bt
not necessarily the false) sigma_1 sentences) is equivalent with
Turing universality.

There is no direct relationship with error correction.

Bruno

>
> Roger Clough, rclo...@verizon.net
> 11/5/2012
> "Forever is a long time, especially near the end." -Woody Allen
>
>
> ----- Receiving the following content -----
> From: Bruno Marchal
> Time: 2012-11-04, 08:56:01
> Subject: Re: The two types of truth
>
>
> On 03 Nov 2012, at 12:45, Roger Clough wrote:
>
>> Hi Bruno Marchal and Stephen,
>>
>> http://www.angelfire.com/md2/timewarp/leibniz.html
>>
>> "Leibniz declares that there are two kinds of truth:
>> truths of reason [which are non-contradictory, are always either
>> true or false],
>
> We can only hope that they are non contradictory.
> And although true or false, they are aslo known or unknown, believed
> of not believed, disbelieved or not disbelieved, etc.
>
>
>
>
>> and truths of fact [which are not always either true or false].
>
> Why? They are contextual, but you can study the relation fact/context
> in the higher structure level.
>
>
>>
>> Truths of reason are a priori, while truths of fact are a posteriori.
>> Truths of reason are necessary, permanent truths. Truths of fact are
>> contingent, empirical truths.
>> Both kinds of truth must have a sufficient reason. Truths of reason
>> have their
>> sufficient reason in being opposed to the contradictoriness and
>> logical inconsistency
>> of propositions which deny them. Truths of fact have their
>> sufficient reason in
>> being more perfect than propositions which deny them."
>
> Unfortunately, this is acceptable below Sigma_1 truth, but doubtable
> above, so even in the lower complexity part of arithmetic, things are
> not that simple.
>
> Bruno
>
>
>
>
>>
>> Roger Clough, rclo...@verizon.net
>> 11/3/2012
>> "Forever is a long time, especially near the end." -Woody Allen
>>
>>
>> ----- Receiving the following content -----
>> From: Bruno Marchal
>> Time: 2012-11-03, 07:13:24
>> Subject: Re: Numbers in the Platonic Realm
>>
>>
>> On 02 Nov 2012, at 23:12, Stephen P. King wrote:
>>
>>> On 11/2/2012 1:23 PM, Bruno Marchal wrote:
>>>>>>> I can understand these symbols because there is at least a way
>>>>>>> to physically implement them.
>>>>>>
>>>>>> Those notion have nothing to do with "physical implementation".
>>>>>
>>>>
>>>> Too much ambiguous. Even staying in comp I can answer "yes" and
>>>> "no".
>>>> Yes, because my human thinking is locally supported by physical
>>>> events.
>>>> No, because the whole couple mind/physical events is supported by
>>>> platonic arithmetical truth.
>>> Dear Bruno,
>>>
>>> Where is the evidence of the existence of a Platonic realm?
>>
>> It is part of the assumption. We postulate arithmetic. I try to avoid
>> the use of "platonic" there, as I used the term in Plato sense. In
>> that sense Platonia = the greek No?, and it is derived from
>> arithmetic and comp.
>>
>> All you need is the belief that 43 is prime independently of "43 is
>> prime".
>>
>>
>>
>>> The mere self-consistency of an idea is proof of existence
>>
>> Already in arithmetic we have the consistence of the existence of a
>> prrof of the false, this certainly does not mean that there exist a
>> proof of the false. So self-consistency is doubtfully identifiable
>> with truth, and still less with existence.
>>
>>
>>
>>> but the idea must be understood by a multiplicity of entities with
>>> the capacity to distinguish truth from falsehood to have any
>>> coherence as an idea!
>>
>> Not at all. 43 is prime might be true, even in absence of universe
>> and
>> observer.
>>
>>
>>
>>> We cannot just assume that the mere existence of some undefined acts
>>> to determine the properties of the undefined. Truth and falsity are
>>> possible properties, they are not ontological aspects of existence.
>>
>> Truth is no more a property than existence. It makes no sense.
>>
>> Bruno
>>
>>
>> http://iridia.ulb.ac.be/~marchal/
>>
>>
>>
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>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
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