> On 20 Feb 2019, at 00:00, Brent Meeker <meeke...@verizon.net> wrote:
> 
> 
> 
> On 2/17/2019 2:10 AM, Bruno Marchal wrote:
>>>> But the machine itself will not believe us, or understand this.
>>> Why not?  It can't prove what algorithm it is, but it can know that we 
>>> know...so why would it disbelieve us.
>> Tha machine becomes inconsistent if it assumes its consistency (cf Rogers’s 
>> sentence). The machine can assume a sort of consistency of its past belief, 
>> like PA can add the axioms that PA is consistent, (or that PA is 
>> inconsistent) without losing its consistency, but in that case it becomes a 
>> new machine, with a similar theology in shape, but a different 
>> content/meaning for the box. She has changed her own code (as we do every 
>> second instinctively).
> 
> I think this is misleading.  When you say it becomes inconsistent if it 
> assumes it's consistency, you mean that if it uses its consistency as an 
> axiom it can lead to proving "false".   But in fact everyone assumes that 
> their beliefs are consistent, they just don't take it as an axiom and neither 
> do they take it as an axiom that they are inconsistent.  If I'm creating an 
> AI I see no reason to have it make any assumption or inference about it's 
> consistency in the sense of Goedel.  It need only be consistent in the sense 
> of avoiding ex quod libet.

Yes. That is what I was explaining.

In fact the machine, PA say, can guess or infer abductively or inductively its 
own consistency, and add it as a new axiom leading to the “new” machine PA + 
con PA (which is different than PA, and indeed much more powerful in the range 
of its theorem, and this makes the length of many proof shorter (cf the 
speeding role of consciousness). If the machine is not cautious, it can lead to 
the theory PA’ = PA + con PA’ (that exists by the diagonal lemma or Kleene’s 
second recursion theorem), and that leads to an inconsistent theory. So there 
is an important nuance between “guessing or inferring one’s consistency, and 
assuming it as part of our belief, without any distinguishing label or 
interrogation mark. It is the same as the difference with the existence of the 
universe or just infinity. PA assumes (or derive from its assumption/axioms) 
the existence of each numbers, but not of the entirety or infinity of the 
natural numbers. Mechanism infer/guess the existence of a physical universe 
(and a doctor, physical computers), but not as part of the mechanist 
assumption, which would make that universe primitive. That again is a nuance 
brought by incompleteness, and plays an important rôle. If a machine 
asserts/assumes its own consistency, it can prove it (in one line like “see the 
assumption”), and get inconsistent by incompleteness. It becomes equivalent to 
a Rogerian sentences, that is, a sentence K such that PA proves (K <-> ~[]~K), 
but then PA proves [](K <-> ~[]~K) <-> [](K <-> f), like PA proves (with [] = 
beweisbar)

 [](K <-> []K) <-> [](K <-> t).   (Löbian sentence)
 [](K <-> ~[]K) <-> [](K <-> <>t)  (Gödelianl sentence)
 [](K <-> []~K) <-> [](K <-> []f). (Jeroslowian sentence)

The moral is that in psychology and theology, many truth go without saying and 
*only* without saying. They are called the Protagorean virtue in my longer 
exposition. 

You need only to recall that I use “understands”,” asserts", “proves", 
“believes” in a deductive sense, as opposed to semantic inductive inference, 
related to the fact that a machine has a body, or code, and can infer truth by 
experience (and not a reasoning, conscious or not conscious). That is reflected 
also in the difference between []p and []p & p, or []p & <>t, etc.

Bruno




> 
> Brent
> 
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