Bruno,
I am glad for the opportunity to discuss these things with someone who
knows something about these issues.
> In my opinion, revision theories are useful when a machine begins to
> bet on an universal environment independent of herself. Above her
> Godel-Lob-Solovay correct self-reference logic, she will have to
> develop a non monotonic surface to be able to handle its errors,
> dreams, etc. It is a bit more close to practical artifiicial
> intelligence engineering than machine theology, but I am ok with that.
I am interested in nonmonotonic logics as an explanation of how we can
have "concepts" that don't just reduce to first-order theories--
specifically, concepts such as "number" that fall prey to Godelian
incompleteness. In other words, I think that "we use nonmonotonic
logic" is at least a partial answer to what I called the little
puzzle.
>> First we have "true" and "false". Dealing with these in an
>> unrestricted manner, we can construct sentences such as "this sentence
>> is false".
>
> I don't think we can really do that. We cannot, I think. (And I can
> prove this making the assumption
> that we are ideally sound universal machines).
I'm not claiming that we can *consistently* construct such sentences,
just that we can try to construct them, and then run into problems
when we try to reason about them. Luckily we have what you called a
"nonmonotonic surface" so we draw back and either give up or try from
different angles.
--Abram
On Wed, Nov 26, 2008 at 10:54 AM, Bruno Marchal <marchal.domain.name.hidden> wrote:
>
> Hi Abram,
>
>
> On 26 Nov 2008, at 00:01, Abram Demski wrote:
>
>>
>> Bruno,
>>
>> Yes, I have encountered the provability logics before, but I am no
>> expert.
>
>
> We will perhaps have opportunity to talk about this.
>
>
>>
>>
>>>> In any given
>>>> generation, the entity who can represent the truth-predicate of the
>>>> most other entities will dominate.
>>>
>>> Why?
>>
>> The notion of the entities adapting their logics in order to better
>> reason about each other is meant to be more of an informal
>> justification than an exact proof, so I'm not worried about stating my
>> assumptions precisely... If I did, I might simply take this to be an
>> assumption rather than a derived fact. But, here is an informal
>> justification.
>>
>> Since the entities start out using first-order logic, it will be
>> useful to solve the halting problem to reach conclusions about what a
>> fellow-creature *won't* ever reach conclusions about. This means a
>> "provable" predicate will be useful. To support deduction with this
>> predicate, of course, the entities will gain more and more axioms over
>> time; axioms that help solve instances of the halting problem will
>> survive, while axioms that provide incorrect information will not.
>> This means that the "provable" predicate has a moving target: more and
>> more is provable over time.
>
>
> All right.
>
>
>
>> Eventually it will become useful to
>> abstract away from the details with a "true" predicate.
>
>
> Here, assuming the mechanist hypothesis (or some weakening), the way
> the "truth predicate" is introduced is really what will decide if the
> soul of the machine will fall in Hell, or get enlightened and go to
> Heaven. The all encompassing "truth" is not even nameable or
> describable by the machines.
>
>
>
>
>
>> The "true"
>> predicate essentially says "provable by some sufficiently evolved
>> system". This allows an entity to ignore the details of the entity it
>> is currently reasoning about.
>
>
> If PA (Peano Arithmetic) deduces "I can prove that I am consistent"
> from "I can prove that ZF (Zermelo Fraenkel Set Theory) proves that I
> am consistent", then PA goes to hell!
> If an entity refers to a more powerful entity, even if "we" trust that
> more powerful entity, it just an invalid "argument of authority".
> Of course if PA begins to *believe* in the axioms of ZF, then PA
> becomes ZF, and can assert the consistency of PA without problem. But
> then, "we" are no more talking *about* PA, but about ZF.
>
>
>
>> This won't always work-- sometimes it
>> will need to resort to reasoning about provability again. But, it
>> should be a useful concept (after all, we find it to be so).
>
>
> Sure. But truth is really an interrogation mark. We can only "search"
> it.
>
>
>
>>
>>
>>>> Of course, this gives rise to an outlandish number of truth-values
>>>> (one
>>>> for each ordinal number), when normally any more than 2 is
>>>> considered
>>>> questionable.
>>>
>>>
>>> Not really, because those truth value are, imo, not really truth
>>> value, but they quantify a ladder toward infinite credibility,
>>> assurance or something. Perhaps security.
>>
>> I agree that the explosion of "truth-values" is acceptable because
>> they are not really truth-values... but they do not go further and
>> further into absolute confidence, but rather further and further into
>> meaninglessness. Obviously my previous explanation was not adequate.
>>
>> First we have "true" and "false". Dealing with these in an
>> unrestricted manner, we can construct sentences such as "this sentence
>> is false".
>
>
> I don't think we can really do that. We cannot, I think. (And I can
> prove this making the assumption
> that we are ideally sound universal machines).
>
>
>
>
>
>> We need to label these somehow as meaningless or
>> pathological. I think either a fixed-point construction or the
>> revision theory are OK options for doing this;
>
>
> In my opinion, revision theories are useful when a machine begins to
> bet on an universal environment independent of herself. Above her
> Godel-Lob-Solovay correct self-reference logic, she will have to
> develop a non monotonic surface to be able to handle its errors,
> dreams, etc. It is a bit more close to practical artifiicial
> intelligence engineering than machine theology, but I am ok with that.
>
>
>
>> perhaps one is better
>> than the other, perhaps they are ultimately equivalent where it
>> matters, I don't know. Anyway, now we are stuck with a new predicate:
>> "meaningless". Using this in an unrestricted manner, I can say "this
>> sentence is either meaningless or false". I need to rule this out, but
>> I can't label it "meaningless", or I will then conclude it is true
>> (assuming something like classical logic). So I need to invent a new
>> predicate, 2-meaningless. Using this in an unrestricted manner again
>> would lead to trouble, so I'll need 3-meaningless and 4-meaningless
>> and finitely-meaningless and countably-meaningless and so on.
>
>
> Indeed. It seems you make the point.
>
> Best,
>
>
> Bruno Marchal
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> >
>
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Received on Wed Nov 26 2008 - 15:24:56 PST