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Re: Kinds of knowledge

From:Yahya Abdal-Aziz <yahya@...>
Date:Tuesday, September 27, 2005, 13:29
Hi all,

On Mon, 26 Sep 2005, "David G. Durand" wrote:
> On 9/26/05, Peter Bleackley <Peter.Bleackley@...> wrote: > > Gdel's Incompleteness Theorem states that any system of logic capable of > > describing itself must either be unable to prove at least one true > > statement, and therefore be incomplete, or be able to prove at least one > > false statement, and therefore be inconsistent. This is because the > > self-describing nature of the system enables you to formulate a proposition > > equivalent to, "This system of logic cannot prove this statement." If the > > statement is true, then it cannot be proven, and the system must be > > incomplete. If not, then the system can prove a falsehood, and therefore > > contains inconsistencies. Obviously, incomplete systems of logic are > > preferable to inconsistent ones. > > It's actually a little more subtle than that. The Goedel sentence says > "There does not exist a number under a certain encoding that > represents the proof that this sentence is false" > That number cannot be constructed by any of the proof rules of the > system.
David, I'm familiar with the proof - a tedious matter!, and I was not aiming at exactness in my initial paraphrase. I had hoped that no-one would be too fussed by that! But you've added some interesting observations:
> ... However, it is possible to decide that the statement is false > and that the system is not incomplete -- but it is only possible by > adding new "magic numbers" (what are called non-standard extensions of > the model),
I'd heard of this some time back, but haven't seen the details. Perhaps wikipedia or mathworld will enlighten me?
> ... which cannot be written down, but whose existence does not > imply any falsehood in the system.
This is actually quite germane to my quest for different kinds of knowledge. One of the levels of my "thepfi" hierarchy, which I mentioned in the original post, was the level "f" standing for "formulable". Having a number which you can't write down is much like having a proposition you can't formulate; the Gödel proof provides an explicit method of constructing a unique (Gödel) number for any proposition or formula of the propositional calculus. However, I would not expect any language arising by natural means to encode this kind of logical knowledge; it's simply too abstract. Tho I suppose as computers become more nearly self-organising systems, we may expect them to begin to create their own languages ad hoc for the purpose of conversing amongst themselves! If they do so, those languages would probably have little reference to the world natural to carbon-based life-forms ... :-)
> While it might seem strange to accept whole numbers that can not be > constructed by repeatedly adding one to zero, such moves are not > unheard of. Some models of calculus involve the addition of "numbers"
.. infinitesimals?
> that are not equal to 0, but are smaller than the difference of any > "ordinary" numbers. > > It's also weird to believe that "this statement is false" is true.
Not weird, simply inconsistent.
> Personally, my favorite short form of Goedel's theorem is: > > There are unprovable statements in any system powerful enough to prove > anything interesting about strings or numbers.
Yeah, I like its brevity, but it leaves the very vague notion of "interesting" wide open ...
> As with much logic, the definition of interesting is pretty low, so > basically any useful logical system is powerful enough to have such > unprovable statements.
Turning "interesting" into "useful" - probably a more "useful" idea!
> OBconlang, natuaral languages do not operate under the same > constraints and regime as logic.
"... of binary or Boolean logic", perhaps. And probably, but not certainly, of a multi-dimensional logic of evidentials.
> ... However, the dream of a calculational > system that will decide true or false for any set of statements fed > into it is unattainable.
True. That, of course, is nothing like I described as my aim.
> ... I tend to be skeptical of loglangs as a tool > because people don't seem to use them correctly. I think that this is > fundamental to the way people use logic, but are not logically > consistent themselves.
We all of us have human limitations, which make it quite impossible for us to recall with any accuracy what we said (and meant) half an hour ago. I don't think a totally logical language can be useful to beings with less than perfect recall; else how could they detect and avoid their own potential inconsistencies? With the best will in the world, we simply can't be consistent. And since we have no evidence that a perfect memory has a significant survival advantage over a merely good memory, I think it unlikely that any species arising naturally will ever have a perfect memory. Even if it did, making that memory accessible in useful timeframes poses a formidable organisational challenge, along the lines of "Build a better brain (than the human one)". So, "Loglangs for robots", say I! Regards, Yahya -- No virus found in this outgoing message. Checked by AVG Anti-Virus. Version: 7.0.344 / Virus Database: 267.11.6/111 - Release Date: 23/9/05