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 <[EMAIL PROTECTED]> 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/ > > > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---