And, one more thing. The use of references in a language guarantees that anything can be slipped into the language - if the language is meant to be more than insipidly representational. Most of the value of the mathematical system is because it can be analyzed through theoretical (or presumed) abstractions.
Jim Bromer On Sun, Aug 24, 2014 at 8:29 PM, Jim Bromer <[email protected]> wrote: > I don't completely understand the paper but the underlying belief that if > you use weighted reasoning or probability reasoning then the problem of > 'understanding' can be resolved by methods of increasing the accuracy of > successive approximations seems a little presumptuous to me. The reliance > of methods which are based on this kind of premise may be the foundation of > some of the applied science for technicians but it is not a > secure foundation for research science or even for engineering. > > While I think their idea is interesting (if I understand it as far as I > have gotten) I don't think that you can deny Godelian paradoxes just by > defining an algorithm which is designed to avoid them. Even assuming that > their use of the Oracle approximator is a stub for unknown processes which > could be used to examine a sentence in the induction process, the denial > that the strings that are being produced by the algorithm has to constitute > a logical Language is a little weak. And if the examination of successive > approximations is based on a poor theoretical analysis then paradoxes be > insipidly inducted into the system. > > When I was first told about paradoxes I am sure the concept went right > over my head. I believe I eventually discovered paradoxes in general rules > of how I was supposed to behave or in trying to find good ways to respond > to situations. Because of the importance of being able to use a language to > consider the nature of paradoxes, any foundational AGI mathematical system > has to be able to deal with paradoxes that can be analyzed even it they > can't be valued as True or False. How could a Language (with a fancy L) be > considered as a good theoretical underpinning for an AGI project if it > cannot hold paradoxes? The Godelian paradox seems to be a feature of any > Language that is strong enough to be dependent on evaluation across > a finite or infinite iteration via a theoretical representation. That > means that the valuation of a sentence may need to be based on more > information that might be represented using a virtual meta-language. And > the typical nature of a logic statement is that its evaluation may require > more than a single simple evaluation step. A logical sentence may be > partially evaluated but the evaluation of the entire sentence will usually > need to take more than one step. > > I think Logical Positivism faded away because the languages of > the Positivists were not strong enough to be used to analyze logical > statements other than those that were already evaluated. I suspect that > you might not be technically able to use a truly Positivist Language to > study logical statements because some logical statements can be paradoxical. > > It is important to be able to work with apparent paradoxes in order to > discover how the paradoxes might be resolved. While their Oracle would be > able to tell you that a resolution was more likely after taking successive > analytical steps, that does not always (or usually) hold for practical > methods of study. > > I am making what is probably my last effort to find an effective solution > to Boolean Satisfiability not because I think that Logic would be > sufficient for an AGI language of thought but because I think that a > polynomial time solution would provide a great deal of leverage for a > program capable of 'thought' to work with. > > > Jim Bromer > > > On Thu, Aug 21, 2014 at 1:02 PM, Ben Goertzel via AGI <[email protected]> > wrote: > >> >> This time some of the MIRI (Singularity Institute) guys have discovered a >> fairly cool mathematical nugget, that relates a bit to OpenCog... >> >> http://intelligence.org/files/DefinabilityTruthDraft.pdf >> >> What they show here is basically that some paradoxes of "reflection" in >> logical systems go away if one considers only statements with (open) >> interval truth values, rather than single-point truth values... >> >> This has direct implications for PLN, which uses imprecise (i.e. >> interval) truth values.... It shows that, in a sense, PLN can be >> reflective without spawning nasty Godel paradoxes.... It shows that one >> can define "truth within PLN" within PLN, without running into >> Godelian/Tarskian limitations.... Kinda cool... >> >> -- Ben >> >> -- >> Ben Goertzel, PhD >> http://goertzel.org >> >> "In an insane world, the sane man must appear to be insane". -- Capt. >> James T. 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