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I'm going to consolidate replies to all three of your comments here and only going to address specific salient points that I think are the crux of disagreements

I'm going to consolidate replies to all three of your comments here and only going to address specific salient points that I think are the crux of disagreements: Re: Bayesian inference and logic To clarify, I was informally referring to the standard correspondence with propositional logic [1], but generalizations of this to include quantifiers also seem possible [2]. That said, I agree that my simple sketch is not a good justification for Solomonoff Induction, so a poor attempt at a summary on my part. Re: A "system" is not a function. I don't know a formal definition of a system, but if you're talking about whatever the human brain is doing that's likely to be more easily modelled as a Turing machine. If you agree that a system can be modelled as a function, then there's a formal equivalence. That's what "is" means to me. Certainly you can insist on a definition for "is" that's more like a small-step operational semantics to more faithfully model internals, but for the purposes of this discussion, I don't think the internal details matter if the observable inputs and outputs are the same. If a human brain "is" equivalent to some Turing machine, then I can also say that it "is" equivalent to some term in the lambda calculus. Re: If, by any chance, you think of the brain as a universal Turing machine (UTM) The brain can be described by a finite state automaton due to the Bekenstein Bound. So it's not a universal Turing machine, strictly speaking. Certainly it would be interesting to know how the internals of the brain work in minute detail, but that's ambitious. Re: They have consistently failed because the pieces of the puzzle don't fit. I'm curious specifically what doesn't fit. I understand there's tension between logic and probability, and learning and human-like reasoning are obviously still burgeoning fields, but the rest seem fairly well explored with plenty of correspondences. Re: I think in modern machine learning people have simply given up on trying to make any coherent image out of the puzzle pieces and just bashed them until they yielded and are now wedged together in incongruous patterns that sort of show some kind of image, there. Which is kinda what you'd expect from evolution by natural selection. [1] https://en.wikipedia.org/wiki/Bayes%27_theorem#Correspondence_to_other_mathematical_frameworks https://en.wikipedia.org/wiki/Bayes%27_theorem#Correspondenc... [2] https://www.lesswrong.com/posts/W8YscokXMiDnLKJ96/bayesian-inference-on-1st-order-logic https://www.lesswrong.com/posts/W8YscokXMiDnLKJ96/bayesian-i...

YeGoblynQueenne 3y ago
>> To clarify, I was informally referring to the standard correspondence with propositional logic [1], but generalizations of this to include quantifiers also seem possible [2]. The order of logic doesn't matter. That "standard" mapping of probabilities to logic doesn't work with classical logical semantics and with the common assumption that all probabilities sum to 1. I pointed out why in my previous comment: mapping true to 1 and false to 0 makes for too many 1's to fit in the open interval (0,1). The wikipedia article (which lacks relevant references) and the LessWrong post do not stop to address this foundational inconsistency and instead plow on with building on top of it, but they're only building a house of cards: one cannot draw safe inferences from inconsistent assumptions. As I say, I get the impression that many such attempts to marry logic and probabilities come from people who understand probabilities, but not logic. They might be considered "standard" in communities that simply wish to do away with logic and only have to think about probabilities, but they are a cop-out supported by sloppy thinking. On the other hand, there are a number of frameworks that avoid the cardinal sin of mapping true and false to 1 and 0, which they do by eschewing classical semantcis, for example Łukasiewicz logic: https://en.wikipedia.org/wiki/%C5%81ukasiewicz_logic https://en.wikipedia.org/wiki/%C5%81ukasiewicz_logic These are the obvious choice of logical semantics if one wishes to combine probability with logic, but somehow I rarely see them used. Possibly because they are not as well-known as propositional logic. In general, I propose to anyone who is interested in the subject to get at least a functional understanding of classical logic, if not as good an understanding as they have of probabilities and statistics. >> I'm curious specifically what doesn't fit. I understand there's tension between logic and probability, and learning and human-like reasoning are obviously still burgeoning fields, but the rest seem fairly well explored with plenty of correspondences. What I mean by "the pieces of the puzzle don't fit" is that the various abstractions we have to describe computation don't work when we try to apply them to human thinking. All the formal systems we have are missing something. Trying to work outside formal systems of course leads away from the realm of science and into the world of wild speculation and wishful thinking. >> Which is kinda what you'd expect from evolution by natural selection. I don't follow. Evolution by natural selection should be expected to make us lose all hope of building consistent theories?

YeGoblynQueenne 3y ago

>> To clarify, I was informally referring to the standard correspondence with propositional logic [1], but generalizations of this to include quantifiers also seem possible [2]. The order of logic doesn't matter. That "standard" mapping of probabilities to logic doesn't work with classical logical semantics and with the common assumption that all probabilities sum to 1. I pointed out why in my previous comment: mapping true to 1 and false to 0 makes for too many 1's to fit in the open interval (0,1). The wikipedia article (which lacks relevant references) and the LessWrong post do not stop to address this foundational inconsistency and instead plow on with building on top of it, but they're only building a house of cards: one cannot draw safe inferences from inconsistent assumptions. As I say, I get the impression that many such attempts to marry logic and probabilities come from people who understand probabilities, but not logic. They might be considered "standard" in communities that simply wish to do away with logic and only have to think about probabilities, but they are a cop-out supported by sloppy thinking. On the other hand, there are a number of frameworks that avoid the cardinal sin of mapping true and false to 1 and 0, which they do by eschewing classical semantcis, for example Łukasiewicz logic: https://en.wikipedia.org/wiki/%C5%81ukasiewicz_logic https://en.wikipedia.org/wiki/%C5%81ukasiewicz_logic These are the obvious choice of logical semantics if one wishes to combine probability with logic, but somehow I rarely see them used. Possibly because they are not as well-known as propositional logic. In general, I propose to anyone who is interested in the subject to get at least a functional understanding of classical logic, if not as good an understanding as they have of probabilities and statistics. >> I'm curious specifically what doesn't fit. I understand there's tension between logic and probability, and learning and human-like reasoning are obviously still burgeoning fields, but the rest seem fairly well explored with plenty of correspondences. What I mean by "the pieces of the puzzle don't fit" is that the various abstractions we have to describe computation don't work when we try to apply them to human thinking. All the formal systems we have are missing something. Trying to work outside formal systems of course leads away from the realm of science and into the world of wild speculation and wishful thinking. >> Which is kinda what you'd expect from evolution by natural selection. I don't follow. Evolution by natural selection should be expected to make us lose all hope of building consistent theories?