I agree, Yvain said it first, and it doesn’t sound like group selection.
Concerning your group selection comment, that does sound plausible… but being relatively unfamiliar with tribal behavior, I would want to be sure that greedy genes were not spreading between groups before concluding that group selection could actually occur.
Is this really the case?
In fuzzy logic, one requires that the real-numbered truth value of a sentence is a function of its constituents. This allows the “solve it” reply.
If we swap that for probability theory, we don’t have that anymore… instead, we’ve got the constraints imposed by probability theory. The real-numbered value of “A & B” is no longer a definite function F(val(A), val(B)).
Maybe this is only a trivial complication… but, I am not sure yet.
The crisp portion of such a self-reference system will be equivalent to a Kripke fixed-point theory of truth, which I like. It won’t be the least fixed point, however, which is the one I prefer; still, that should not interfere with the normal mathematical reasoning process in any way.
In particular, the crisp subset which contains only statements that could safely occur at some level of a Tarski hierarchy will have the truth values we’d want them to have. So, there should be no complaints about the system coming to wrong conclusions, except where problematically self-referential sentences are concerned (sentences which are assigned no truth value in the least fixed point).
So; the question is: do the sentences which are assigned no truth value in Kripke’s construction, but are assigned real-numbered truth values in the fuzzy construction, play any useful role? Do they add mathematical power to the system?
For those not familiar with Kripke’s fixed points: basically, they allow us to use self-reference, but to say that any sentence whose truth value depends eventually on its own truth value might be truth-value-less (ie, meaningless). The least fixed point takes this to be the case whenever possible; other fixed points may assign truth values when it doesn’t cause trouble (for example, allowing “this sentence is true” to have a value).
If discourse about the fuzzy value of (what I would prefer to call) meaningless sentences adds anything, then it is by virtue of allowing structures to be defined which could not be defined otherwise. It seems that adding fuzzy logic will allow us to define “essentially fuzzy” structures… concepts which are fundamentally ill-defined… but in terms of the crisp structures that arise, correct me if I’m wrong, but it seems fairly clear to me that nothing will be added that couldn’t be added just as well (or, better) by adding talk about the class of real-valued functions that we’d be using for the fuzzy truth-functions.
To sum up: reasoning in this way seems to have no bad consequences, but I’m not sure it is useful...
YKY,
The problem with Kripke’s solution to the paradoxes, and with any solution really, is that it still contains reference holes. If I strictly adhere to Kripke’s system, then I can’t actually explain to you the idea of meaningless sentences, because it’s always either false or meaningless to claim that a sentence is meaningless. (False when we claim it of a meaningful sentence; meaningless when we claim it of a meaningless one.)
With the fuzzy way out, the reference gap is that we can’t have discontinuous functions. This means we can’t actually talk about the fuzzy value of a statement: any claim “This statement has value X” is a discontinuous claim, with value 1 at X and value 0 everywhere else. Instead, all we can do is get arbitrarily close to saying that, by having continuous functions that are 1 at X and fall off sharply around X… this, I admit, is rather nifty, but it is still a reference gap. Warrigal refers to actual values when describing the logic, but the logic itself is incapable of doing that without running into paradox.
“Shouldness” refers to a particular very specific way of presenting the system’s >behavior, and it’s not free energy. Notice that you can describe AI’s or man’s behavior >with physical variational principles as well, but that will have nothing to do with their >preference.
It seems to me that what SilasBarta is asking for here is a definition of shouldness such that the above statement holds. Why is it invalid to think that the system “wants” its physics? All you are indicating is that such is not what’s intended (which I’m sure SilasBarta knows)...
I agree. The idea that low-entropy pockets that form are totally immune to a simplicity prior seems unjustified to me. The universe may be in a high-entropy state, but it’s still got physical laws to follow! It’s not just doing things totally at random; that’s merely a convenient approximation. Maybe I am ignorant here, but it seems like the probability of a particular low-entropy bubble will be based on more than just its size.
The reason is simply that, in the multiple worlds interpretation, we do survive—we just also die. If we ask “Which of the two will I experience?” then it seems totally valid to argue “I won’t experience being dead.”
I agree with all of your summaries, so any readers should partially discount the many occurrences of “so far as I can tell.” :)
How do you respond to the claim that the few researchers who are working directly on very general, Solomonoff-approximating AI systems are, when put together, dangerous?
This cry of “was it ever done any other way?” strikes me as historically naive… arranged marriages happened, after all, and still happen. During certain space-time periods I understand it is/was customary to have much younger brides than grooms, in which case it seems more reasonable to surprise rather than discuss (since the groom may not have a great desire for the young bride’s opinions in the matter).
In any case, it seems the question should be answered by a historical sociologist...
“Society is addicted to X?” Alcohol, tobacco, & caffeine come to mind. Of course, our present culture is trying hard to overcome its former tobacco addiction.
Sorry for the slow response. Solomonoff, until he died recently. Marcus Hutter is implementing an AIXI approximation last I heard. Eray Ozkural, implementing Solomonoff’s ideas. Sergey Pankov, implementing an AIXI approximation.
Apologies!
It’s an interesting experience to learn formal logic and then take a higher-level math class (any proof-intensive topic). During the process of finding a proof, we ask all sorts of questions of the form “does that imply that?”. However, since we’re typically proving something which we already know is a theorem, we could logically answer: “Yes: any two true statements imply one another, and both of those statements are true.” This is a silly and unhelpful reply, of course. One way of seeing why is to point out that although we may already be willing to believe the theorem, we are trying to construct an argument which could increase the certainty of that belief; hence, the direction of propagation is towards the theorem, so any belief we may have in the theorem cannot be used as evidence in the argument.
What do you think, am I overstepping my bounds here? I feel like the probabilistic case gives us something more. In classical logic, we either believe the statement already or not; we don’t need to worry about counting evidence twice because evidence is either totally convincing or not convincing at all.
I actually cited the Wolfram article because I preferred it, but I went ahead and added a link to the wikipedia article for those whose taste is closer to yours! Thanks.
The Risch algorithm for symbolic integration is what first gave me a hunger to learn “the actually good ways of doing things” in this respect, and a sense that I might have to search for them beyond the classroom. However, I never did learn to use the Risch algorithm itself! I don’t really know whether it turns out to be good for human use.
Referring to belief propagation? The actual procedure does something really simple: ignore the problem. Experimentally, this approach has shown itself to be very good in a lot of cases. Very little is known about what determines how good an approximation this is, but if I recall correctly, it’s been proven that a single loop will always converge to the correct values; it’s also been proven that if all the local probability distributions are Gaussian, then the estimated means will also converge correctly, but the variances might not.
Many things can be done to improve the situation, too, but I’m not “up” on that at the moment.
I’ll second and constraint-solving algorithms; specifically, the variable-ordering heuristics seem helpful to me. Choose the variables which most constraint the problem first! Note, the constraint propagation step is another instance of the sum-product algorithm. :)
Algorithms I find useful that I didn’t put in the article:
--Find decent solutions to packing problems by packing the largest items first, then going down in order of size
--Minimum-description-length ideas (no surprise to rationalists! Just occam’s razor)
--Binary search (just for finding a page in a book, but still, learning the algorithm actually improved my speed at that task :p)
--Exploration vs exploitation trade-off in reinforcement learning (I can’t say I’m systematic about it, but learning the concept made me aware that it is sometimes rational to take actions which seem suboptimal from what I know, just to see what happens)
I suspect your intuition about computer programming is also based on the way it forces a certain amount of clear thinking to be done.
The following argument comes from an intro sociology text:
If there are three people competing, all of different strengths, it is worthwhile for the two weakest people to ban together to defeat the strongest person. This takes out the largest threat. (Specific game-theoretic assumptions were not stated.)
Doesn’t this basically explain the phenomenon? If Zug kills Urk, I might be next! So I should ban together with Urk to defeat Zug. Even if Urk doesn’t reward me at all for the help, my chances against Urk are better than my chances against Zug. (Under certain assumptions.)