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gworley comments on GroupThink, Theism ... and the Wiki - Less Wrong
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Interesting thoughts.
I have tried in the past to follow the mathematical definitions I learned regarding rationality. Unfortunately they haven't always served well in discussions. They are:
Rationality: the application of Bayes rule
Well-informed: the possession of evidence with correct confidence
Prudence: the result of combining rationality with well-informed-ness
But, like I said, this doesn't work well in normal speech because each of those words doesn't have such a clean and precise meaning in the natural language. Instead, if we must choose words so that we may speak with mathematical precision, I propose we don't spend time choosing the right word to impose our definition on and instead follow a simple rule that has helped many software developers avoid time wasting conflicts: a construct will be given the worst name someone thinks up. The only way you can change the name of a construct is to propose a worse name. It stops the fighting about choosing the right word and let's us all get on with the discussion.
Not sure how well that will work here, but I thought it was worth mentioning.
Rationality is surely bigger than Bayes - since it incudes deductive reasoning.
Well solomonoff induction and systems like AIXI are bigger than Bayes as they use it as a part of themselves. They are intractable.
And I'd guess there's a link between those and rationality. Epistemical and instrumental rationality respectively, pushed to their theoretical limits of optimality.
this can be viewed the other way around, deductive reasoning as a special case of Bayes
Exactly: the special case where the conditional probabilities are (practically) 0 or 1.
yes, exactly
And induction is a special case of deduction, since probability theory itself is a logic with theorems: what a given prior updates to, on given evidence, is a deductive mathematical fact.
Besides, I'm informed that I just use duction.
No, see:
http://en.wikipedia.org/wiki/Problem_of_induction
Tim--- To resolve your disagreement: Induction is not purely about deduction, but it nevertheless can be completely modelled by a deductive system.
More specifically, I agree with your claim about induction (see point 4 above). However, in defense of Eliezer's claim that induction is a special case of deduction, I think you can model it in a deductive system even though induction might require additional assumptions. For one thing, deduction in practice seems to me to require empirical assumptions as well (i.e., the "axioms" and "inference rules" are chosen based on how right they seem), so the fact that induction needs some axioms should not itself prevent deductive style proofs using an appropriately formalized version of it. So, once one decides on various axioms, such as the various desiderata I list above for a Solomonoff-like system, you CAN describe via a mathematical deduction system how the process of induction would proceed. So, induction can be formalized and proofs can be made about the best thing for an agent to do; the AIXI model is basically an example of this.
If that is a defense of induction being a special case of deduction, then it's a defense of anything being a special case of deduction - since logic can model anything.
The golden gate bridge is a special case of deduction, in this sense.
I am not impressed by the idea that induction is a special case of deduction - I would describe it as being wrong. You need extra axioms for induction. It is not the same thing at all.
Yes, the golden gate bridge is a special case of deduction in the sense meant here. I have no problem with anything in your comment, I think we agree.
Induction tells us whether something is probable; based on past experience we can make a prediction about the future. But to apply induction to decide something is a deduction:
First, make the assumption that induction can be applied to infer truth. Then, apply induction. The result is a valid conclusion deduced using (1) induction and (2) the belief that you can use induction.
To recap... induction is not a purely deductive principle - since it relies on an axiom known as "The Principle of Uniformity of Nature" - http://en.wikipedia.org/wiki/Principle_of_uniformity which states that the laws of physics are the same from place to place and that the past is a useful guide to the future.
That axiom is not available as a result of any deduction - and attempts to justify it always seem to be circular - i.e. they use induction.
According to http://en.wikipedia.org/wiki/Problem_of_induction#Ancient_origins this problem has been known about for over 2,000 years.
It looks to me like those uniformity of nature principles would be nice but that induction could still be a smart thing to do despite non-uniformity. We'd need to specify in what sense uniformity was broken to distinguish when induction still holds.
Right. In case anyone thinks this thread is an argument, it's not -- the assumption of induction would need to be added to deduce anything about the empirical world. The definition above didn't say how deductions would be made... You just make assumptions and then keep track of what your conclusions would be given those assumptions (that's deduction). I'm not sure if we could or would start listing the assumptions. I made the mistake of including (1), which is the only explicit assumption, but AndySimpson and ALexU have pointed out that elevating that assumption is empiricism.
By "Bayes" I meant this: http://en.wikipedia.org/wiki/Bayes'_theorem - a formalisation of induction.
If you think "Bayes" somehow includes deductive reasoning, can you explain whether it supposedly encapsulates first-order logic or second-order logic?
I think we're probably using some words differently, and that's making you think my claim that deductive reasoning is a special case of Bayes is stronger than I mean it to be.
All I mean, approximately, is:
Bayes theorem: p(B|A) = p(A|B)*p(B) / p(A)
Deduction : Consider a deductive system to be a set of axioms and inference rules. Each inference rule says: "with such and such things proven already, you can then conclude such and such". And deduction in general then consists of recursively turning the crank of the inference rules on the axioms and already generated results over and over to conclude everything you can.
Think of each inference rule "i" as i(A) = B, where A is some set of already established statements and B corresponds to what statements "i" let's you conclude, if you already have A.
Then, by deduction we're just trying to say that if we have generated A, and we have an inference rule i(A) = B, then we can generate or conclude B.
The connection between deduction and Baye's is to take the generated "proofs" of the deductive system as those things to which you assign probability of 1 using Bayes.
So, the inference rule corresponds to the fact that p(B | A) = 1. The fact that A has been already generated corresponds to p(A) = 1. Also, since A has already been generated independently of B, p(A | B) = 1, since A didn't need B to be generated. And we want to know what p(B) is.
Well, plugging into Bayes:
p(B|A) = p(A|B)*p(B) / p(A) i.e. 1 = 1 * p(B) / 1 i.e. p(B) = 1.
In other words, B can be generated, which is what we wanted to show.
So basically, I think of deductive reasoning as just reasoning with no uncertainty, and I see that as popping out of bayes in the limiting case. If a certain formal interpretation of this leads me into Godelian problems, then I would just need to weaken my claim somewhat, because some useful analogy is clearly there in how the uncertain reasoning of Bayes reduces to certain conclusions in various limits of the inputs (p=0, p=1, etc.).
I think I would describe what you are talking about as being Bayesian statistics - plus a whole bunch of unspecified rules (the "i" s).
What I was saying is that there isn't a standard set of rules of deductive reasoning axioms that is considered to be part of Bayesian statistics. I would not dispute that you can model deductive reasoning using Bayesian statistics.
Tim-Good, your distinction sounds correct to me.