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Zed comments on Take heed, for it is a trap - Less Wrong
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Comments (187)
You're wrong in labeling me a level-0 rationalist, for I have never believed in that fallacy. I'm familiar with MML, Kolmogorov complexity, and so forth.
What I meant is that given a proposition P, and its negation non-P, that if both have the same level of complexity and no evidence is provided for either then the same probability must be assigned to both (because otherwise you can be exploited, game theory wise).
The unplausbility of a proposition, measured in complexity, is just the first piece of counter-evidence that it must overcome.
If you don't know what the proposition is, but just that there IS a proposition named P, then there is no way to calculate its complexity. But even if you don't know what the statement is and if you don't know what its complexity is and even if you don't know what kind of evidence there is that supports it you must still be willing to assign a number to it. And that number is 50%.
Okay, it seems to me we're simply talking about different things. "Any statement" communicated to me that the particulars of the statements must be taken into account, just not the evidential context. So when you say "any statement starts out with a 50% probability of being true", this communicated to me that you mean the probability AFTER the sentence's complexity has been calculated.
But you basically meant what I'd have understood by "truth-claim of any unknown statement*". In short not evaluating the statement "P" but rather to evaluate "The unknown statement P is true".
At this point your words improve to being from "simply wrong" to "most massively flawed in their potential for miscommunication" :-)
I think you were too convinced I was wrong in your previous message for this to be true. I think you didn't even consider the possibility that complexity of a statement constitutes evidence and that you had never heard the phrasing before. (Admittedly, I should have used the words "total ignorance", but still)
Your previous post strikes me as a knee-jerk reaction. "Well, that's obviously wrong". Not as an attempt to seriously consider under which circumstances the statement could be true. You also incorrectly claimed I was an ignoramus rationalist (for which you didn't apologize) which only provides further evidence you didn't really think before you started writing your critique (because who seriously considers the opinions of an ignoramus?).
And now, instead of just saying "Oops" you shift the goalpost from "false no matter what way you look at it" to something fuzzy where we're simply talking about different things.
This is blatant intellectual dishonesty.
You are probably right, but I would suggest you to phrase your reaction less combatively. Especially the last sentence is superfluous; it doesn't contain any information and only heats up the debate.
"Any proposition starts out with a 50% probability of being true" is still utterly wrong. Because "any" indicates multiplicity. At the point where you claim these proposition "begin", they aren't even differentiated into different propositions; they're nothing but the abstraction of a letter P as in "the unknown proposition P".
I've conceded that you were instead talking about an abstraction of statements, not any actual statements. At this point, if you want to duel it out to the end, I will say that you failed at using language in order to communicate meaning, and you were abstracting words to the point of meaninglessness.
edit to add: And as a sidenote, even unknown statements can't be divided into 50% chance of truth and 50% falsehood, as there's always the chances of self-referential contradiction (e.g. the statement "This statement is wrong", which can never be assigned a True/False value), self-referential validity (e.g. The statement "This statement is true", which can be assigned either a true or false value), confusion of terms (e.g. The statement "A tree falling in the woods makes a sound." which depends on how one defines a "sound".), utter meaninglessness ("Colorless green ideas sleep furiously") etc, etc.
Now that I've cooled off a bit, let me state in detail my complaint against this comment of yours.
You seem to be asking for the highest amount of charity towards your statements. To the point that I ought strive for many long minutes to figure out a sense in which your words might be correct, even if I'd have to fix your claim (e.g. turn 'statement' into 'proposition' -- and add after 'any proposition starts out' the parenthetical 'before it is actually stated in words') before it actually becomes correct.
But in return you provide the least amount of charity towards my own statements: I kept using the word "seems" in my original response to you (thus showing it may just be a misunderstanding) and I did NOT use the word 'ignoramus' which you accuse me of claiming you to be -- I used the term 'Level-0 rationalist'. You may think it's okay to paraphrase Lesswrong beliefs to show how they might appear to other people, but please don't paraphrase me and then ask for an apology for the words you put in my mouth. That's a major no-no. Don't put words in my mouth, period.
No, I did not apologize for calling you a Level-0 rationalist; I still do not apologize for putting you in that category, since that's where your badly chosen words properly assigned you (the vast majority of people who'd say something like "all statements begin with a 50% probability" would truly be Level-0), NOR do I apologize for stating I had placed you in that category: would you prefer if everyone here had just downvoted your article instead of giving you a chance to clarify that (seemingly) terribly wrong position first?
Your whole post was about how badly communicated beliefs confer us low status in the minds of others. It was only proper that I should tell you what a status you had achieved in my mind.
I don't consider you a Level-0 rationalist anymore. But I consider you an extremely low-level communicator.
I'm not sure why you'd assume that the MML of a random proposition is only one bit...
A complex proposition P (long MML) can have a complex negation (also with long MML) and you'd have no reason to assume you'd be presented with P instead of non-P. The positive proposition P is unlikely if its MML is long, but the proposition non-P, despite its long MML is then likely to be true.
If you have no reason to believe you're more likely to be presented with P than with non-P, then my understanding is that they cancel each other out.
But now I'm not so sure anymore.
edit: I'm now pretty sure again my initial understanding was correct and that the counterarguments are merely cached thoughts.
I think often "complicated proposition" is used to mean "large conjunction" e.g. A&B&C&D&...
In this case its negation would be a large disjunction, and large disjunctions, while in a sense complex (it may take a lot of information to specify one) usually have prior probabilities close to 1, so in this case complicated statements definitely don't get probability 0.5 as a prior. "Christianity is completely correct" versus "Christianity is incorrect" is one example of this.
On this other hand, if by 'complicated proposition' you just mean something where its truth depends on lots of factors you don't understand well, and is not itself necessarily a large conjunction, or in any way carrying burdensome details, then you may be right about probability 0.5. "Increasing government spending will help the economy" versus "increasing government spending will harm the economy" seems like an example of this.
My claim is slightly stronger than that. My claim is that the correct prior probability of any arbitrary proposition of which we know nothing is 0.5. I'm not restricting my claim to propositions which we know are complex and depend on many factors which are difficult to gauge (as with your economy example).
I think I mostly agree. It just seemed like the discussion up to that point had mostly been about complex claims, and so I confined myself to them.
However, I think I cannot fully agree about any claim of which we know nothing. For instance, I might know nothing about A, nothing about B|A, and nothing about A&B, but for me to simultaneously hold P(A) = 0.5, P(B|A) = 0.5 and P(A&B) = 0.5 would be inconsistent.
"B|A" is not a proposition like the others, despite appearing as an input in the P() notation. P(B|A) simply stands for P(A&B)/P(A). So you never "know nothing about B|A", and you can consistently hold that P(A) = 0.5 and P(A&B) = 0.5, with the consequence that P(B|A) = 1.
The notation P(B|A) is poor. A better notation would be P_A(B); it's a different function with the same input, not a different input into the same function.
Fair enough, although I think my point stands, it would be fairly silly if you could deduce P(A|B) = 1 simply from the fact that you know nothing about A and B.
Well, you can't -- you would have to know nothing about B and A&B, a very peculiar situation indeed!
EDIT: This is logically delicate, but perhaps can be clarified via the following dialogue:
-- What is P(A)?
-- I don't know anything about A, so 0.5
-- What is P(B)?
-- Likewise, 0.5
-- What is P(C)?
-- 0.5 again.
-- Now compute P(C)/P(B)
-- 0.5/0.5 = 1
-- Ha! Gotcha! C is really A&B; you just said that P(A|B) is 1!
-- Oh; well in that case, P(C) isn't 0.5 any more: P(C|C=A&B) = 0.25.
As per my point above, we should think of Bayesian updating as the function P varying, rather than its input.
I believe that this dialogue is logically confused, as I argue in this comment.
This is the same confusion I was originally having with Zed. Both you and he appear to consider knowing the explicit form of a statement to be knowing something about the truth value of that statement, whereas I think you can know nothing about a statement even if you know what it is, so you can update on finding out that C is a conjunction.
Given that we aren't often asked to evaluate the truth of statements without knowing what they are, I think my sense is more useful.
If you know nothing of A and B then P(A) = P(B) = 0.5, P(B|A) = P(A|B) = 0.5 and P(A & B) = P(A|B) * P(B) = 0.25
You do know something of the conjunction of A and B (because you presume they're independent) and that's how you get to 0.25.
I don't think there's an inconsistency here.
How do you know something about the conjunction? Have you manufactured evidence from a vacuum?
I don't think I am presuming them independent, I am merely stating that I have no information to favour a positive or negative correlation.
Look at it another way, suppose A and B are claims that I know nothing about. Then I also know nothing about A&B, A&(~B), (~A)&B and (~A)&(~B) (knowledge about any one of those would constitute knowledge about A and B). I do not think I can consistently hold that those four claims all have probability 0.5.
If you know nothing about A and B, then you know something about A&B. You know it is the conjunction of two things you know nothing about.
Since A=B is a possibility the uses of "two things" here is bit specious. You're basically saying you know A&B but that could stand for anything at all.
Yeah, and I know that A is the disjunction of A&B and A&(~B), and that it is the negation of the negation of a proposition I know nothing about, and lots of other things. If we reading a statement and analysing its logical consequences to count as knowledge then we know infinitely many things about everything.
In that case it's clear where we disagree because I think we are completely justified in assuming independence of any two unknown propositions. Intuitively speaking, dependence is hard. In the space of all propositions the number of dependent pairs of propositions is insignificant compared to the number of independent pairs. But if it so happens that the two propositions are not independent then I think we're saved by symmetry.
There are a number of different combinations of A and ~A and B and ~B but I think that their conditional "biases" all cancel each other out. We just don't know if we're dealing with A or with ~A, with B or with ~B. If for every bias there is an equal and opposite bias, to paraphrase Newton, then I think the independence assumption must hold.
Suppose you are handed three closed envelopes each containing a concealed proposition. Without any additional information I think we have no choice but to assign each unknown proposition probability 0.5. If you then open the third envelope and if it reads "envelope-A & envelope-B" then the probability of that proposition changes to 0.25 and the other two stay at 0.5.
If not 0.25, then which number do you think is correct?
Okay, in that case I guess I would agree with you, but it seems a rather vacuous scenario. In real life you are almost never faced with the dilemma of having to evaluate the probability of a claim without even knowing what that claim is, it appears in this case that when you assign a probability of 0.5 to an envelope you are merely assigning 0.5 probability to the claim that "whoever filled this envelope decided to put a true statement in".
When, as in almost all epistemological dilemmas, you can actually look at the claim you are evaluating, then even if you know nothing about the subject area you should still be able to tell a conjunction from a disjunction. I would never, ever apply the 0.5 rule to an actual political discussion, for example, where almost all propositions are large logical compounds in disguise.
This can't be right. An unspecified hypothesis can be as many sentence letters and operators as you like, we still don't have any information about it's content and so can't have any P other than 0.5. Take any well-formed formula in propositional logic. You can make that formula say anything you want by the way you assign semantic content to the sentence letters (for propositional logical, not the predicate calculus where can specify indpendence). We have conventions where we don't do silly things like say "A AND ~B" and then have B come out semantically equivalent to ~A. It is also true that two randomly chosen hypotheses from a large set of mostly independent hypotheses are likely to be independent. But this is a judgment that requires knowing something about the hypothesis: which we don't, by stipulation. Note, it isn't just causal dependence we're worried about here: for all we know A and B are semantically identical. By stipulation we know nothing about the system we're modeling- the 'space of all propositions' could be very small.
The answer for all three envelopes is, in the case of complete ignorance, 0.5.
The number of possible probability distributions is far larger than the two induced by the belief that P, and the belief that ~P.
If at this point you don't agree that the probability is 0.5 I'd like to hear your number.
P(A) = 2^-K(A).
As for ~A, see: http://lesswrong.com/lw/vs/selling_nonapples/ (The negation of a complex proposition is much vaguer, and hence more probable (and useless))