potato comments on Science as Attire - Less Wrong

48 Post author: Eliezer_Yudkowsky 23 August 2007 05:10AM

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Comment author: Eliezer_Yudkowsky 23 August 2007 05:54:26AM 24 points [-]

The probability for anything is non-zero. But when we see something to which our hypothesis assigns a sufficiently infinitesimal probability, we call it falsified - because even maximum entropy does better.

Comment author: potato 15 November 2011 12:32:48AM 0 points [-]

What about the probability of "2+2=4 and not 2+2=4." Pretty sure that's zero, and if not we got some problems I believe.

Comment author: potato 15 November 2011 07:28:29PM 0 points [-]

So is the conjunction of two contradictory phenomena not zero? I am confused. I believe if that is so the rest of bayes falls apart, no? Bayes requires that you give zero probabilities to contradictions, if you do not then you can be dutch booked, right? It also requires that you give a probability of one to logical tautologies, if you give more or less, then you can also be dutch booked. What am I messing up? Really, please help if you understand.

Comment author: dlthomas 15 November 2011 07:40:03PM * 7 points [-]

I expect the problem is not that you are wrong (that's more or less open), but that there has been similar discussion in many places (one is here) on this site and building another tree with pretty much the same starting point doesn't really make sense.

Comment author: potato 15 November 2011 09:04:00PM * 2 points [-]

Ah i see. Thanks

(edit): shoud i just take my posts down?

Comment author: orthonormal 08 April 2012 06:27:41PM 4 points [-]

(edit): shoud i just take my posts down?

No— it's more helpful if you edit your first comment to say "OK, this is already discussed here", so that if anyone else reads Eliezer's comment and has the same objection as you, they know where to go to discuss it rather than opening another instance of the same comment thread...

Comment author: Odinn 04 December 2013 09:16:09AM -1 points [-]

Even the simplest expression such as 2+2=4 should not be literally tautological. There is an infinitesimal possibility that the human brain has a fundamental flaw that causes us to read the expression incorrectly, or that every person or program that has ever attempted to calculate the sum of 2 and 2 has erroneously provided an incorrect answer, or that our universe is actually configured in a way that isn't mathematically accurate (despite what those pictures of apples in first grade textbooks claim.) Conjugate all of these extremely strange possibilities and your odds of 2 and 2 not equaling 4 are so small, so imaginary, that we can confidently and adequately use probability 1 for 2+2=4, but a perfect Bayesian formula still has no room for entries of P=1 or 0, especially since the formula just doesn't provide useful data that way.

Comment author: [deleted] comment score below threshold [+] (17 children)

Comment author: potato 08 April 2012 05:50:57PM 1 point [-]

Any (A & ~A) works just as well. How can you assign greater than zero probability to a contradiction? Doesn't the whole system fall apart? Same thing for less than one probability to a tautology. If there are probability theories that do that, I would like to know of them.

Comment author: [deleted] comment score below threshold [+] (15 children)

Comment author: Polymeron 08 April 2012 06:28:43PM 2 points [-]

Now I have seen some interesting papers that make expanded probability theories that include 0 and 1 as logical falsehood and truth respectively. But that still does not include a special value for contradictions.

Except, contradictions really are the only way you can get to logical truth or falsehood; anything other than that necessarily relies on inductive reasoning at some point. So any probability theory employing those must use contradictions as a means for arriving at these values in the first place.

I do think that there's not much room for contradictions in probability theories trying to actually work in the real world, in the sense that any argument of the form A->(B & ~B) also has to rely on induction at some point; but it's still helpful to have an anchor where you can say that, if a certain relationship does exist, then a certain proposition is definitely true.

(This is not like saying that a proposition can have a probability of 0 or 1, because it must rely, at least somewhere down the line, on another proposition with a probability different from 0 and 1).

Comment author: orthonormal 08 April 2012 06:30:31PM 1 point [-]

I think you and potato are talking about different things; potato's criticism is close to what's discussed in this post.

But for what it's worth, Bayesians with finite (and flawed) computational powers can meaningfully assign probabilities to mathematical statements, and update as they prove more theorems.

Comment author: potato 08 April 2012 06:40:47PM 0 points [-]

More over, we would make little progress in prime analysis if we could not use probability theory to restrict expected experience in purely formal environments.

Comment author: potato 08 April 2012 06:31:01PM * 2 points [-]

Um, you can derive from Komolgorov that (A & ~A) has prob zero. Very easily.

If (A & B) = nullset, then P(A & B) = 0.
(A & ~A) = nullset
so: P(A & ~A) = 0.

In other words, if two sets are disjoint, the probability of their intersection is zero. Any set and its compliment are disjoint, so the probability of their conjunction is zero.

It also helps if you think of the boolean variables as fractions of a dart board, and the probability as the area of that fraction. The formalism is perfectly isomorphic. Obviously, the intersection of any fraction and its compliment will have an area of zero.

See, I believe I have a proof that the probability of any contradiction is zero. So I am going to have to ask for a proof to change my mind, or a problem with my proof (which I doubt, it's very short, and I math often).

Comment author: [deleted] 08 April 2012 06:37:32PM * -1 points [-]

Oh I see, you are talking of exclusive outcomes, not contradictions. Yes you are entirely right, exclusive outcomes work exactly that way. The probability of both Heads and Tails occuring at the same time on a coinflip is zero.

Contradictions in a system isomorphic with Propositional Logic do not and are an entirely separate mathematical object.

Comment author: potato 08 April 2012 06:42:53PM * 0 points [-]

I would like to read more on that, because I believed them to be exactly equivalent.

A set's intersection with its compliment
has a perfect isomorphism with
A propositions conjunction with its negation.

Comment author: [deleted] 08 April 2012 07:12:20PM -2 points [-]

The Principle of explosion tells us that a if one supposes both truth and falsehood as base premises, one can derive any conclusion, by means of Propositional Logic theorems.

The intersection of a set with it's complement is the empty set.

A consistent isomorphism of ZF Set theory and Propositional Logic is achieved by letting truth be a non-empty set and falshood be an empty set. Then intersection becomes logical conjunction and complement with respect to the truth set becomes logical inversion.

Now, the reason I can argue for such things is actually backed up by Godel's incompleteness theorem. Propositional Logic and ZF Set Theory can both implement Peano Arithmetic, Bayesian Probability cannot. Thus Propositional Logic and ZF Set Theory are Complete but not Consistent, while Bayes is Consistent but not Complete.

Comment author: potato 08 April 2012 07:37:58PM * 0 points [-]

argue for what things? I have no clue what the POE or Gtheorem have to do with komologrov provably assigning zero probability to contradictions.

What is the difference between exclusive outcomes and contradictions? How are they not the same mathematical object? If two exclusive outcomes end up resulting, you can also explode the universe.

For A to be exclusive with B, means that if A happened B did not. So: If A and B, then ~B and B, then B or P, then ~B, so P. The POE is not something unique to contradictions which exclusive outcomes lack.

continue this thread »

Comment author: [deleted] 08 April 2012 07:10:47PM * 1 point [-]

ISTM that you're using the word contradiction in a non-standard way: in the usual sense they are logical falsehoods. What do you actually mean? (ETA: I guess paradoxes such as “This sentence is false”?)

Comment author: [deleted] 08 April 2012 07:14:31PM 0 points [-]

I use contradiction in the completely ordinary sense as seen in propositional logic. (P & ~P)

Comment author: [deleted] 08 April 2012 07:18:05PM 1 point [-]

How is that not a logical falsehood?

Comment author: [deleted] 08 April 2012 08:27:28PM 0 points [-]

It is not a logical falsehood for several reasons. What I actually mean by using the traditional notation of propositional calculus is that the statement A is a true statement. Were it a false statement I would write ~A. Similarly i write (P & ~P) to mean "It is true that both P and not P" while I write ~(P & ~P) to mean "It is true that not both P and not P."

Solving the latter as the equation ~(P & ~P) = TRUE for the variable P gives the trivial solution set of {TRUE, FALSE}, solving the former equation (P & ~P) = TRUE for the variable P gives the empty solution set {}

This is simple convention of notation, I am sorry if that wasn't clear. Yes, evaluating the logic arithmetic statement (P & ~P) for any given boolean value of P yields false.

Comment author: potato 08 April 2012 07:40:21PM 0 points [-]

Logical falsehoods, and disjoint events, all reduce to contradictions in boole. There is no difference in propositional logic. Where are you getting this from?

Comment author: [deleted] 08 April 2012 08:30:01PM 0 points [-]

See this comment

Comment author: orthonormal 08 April 2012 06:23:12PM 2 points [-]

Short answer: I can never assign probability exactly 0, because there's a tiny chance that I'm currently hallucinating the "obvious" contradiction, etc.

Longer answer: How To Convince Me That 2+2=3, and my comment on it.

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