Let me ask you in reply, Paul, if you think you would refuse to change your mind about the "law of non-contradiction" no matter what any mathematician could conceivably say to you - if you would refuse to change your mind even if every mathematician on Earth first laughed scornfully at your statement, then offered to explain the truth to you over a couple of hours... Would you just reply calmly, "But I know I'm right," and walk away? Or would you, on this evidence, update your "zero probability" to something somewhat higher?

Why can't I repose a very tiny credence in the negation of the law of non-contradiction? Conditioning on this tiny credence would produce various null implications in my reasoning process, which end up being discarded as incoherent - I don't see that as a killer objection.

In fact, the above just translates the intuitive reply, "What if a mathematician convinces me that 'snow is white' is both true and false? I don't consider myself entitled to rule it out absolutely, but I can't imagine what else would follow from that, so I'll wait until it happens to worry about it."

As for Descartes's little chain of reasoning, it involves far too many deep, confusing, and ill-defined concepts to be assigned a probability anywhere near 1. I am not sure anything exists, let alone that I do; I am far more confident that angular momentum is conserved in this universe than I am that the statement "the universe exists" represents anything but confusion.

The one that I confess is giving me the most trouble is P(A|A). But I would prefer to call that a syntactic elimination rule for probabilistic reasoning, or perhaps a set equality between events, rather than claiming that there's some specific proposition that has "Probability 1".

Eli said:

Peter de Blanc has an amusing anecdote on this point, which he is welcome to retell in the comments.

Here's the anecdote.

Gray Area said: "Amusingly, this is one of the more controversial tautologies to bring up. This is because constructivist mathematicians reject this statement."

Actually constructivist mathematicians reject the law of the excluded middle, (P v ~P), not the law of non-contradiction (they are not equivalent in intuitionistic logic, the law of non-contradiction is actually equivalent to the double negation of the excluded middle).

We can go even stronger than mathematical truths. How about the following statement?

~(P &~P)

I think it's safe to say that if anything is true, that statement (the flipping law of non-contradiction) is true. And it's the precondition for any other knowledge (for no other reason than if you deny it, you can prove anything). I mean, there are logics that permit contradictions, but then you're in a space that's completely alien to normal reasoning.

So that's lots stronger than 2+2=4. You can reason without 2+2=4. Maybe not very well, but you can do it.

So Eliezer, do you have a probability of 1 in the law of non-contradiction?

Ben, you're making an obvious error: you are taking the statement that "P never equals 1" has a probability of less than 1 to mean that in some proportion of cases, we expect the probability to equal 1. This would be the same as supposing that assigning the light-speed limit a probability of less than 1 implies that we think that the speed of light is sometimes exceeded.

But it doesn't mean this, it means that if we were to enunciate enough supposed physical laws, we would sometimes be mistaken. In the same way, a probability of less than 1 for the proposition that we should never assign a probability of 1 simply means that if we take enough supposed claims regarding mathematics, logic, and probability theory, each of which we take to be as certain as the claim rejecting a probability of unity, we would sometimes be mistaken. This doesn't mean that any proposition has a probability of unity.

Thanks, Eliezer. Helpful post.

I have personally witnessed a room of people nod their heads in agreement with a definition of a particular term in software testing. Then when we discussed examples of that term in action, we discovered that many of us having agreed with the words in the definition, had a very different interpretation of those words. To my great discouragement, I learned that agreeing on a sign is not the same as agreeing on the interpretant or the object. (sign, object, and interpretant are the three parts of Peirce's semiotic triangle)

In the case of 2+2=4, I think I know what that means, but when Euclid, Euler, or Laplace thought of 2+2=4, were they thinking the same thing I am? Maybe they were, but I'm not confident of that. And when someday a artificial intelligence ponders 2+2=4, will it be thinking what I'm thinking?

I feel 100% positive that 2+2=4 is true, and 100% positive that I don't entirely know what I mean by "2+2=4". I am also not entirely sure what other people mean by it. Maybe they mean "any two objects, combined with two objects, always results in four objects", which is obviously not true.

In thinking about certainty, it helps me to consider the history of the number zero. That something so obvious could be unknown (or unrecognized as important) for so long is sobering. The Greeks would also have sworn that the square root of negative one has no meaning and certainly no use in mathematics. 100% certain! The Pythagoreans would have sworn it just before stoning you to death for math heresy.

Also (and sorry for the rapid-fire commenting), do you accept that we can have conditional probabilities of one? For example, P(A|A)=1? And, for that matter, P(B|(A-->B, A))=1? If so, I believe I can force you to accept at least probabilities of 1 in sound deductive arguments. And perhaps (I'll have to think about it some more) in the logical laws that get you to the sound deductive arguments. I'm just trying to get the camel's nose in the tent here...

Why is the uncertainty fetish so appealing that people will entertain such weird ideas to retain it?

Why is the certainty fetish so appealing that people will ignore the obvious fact that all conclusions are contingent?

Mr. Bach,

I think you're right to point out that "number" meant a different thing to the Greeks; but I think that should make us more, not less, confident that "2+2=4." If the Greeks had meant the same thing by number as modern mathematicians do, than they were wrong to be very confident that the square root of negative one was not a number. However, the square root of negative one does in fact fall short of being a simple, definite multitude -- what Euclid, at least, meant by number. So if they were in error, it was the practical error of drawing an unnecessary distinction, not a contradictory one.

Perhaps "100% certain" or "P=1" could mean that I believe something to be true with the same level of certainty as that by which I believe certainty and probability to be coherent terms. We can only evaluate judgments if we accept "judgment" as a valid kind of thought anyway.

If you say 99.9999% confidence, you're implying that you could make one million equally fraught statements, one after the other, and be wrong, on average, about once.

Excellent post overall, but that part seems weakest - we suffer from an unavailability problem, in that we can't just think up random statements with those properties. When I said I agreed 99.9999% with "P(P is never equal to 1)" it doesnt't mean that I feel I could produce such a list - just that I have a very high belief that such a list could exist.

An intermediate position would be to come up with a hundred equally fraught statements in a randomly chosen narrow area, and extrapoltate from that result.

If you get past that one, I'll offer you another.

"There is some entity [even if only a simulation] that is having this thought." Surely you have a probability of 1 in that. Or you're going to have to answer to Descartes's upload, yo.

I don't think you could get up to 99.99% confidence for assertions like "53 is a prime number". Yes, it seems likely, but by the time you tried to set up protocols that would let you assert 10,000 independent statements of this sort - that is, not just a set of statements about prime numbers, but a new protocol each time - you would fail more than once.

If you forced me to come up wit 10,000 statements I knew to >=99.99% I would find it easy, given sufficient time. Most of them would be probability much much more than 99.99% however.

Here is a sample of the list: I am not the Duke of Edinburgh. Ronald Mcdonald is not on my roof I am not currently in a bath I am currently making a list of things I believe are highly likely Eliezer Yudowsky is not a paperclip maximising AI I am not the 10,000th sentient being ever to have existed. The Queen is not a cockerspaniel in disguise. I am not a P-zombie.

53 has no prime factors other than itself. (this is much greater certainty; as I can hold in my mind the following facts "the root of 53 is less than 8. 53 is not in the 7 times table. 53 is not in the 5 times table. 53 is not in the 3 times table and 53 is odd" simultaneously. For 53 not to be prime would require, as for 2+2 not to equal 4, that I be very insane. My probability of being that insane is less than 1 in 10,000, and of having that specific insanity is lower still.)

The difficult part is in finding 10,000 statements with precisely 1 in 10,000 odds; not finding 10,000 statements with less than 1 in 10,000 odds.

The ratio of these two probabilities may be 1, but I deny that there's any actual probability that's equal to 1. P(|) is a mere notational convenience

I'd have to diagree with that. The axioms I've seen of probabilty/measure theory do not make the case that P() is a probability while P(|) is not - they are both, ulitmately, the same type of object (just taken from different measurable sets).

However, you don't need to appeal to this type of reasoning to get rid of P(A|A) = 1. Your probability of correctly remembering the beginning of the statement when reaching the end is not 1 - hence there is room for doubt. Even your probability of correctly understanding the statement is not 1.

P(P is never equal to 1) = ?

I know, I know, 'this statement is not true'.

Would this be an argument for allowing "probabilities of probabilities"? So that you can assign 99.9999% (that's enough 9's I feel) to the statement "P(P is never equal to 1)".

P(P is never equal to 1) = ?

I know, I know, 'this statement is not true'. But we've long since left the real world anyway. However, if you tell me the above is less than one, that means that in some cases, infinite certainty can exist, right?

Get some sleep first though Eliezer and Paul. It's 9.46am here.

Huh, I must be slowed down because it's late at night... P(A|A) is the simplest case of all. P(x|y) is defined as P(x,y)/P(y). P(A|A) is defined as P(A,A)/P(A) = P(A)/P(A) = 1. The ratio of these two probabilities may be 1, but I deny that there's any actual probability that's equal to 1. P(|) is a mere notational convenience, nothing more. Just because we conventionally write this ratio using a "P" symbol doesn't make it a probability.

Good point about infinite certainty, poor example.

Assert 99.9999999999% confidence, and you're taking it up to a trillion. Now you're going to talk for a hundred human lifetimes, and not be wrong even once?

Leaky induction. Didn't that feel a little forced?

evidence that convinced me that 2 + 2 = 4 in the first place.

"(the sum of) 2 + 2" means "4"; or to make it more obvious, "1 + 1" means "2". These aren't statements about the real world*, hence they're not subject to falsification, they contain no component of ignorance, and they don't fall under the purview of probability theory.

*Here your counter has been that meaning is in the brain and the brain is part of the real world. Yet such a line of reasoning, even if it weren't based on a category error, proves too much: it cuts the ground from under your absolute certainty in the Bayesian approach - the same certainty you needed in order to make accurate statements about 99.99---% probabilities in the first place.

The laws of probability are only useful for rationality if you know when they do and don't apply.

I'm totally missing the "N independent statements" part of the discussion; that seems like a total non-sequitur to me. Can someone point me at some kind of explanation?

-Robin

First, an individual particle can briefly exceeed the speed of light; the group velocity cannot. Go read up on Cerenkov radiation: It's the blue glow created by (IIRC) neutrons briefly breaking through c, then slowing down. The decrease in energy registers as emitted blue light.

Breaking through the speed of light in a medium, but remaining under c (the speed of light in a vacuum).

For one reason, again, if we're in any conventional (i.e. not paraconsistent) logic, admitting any contradiction entails that I can prove any proposition to be true.

Yes, but conditioned on the truth of some statement P&~P, my probability that logic is paraconsistent is very high.

Bayesianism is all about ratios of probabilities, yes, but we can write these ratios without ever using the P(|) notation if we please.

Wait a second, conditional probabilities aren't probabilities? Huhhh? Isn't Bayesianism all conditional probabilities?

Hah, I'll let Decartes go (or condition him on a workable concept of existence -- but that's more of a spitball than the hardball I was going for).

But in answer to your non-contradiction question... I think I'd be epistemically entitled to just sneer and walk away. For one reason, again, if we're in any conventional (i.e. not paraconsistent) logic, admitting any contradiction entails that I can prove any proposition to be true. And, giggle giggle, that includes the proposition "the law of non-contradiction is true." (Isn't logic a beautiful thing?) So if this mathematician thinks s/he can argue me into accepting the negation of the law of non-contradiction, and takes the further step of asserting any statement whatsoever to which it purportedly applies (i.e. some P, for which P&~P, such as the whiteness of snow), then lo and behold, I get the law of non-contradiction right back.

I suppose if we wanted to split hairs, we could say that one can deny the law of non-contradiction without further asserting an actual statement to which that denial applies -- i.e. ~(~(P&~P)) doesn't have to entail the existence of a statement P which is both true and false ((∃p)Np, where N stands for "is true and not true?" Abusing notation? Never!) But then what would be the point of denying the law?

(That being said, what I'd actually do is stop long enough to listen to the argument -- but I don't think that commits me to changing my zero probability. I'd listen to the argument solely in order to refute it.)

As for the very tiny credence in the negation of the law of non-contradiction (let's just call it NNC), I wonder what the point would be, if it wouldn't have any effect on any reasoning process EXCEPT that it would create weird glitches that you'd have to discard? It's as if you deliberately loosened one of the spark plugs in your engine.

Eliezer, what could convince you that Baye's Theorem itself was wrong? Can you properly adjust your beliefs to account for evidence if that adjustment is systematically wrong?

If you put a chair next to another chair, and you found that there were three chairs where before there was one, would it be more likely that 1 + 1 = 3 or that arithmetic is not the correct model to describe these chairs? A true mathematical proposition is a pure conduit between its premises and axioms and its conclusions.

But note that you can never be quite completely certain that you haven't made any mistakes. It is uncertain whether "S0 + S0 = SS0" is a true proposition of Peano arithmetic, because we may all coincidentally have gotten something hilariously wrong.

This is why, when an experiment does not go as predicted, the first recourse is to check that your math has been done correctly.