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Eliezer_Yudkowsky comments on Infinite Certainty - Less Wrong

32 Post author: Eliezer_Yudkowsky 09 January 2008 06:49AM

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Comment author: Eliezer_Yudkowsky 09 January 2008 09:06:34AM 12 points [-]

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".

Comment author: Tyrrell_McAllister 16 December 2010 09:28:25PM *  1 point [-]

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?

This seems to me to be a very different question. "Do I doubt A?" and "Could any experience lead me to doubt A?" are different questions. They are equivalent for ideal reasoners. And we approximate ideal reasoners closely enough that treating the questions as interchangeable is typically a useful heuristic. Nonetheless, if absolute certainty is an intelligible concept at all, then I can imagine

  1. being absolutely certain now that A is true, while

  2. thinking it likely that some stream of words or experiences in the future could so confuse or corrupt me that I would doubt A.

But, if I allow that I could be corrupted into doubting what I am now certain is true, how can I be certain that my present certainty isn't a result of such a corruption? At this point, my recursive justification would hit bottom: I am certain that my evaluation of P(A) as equal to 1 is not the result of a corruption because I am certain that A is true. Sure, the corrupted future version of myself would look back on my present certainty as mistaken. But that version of me is corrupted, so why would I listen to him?

ETA:

In your actual scenario, where all other mathematicians scorn my belief that ~(P&~P), I would probably conclude that everyone is doing something very different with logical symbols than what I thought that they were doing. If they persisted in not understanding why I thought that ~(P&~P) followed from the nature of conjunction, I would conclude that my brain works in such a different way that I cannot even map my concepts of basic logical operation into the concepts that other people use. I would start to doubt that my concept of conjunction is as useful as I thought (since everyone else apparently prefers some alternative), so I would spend a lot of effort trying to understand the concepts that they use in place of mine. I would consider it pretty likely that I would choose to use their concepts as soon as I understood them well-enough to do so.

Comment author: pnrjulius 27 May 2012 03:38:13AM 1 point [-]

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.

I don't know what the above sentence means. You must be using the word "exist" differently than I do.