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EHeller comments on 0 And 1 Are Not Probabilities - Less Wrong
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Comments (128)
Cromwell's Rule is not EY's invention, and relatively uncontroversial for empirical propositions (as opposed to tautologies or the like).
If you don't accept treating probabilities as beliefs and vice versa, then this whole conversation is just a really long and unnecessarily circuitous way to say "remember that you can be wrong about stuff".
The part that is new compared to Cromwell's rule is that Yudkowsky doesn't want to give probability 1 to logical statements (53 is a prime number).
Because he doesn't want to treat 1 as a probability, you can't expect complete sets of events to have total probability 1, despite them being tautologies. Because he doesn't want probability 0, how do you handle the empty set? How do you assign probabilities to statements like "A and B" where A and B are logical exclusive? (the coin lands heads AND the coin lands tails).
Removing 0 and 1 from the math of probability breaks most of the standard manipulations. Again, it's best to just say "be careful with 0 and 1 when working with odds ratios."