Comment author:player_03
07 July 2011 07:52:50AM
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0 points
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Thanks for the analysis, MathijsJ! It made perfect sense and resolved most of my objections to the article.

I was willing to accept that we cannot reach absolute certainty by accumulating evidence, but I also came up with multiple logical statements that undeniably seemed to have probability 1. Reading your post, I realized that my examples were all tautologies, and that your suggestion to allow certainty only for tautologies resolved the discrepancy.

The Wikipedia article timtyler linked to seems to support this: "Cromwell's rule [...] states that one should avoid using prior probabilities of 0 or 1, except when applied to statements that are logically true or false." This matches your analysis - you can only be certain of tautologies.

Also, your discussion of models neatly resolves the distinction between, say, a mathematically-defined die (which can be certain to end up showing an integer between 1 and 6) and a real-world die (which cannot quite be known for sure to have exactly six stable states).

Eliezer makes his position pretty clear: "So I propose that it makes sense to say that 1 and 0 are not in the probabilities; just as negative and positive infinity, which do not obey the field axioms, are not in the real numbers."

It's true - you cannot ever reach a probability of 1 if you start at 0.5 and accumulate evidence, just as you cannot reach infinity if you start at 0 and add integer values. And the inverse is true, too - you cannot accumulate evidence against a tautology and bring its probability down to anything less than 1. But this doesn't mean a probability of 1 is an incoherent concept or anything.

Eliezer: if you're going to say that 0 and 1 are not probabilities, you need to come up with a new term for them. They haven't gone away completely just because we can't reach them.

Edit a year and a half later: I agree with the article as written, partially as a result of reading How to Convince Me That 2 + 2 = 3, and partially as a result of concluding that "tautologies that have probability 1 but no bearing on reality" is a useless concept, and that therefore, "probability 1" is a useless concept.

## Comments (128)

Old*0 points [-]Thanks for the analysis, MathijsJ! It made perfect sense and resolved most of my objections to the article.

I was willing to accept that we cannot reach absolute certainty by accumulating evidence, but I also came up with multiple logical statements that undeniably seemed to have probability 1. Reading your post, I realized that my examples were all tautologies, and that your suggestion to allow certainty only for tautologies resolved the discrepancy.

The Wikipedia article timtyler linked to seems to support this: "Cromwell's rule [...] states that one should avoid using prior probabilities of 0 or 1, except when applied to statements that are logically true or false." This matches your analysis - you can only be certain of tautologies.

Also, your discussion of models neatly resolves the distinction between, say, a mathematically-defined die (which can be certain to end up showing an integer between 1 and 6) and a real-world die (which cannot quite be known for sure to have exactly six stable states).

Eliezer makes his position pretty clear: "So I propose that it makes sense to say that 1 and 0 are not in the probabilities; just as negative and positive infinity, which do not obey the field axioms, are not in the real numbers."

It's true - you cannot ever reach a probability of 1 if you start at 0.5 and accumulate evidence, just as you cannot reach infinity if you start at 0 and add integer values. And the inverse is true, too - you cannot accumulate evidence against a tautology and bring its probability down to anything less than 1. But this doesn't mean a probability of 1 is an incoherent concept or anything.

Eliezer: if you're going to say that 0 and 1 are not probabilities, you need to come up with a new term for them. They haven't gone away completely just because we can't reach them.

Edit a year and a half later: I agree with the article as written, partially as a result of reading How to Convince Me That 2 + 2 = 3, and partially as a result of concluding that "tautologies that have probability 1 but no bearing on reality" is a useless concept, and that therefore, "probability 1" is a useless concept.