All of MathijsJ's Comments + Replies

I'm kinda surprised that it's only been mentioned once in the comments (I only just discovered this site, really really great, by the way) and one from 2010 at that, but it seems to me that "a magical symbol to stand for "all possibilities I haven't considered" " does exist: the symbol "~" (i.e. not). Even the commenter who does mention it makes things complicated for himself: P(Q or ~Q)=1 is the simplest example of a proposition with probability 1.

The proposition is of course a tautology. I do think (but I'm not sure) that th... (read more)

Jaynes avoids P(A|B) for "probability of A given evidence B" and P(B) for "probability of B", preferring P(A|BX) and P(B|X) where X is one's background knowledge. This and the above leads naturally to the question of ~X: the situation in which one's "background knowledge" is false. Assume that background knowledge X is the conjunction of a finite number of propositions. ~X is true if any of these propositions is false. If we can factor X into YZ where Y is the portion we suspect of being false — that is, if we can isolate for testing a portion of those beliefs we previously treated as "background knowledge" — then we can ask about P(A|BYZ) and P(A|B·~Y·Z).
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