Comment by MathijsJ on 0 And 1 Are Not Probabilities · 2010-12-02T02:24:59.536Z · LW · GW

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 that is the only sort of statement that receives probability 1. This is in sync with Eliezer's "amount of evidence" interpretation. A bayesian update can only generate 1 if the initial proposition was of probability 1 or if the evidence was tautological (i.e. if Q then Q or, slightly less lame, if "Q or R" and "~R" then Q, where "Q or R" and "~R" are the evidence).

Skimming the comments, I saw two other proposals for "sure bets", the runner who clocked a negative time and the golf ball landing in a particular spot. That last one degenerated pretty quickly into a discussion about how many points there are in a field and on a ball. I think that's typical of such arguments: it depends on your model. Once you have your model specified the probability becomes 1 (or not) if the statement is (or isn't) tautological in the model. If the model isn't specified, then neither is the statement (what is a precise point?) and hence the probability. Ask the next man what the probability is of a runner clocking a negative time and he'll rightly respond: "Huh?" (unless he is a particularly obfuscatory know-it-all, in which case he might start blabbering about the speed of light. But then too, he makes a claim because he can ascribe meaning to the question, that is, he picks his model). So these are also tautological examples.

I think Eliezer's hold up pretty well for proposition that aren't tautological and hence empirical in nature: they require evidence and only tautological evidence will suffice for certainty.

About the problem of inserting 0's in certain standard theorems: I don't see a problem with Bayes' theorem (I'm curious about other examples). Dividing by 0 is not defined, so the probability of it raining when hell freezes over is not defined. That seems like a satisfactory arrangement.