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Vladimir_Nesov comments on You can't believe in Bayes - Less Wrong
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Well, PhilGoetz is claiming (if I am finally understanding him) that casting things in the light of believe/disbelieve loses information. To me--and to you also, it would seem--it gains information. It could be context dependent, but I can't think of a context* in which I would take it to mean something other than a statement about how probable something is, including the examples Phil gave in is post. We can't all be right...
In general I agree with the premise that things can be forced into bad terms by a less-than-helpful question, but I'm not at all convinced that this is a good example. However, I know that when I think to, I use the word "think" instead of "believe" because I think it's clearer, so on some level I must agree that "believe" leaves some sort of ambiguities.
*I'm completely excluding religious usages from consideration and will not mention this caveat again.
I agree. Compare this with computation of a factorial function. You start with knowing that the function is f(n)=if(n>1) n*f(n-1) else 1. Then you find out that f(1)=1, then that f(2)=2, etc. With each step, you are not taking new data out of environment, you are working from what you already have, simply juggling the numbers, but you gain new information.
For more on this view, see S. Abramsky (2008). `Information, processes and games' (PDF). In P. Adriaans & J. Benthem (eds.), Handbook of the philosophy of information. Elsevier Science Publishers.
That's an invalid comparison. That's a mathematical operation that doesn't involve information loss, and hence has nothing to do with this discussion.