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.

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.