The difference was very intentional. I wanted to make clear the extra level of indirection between the two phrases. In the second case John may not actually have a bridge on his map at the indicated point, all we know is that he has the note saying that he believes there is a bridge there. It should logically follow that he should only say he believe something to be on his map if is it actually on his map. The point I was trying to make is that sometimes these things do not follow.
Gotcha, and thanks for the clarification. I see the difference, and in the first case you seem to be highlighting the difference between the map and the territory, while in the second case, you are highlighting the difference between the actual map and a belief about what's on one's map (in other words, one more "meta level" removed).
Now knowing that this was intentional, my suggestion is that you might have wanted to hold back on that first phrase until you made your clarification about the three types of statement meanings. Then perhaps highli...
This is my attempt to provide examples and a summarised view of the posts on "Against Doublethink" on the page How To Actually Change Your Mind.
What You Should Believe
Lets assume I am sitting down with my friend John and we each have incomplete and potentially inaccurate maps of a local mountain. When John says "My map has a bridge at grid reference 234567", I should add a note to my map saying "John's map has a bridge at grid reference 234567" *not* actually add the bridge to my map.
The same is true of beliefs. If Sarah tells me "the sky is green" I should, assuming she is not lying, add to my set of beliefs "Sarah believes the sky is green". What happens too often is that we directly add "The sky is green" to our beliefs. It is an overactive optimisation that works in most cases but causes occasional problems.
Taking the analogy a step further we can decide to question John about why he has drawn the bridge on his map. Then, depending on the reason, we can choose to draw the bridge on our map or not.
We can give our beliefs the same treatment. Upon asking Sarah why she believed the sky is green, if she said "someone told me" and couldn't provide further information I wouldn't choose to believe it. If, however, she said "I have seen it for myself" then I may choose to believe it, depending on my priors.
I Believe You Believe
The curious case is when someone says "I believe X". This can be meant a few ways:
The first case I do not have a problem with. It means your probability density has not yet shown a clear winner but you are giving me the answer that is in the lead at the moment. In this case I should add a note saying "You believes there is a bridge here, you are not very confident in the belief".
I don't have a problem with the second case either. I can have the belief "Angelina Jolie is attractive", someone else not have that belief, and we both be rational. This is because we are using different criteria for attractive. If I were to change it to a consistent definition of attractive it wouldn't be a problem e.g. The phrase "Angelina Jolie is regularly voted in the top 100 most attractive people in the world" doesn't require the phrase 'I believe...'.
The last case is even more curious. Lets assume that John (from our first example) says "I believe there is a bridge at grid reference 234567" but means it in the third case. I should add a note to my map saying "John has the following note on his map: 'I believe there is a bridge at grid reference 234567'". You would hope that the reason he has that note is because there is actually a bridge on his map. Unfortunately people are not that rational. You can have a cached belief that says "I believe X" even if you do not have "X" as a belief. By querying why they have that belief you should be able to work out if you should believe it, or even if they should.
To use the example from religion you can have the belief "I believe god exists" even if you do not have the belief "god exists".
Recommendations
I'm going to put myself on the line and give some recommendations: