I can't think of this theorem as anything other than silly.
Let's look at what two people might disagree over. It is quite likely a "should" statement, rather than something simply factual. If it were cleanly factual and measurable, they could just devise a test and observe it. So it's probably a "should" statement.
A conclusion is derived from premises that logically lead to the conclusion. Those premises are in turn derived from deeper premises, and so on.
It's known that there are only three possible outcomes to such an exercise. You either delve to infinity, which is not reasonably acceptable, or you engage in circular reasoning, which is not rational, or you eventually arrive at axioms.
The axiomatic approach is the one commonly accepted. The problem is that it is pretty well-accepted that you cannot derive an "ought" statement (a "should" statement) from "is" statements alone. Any "ought" statement is going to partially rely on "ought" premises. And this means that you're eventually going to drill down to "ought" axioms.
Blowhards like Sam Harris like to argue otherwise, but really all they are doing about with their "moral science" talk is strenuously arguing that we should accept certain "ought" axioms. They couch it in language such that we should consider these axioms so basic and obvious that they somehow don't become axioms anymore, but that's silly - they are still "ought" axioms.
Anyway, the point is that we do not all have the same "ought" axioms. That's what the entire field of philosophy is all about - the different collections of "ought" axioms we accept for ourselves.
And since two people can have different "ought" axioms, it is then possible for them to have entire world views that conflict with others, and yet are still internally consistent.
In other words, it's possible for two people to come to two different conclusions while still both being perfectly and completely logical, rational, and internally consistent.
This thing about different "ought" axioms might be what the theorem means about "different priors", but if so, that would basically make the theorem meaningless - and it seems to be a different meaning than what "common knowledge" implies.
Let's look at what two people might disagree over. It is quite likely a "should" statement, rather than something simply factual. If it were cleanly factual and measurable, they could just devise a test and observe it. So it's probably a "should" statement.
A conclusion is derived from premises that logically lead to the conclusion. Those premises are in turn derived from deeper premises, and so on.
It's known that there are only three possible outcomes to such an exercise. You either delve to infinity, which is not reasonably acceptable, or you engage in circular reasoning, which is not rational, or you eventually arrive at axioms.
The axiomatic approach is the one commonly accepted. The problem is that it is pretty well-accepted that you cannot derive an "ought" statement (a "should" statement) from "is" statements alone. Any "ought" statement is going to partially rely on "ought" premises. And this means that you're eventually going to drill down to "ought" axioms.
Blowhards like Sam Harris like to argue otherwise, but really all they are doing about with their "moral science" talk is strenuously arguing that we should accept certain "ought" axioms. They couch it in language such that we should consider these axioms so basic and obvious that they somehow don't become axioms anymore, but that's silly - they are still "ought" axioms.
Anyway, the point is that we do not all have the same "ought" axioms. That's what the entire field of philosophy is all about - the different collections of "ought" axioms we accept for ourselves.
And since two people can have different "ought" axioms, it is then possible for them to have entire world views that conflict with others, and yet are still internally consistent.
In other words, it's possible for two people to come to two different conclusions while still both being perfectly and completely logical, rational, and internally consistent.
This thing about different "ought" axioms might be what the theorem means about "different priors", but if so, that would basically make the theorem meaningless - and it seems to be a different meaning than what "common knowledge" implies.