I think what cousin_it was asking (and I would also like to know) is: what problem with the Axiom of Independence does the indexical uncertainty in your example (or cousin_it’s rephrasing) illustrate?
Let A = “I’m immunized with vaccine A”, B = “I’m immunized with vaccine B”, p = probability of being the original. The Axiom of Independence implies
p A + (1-p) A > p B + (1-p) A iff p A + (1-p) B > p B + (1-p) B
To see this, substitute A for C in the axiom, and then substitute B for C. This statement says that what I prefer to happen at one location doesn’t depend on what happens at another location, which is false in the example. In fact, the right side of the iff statement is true while the left side is false.
That is based on the unspoken assumption that you prefer A to B. You yourself explained that such a preference is nonsense:
If my counterpart is vaccinated with A, then I’d prefer to be vaccinated with B, and vice versa. “immunizes me with vaccine A” by itself can’t be assigned an utility.
If an axiom or theorem has the form “If X then Y”, you should demonstrate X before invoking the axiom or theorem.
I guess it’s not, unless you’re already interested in figuring out the nature of indexical uncertainty. If you’re not sure what’s interesting about indexical uncertainty, take a look at http://www.simulation-argument.com/ and http://en.wikipedia.org/wiki/Doomsday_argument.
I think what cousin_it was asking (and I would also like to know) is: what problem with the Axiom of Independence does the indexical uncertainty in your example (or cousin_it’s rephrasing) illustrate?
Let A = “I’m immunized with vaccine A”, B = “I’m immunized with vaccine B”, p = probability of being the original. The Axiom of Independence implies
p A + (1-p) A > p B + (1-p) A iff p A + (1-p) B > p B + (1-p) B
To see this, substitute A for C in the axiom, and then substitute B for C. This statement says that what I prefer to happen at one location doesn’t depend on what happens at another location, which is false in the example. In fact, the right side of the iff statement is true while the left side is false.
Does this explanation help?
That is based on the unspoken assumption that you prefer A to B. You yourself explained that such a preference is nonsense:
If an axiom or theorem has the form “If X then Y”, you should demonstrate X before invoking the axiom or theorem.
Yes, I meant to ask exactly that.