First, I did study mathematical logic, and please avoid such kind of ad hominem.
Fair enough
That said, if what you're referring to is the whole world state, the outcomes are, in fact, always different. Even if only because there is somewhere in your brain the knowledge that the choice is different.
I thought this would be your reply, but didn't want to address it because the comment was too long already.
Firstly, this is completely correct. (Well, technically we could imagine situations where the outcomes removed your memory of there ever having been a choice, but this isn't usually the case). Its pretty much never possible to make actually useful deductions just from pure logic and the axiom of independence.
This is much the same as any other time you apply a mathematical model to the real world. We assume away some factors, not because we don't think they exist, but because we think they do not have a large effect on the outcome or that the effect they do have does not actually affect our decision in any way.
E.g. Geometry is completely useless, because perfectly straight lines do not exist in the real world. However, in many situations they are incredibly good approximations which let us draw interesting non-trivial conclusions. This doesn't mean Euclidean Geometry is an approximation, the approximation is when I claim the edge of my desk is a straight line.
So, I would say that usually, my memory of the other choice I was offered has quite small effects on my satisfaction with the outcome compared to what I actually get, so in most circumstances I can safely assume that the outcomes are equal (even though they aren't). With that assumption, independence generates some interesting conclusions.
Other times, this assumption breaks down. Your cholera example strikes me as a little silly, but the example in your original post is an excellent illustration of how assuming two outcomes are equal because they look the same as English sentences can be a mistake.
At a guess, a good heuristic seems to be that after you've made your decision, and found out which outcome from the lottery you got, then usually the approximation that the existence of other outcomes changes nothing is correct. If there's a long time gap between the decision and the lottery then decisions made in that time gap should usually be taken into account.
Of course, independence isn't really that useful for its own sake, but more for the fact that combined with other axioms it gives you expected utility theory.
The cholera example was definitely a bit silly - after all, "cholera" and "apple vs orange" are usually really independent in the real world, you've to make very far-fetched circumstances for them to be dependent. But an axiom is supposed to be valid everywhere - even in far-fetched circumstances ;)
But overall, I understand the thing much better now: in fact, the independence principle doesn't strictly hold in the real world, like there are no strictly right angle in the real world. But yet, like we do use the Pythagoras theorem in the ...
Followup to : Is risk aversion really irrational?
After reading the decision theory FAQ and re-reading The Allais Paradox I realized I still don't accept the VNM axioms, especially the independence one, and I started thinking about what my true rejection could be. And then I realized I already somewhat explained it here, in my Is risk aversion really irrational? article, but it didn't make it obvious in the article how it relates to VNM - it wasn't obvious to me at that time.
Here is the core idea: information has value. Uncertainty therefore has a cost. And that cost is not linear to uncertainty.
Let's take a first example: A is being offered a trip to Ecuador, B is being offered a great new laptop and C is being offered a trip to Iceland. My own preference is: A > B > C. I love Ecuador - it's a fantastic country. But I prefer a laptop over a trip to Iceland, because I'm not fond of cold weather (well, actually Iceland is pretty cool too, but let's assume for the sake of the article that A > B > C is my preference).
But now, I'm offered D = (50% chance of A, 50% chance of B) or E = (50% chance of A, 50% chance of C). The VNM independence principle says I should prefer D > E. But doing so, it forgets the cost of information/uncertainty. By choosing E, I'm sure I'll be offered a trip - I don't know where, but I know I'll be offered a trip, not a laptop. By choosing D, I'm no idea on the nature of the present. I've much less information on my future - and that lack of information has a cost. If I know I'll be offered a trip, I can already ask for days off at work, I can go buy a backpack, I can start doing the paperwork to get my passport. And if I know I won't be offered a laptop, I may decide to buy one, maybe not as great as one I would have been offered, but I can still buy one. But if I chose D, I've much less information about my future, and I can't optimize it as much.
The same goes for the Allais paradox: having certitude of receiving a significant amount of money ($24 000) has a value, which is present in choice 1A, but not in all others (1B, 2A, 2B).
And I don't see why a "rational agent" should neglect the value of this information, as the VNM axioms imply. Any thought about that?