I am a professional sign language interpreter, moving meaning between English and American Sign Language. Sometimes close enough is close enough, and sometimes close enough is worse than not at all. When meaning matters most, such as your desire to differentiate between 1x20 and 20x1, here is what I do:
Do not say "I estimate 5% odds of X happening" at all. Say either "I have about 1-in-20 confidence that all future timelines from this point contain X, and about 19-in-20 confidence that none do" or say "I estimate about 1-in-20 future timelines contain X, and 19-in-20 future timelines don't" but do not say both and do not say nothing. Both of these choices use up more time to convey than the original sentence, but in an interpretation there will be other times that the target language will be more economical with time and over the course of an interpretation it can even out.
I agree with everything you say, as long as one assumption holds: the language(s) being used are static, in the domain of the topic being discussed. If the only available method to express 1x20 vs 20x1 is colloquial English or arcane math, then the method you describe seems to be ideal.
On the other paw, I've already gotten good use out of taking an existing language structure (the subset of Lojban sometimes called Cniglic, which can be bolted-on to English), and extending it for my own purpose. Since it worked so well before, I'm willing to try it again, if the next-best option is what you describe.
Or, looking at it another way, I'm drawing on the Sapir-Whorf hypothesis, and am more interested in improving language as a tool of thought rather than a tool of communication.
Quantum mechanics has an already-existing --and quite general and powerful-- formalism for dealing with cases when we have at the same time quantum uncertainty about the results of measurements done on a quantum state, and ordinary non-quantum uncertainty about which quantum state we have. This seems to be what you are talking about, am I right? Keywords: "mixed state", "density matrix", "quantum statistical mechanics".
That does seem to be the more formal version of the idea I tried to describe.
One of my assumptions going into this is that human brains find it easier to add than to multiply, and easier to multiply than deal with vectors; and that the easier it can be made for a brain to handle the math involved, the more likely that brain will be able to develop accurate intuitions. I'm not going to try to turn the complete formalism of quantum statistical mechanics into language - I want something that's just slightly more useful than a placeholder word for "insert math I don't understand here".
Eg, with bei'e, the numbers can be added and subtracted to give useful confidence numbers; I wouldn't mind a similar word with attached fractions that could be added to each other. (And then, possibly, multiplied by relevant confidence probabilities, but at this point that's counting unhatched chickens.)
Except for very simple quantum systems, I wouldn't be able to tell you if uncertainty is in my mind or a consequence of physics.
The weather forecast for November 25 2014 is a 5% chance of rain. Will all future timelines have rain on that day, or is one years long enough to amplify quantum randomness to the scale of global weather patterns? Do you know? I don't know; the physics to determine that is beyond me. Most humans can't distinguish between the two cases when estimating.
All-or-nothing events feel riskier because they indicate a greater chance of catastrophe. A 5% chance of losing your house to a local fire is much better than a 5% chance of losing your house to a city-destroying conflagration. Life insurance claims come steadily; flood insurance claims do not.
These seem like they correspond respectively to epistemic and ontological uncertainties, perhaps not exactly but in many cases.
I still don't know a standard way to distinguish them, but improvising isn't hard. "I'm 99.9% confident, epistemically, that the cointoss I'm about to make has 50% ontological probability of heads."
It's not like area of a rectangle. It's like area under a curve. You're x% sure that something will happen in f(x)% of Everett branches. The curve being a rectangle is just a special case. You're not going to find an easy way to say it, because there's an infinite number of ways you can end up with 5% probability, and you can't quickly distinguish one way in particular.
If you had some reason you wanted to distinguish those two particular cases, you could make up a word for it, but since neither comes up often, I see no point.
You are quite right about a better analogy being the area under a curve.
Over in the Lotteries and MWI thread, there's a few pages of discussion in which serious decisions about how to live one's life - using the archetypical example of whether or not a lottery ticket could be worth buying - depend on the differences between those two cases. Even if nobody else sees any point in the exercise, I'm entirely willing to work with whatever mental tricks I can come up with to help my own imagination and intuition better handle the ideas involved.
If I do go for a new word again, then one possibility is a word meaning "X fraction of timelines in the future of the designated time; if no time is designated, then it is assumed to be the beginning of the sentence.", and combine it with bei'e for the classical-probability measurement. This could make expressing, "If I'm cryonically preserved tomorrow, I'm 30% confident that I'll be revived in 10% of future timelines, 60% confident that I'll be revived in 1% of future timelines, and 10% confident that I'll be revived in near-0% of future timelines" much easier.
Logical probabilities. This is precisely the distinction between logical uncertainty and the kind you're stuck with after achieving logical omniscience.
No, it's not. It is the most obvious case of being 5% sure that something is true in every Everett branch, but it's not the only one. For example, I think a particle has the same mass in every Everett branch, but not even logical omniscience will tell you the exact mass.
Well, yes. The point is that there's a set of common problems that distinguish between the extreme cases, and therefore justify names.
"I estimate 5% odds of X happening" can mean at least two things:
* I have about 1-in-20 confidence that all future timelines from this point contain X, and about 19-in-20 confidence that none do.
* I estimate about 1-in-20 future timelines contain X, and 19-in-20 future timelines don't.
Looked at this way, the usual way of quantifying probability seems to be a lot like quantifying area - the first bullet-point by having a 1x20 rectangle, the second by having a 20x1 one. (This also seems valid for having, say, 50% confidence that 1-in-10 future timelines contain X.) It seems like it might be worth having an easy and understandable way to differentiate between these different forms of '5% odds', but any easy way I've been able to think of is barely understandable, and vice versa. Are there any existing standard ways to do this that I'm unaware of? If not, does anyone reading this have any decent answers?
I'm not opposed to coming up with a new word for personal use to help get in the habit of thinking in certain ways; such as bei'e in Lojban to remind myself to think of probability logarithmically. I don't mind doing the same with a word meaning 'such-and-such a fraction of future MWI branches', if that's the best solution, or even just a useful tool; I'd just like to know what the full range of useful approaches really are, first, and any potential loopholes therein or drawbacks thereof.