Okay, so I got your fractions mostly right.

Well, when you use a computer to make a probabilistic calculation, you have at some point to feed it with the relevant information, or it won't know what to make of the data. The data alone is not enough. And if you are absolutely certain your program is (i) correct, and (ii) fed with all the relevant information you know about, then you should definitely believe the result. (Of course, this absolute certainty is not attainable, so any sufficiently surprising result should lead you to think that something went wrong.) Assuming common priors on the other hand, seems unreasonable, unless we're only talking about "fully uninformed" priors such as those based on Kolmogorov complexity.

Yes, I would say you're half right. The example of the temperature given by the Wikipedia is really good. The "degree of coldness" is a worthy notion. What bothers me is that fuzzy logicians don't seem to run probability distributions over those degrees of truths. I mean, I can surely make up a probability distribution on the outside temperature for tomorrow at 10AM. Assuming that temperature maps to degrees of truth to the sentence "it's cold" (say, 100% true below 10°C, 0% true beyond 20°C, and a linear interpolation in between), then I naturally come up with a probability distribution over the degrees of truth (I should get a Dirac spike at both extremities).

Your last one is exactly right. I'm only 16% from empty to 75% full (excluded). That's the punishment I get for not producing a theory of the fullness of the bottle which describes my actual beliefs about it.

I assume that your ultimate goal in life is to make decisions, or to create a machine that will take decisions. To make sound decisions, you need to asses their consequences. Take the example of the bottle: I need a full bottle of water to hike today. Let's assume for the sake of the argument that it is either empty, or full. If it's empty, I need to re-fill it, or I would faint from the thirst (not good). If it's full, checking is only a nuisance (not good either). Now the question is, how can I minimize my discomfort, on average?

Well I need two things: first, a probability distribution. Let's say I'm pretty sure I filled the bottle yesterday, let's call it a 90% chance. Second, a utility function. Baseline, the bottle is full, and I did not bother checking it, utility zero. Checking costs me 1 utility point, and fainting from thirst costs me 50 points (let's assume my brain is actually capable of performing those utility assessments). Should I check the bottle?

Oh yes. Not checking means a 10% chance of losing 50 points, or 5 on average. Checking means a certainty of losing only one point, which costs much less. Conclusion: check your water supply before you go hiking. And re-check just to be sure.

Now this problem didn't call for fuzzy logic. We can however come up with a full probability distribution over the quantity of water in the bottle instead of just two possibilities. From then, fuzzy logic should be able to naturally step in. But frankly, I prefer to run a a separate (dis)utility function over the state of thirst that the lack of water will provoke (from light thirst to actually faint), and combine it with my probability distribution to make my decision. (Though at that point, I'd rather just check than frying my brain trying to solve this cost-benefit calculation.)