How does one assign a probability to statements that are themselves probabilistic? For example, how would one assign a probability to the statement, "There is an 80% chance that it will rain on September 16, 2017"? And how would one assign a probability of THAT statement? etc. Is there a paper where someone talks about this problem?
You're talking about the mean (first moment), as opposed to the variance (second moment), etc, and perhaps more directly about quantiles. See https://en.wikipedia.org/wiki/Moment_%28mathematics%29#Significance_of_the_moments and https://en.wikipedia.org/wiki/Quantile_function or just google "statistics moments quantiles" for detailed discussions.
My analysis would be that such statements represent propositions that are about other propositions. The example you gave proposes that the proposition [it will rain on September 16, 2017] has an 80% likelihood of being true. I would claim that such propositions will always be probabilistic in nature because there is nothing to be said about a proposition except its truth value.
If I were to make a 'higher order' proposition about the likelihood of the probabilistic assertion, I claim that the same meaning, with respect to the non-probabilistic proposition, can be achieved by revising the lower order claim. For instance, if I were to say [There is a 20% chance the exemplary proposition is true], all I have to do is multiply .2 and .8 to get the revised probability that [it will rain on September 16, 2017], by the multiplicative rule of probabilities. I have no real conclusion about this kind of proposition, but I've described what they are and how I think they work. Check out Bayes' Rule if you are interested in reading more about revising probabilities.
There is much more to so called higher-order probabilities than taking the mean of a random variable.
Yes, there are a bunch of papers which talk about this problem. A good starting point is:
Do We Need Higher-Order Probabilities and, If So, What Do They Mean? https://arxiv.org/abs/1304.2716
The takeaway by Pearl and others is that yes you can talk about higher-order probabilities (people do it all the time), but that typically they are derived from an existing causal model.