Everything I read about confidence intervals warns against the following interpretation: If I construct a 95% confidence interval (a,b) for some parameter p, it is incorrect to say that the probability that p is in (a,b) is .95.
Mathematically, I totally get it. a, b, and p are constants, so the probability is either 1 or 0.
However, philosophically, I don't see the problem with constructing an interval based on a sample and saying that the probability that your interval contains the unknown parameter is .95 since probability is, after all, a consequence of uncertainty in the face of imperfect information.
For example, before I flip a coin, there is a 50% chance it lands on heads. Suppose I flip the coin but I don't look at the outcome. Is it wrong to say that the probability that the coin is currently heads up is 50%, even though it's and 'unknown constant?'