I just read the following sentence from Wikipedia:
A 95% confidence interval does not mean that for a given realized interval there is a 95% probability that the population parameter lies within the interval (i.e., a 95% probability that the interval covers the population parameter).
Instead, the correct interpretation would be
The confidence interval can be expressed in terms of samples (or repeated samples): "Were this procedure to be repeated on numerous samples, the fraction of calculated confidence intervals (which would differ for each sample) that encompass the true population parameter would tend toward 90%.
But then my question is: if the current confidence intervals are an instance of an procedure that contains the true parameter value 90% of the time when time tends to infinity, like the second quote implies, then why can't we say that the current intervals contains the true parameter with 90% probability (the probability referring to the CI calculation, not to the parameter)? Isn't that the frequentest definition of probability?