> What does this actually mean? It it a proposition? If so, is it falsifiable? And how?
If you saw a sequence of 1000 coin tosses at say 99% heads and 1% tails, you were convinced that the same process is being used for all the tosses and you had an opportunity to bet on tails with 50% stakes, would you do it?
This is a pragmatic answer which rejects P(X)=0.5. We can try to make sense of this pragmatic decision with some theory. (Incidentally, being exactly 0.5 is almost impossible, it makes more sense to verify if it is an interval like (0.49,0.51)).
The CLT says that probability of X can be obtained by conducting independent trials and the in limit, the average number of times X occurs will approach p(X).
However, 'limit' implies an infinite number of trials, so any initial sequence doesn't determine the limit. You would have to choose a large N as a cutoff and then take the average.
But, is this unique to probability? If you take any statement about the world, "There is a tree in place G", and you have a process to check the statement ("go to G and look for a tree"), can you definitely say that the process will successfully determine if the statement is true? There will always be obstacles("false appearances of a tree" etc.). To rule out all such obstacles, you would have to posit an idealized observation process.
For probability checking, an idealization which works is infinite independent observations which gives us p(X).
PS: I am not trying to favour frequentism as such, just that the requirement of an ideal of observation process shouldn't be considered as an overwhelming obstacle. (Sometimes, the obstacles can become 'obstacles in principle' like position/momentum simultaneous observation in QM and if you had such obstacles, then indeed one can abandon the concept of probability).