If the universe were deterministic, then it must be possible to, given sufficient data about the universe and its current state, predict what the state of it will be at some point in the future. We shall confine our predictions for this purpose only to the outcome of a single experiment, and let us take a black box that can tell us what the outcome of the experiment will be. If such a black box cannot exist for any reason, then it follows that there are at least some factors in the universe that are non-deterministic.
The experiment is designed as follows: One first designs a deterministic system that outputs the opposite of its input. For example, let us say there are two switches labelled A and B, and if A is pressed then it outputs B and if B is pressed it outputs A. There is no reason that such a system could not be built. One then arranges for the input of this system to be the output of the aforementioned black box, which if it were possible for the black box to exist then there is no reason why this could not be done.
However, one quickly realizes in this experiment that a contradiction occurs, anything that the black box can possibly predict about the outcome of the experiment is wrong, and so either the black box cannot exist or else it is impossible for the black box to communicate to the outside world such information about the future state, which suggests that the universe is either non-deterministic, or else it may be treated as non-deterministic for *all* experimental purposes.