Quantum physics does indeed reveal that the universe on a submicroscopic scale does operate according to fuzzy logic, in which effects don't need to have causes and where things exist only in probabilities, but the averages of all these probabilities over macroscopic space gives rise to a very reliable set of physics called Newtonian physics which dictates the activities of pretty much everything we deal with on a day-to-day basis.

The real reason why we can't prove anything 100% is because we are all beings experiencing the objective world subjectively. I can't even prove to myself that the external world exists, but for all practical purposes the assumption works just fine and the alternative explanation seems extremely unlikely. Even if it were true it would not have much effect on things.

But while it is impossible to prove things 100%, we can prove things to be very likely true, and that is the basis of science. By observing the universe carefully and using the rules of logic we can uncover theories, that is, working models of how the universe operates, that are extremely likely but still potentially falsifiable. That means that any scientifically valid theory must be constructed in such a way that it could be proven wrong if it is. It is only by this process that we can progress in our knowledge and enjoy all the advances that science has given us.