The key, he knows is to eliminate the impossible. Once you do that what remains, no matter how improbable must be the truth. Holmes, as it comes out might have been right. Oliver Wendell was, but how can you know when you’ve eliminated all impossibilities? Doyle (Roddy perhaps) would note that improbabilities can look a great deal like impossibilities, but may nevertheless prove to be the truth. We could enlist Watson’s superb mind, but we know just how possessive Gates can be, and it could swing shut on us at any moment.