Imagine that you are a detective and you are investigating a crime. Suppose you can't point to a criminal yet. However, you can name him in your reasoning. You can call him the criminal, the murderer, Mr. X, and so on. So the question is, why can we do that? What does the word criminal mean in this case? Don't we need to know which object the name is pointing to?
Similar reasoning can be found in mathematics. For example, if you have some predicate P and you know that there is at least one value that satisfies the predicate, then you can denote this value with a letter (for example, n) and use it in further reasoning. For example, if P = "be an even prime number", then n denotes a specific number 2. If P = "be a prime number", then n is not the name of a specific object, but a variable that takes a value on prime numbers. A completely strange situation occurs if we prove the non-existence of an object by contradiction. Then we believe that such an object still exists and denote it by a letter, but in the course of reasoning we come to a contradiction. In this situation, it turns out that n does not mean anything at all. I am interested on what philosophical principles such reasoning is based