- A king of France doesn't exist.
It makes sense because there's no king of France.
- The king of France doesn't exist.
Does it make sense? Because if "the" implies the existence, this sentence is paradoxical (the king of France who exists doesn't exist).
According to Russell's theory of descriptions, indefinite descriptions (e.g. "an F") contribute only a bare existence claim to the truth-conditions of the sentences in which they appear, whereas definite descriptions (e.g. "the F") contribute both existence and uniqueness claims.
When it comes to demonstrative determiners (this, that, etc.), they certainly imply the existence of something. So "this cat doesn't exist" is self-contradictory. So what about "the"?