Is what he says true or false?

Liar paradox. This paradox takes several forms. The simplest is that of the man who says "I am lying"; if he lies, he is speaking the truth, and vice versa (Curry). Epimenides the Cretan said that all Cretans lie; did he tell the truth, or not? Let us assume, for the sake of argument , that every Cretan, except possibly Epimenides himself, was in fact a liar; but what then of Epimenides?

In effect, he says he himself lies; but if he is lying, then he is telling the
truth ; and if he is telling the truth , then he is lying ! Which then is it?
The same conundrum arises from the following sentence:
"This sentence is false."
That sentence is known as the "Liar Paradox", or "pseudomenon".

By a sentence is to be understood any expression concerning which it is meaningful to say that its content is true or false. In 1933, the Polish logician Alfred Tarski defined the concept of truth in a general setting and pointed out what is known as Tarski’s Undefinability of Truth argument: no language can define its own truth, owing to the Liar Paradox, namely to the sentence “This sentence is false.” This sentence is neither true nor false, contrary to the Law of Excluded Middle, which Tarski took for granted.

Contradiction - "This statement is false"—the statement cannot be false and true at the same time.

In the sentence "This statement is false" is just said that "This statement is false" is false. Thus, the condition is satisfied that "the statement cannot be false and true at the same time", so there is no contradiction.

A man says that he is lying. Is what he says true or false?
This is similar to how a man sees a naked king and asks: "Is the King Naked"? Do not believe your eyes!