How did the liar paradox challenge ancient logicians?

The liar paradox challenged ancient logicians by creating a fundamental conflict in determining the truth value of a statement. The paradox arises when a person asserts, "I am lying." If the statement is true, then the speaker is lying, which makes the statement false; conversely, if the statement is false, then the speaker is not lying, which makes the statement true.123

This puzzle is often attributed to the Cretan seer Epimenides, who famously declared, "All Cretans are liars," and it was further developed by Eubulides of Miletus of the Megarian school. The Megarians and later the Stoics were deeply interested in such logical puzzles and fallacies, which they used to test the limits of their theories on truth and propositional logic.123

Because the liar paradox creates severe difficulties for logically rigorous theories of truth, it remained a significant challenge for centuries, as ancient logicians lacked the tools to resolve the self-referential nature of such statements.14