Engaging in an argument with a foolish person carries a hidden danger. It is not just a waste of time, but a trap that can drag a wise person down to a lower level. The primary approach among commentators is that one should remain silent when a fool curses [אבן עזרא] or tries to pick a fight [רש״י, מצודת דוד]. Arguing with someone who lacks reason forces a person to adopt their flawed language, style, and foolishness [ביאור שטיינזלץ, עמנואל הרומי]. It is beneath a person's dignity to listen to or become emotionally invested in such nonsense [רלב״ג]. This is because a fool does not question the laws of wisdom and faith out of a genuine desire to learn. Instead, his doubts are merely excuses to continue chasing his own physical desires. Because of this, any rational attempt to disprove his claims is useless, as he will simply refuse to listen [מלבי״ם].
The danger of responding unfolds on both a social and a personal level. Socially, bystanders will have a hard time telling the two arguers apart. Since wise individuals do not typically engage in petty quarrels, the mere act of participating makes onlookers assume that the responder is just as foolish [מצודת דוד]. Furthermore, engaging with these empty arguments might make the responder appear as though he secretly shares the fool's doubts and is searching for answers [מלבי״ם]. On a personal level, making a habit of answering foolishness with more foolishness gradually lowers a person's own standards, leading them to mistakenly believe that both approaches are equally valid [אלשיך].
However, avoiding a foolish argument does not always mean absolute silence. The warning is specifically against adopting the fool's flawed methods. It is completely acceptable, and sometimes necessary, to respond with true wisdom to correct the error and prevent the fool from thinking he is actually wise [אבן עזרא, רלב״ג, עמנואל הרומי]. The accepted distinction is based on the topic at hand. When dealing with personal quarrels or everyday matters, it is best to ignore the provocation entirely. But if the fool's claims could lead others to sin or cause disgrace to the Torah, there is a strict obligation to speak up and silence him [עמנואל הרומי].