Proverbs has much to say about fools (Note 1). The futility of trying to impart wisdom to a fool is the basis of Proverbs 26:4-5, which tell us how to answer a fool:
4 Don’t answer a fool according to his folly, or you will become as foolish as him. 5 Be sure to answer a fool according to his folly, lest he be wise in his own eyes. Proverbs 26:4-5
These seemingly contradictory verses are actually a common form of parallelism found in the Old Testament, where one idea builds upon another. Verse 4 warns against arguing with a fool on his own terms, lest we stoop to his level and become as foolish as he is. Because he despises wisdom and correction, the fool will not listen to wise reason and will try to draw us into his type of argument, whether it is by using deceit, scoffing at our wisdom, or becoming angry and abusive. If we allow him to draw us into this type of discourse, we are answering him “according to his folly” in the sense of becoming like him.
On the other hand, the phrase “according to his folly” in verse 5, tells us that there are times when a fool has to be addressed so that his nincompoopery will not go unchallenged. In this sense answering him according to his folly means to expose the foolishness of his words, rebuking him on the basis of his folly so he will see the idiocy of his words and reasoning. Our answer in this case is to be one of reproof, showing him the truth so he might see the foolishness of his words in the light of reason. Even though he will most likely despise and reject the wisdom offered to him, we are to make the attempt, both for the sake of the truth which is always to be declared, and for the sake of those listening, that they may see the difference between wisdom and folly and be instructed.
Whether we use the principle of verse 4 and deal with a fool by ignoring him, or obey verse 5 and reprove a fool depends on the situation. In matters of insignificance, it’s probably better to disregard him. In more important areas, such as when a fool denies the existence of God (Psalm 14:1), verse 5 tells us to respond to his foolishness with words of rebuke and instruction. To let a fool speak his nonsense without reproof encourages him to remain wise in his own eyes and possibly gives credibility to his folly in the eyes of others.
In short, in negligible issues we should just ignore fools, but in issues that matter, they must be dealt with so that credence will not be given to what they say.
Note 1: Fools despise wisdom (Proverbs 1:7, 22, 10:21, 23:9); they are right in their own eyes (Proverbs 12:15); they are deceitful (Proverbs 14:8) and scornful (Proverbs 10:23, 14:9). The wise are also given instruction on how to deal with fools in Proverbs. Instructing a fool is pointless because his speech is full of foolishness (Proverbs 15:2, 14) and he does not want wisdom and understanding (Proverbs 18:2).