ABC:Proverbs 26

Verse 4
Jim Meritt of Infidels.org and TheThinkingAtheist.com accuses the Bible of a contradiction here, asking "Do you answer a fool?"

Proverbs 26:4 Answer not a fool according to his folly, lest thou also be like unto him. 5 Answer a fool according to his folly, lest he be wise in his own conceit.

The best explanation I have seen for this is in the Scofield Study Bible III:

"26:5: Verses 4-5 illustrate Ecc. 3:7. The apparent contradiction between these verses is best resolved by remembering these two things: (1) There is nothing to be gained in answering a fool in his own manner. And (2) there may be occasions when to permit a fool to go unrefuted would confirm him in his conceit. Examples: (1) see 2Ki. 18:36; (2) see Ne. 6:8; Job 2:9-10." -The Scofield Study Bible III, p. 844.

Similar answers likewise exist at Tektonics.org and GotQuestions.org. The latter advises against "arguing with a fool on his own terms" because a fool will not care about wisdom and use deceit or insults (see logical fallacies such as the strawman or ad hominem) rather than pursuing wisdom, but agrees with the Scofield that there are times a fool must be challenged so their foolishness is not assumed correct (thus the purpose of this website).

In other words, there are times not to answer a fool's foolishness because he is only using insults or distorting your arguments through logical fallacies dishonestly because he cares nothing for truth, only appearing to be right for reputation's sake, and making his opinions seem correct. However, there are also times when to let a fool's claims go unchallenged would allow the unlearned to think a fool is correct, and thus one must confront publicly the claims of a fool. To phrase it another way, we should not sink to the level of a fool in using insults and distortions, dishonest arguments, their "folly" or foolishness in other words, yet to let them go unrefuted would be to give credence to their claims; thus one must publicly rebuke them, making their foolishness apparent that the easily misled are not deceived by their false claims.