The first verse says not to answer a fool according to his folly. The second verse says you should answer a fool according to his folly. On the surface, this seems contradictory.
If these verses were widely separated in different parts of the Bible, then the apparent contradiction might be hard to explain. But these verses occur together, one right after the other. So they are meant to be taken together. You must dig a little deeper to get to the meaning.
You cannot reason with a fool, because fools are unreasonable. The fool will either mock or get angry at the reasoning of the wise. Either way, there will be no peaceful resolution of the disagreement. But that does not mean you should leave a fool unanswered, because then they will smugly think they are right.
Since a fool cannot be reasoned with, you cannot win an argument with one. If you try, you are likely to appear foolish yourself as the fool drags you down to his level, trying to answer his erroneous arguments.
You do not know just how foolish someone is
So does that mean we should always follow the advice of Proverbs 26:4 by not answering the fool? The answer depends.
If the dispute is about something unimportant or unprovable, the wisest course may be to let the ignorant remain ignorant and avoid the quarrel. As the apostle Paul told Timothy, “Avoid foolish and ignorant disputes, knowing that they generate strife” (2 Tim. 2:23 / NKJV).
In important matters, it may be best to reason with the fool in hopes of bringing them to the truth
In important matters, it may be best to reason with the fool in hopes of bringing them to the truth. This is particularly true of the gospel. We have been commanded to “preach the gospel to every creature” (Mark 16:15). No exception was made for fools.
And a servant of the Lord must not quarrel but be gentle to all, able to teach, patient, in humility correcting those who are in opposition, if God perhaps will grant them repentance, so that they may know the truth, and that they may come to their senses and escape the snare of the devil, having been taken captive by him to do his will. (2 Tim. 2:24-26 / NKJV)
We should try to reason with the fool and lead them to the truth, “lest they be wise in their own eyes.” And you do not know just how foolish someone is. If they are merely ignorant or deluded, you may be able to reason with them.
Proverbs 31- The Virtuous Christian