Do not answer a fool according to his folly, or you will be like him yourself. (Prov 26:4) and right after that: Answer a fool according to his folly, or he will be wise in his own eyes. (Prov 26:5)
It seems at first glance that the Bible is contradictory in this sense. 2 verses contradicting each other, and one following the other.
One easy way to explain it is to use 2 common English sayings:
Look before you leap and He who hesitates is lost.
Both are contradictory, yet we know they are true. It's the wisdom to know the difference that is important.
The writer of Proverbs knew exactly what he was doing by putting those verses next to one another. He knew that life had paradoxes and was using these two sayings to illustrate the point. Both sayings are right. It's the circumstances that dictate how to apply it. The meta meaning is that "somehow, you'll still lose when talking to a fool". It's something like a Catch-22 situation.
Do a search for paradox in my blog, and you will find many examples that I have observed. It's because life isn't one dimensional.
So then, is the best way to "win" an argument with a fool is to try to avoid one much as possible?