Do we answer the arguments of fools or not? How are Proverbs 26:4-5 to be understood? Proverbs 26:4-5 (NLT)   
  
  4 Don’t answer the foolish arguments of fools,
    &nbsp; &nbsp; &nbsp; or you will become as foolish as they are.  
  
  5 Be sure to answer the foolish arguments of fools,
    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; or they will become wise in their own estimation.  


At first read, these seem to be contradicting one another. Verse 4 tells us not to answer fools' arguments and verse 5 tells us do answer fools' arguments. With such a seeming contradiction, there is certainly some sort of official material out there that resolves this issue. According to the Wesleyan denomination (or as close as possible), how is this apparent contradiction resolved?
The common understanding of this apparent conflict is that the two verses describe two situations. 


One involves answering foolish questions...  The ones that are so foolish they don't deserve an answer.  The ones that people ask to get a rise out of you, or make a point, when there's no real possibility of an intelligent conversation.  The kind we quickly vote to close, downvote, and delete. Don't waste your time on them. 
The other addresses rebuking foolishness where necessary. When someone is honestly confused, or their foolishness is causing another to stumble or fall into error.


There's a relevant article on it at Ask the Pastors that spells this out better than I, with the following points.


Wisdom is only acquired by those who want it bad enough to work for it 
Every issue has many sides to it 
There are exceptions to some rules


and the conclusion of that article:


  In the proverb you cite, there is a time to answer a fool according to
  his folly and a time not to. Jesus talked about not casting our pearls
  before swine (Matthew 7). This describes a situation where the
  receiver of truth despises truth and it is not worth telling it to him
  and may even be dangerous to. Paul is an example, however, of
  answering a fool according to his folly when in 2 Corinthians he brags
  and boasts of his “accomplishments” to the Corinthians because it was
  the only way they could receive him (11:16; 12:11). The wise person
  will learn when and how to respond to the fool in the best way
  possible, though even then he or she may not know which way was best
  until he or she got into it.