The apostle continued to deal with the general subject of discipline in the church that he began in 1 Corinthians 5:1. He proceeded to point out some other glaring instances of inconsistency that had their roots in the Corinthians" lax view of sin. Rather than looking to unsaved judges to solve their internal conflicts, they should have exercised discipline among themselves in these cases. Gallio had refused to get involved in Jewish controversies in Corinth and had told the Jews to deal with these matters themselves ( Acts 18:14-16). Paul now counseled a similar approach for the Christians. [source][source][source]
"In this section Paul is dealing with a problem which specially affected the Greeks. The Jews did not ordinarily go to law in the public law-courts at all; they settled things before the elders of the village or the elders of the Synagogue; to them justice was far more a thing to be settled in a family spirit than in a legal spirit.... The Greeks were in fact famous, or notorious, for their love of going to law." [1][source]
"Roman society was notoriously litigious, and Corinth, with its rising class of nouveau riche, was even more so." [2][source]
". . . the congregation"s root problem lies in its lack of theological depth. It shames itself by not understanding itself as an eschatological community ("Do you not know that we are to judge angels?") and as a community redeemed by Christ." [3][source]
"Paul has not finished with the theme of church discipline in regard to sexual life; see vi12and chapter vii; but in 1 Corinthians 6:12 f. he had spoken of judgement [4], and this brings to his mind another feature of Corinthian life of which he had heard ..." [5][source]