The two halves of this verse are rhetorical questions. These questions are designed to show emphasis and not to elicit an answer, as we have already mentioned. So these questions may be translated as strong positive statements.
Do you not know…?: this rhetorical question is equivalent to a strong statement, “Surely you know,” or “You have been taught that,” or “I am certain that you are aware of the fact that” (see comments on 5.6). Here, and perhaps also in verse 3, Paul is referring to a teaching that his readers have already received. The belief that God’s people would share in the final judgment is mentioned in Daniel 7.22, and more clearly in Matthew 19.28.
For a discussion of saints or “God’s people” (Good News Bible), see comments on 1 Cor. 6.1.
Will judge, here and in verse 3, can be a present tense in the Greek, “judge,” meaning “have the function of judging.” But the context, especially the verb “will inherit” in verses 9-10, suggests a future event.
World probably has here its neutral sense of “universe,” including angels (verse 3).
Revised Standard Version‘s And if does not show the logical connection between these two rhetorical questions. Good News Bible clearly shows this connection with “Well then, if….” One could also render this as “So, if…” or “In that case, if….”
In languages that do not naturally use passives it will be necessary for the translator to render if the world is to be judged by you as “if you are to judge the world,” as Good News Bible has done.
Are you incompetent is literally “are you unworthy?” The Greek suggests not only “unqualified to judge” but also “not having the moral capacity for judging.” A possible translation of this clause would be “Are you unqualified to judge small matters?” This rhetorical question could also be rendered as a statement; for example, “So then, if you are to judge the world, you are certainly qualified to judge small matters.”
To try trivial cases: the same word for cases is used in verse 4. The meaning may be either “to form even the least important law courts” (New American Standard Bible, and Barrett) or, more probably, “to try even the most unimportant lawsuits” (so Fee and all other translations consulted). One could therefore render the final clause as “you are certainly qualified to judge trivial (or, small) lawsuits.”
Quoted with permission from Ellingworth, Paul and Hatton, Howard A. A Handbook on Paul’s First Letter to the Corinthians, 2nd edition. (UBS Handbook Series). New York: UBS, 1985/1994. For this and other handbooks for translators see here .