What do these passages teach? 1 Cor. 11:28-3128But let a man examine himself, and so let him eat of that bread, and drink of that cup. 29For he that eateth and drinketh unworthily, eateth and drinketh damnation to himself, not discerning the Lord's body. 30For this cause many are weak and sickly among you, and many sleep. 31For if we would judge ourselves, we should not be judged. (1 Corinthians 11:28‑31), the duty, need and value of each Christian testing himself by the solemn truth of the Lord's death expressed and confessed and enjoyed in His supper. How slur over sin of any kind, were it but levity in word or deed, in presence of that death in which it came under God's judgment unsparingly for our salvation? Nor is it enough to confess our faults to God or man, as the case may require; but as on the one hand we discern the body, the Lord's body, in that holy feast of which we are made free and which we can never neglect without dishonoring Him who thus died for us, so on the other hand we are called to examine ( or, discern) ourselves, scrutinizing the inward springs and motives of all, and not merely the wrong which appears to others. But this intimate self-searching, to which we are each called who partake of the Lord's Supper, is on the express ground of faith, and has no application whatever to an unbeliever. This last thought has been mischievously helped on by the rendering of "damnation" in verse 29, which verses 30-32 clearly refute-proving that the judgment in question is the discipline of sickness or death which the Lord wields over careless or faulty saints in positive contrast with the condemnation of the world.
As for the passage in 2 Cor. 13, the statement, "Examine yourselves, whether ye be in the faith," derives all its force from the certainty that those appealed to were in the faith, not in the least that they were uncertain. That they were in the faith through Paul's preaching ought to have been an unanswerable proof that Christ spoke through him; if Christ was not in them, they were reprobate; and was it for such to question his apostleship? Scripture never calls a soul to doubt, always to believe. But self-judgment is ever a Christian's duty; and our privileges, we being in ourselves what we are, only deepen the importance, as representing Christ, of dealing with ourselves truly and intimately before God, as well as reminding our souls habitually of the Lord's death and of its infinite and solemn import as shown forth in His Supper.
(From “Notes on 2 Corinthians,” by William Kelly. G. Morrish, London; 1882. P. 241-242)