For by the law is the knowledge of sin: lest any should think that the law hereupon is useless, he goes on to show its use, but a quite contrary one to what they intended. It convinceth us of our guilt, and therefore is far from being our righteousness, Romans 7:7 1 Corinthians 15:56. Matthew Poole

That every mouth may be stopped - This is a proverbial expression, Job 5:15; Psalm 107:42. It denotes they would be thoroughly convinced; the argument would be so conclusive they would have nothing to reply. This is the conclusion of Paul's argument. The expressions refer not to the Jews only, but to all the world. Its meaning may be expressed, 'the Gentiles are proved guilty by their own deeds, and by a violation of the laws of nature. They sin against their own conscience. All the objections of the Jews have been silenced by appeals to their own Law. The mouths of both are stopped. The whole world becomes guilty before God.' Barnes Notes How can the whole world be guilty before God? The failure of the Gentiles is obvious (1:18-32), but here Paul is indicting the self-righteous Jews. Barnes Notes on the Bible

By the deeds of the law - The apostle had not been discussing the question whether they had yielded obedience to their ceremonial law, but whether they had obeyed the moral law. The conclusion was, that in all this they had failed, and therefore could not be justified by that Law. Barnes Notes on the Bible

Be justified in his sight; or be discharged in the court of heaven: the phrase is taken from Psalm 143:2. Matthew Poole