BookMark,
I was looking in Martyn Lloyd-Jones' book Sermon on the Mount about this question of the law and the Christian. Chapter 19 covers the area you are struggling with. You might want to get a used copy to read more.
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Matthew 5: 17-20 5:17 Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.

5:18 For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.

5:19 Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.

5:20 For I say unto you, That except your righteousness shall exceed the righteousness of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven.



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...His teaching is in no way inconsistent with that of the law and the prophets; but secondly, it is very different from the teaching of the scribes and Pharisees. Our Lord has not come to make it easier for us or to make it in any sense less stringent in its demands upon us. His purpose in coming was to enable us to keep the law, not to abrogate it. So He emphasizes here that we must know what the law is, and then must keep it: "Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.' p. 199-200

...Now let me ask the question that is probably in your mind at this point. What then is our Lord teaching? Is He teaching salvation by works? Is He saying that we have to live a life better than that of the Pharisees in order to enter the kingdom? Patently not, because ' there is none righteous, no, not one'. The law of God given to Moses condemned the whole world; 'every mouth has been stopped'; all are 'guilty before God' and have 'come short of the glory of God'. Our Lord did not come to teach justification or salvation by works, or by our own righteousness. ' Very well,' says the opposite school; 'is He not teaching that salvation is by means of the rithteousness of Christ alone, so that it does not matter at all what we may do? He has done it all and therefore we have nothing to do.' Now that is the other extreme, and the other error. That, I argue, is an imposible exposition of this verse because of the little word 'for' at the beginning of verse twenty. It links with verse nineteen where He said, 'Whosoever therefore should break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.' He is emphasizing the practical carrying out of the law. That is the whole purpose of the paragraph. It is not to make it easy for us or to enable us to say, 'Christ has done it all for us and therefore it matters not what we do.' We always tend in our folly to consider things as antitheses which are meant to be complementary. Our Lord is teaching that the proof of our having truly received the grace of God in Jesus Christ is that we are living a righteous life.You know the old argument of course about faith and works. Some say the one is all important, some say the other. The Bible teaches that both these views are wrong: it is faith showing itself by works which is the mark of the true Christian.

Now lest you may think this is my doctrine, let me quote the apostle Paul, who of all others is the apostle of faith, and of grace. "Be not deceived', he says--not to the world, but to church members at Corinth--'be not deceived: neither fornicators, nor idolaters, nor adulterers...nor extortioners, shall inherit the kingdom of God.' It is no use saying, "Lord, Lord," unless you do the things that I command you", says Christ. It comes to this, that unless my life is a righteous life, I must be very careful before I claim that I am covered by the grace of God in Jesus Christ. For to receive the grace of God in Jesus Christ means not only that my sins are forgiven because of His death for me on the cross at Calvary's hill, but also that I have been given a new life and a new nature. It means that Christ is being formed in me, that I have become a partaker of the divine nature, that old things have passed away and all things have become new. It means that Christ is dwelling in me, and that the Spirit of God is in me. The man who has been born again, and who has the divine nature within him, is a man who is righteous and his righteousness does exceed that of the scribes and Pharisees. He is no longer living for self and his own attainments, he is no longer self-righteous and self-satisfied. He has become poor in spirit, meek, and merciful; he hungers and thirsts after righteousness; he has become a peacemaker. His heart is being purified. He loves God, yes unworthily, alas, but he loves Him and longs for His honour and glory. His desire is to glorify God and to keep and honour and fulfill His law. The commandments of God to such a man 'are not grievous'. He wants to keep them, for he loves them. He is no longer at enmity against God; but he now sees the holiness of the law and nothing so appeals to him as the living of this law and the exemplifying of it in his daily life. It is a righteousness that far exceeds the righteousness of the scribes and Pharisees. pp. 207-209