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The Third Commandment states: "You shall not take the name of the Lord your God in vain, for the Lord will not leave him unpunished who takes His name in vain" (Exodus 20:7). Although this commandment is typically understood to mean how we use the name of God in our speech, this is not primarily what is being commanded. Notice that the verse says "take" not "use" the name of the Lord in vain. While "using" God's name improperly is certainly implied, it is not the main point of the Third Commandment.
When we "take" the name of the Lord, we are making a vow. Similar to the language of modern wedding ceremonies, when we "take" God's name, as we would "take" our spouse, we are entering a binding agreement. At a wedding, the father "gives" a daughter and the groom "takes" a wife. Likewise, when we covenant with God, we are giving up our own name and taking His; we are literally taking God's name. When we do this, we are agreeing to certain covenantal stipulations. In fact, the verse indicates that God "will not leave him unpunished who takes His name in vain." This is serious business, so we better make sure we understand what it means to "take God's name in vain."
Words have power because they are infused with meaning. We might even say that words have "imputed" meaning. When Jesus spoke as "one having authority" (Matthew 7:28-29), He was speaking as one who had the power to impute meaning into the words that He was speaking. The Word of God has power because the God of the Word has power and authority. There is nothing magical or mystical about the words in the Bible, they are powerful and authoritative because of their transcendent source.
The same thing can be said about the "name" of God. God names Himself. Naming implies dominion and status. When God told Adam to name all of the animals, He was teaching Adam that the animals were under his charge. When parents name their children, they are showing that this child belongs to them and is a welcome part of the family. When we give our children a "first" name, we are assigning a unique identity to him or her. The last name, the one we don't get to choose, is the family name.
Historically, the Church has added a Christian name at baptism to be placed in front of the surname [family name], indicating that the covenantal tie supersedes the blood bond...This change in name means the covenant member has been transferred into a new family. It is called adoption. 
This means that when we "take" the name of the Lord, we are being adopted into His family. The tradition among the Jews to not speak the name of the Lord comes from the idea that He is so holy that even speaking His name is a blasphemy. They devised ways around this tradition however, as the sinful heart of man always does. You get a hint of this in Jesus' words in the Sermon on the Mount when He draws attention to their practice of making vows and oaths in the name of heaven, or earth, or Jerusalem. Jesus condemns this supposed "loophole" by commanding that they let their "statement be 'Yes, yes' or 'No, no'; anything beyond these is of evil." (Matt. 5:33-37). Completely missing the point of Jesus' teaching, modern Christianity has turned the Third Commandment into a prohibition against using God's name as a swear word.
Jesus' statement to let your yes be yes and your no be no is picked up two other New Testament authors. In 2 Corinthians, Paul writes: "But as God is faithful, our word to you is not yes and no" (2 Cor. 1:18). And in his short epistle, James writes: "But above all, my brethren, do not swear, either by heaven or by earth or with any other oath; but your yes is to be yes, and your no, no, so that you may not fall under judgment" (James 5:12).
We must realize that the Third Commandment is not only referring to words that proceed out of our mouths. We can also "take God's name in vain" without ever speaking a single word; in fact, this is probably more often the case. When we "take" God's name, we are taking an oath of allegiance to love Him with all our heart, soul, strength, and mind. When we violate this oath of allegiance, we are "taking God's name in vain."
Knowing that His audience had turned the law of God into a list of dos and don'ts, Jesus cuts to the real issue in the Sermon on the Mount. Throughout His monologue, Jesus contrasts the tradition of the first-century legalists (using the phrase "You've heard it said...") with the actual intention of the law (using the phrase "But I say to you..."). When He gets to the part about "not making false vows" (Mt. 5:33-37), He directly attacks their tradition of using heaven, earth, or Jerusalem as the authoritative object of their faithfulness to do what they agreed to do. In other words, the Jews had gotten in the habit—out of their fear of misusing God's name and violating the Third Commandment—of making vows in the name of heaven, earth, Jerusalem, etc. Similar to modern children saying "I swear on my mother's grave I will not tell this secret," the Jews had developed a tradition that appeared to not violate the Third Commandment, yet still gave their vow some credibility. But Jesus exposes their tradition for the folly that it was by making the point that heaven, earth, and Jerusalem all belong to God anyway.
The primary point of Jesus' teaching is actually two-fold. He first draws attention to the fact that by swearing on these other objects, his hearers were essentially admitting that their own words were worthless. If the tradition of swearing on these objects was prevalent enough to be included in His sermon, it must mean that a general sense of suspicion was present among that society. In an attempt to have some semblance of honesty in a dishonest culture, they began swearing on the "higher" things that they held in common.
The second aspect of Jesus' teaching however is the more important one. After condemning their "heaven, earth, and Jerusalem" oaths in verses 34 and 35, He moves into the personal space of each of his listeners (and us as readers). "Nor shall you make an oath by your head, for you cannot make one hair white or black. But let your statement be, 'Yes, yes' or 'No, no'; anything beyond these is of evil" (vv. 36-37). The issue of authority looms large here. Jesus isn't about to leave the door open for his hearers to come up with a different object to swear upon. Bringing it to the individual level, Jesus makes the point that we do not even own ourselves, which means that we don't even possess the authority to swear on our own lives. God not only owns heaven, earth, and Jerusalem, He owns us too. It is for this reason that the only option we have is to let our "yes be yes," and our "no be no;" we have no authority to say anything otherwise. James makes this point clear when he writes:
Do not speak evil of one another, brethren. He who speaks evil of a brother and judges his brother, speaks evil of the law and judges the law. But if you judge the law, you are not a doer of the law but a judge. There is one Lawgiver, who is able to save and to destroy. Who are you to judge another? Come now, you who say, “Today or tomorrow we will go to such and such a city, spend a year there, buy and sell, and make a profit”; whereas you do not know what will happen tomorrow. For what is your life? It is even a vapor that appears for a little time and then vanishes away. Instead you ought to say, “If the Lord wills, we shall live and do this or that.” But now you boast in your arrogance. All such boasting is evil. Therefore, to him who knows to do good and does not do it, to him it is sin. (James 4:11-17)
Notice that James connects boasting with evil, just as Jesus connected anything beyond yes and no as being evil in His sermon. At first glance, it may seem that Paul—in the 2 Corinthians passage—is violating this principle by tying what he said to the faithfulness of God, but he is actually affirming it. When the passage is read in its context, it is clear that Paul is speaking about the words that God gave him to write and speak. What Paul is really saying is that he is delivering the message that God gave to him, not words and meanings of his own creation. In other words, Paul is saying God's words are faithful because God Himself is faithful. In reality, Paul is making a clear distinction from his own words and the Words of God.
And in this confidence I intended to come to you before, that you might have a second benefit—to pass by way of you to Macedonia, to come again from Macedonia to you, and be helped by you on my way to Judea. Therefore, when I was planning this, did I do it lightly? Or the things I plan, do I plan according to the flesh, that with me there should be Yes, Yes, and No, No? But as God is faithful, our word to you was not Yes and No. For the Son of God, Jesus Christ, who was preached among you by us—by me, Silvanus, and Timothy—was not Yes and No, but in Him was Yes. (2 Cor. 1:15-19)
Christians should be known as people of their word, just as God is of His. We should never have to resort to dragging God into our promises to other men. We have no authority to do this in the first place; and in the second, if we feel compelled to do this, it is a judgment against us that we are not faithful and true. May God be pleased to make us faithful to His Word, which will in turn make us faithful to our own.