Mother of God
A woman becomes a mother when she gives the child her genes, bears the child in her womb, and gives birth to the child. Since this was the case with Mary and Jesus, Mary is the mother of Jesus. Since Mary is the mother of Jesus, and Jesus is God, Mary is the mother of God.
Some Protestants argue against this syllogism by offering a syllogism of their own: Mary is the Mother of God, and God is a Trinity; therefore, Mary is the Mother of the Trinity. This, they say, is a similar syllogism, but since it is false, it offers proof that the Catholic syllogism is false. Of course, this syllogism fails because the premises do not connect. They use "God" in one premise to refer to the second person of the Trinity--Jesus. In the other premise, they use "God" to refer to Father, Son and Holy Spirit. Therefore, instead of the Catholic syllogism, where "God" in both cases refers to Jesus alone and you have an "A is B, B is C, therefore A is C" scenario, you have, in the Protestant syllogism an "A is B, C is D" scenario, which proves the syllogism false.
Protestants argue by saying that since Mary only gave Jesus His human nature but not His divine nature, she cannot be called the mother of God. But women do not give birth to natures, they give birth to persons. Mary gave birth to a person, a divine person, therefore she is the mother of that person–God, the Second Person of the Trinity. After all, Elizabeth called Mary "the mother of my Lord," not "the mother of my Lord's human nature" (Luke 1:43). Holding the Protestant position amounts to Nestorianism, which claimed that Mary was only the mother of Jesus and not the mother of God, because Jesus was a fusion of two persons–one human and one divine.
Protestants also argue that Mary cannot be the mother of God because God is older. They also say that if Mary is God's mother, than Anne and Joachim are God's grandparents, while David would be God's forefather. This, they say, is ridiculous.
First of all, Mary is not the mother of God in the sense that she is older, or that she gave Jesus his divinity. She is God's mother only in the sense that she bore God in the womb, and gave Jesus His human nature. Secondly, it is not ridiculous to say that Joachim and David are forefathers of God. Since they are forefathers of Jesus, and Jesus was God, this means they were forefathers of God. Jesus "was descended from David according to the flesh" (Rom. 1:3), Mary being descended from David. This also does not mean that David and Joachim are greater or older than God–rather, it describes their blood relationship to Jesus Christ as the human manifestation of God.
The list of Church Fathers that called Mary "Mother of God" is remarkable. These include Irenaeus (Against Heresies, 5:19:1 [A.D. 189]), Hippolytus (Discourse on the End of the World 1 [A.D. 217]), Gregory the Wonderworker (Four Homilies 1 [A.D. 262]), Cyril of Jerusalem (Catechetical Lectures 10:19 [A.D.350]), Ephraim (Songs of Praise 1:20 [A.D. 351]), Athanasius (The Incarnation of the Word of God 8 [A.D. 365]), Ambrose of Milan (The Virgins 2:2 [A.D. 377]), Jerome (Against Rufinus 2:10 [A.D. 401]), Gregory of Nazianz (Letter to Cledonius the Priest 101 [A.D. 382]), Cyril of Alexandria (Third Letter to Nestorius [A.D. 430]).