The second premise is ambiguous in the sense that the word, “God” implies the totality of Jehovah, who is one in essence, and three in Person, not just, "the Second Person of the Trinity", and it also implies His eternal power and divine nature.
That ambiguity is also implicit in the third premise so that the conclusion “mother of God” gives the misleading impression that Mary preceded the Eternal God, which is impossible.
Jesus is God, who became man. Mary is the mother of Jesus according to the flesh (Romans 9:5) However, according to His divine nature He did not have a physical mother.
If Mary is the "Mother of God", how then is Jesus a descendant of David as it says in Matthew 1 and Luke 3 and Romans 9:5?
The conclusion of the syllogism is superficial in that it states only a partial truth and consequently it is misleading. If the syllogism added the word, "incarnate" after the word, "God" in both instances I wouldn't have a problem with it because it would eliminate some of the ambiguity.
Scripture itself is much clearer on the subject than the syllogism.
Cordially,
The precision of subtlety of what we believe is presented in a prayer recited most Sundays of the year:
"It is truly meet and just, right for our salvation, that we should at all times and in all places, give thanks unto Thee, O holy Lord, Father almighty, everlasting God; Who, together with Thine only-begotten Son, and the Holy Ghost, art one God, one Lord: not in the oneness of a single Person, but in the Trinity of one substance. For what we believe by Thy revelation of Thy glory, the same do we believe of Thy Son, the same of the Holy Ghost, without difference or separation. So that in confessing the true and everlasting Godhead, distinction in persons, unity in essence, and equality in majesty may be adored."
That is what Catholics believe. Now you know.
Another way to present it. HTH:
Whosoever will be saved, before all things it is necessary that he hold the Catholic faith. Which faith except every one do keep whole and undefiled; without doubt he shall perish everlastingly. And the Catholic faith is this: That we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons; nor dividing the Essence. For there is one Person of the Father; another of the Son; and another of the Holy Ghost. But the Godhead of the Father, of the Son, and of the Holy Ghost, is all one; the Glory equal, the Majesty coeternal. Such as the Father is; such is the Son; and such is the Holy Ghost. The Father uncreated; the Son uncreated; and the Holy Ghost uncreated. The Father unlimited; the Son unlimited; and the Holy Ghost unlimited. The Father eternal; the Son eternal; and the Holy Ghost eternal. And yet they are not three eternals; but one eternal. As also there are not three uncreated; nor three infinites, but one uncreated; and one infinite. So likewise the Father is Almighty; the Son Almighty; and the Holy Ghost Almighty. And yet they are not three Almighties; but one Almighty. So the Father is God; the Son is God; and the Holy Ghost is God. And yet they are not three Gods; but one God. So likewise the Father is Lord; the Son Lord; and the Holy Ghost Lord. And yet not three Lords; but one Lord. For like as we are compelled by the Christian verity; to acknowledge every Person by himself to be God and Lord; So are we forbidden by the catholic religion; to say, There are three Gods, or three Lords. The Father is made of none; neither created, nor begotten. The Son is of the Father alone; not made, nor created; but begotten. The Holy Ghost is of the Father and of the Son; neither made, nor created, nor begotten; but proceeding. So there is one Father, not three Fathers; one Son, not three Sons; one Holy Ghost, not three Holy Ghosts. And in this Trinity none is before, or after another; none is greater, or less than another. But the whole three Persons are coeternal, and coequal. So that in all things, as aforesaid; the Unity in Trinity, and the Trinity in Unity, is to be worshipped. He therefore that will be saved, let him thus think of the Trinity.
Furthermore it is necessary to everlasting salvation; that he also believe faithfully the Incarnation of our Lord Jesus Christ. For the right Faith is, that we believe and confess; that our Lord Jesus Christ, the Son of God, is God and Man; God, of the Essence of the Father; begotten before the worlds; and Man, of the Essence of his Mother, born in the world. Perfect God; and perfect Man, of a reasonable soul and human flesh subsisting. Equal to the Father, as touching his Godhead; and inferior to the Father as touching his Manhood. Who although he is God and Man; yet he is not two, but one Christ. One; not by conversion of the Godhead into flesh; but by assumption of the Manhood by God. One altogether; not by confusion of Essence; but by unity of Person. For as the reasonable soul and flesh is one man; so God and Man is one Christ; Who suffered for our salvation; descended into hell; rose again the third day from the dead. He ascended into heaven, he sitteth on the right hand of the God the Father Almighty, from whence he will come to judge the living[17] and the dead. At whose coming all men will rise again with their bodies; And shall give account for their own works. And they that have done good shall go into life everlasting; and they that have done evil, into everlasting fire. This is the catholic faith; which except a man believe truly and firmly, he cannot be saved.
The term “God” in the title “Mother of God” has ALWAYS meant “the Second Person of the Trinity.”
ALL of the “ambiguity” you write about is totally imaginary. It stems from ignorance, and from having been subjected to propaganda.
Christians have always called Jesus “God.” They never meant that Jesus is the Trinity. Christians have always called Jesus “the Son of God,” by which they never meant that he is the Son of the Trinity. They meant that he is the Son of the Father.
Mary is the mother of God because she is the mother of Jesus, who is God.
To object to calling Mary “Mother of God” on the grounds that she is not the mother of the Trinity is as relevant as objecting: “Mary cannot be the ‘Mother of God’ because fire engines are red.”