Why can't Tritheism or Social Trinitarianism be true, where 'God' is a generic abstract essence while there are three personal essences ?
A Catholic site speaks against this:
It Rejects the Fourth Lateran Council (1215)
The Fourth Lateran Council (Twelfth Ecumenical) declared:
We, however, with the approval of this sacred and universal council, believe and confess with Peter Lombard that there exists a certain supreme reality, incomprehensible and ineffable, which truly is the Father and the Son and the Holy Spirit, the three persons together and each one of them separately. Therefore in God there is only a Trinity, not a quaternity, since each of the three persons is that reality — that is to say substance, essence or divine nature-which alone is the principle of all things, besides which no other principle can be found. This reality neither begets nor is begotten nor proceeds; the Father begets, the Son is begotten and the holy Spirit proceeds. Thus there is a distinction of persons but a unity of nature. Although therefore the Father is one person, the Son another person and the holy Spirit another person, they are not different realities, but rather that which is the Father is the Son and the holy Spirit, altogether the same; thus according to the orthodox and catholic faith they are believed to be consubstantial. For the Father, in begetting the Son from eternity, gave him his substance, as he himself testifies: What the Father gave me is greater than all. It cannot be said that the Father gave him part of his substance and kept part for himself since the Father’s substance is indivisible, inasmuch as it is altogether simple. Nor can it be said that the Father transferred his substance to the Son, in the act of begetting, as if he gave it to the Son in such a way that he did not retain it for himself; for otherwise he would have ceased to be substance. It is therefore clear that in being begotten the Son received the Father’s substance without it being diminished in any way, and thus the Father and the Son have the same substance. Thus the Father and the Son and also the holy Spirit proceeding from both are the same reality. … If anyone therefore ventures to defend or approve the opinion or doctrine of the aforesaid Joachim on this matter, let him be refuted by all as a heretic.7
and brings this argument:
It Entails a Tritheism which Reduces to Atheism or Unitarianism
Plantinga denies that each of the three divine persons is that reality (i.e. substance, essence or divine nature) which is God. Rather, according to Plantinga, each of the persons participates in the divine nature, which is abstract. For Plantinga, each person is God by participation in the generic divine nature, but that generic divine nature in which they each participate is not a being. Three non-beings, however, cannot together compose or constitute one whole being. Therefore, Plantinga’s model entails that the one communal being formed by the loving union of the three persons, is formed by the loving union of three beings. The three persons are three beings, which by their loving union compose a social unity. That raises the problem of tritheism, but that problem reduces to another.
Because they are three immaterial beings, they cannot have the same essence, for the same reason explained above regarding the angels(1). And that sets up the following dilemma: either none of the three persons has existence as his essence, or only one of them, at most, has existence as His essence. If none of them has existence as his essence, then all three are created, and hence none of them is God; each is an angel. Any position in which something that is treated as the entirety of God is actually a creature, is an idolatrous form of atheism.
But, if only one of the three persons has existence as his essence, then since, according to Plantinga, each of the three persons is a distinct being, the being of the one whose essence is existence is not the being of the other two persons. Therefore, since the other two persons have being, and since their essence is not existence, and since their being is not the being of the one whose essence is existence, it follows that they have their being from another ex nihilo. Hence, if only one of the three persons has existence as his essence, it follows that the other two persons were created ex nihilo, and are thus creatures. Tritheism in this way entails either atheism or unitarianism.
Basically, why must necessary "existence" be absolutely one? Can you demonstrate this? I do not accept absolute divine simplicity, so please don't go that route.
Why is the Catholic view correct over the social model?
It Makes Each Divine Person Differ in Species
When two beings each have the same nature, something else must differentiate them, because otherwise they would be one being, instead of two beings. For Aquinas, what differentiates one human being from another, is matter. Each human being has a distinct material principle which differentiates his instantiation of human nature from every other instantiation of human nature. Since angels are immaterial, there cannot be two or more angels of the same species; otherwise nothing would differentiate them. That is, nothing would make them two instead of one. Hence, according to Aquinas, each angel must differ in species from every other angel.4
The notion of sharing part of the same essence with some other being is equivalent to being members of the same genus, while differing in species. Take, for example, Aristotle's claim that the human is a rational animal. The genus, in that case, is ‘animal.’ The differentia [i.e. that which differentiates the species from every other species in that genus] is rationality, because rationality differentiates humans from everything else in the genus ‘animal.’ Every other animal species shares this genus with humans, but has something other than rationality as that which differentiates it from every other species in this genus. For example, saying that both humans and prairie dogs share part of the same nature or essence translates into humans and prairie dogs sharing the same genus, but differing in species. Being one in partial-essence is equivalent to being one in genus but distinct in species.
So Plantinga’s claim that each divine person has a two-part essence: a part that is shared with the other two divine persons, and a part that is unique to that person, is equivalent to each of the three persons being members of the same genus, but each differing from the other two persons in species. In other words, it places all three persons in the genus ‘deity,’ but places each of the three persons in a different species from that of the other two persons. In Catholic doctrine, by contrast, the Son is the offspring of the Father, God from God. And since like begets like, the Son is the perfect image of the Father, the very same numerical essence (not part of an essence) and being as the Father. And the Spirit likewise, proceeding from the Father and the Son, is the very same numerical essence and being with the Father and the Son, not sharing only a partial essence with the Father and the Son, but sharing the full and numerically same essence. For this reason, the implication that each of the three persons is a different species is incompatible with Catholic doctrine.
I am not convinced of the above (1), not because its false per se but I just don't understand it so well.
Any help would be appreciated.
Any other reasons to why necessary existence must be one, instead of three like tritheism or social models of the Trinity would help a lot.
This question is only for Catholics; I don't want a debate on whether the Catholic model is true or not, just for Catholics to demonstrate why they believe their model is true where God the Father, the Son and the Holy Ghost have numerically one intelligence, essence and will.