Jimmy Stephens was nice enough to share his outline on the sense of eternal generation and I think it is useful to illustrate the problem:
Sounds like you’re looking for the conceptual reduction argument, as it might be called.
There are senses of the word “begotten.”
Some of them overlap, or at least can overlap.
We can list those senses. We can then move down the list in order of abstraction, and discover that none of the typical meanings of begotten or sonship work.
It could refer to procreation.
It could refer to adoption.
It could refer to creation.
Senses 1 and 3 share the same basic problem. God is a se, self-existent, metaphysically independent. God cannot be created or birthed. Furthermore, the notion of procreation carries with it motherhood, which most Trinitarians would concede is a problem.
Sense 2, adoption, might imply that the Son is not intrinsically the Son but only economically the Son. We then run into a WLC style unidentified (unidentifiable?) Trinity. Even if it does not, this case is not supportable since the text indicates no legal or social framework in which the Son takes on the legal/social status of childhood.
What sense are we then left with to understand “begotten?”
There is a more abstract – in the sense that it is a more common reality not tied down by specifics of biology or law – but also more concrete – in the sense that it touches more deeply the heart of mankind – sense of sonship. That is the spiritual reality of filial love.
A biological father need not love his son. He can be, so to speak, a sperm donor, and just that. An adoptive father, likewise, can neglect, even abuse, his son. And of course, God cannot be created. But a filial devotion to the Father: that is the essence of sonship. Biological and legal sonship are mere occasions for this greater reality of familial intimacy.
The family unit in this way analogizes the Trinity. When we experience biological fathers loving their sons, that is good. When we experience legal fathers loving their sons, that is also good. When we experience spiritual fathers, people in God’s church who have taken to sons by God’s providential grace- that is best, greater than the aforementioned loves. The love is not identical to the Father’s love for the Son, but it is the closest earthly reflection, and there is no other way – as far as I can tell – to apprehend it.
This is where EG proponents drop the ball so badly. They make out the Trinity to be a logical conundrum to be solved with biology or legal guardianship. You can’t know the Trinity through trite definitions. You know the Trinity by interpreting through the mysterious affirmations of Scripture what experiential content God offers us in the world. Love is irreducible and can only be known by facing it with eyes enlightened by the Word, illuminated by the Spirit.
Observe the demonstration of God’s love at Calvary. The Father adopts sinners as His sons because we are bought with the blood of the original Son. The biology and law derive from the original love the Father and Son share. EG proponents have gotten the order entirely backwards. Procreation and adoption are instruments; they are the natural occasions God inbuilt by which all men are obliged to act like fathers/sons. But these are shadows in comparison to the love of spiritual fathers in the church who take on and disciple spiritual sons. And even this pales in light and depends wholly on the heavenly reality of the Son dying so many sons can see glory.
In this way, EG is like reading that God is like a hen and thinking that God’s protectiveness must be conceptual tied to beaks and called “beakdom.” It’s preposterous.