Here are some recent thoughts that I have had about the topic of eternal generation. I was dialoguing with a Latin trinitarian. He stated that the Father possesses a property that causes him to emanate the Son.  So, naturally, I asked if the Son possesses that same property then it seems like he should emanate a son aswell.  This was his thoughts about that problem:

Every property of the divine essence is firstly a hypostatic property of the Father; but each property which is communicated is instantiated distinctly by the distinct hypostases.

Ergo, when you say “is THIS life-givingness, etc.” — it sounds as though you are presuming that each of the hypostases instantiates essential properties in exactly the same way. If you are not saying that, please clarify.

Again, it’s both. They possess generically the same essence, therefore generically the Same in Being; but the essential being of each Person is inextricable from the being of each other Person, therefore also numerically One in Being; *but not in some formless way with no Order of Subsistence*.

The problem with that answer is that it leaves the possibility that the Son can generate another person. The Trinity thus becomes contingently true. Jimmy Stephens successfully argued this in my mind:

If the Son avoids necessitating a Second Son because He instantiates the Father’s life differently than the Father, it still follows that the Second Son is made possible by the Son’s instantiation of the Father’s life.

P1. If the Son involuntarily emerges from the Father’s life, then the Father’s life entails the possibility of a Son. (actual -> possible)
P2. If the Son possesses the Father’s life, then the Father’s life possessed by the Son entails the possibility of a Second Son (quadrinity). (communication of numerically one property)
P3. If the Son involuntarily emerges from the Father’s life, then the Father’s life possessed by the Son entails the possibility of a Second Son. (From P1 & P2)
P4. If the Father’s life possessed by the Son entails the possibility of a Second Son, then God is possibly a non-Trinity.
P5. If God is possibly a non-Trinity, Christianity is false.
P6. If the Son involuntarily emerges from the Father’s life, then Christianity is false.
P7. If EG (version “nuh uh, that’s not EG”), then Christianity is false.

Another slight pass at an answer was that the Holy Spirit fulfills that for the Son. The Son and the Father are what the Holy Spirit proceed from. It is debatable whether generation is meant to be the same as spiration but it leaves the problem the dilemma still applies to the Holy Spirit. He has both the eternal life of the Father and therefore the ability to generate or spirate another person. It also seems like the doctrine of God being pure act also implies that this infinite regress does imply an infinite amount of persons for the Godhead.