Possibility of the Anselmian God

Dx ≝ x is ontological determinate

Gxy ≝ x is greater than y

©… ≝ it is conceivable that…

g ≝ (ɿx)~©(∃y)Gyx

1. (∀x)[~♢(∃y)(y=x) ⊃ ~Dx] (premise)
2. (∀x)(~Dx ⊃ ©(∃y)Gyx) (premise)
3. ~Dg (IP)
4. ©(∃y)Gyg) (2,3 MP)
5. (∃x){~©(∃y)Gyx ∧ {[(∀z)~©(∃y)Gyz ⊃ (z = x)] ∧ ©(∃y)Gyx}} (4 theory of descriptions)
6. ~©(∃y)Gyμ ∧ {[(∀z)~©(∃y)Gyz ⊃ (z = μ)] ∧ ©(∃y)Gyμ} (5 EI)
7. ~©(∃y)Gyμ ∧ {©(∃y)Gyμ ∧ [(∀z)~©(∃y)Gyz ⊃ (z = μ)]} (6 Comm)
8. {~©(∃y)Gyμ ∧ ©(∃y)Gyμ} ∧ [(∀z)~©(∃y)Gyz ⊃ (z = μ)] (7 Assoc)
9. ~©(∃y)Gyμ ∧ ©(∃y)Gyμ (8 Simp)
10. ~~Dg (3-9 IP)
11. ~♢(∃y)(y=g) ⊃ ~Dg (1 UI)
12. ~~♢(∃y)(y=g) (10,11 MT)
13. ♢(∃y)(y=g) (12 DN)

Premise 1: Basic idea would be that if something is not possible, then it lacks a determinate nature, since, by explosion, anything could be asserted to be true of it, or not true of it.

Premise 2: Again, if something is not ontologically determinate, then it is always conceivable that there be something greater than it, since one could conceive of something similar to it but with some other determination of properties.

Thus, the Anselmian God is metaphysically possible, Q.E.D.