# The Ontological Argument From Transcendence 2.0

I’ve presented my own version of Anselm’s ontological argument here and I’ve also argued for an ontological argument using “more transcendent” rather than “greater” here. Combining the two, and refining the argument, I got this:

1. Something is an Anselmian God if and only if it is conceivable, nothing can be conceived of which is more transcendent, and it necessarily exists (definition Θ).

2. There is something conceivable such that nothing can be conceived of which is more transcendent (premise).

3. For all x, if the possibility of failing to conceive of x implies the possibility that x doesn’t exist, x is mentally dependent (premise).

4. For all x, if x is mentally dependent, there is something conceivable that is more transcendent than x (premise). Therefore,

5. An Anselmian God exists.

Let

Cx – x is conceived
Mx – x is mentally dependent
Txy – x is more transcendent than y
Θx- x is an Anselmian God, that is: (∀x){Θx ≝ ([♢Cx & ~(∃y)(Tyx & ♢Cy)] & ☐(∃z)(z=x))} (Def Θ)

1. (∃x)[♢Cx & ~(∃y)(Tyx & ♢Cy)] (premise)
2. (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx} (premise)
3. (∀x){Mx ⊃ [(∃y)(Tyx & ♢Cy)]} (premise)
4. (∀x){[♢Cx & ~(∃y)(Tyx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (IP)
5. ♢Cu & ~(∃y)(Tyu & ♢Cy) (1 EI)
6. [♢~Cu ⊃ ♢~(∃z)(z=u)] ⊃ Mu (2 UI)
7. Mu ⊃ [(∃y)(Tyu & ♢Cy)] (3 UI)
8. [♢~Cu ⊃ ♢~(∃z)(z=u)] ⊃ [(∃y)(Tyu & ♢Cy)] (6,7 HS)
9. [♢Cu & ~(∃y)(Tyu & ♢Cy)] ⊃ [♢~Cu ⊃ ♢~(∃z)(z=u)] (4 UI)
10. ♢~Cu ⊃ ♢~(∃z)(z=u) (5,9 MP)
11. (∃y)(Tyu & ♢Cy) (8,10 MP)
12. Tvu & ♢Cv (11 EI)
13. ~(∃y)(Tyu & ♢Cy) (5 Simp)
14. (∀y)~(Tyu & ♢Cy) (13 QN)
15. ~(Tvu & ♢Cv) (14 UI)
16. (Tvu & ♢Cv) & ~(Tvu & ♢Cv) (12,15 Conj)
17. ~(∀x){[♢Cx & ~(∃y)(Tyx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (4-16 IP)
18. (∃x)~{[♢Cx & ~(∃y)(Tyx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (17 QN)
19. (∃x) ~{~[♢Cx & ~(∃y)(Tyx & ♢Cy)] ∨ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (18 Impl)
20. (∃x){~~[♢Cx & ~(∃y)(Tyx & ♢Cy)] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (19 DeM)
21. (∃x){[♢Cx & ~(∃y)(Tyx & ♢Cy)] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (20 DN)
22. (∃x){[♢Cx & ~(∃y)(Tyx & ♢Cy)] & ~[~♢~Cx ∨ ♢~(∃z)(z=x)]} (21 Impl)
23. (∃x){[♢Cx & ~(∃y)(Tyx & ♢Cy)] & ~[☐Cx ∨ ♢~(∃z)(z=x)]} (22 ME)
24. (∃x){[♢Cx & ~(∃y)(Tyx & ♢Cy)] & [~☐Cx & ~♢~(∃z)(z=x)]} (23 DeM)
25. (∃x){[♢Cx & ~(∃y)(Tyx & ♢Cy)] & [~☐Cx & ☐(∃z)(z=x)]} (24 ME)
26. [♢Cu & ~(∃y)(Tyu & ♢Cy)] & [~☐Cu & ☐(∃z)(z=u)] (25 EI)
27. ~☐Cu & ☐(∃z)(z=u) (26 Simp)
28. ☐(∃z)(z=u) (27 Simp)
29. [♢Cu & ~(∃y)(Tyu & ♢Cy)] (26 Simp)
30. [♢Cu & ~(∃y)(Tyu & ♢Cy)] & ☐(∃z)(z=u) (28,29 Conj)
31. Θu (30 Def Θ)
32. (∃x)Θx (31 EG)

[Update 11/9/204] I’ve noticed that some did not understand why if possibility that failing to conceive x implied that x possibly didn’t exist, then a greater could be conceived than x.  I’ve tried to make this more explicit by explaining this in terms of mental dependence.  Here, a concept is not an abstract object, but an object in the mind.