Revised Meta Modal Ontological Argument in QML
Several weeks ago I posted an ontological argument that I had hoped would overcome many of the powerful parody objections employed against the various Hartshorne-Malcolm-Plantinga modal arguments. Unfortunately, when I tried to symbolize it, an error became quite obvious. The lesson is that I should symbolize the arguments before posting them. This new version essentially reaches the same conclusion I had sought, but it makes use of a two-place predicate. It’s a simple solution, but the results are quite compelling, at least as far as I can tell (I look forward to potential criticisms).
But before I delve into the argument, I would like to address a worry recently raised on an excellent blog, Third Millennial Templar (the post is linked here). The concern is whether this argument somehow confuses or conflates epistemic possibility with the relevant modal possibility (metaphysical or broad-logical) needed in a modal ontological argument. And this worry is well-founded. After all, the motivation behind (1) is based upon Graham Oppy’s admission that it is an open question whether there is a sound ontological argument that is yet to be found. In other words, it is at least an epistemic possibility that a sound ontological argument exists.
Nonetheless, I don’t think that this meta modal argument conflates epistemic possibility with other modal possibilities. Rather, it makes use of epistemic possibility as only part of the background justification for (1). The rest of the justification comes from the fact that while assertions bear an onus, an epistemic possibility receives the benefit of the doubt for being logically possible. It would be ruinous to our modal epistemology if we demand positive proof for every logical possibility. Therefore, we default to the position that something is logically possible unless and until it can be shown to be impossible. The onus is on the one claiming impossibility, as that is the stronger claim. The problem with earlier versions of the ontological argument is that, while granting logical possibility to God leads to the conclusion that God exists, maintaining the proof commits one to special pleading so as to not permit the logical possibility that God does not exist. The benefit of the doubt must be granted to both the existence and non-existence of something, unless there are independent reasons to think that thing is necessary or impossible. Granting both possibilities to God results in a symmetrical anti-ontological argument that just as easily disproves God’s existence. Consequently, there is no reason to prefer the conclusion of the ontological argument to that of the anti-ontological argument from reason alone. So to claim that these earlier ontological arguments serve as disproof for their anti-ontological counterparts is unfounded. Had one started from the assumptions of the anti-ontological arguments, they would have found equal disproof for the premises of the ontological argument. But which argument holds logical priority? Neither!
This argument is different. While it still grounds the logical possibility of the first premise on the benefit of the doubt, it is not disrupted by symmetrically granting the possibility of its contradictory. Fair is fair! To push the argument further two varieties of parody arguments are considered, possibility is granted, and they are shown to be consistent with the conclusions of the meta modal ontological argument. So while older versions of the ontological arguments are blocked by symmetry, this argument is not. Again, we might be tempted to restrict the benefit of the doubt, or presumption of possibility, in light of this proof, but such a restriction seems a bit ad hoc and potentially destructive to modal reasoning generally. We’d have to propose a principle that prevents this argument, but still permits other varieties of modal proofs that rely upon benefit of the doubt based modal reasoning. But struggling to manufacture just the right restrictions is merely to assume a conclusion and to work backwards to prevent an argument that conflicts with that conclusion! This would be reminiscent of the old logical positivist project of developing a working verification principle that restricts meaning to only those propositions that positivists thought should be meaningful. Any restrictive principle they developed was either too restrictive, not restrictive enough, or self-defeating. All of this is to say that it is legitimate to assume the logical possibility of something that is epistemically possible, so long as one is also willing to grant the logical possibility of the contradictory. That is, one should presume possibility of something not known to exist actually or necessarily only if the possibility is granted to both the existence and non-existence of the possibility in question and there are no known reasons to think it impossible.
The argument is as follows…
Ax =df x is a sound modal ontological argument concluding to the existence of a sufficiently defined divine being.
Cx =df x is the conclusion of a sound modal ontological argument that asserts the existence of a sufficiently defined divine being.
Θxy =df x is sufficiently defined as a divine being by proposition y.
T1. ◊(∃x)Fx → (∃x)◊Fx (Barcan Formula)
T2. ◊□(∃x)Fx → □(∃x)Fx (S5 axiom)
1. ◊(∃x)Ax (premise)
2. ◊(∃x)Ax → ◊(∃y)Cy (premise)
3. (∀y)(◊Cy → ◊□(∃z)Θzy) (premise)
4. ◊(∃y)Cy (1,2 MP)
5. ◊(∃y)Cy → (∃y)◊Cy (T1)
6. (∃y)◊Cy (4,5 MP)
7. ◊Cu (6 EI)
8. ◊Cu → ◊□(∃z)Θzu (3 UI)
9. ◊□(∃z)Θzu (7,8 MP)
10. ◊□(∃z)Θzu → □(∃x)Θzu (T2)
11. □(∃x)Θzu (9,10 MP)
12. (∃z)Θzu (11, NE)
To prove no disruption on first level symmetry, we grant
13. ◊¬(∃x)Ax (premise)
And (13) is shown to be consistent insofar as one might defend this premise,
14. ◊¬(∃x)Ax → (∀y)(◊¬(∃z)Θzy ∨ ◊(∃z)Θzy) (premise)
To prove there is no disruption what might be called a second level symmetry, consider the following additions:
Bx =df x is a sound modal anti-ontological argument concluding to the non-existence of a sufficiently defined divine being.
Dx =df x is the conclusion of a sound modal anti-ontological argument that asserts the non-existence of a sufficiently defined divine being.
15. ◊(∃x)Bx (premise)
16. ◊(∃x)Bx → ◊(∃y)Dy (premise)
17. (∀y)(◊Dy → ◊□¬(∃z)Θzy) (premise)
18. ◊(∃y)Dy (15,16 MP)
19. ◊(∃y)Dy → (∃y)◊Dy (T1)
20. (∃y)◊Dy (18,19 MP)
21. ◊Dv (20 EI)
22. ◊Dv → ◊□¬(∃z)Θzv (17 UI)
23. ◊□¬(∃z)Θzv (21,22 MP)
24. ◊□¬(∃z)Θzv → □¬(∃x)Θzv (T2)
25. □¬(∃x)Θzv(23,24 MP)
26. ¬(∃z)Θzv (25, NE)
27. ¬(∃z)Θzv · (∃z)Θzu (12,26 Conj)
It is not evident that (27) is a direct contradiction without the further premise,
28*. v = u (premise)
But there is no reason to presume this identity, at least there is no more reason to assume (28*) than there is to presume,
28′. v ≠ u (premise)
Further, we should take the fact that a contradiction would be generated by claiming identity between v and u as prima facie evidence that they are not identical. And so, this version of the meta modal ontological argument successfully demonstrates that there exists a being that is sufficiently defined as divine by some proposition. In other words, a divine being exists, which is shown by these last two steps,
29. (∃z)Θzu (27, Simp)
30. (∃x)(∃z)Θzx (29, EG)
That is, by (30) we have proved that there exist some x and some z such that z is sufficiently defined as a divine being by proposition x. Or, to put it simply, there exists some such being that can be sufficiently defined as a divine being. As far as I can tell, that means that God exists.
In future posts, I hope to defend the key premises in a bit more detail, along with my use of the S5 axiom and Barcan Formula. In the meantime, I would appreciate any challenges, criticisms, questions, or suggestions.