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.

Posted on July 21, 2012, in Arguments for God and tagged , . Bookmark the permalink. 5 Comments.

  1. [Two comments copy-pasted from Facebook.]


    Actually, the more I look at this argument, the more convinced I am that it is correct to object that premise (1) conflates epistemic with modal possibility: the possibility of the existence of a sound ontological argument is the assertion that there exists a possible world where there is such an argument, but to the extent that we contemplate such a possibility, it is because *for all we know* there could be such an argument (and not that the existence of a sound argument is a contingent truth). Presumably, if there is a sound argument, it exists in all worlds–or there is no such argument in any. This is not a question of demanding ‘positive proof’ of a possibility, but of acknowledging the type of possibility it is.

    The author writes: “Therefore, we default to the position that something is logically possible unless and until it can be shown to be impossible,” but even if I grant this, it does not follow I should treat this possibility “alethically”, as it were. We don’t suppose just any conjecture that would be necessarily true, if true at all, to be ipso facto possibly true in any alethic sense, but only in a restricted epistemic sense.

    For example, when I say that it is possible that there exists is a proof of the continuum hypothesis or the Goldbach conjecture, it is perfectly evident I am making an epistemic claim. Otherwise, by parity of reasoning, I should be able to take up this argument schema to ‘prove’ that there exists a truth-maker to these theorems. But this is clearly wrong.

    Possible mathematical theorems do not represent ‘concrete’ contingents that could be:* if they are true, they are necessarily true, and if false, necessarily false. In other words, to say of a proposition that is either necessarily true or necessarily false, that it is possible, is to say that we don’t know its status yet; otherwise we could say of any candidate necessary truth that has not been validated yet that it is a possible necessary truth, and therefore, by S5, a necessary truth. What a boon to mathematics! But what goes for Goldbach goes for Anselm. Perhaps I am wrong here: in what respects does the possibility of the existence of a proof of Goldbach differ from the possibility of the existence of a proof of God? How would you allow the latter to license the existence of God, but block the the former from showing that there exists two primes that sum to every integer?

    (*Qualifier: of course there is an alethic sense, too: for example, a mathematician could say, for instance, that it would be possible to derive a certain result using a different technique: “it is possible to prove this via constructive techniques, but the reductio is more efficient.” Here we have something like a genuinely S5 possible theorem. For an open question, as Oppy terms it, it remains unseemly, does it not?)


    Hi Daniel,

    Thanks for the response. I still have some worries, but I am willing to countenance the possibility of a principled distinction between the Goldbach and ontological cases, pending further fleshing out of the kind of restriction you have in mind! But really I had two separate problems, and I didn’t do a very good job of articulating them. Let me try again, with your kind indulgence.

    The first had to do with the nature of possibility in possible worlds semantics. For a possibility claim to be non-epistemic requires the postulated state of affairs to be genuinely contingent. Kripke’s examples are good indicators of the kind of thing he had in mind: Nixon might not have won the election; a throw of two die has 36 possible outcomes; Aristotle might not have been Alexander’s teacher; and so on. So to say that it is possible that the dice show ’12’ is to say that there is a possible world where both land 6, etc.

    But logical truths behave differently. It is of course true that because the Pythagorean theorem is necessary, it must be also possible: the truth of this theorem in all possible worlds implies its truth in at least one world. But the crucial difference is that the truth or falsity of necessary propositions can never be ‘local’: I can say, with my metaphysical hat on, that it is possible that the dice land 12, and it is possible that the dice do not land 12. If I look at the space of possible worlds, some are worlds where they dice land 12, some are worlds where they do not. Not so with necessary truths: metaphysically, it is not the case that in some worlds the Goldbach conjecture or Pythagorean theorem happens to be true, and in some worlds, it is not. And likewise with ontological arguments: if there exists a sound ontological argument, in exists (and is valid, sound, etc.) in all worlds or in none.

    So to say that it is possible that there exists a sound ontological argument is to say that there exists an a a necessary truth, that is, a truth in all possible worlds, for if there is a world with a sound argument it must exist in all worlds: otherwise it would not be a sound argument. But then the only way to truly say that it is possible that there exists such an argument is to be implicitly commited to the premise that there necessarily exists such an argument, since if it is (metaphysicaly) possible that it exists, then, because it is an a priori theorem, it must necessarily exist, that is, be available in all worlds. Otherwise we’d have worlds where the successful ontological argument is valid and worlds where it is not valid. Surely we would resist this conclusion!

    This is why I cannot but see the possibility claim as epistemic: it is ‘conceivable’ there could be a sound ontological argument in the sense that it is ‘conceivable’ that Cicero isn’t Tully, or that Hesperus isn’t Phosphorus, to someone not acquainted with the relevant facts. But that ‘conceivability’ does not answer to a genuine possible world.

    The second worry was that if I granted that the possibility in question was metaphysical, it would be difficult to block a general class of arguments that trade on the possibility of proofs to deduce the truth of conjectures. I chose the Goldbach conjecture because it makes a straight-forward existence claim: namely, the existence of a certain mapping between the set of prime numbers and the natural numbers. Define the predicates as follows:

    Ax: x is a mathematical argument concluding to the existence of a mapping between the prime numbers and the natural numbers.
    Cx: x is the conclusion of a mathematical argument that asserts the construction of a sufficiently defined function between the primes and the natural numbers.
    Oxy: x is sufficiently defined as a function between the primes and the natural numbers by proposition y.

    The trick is, of course, to distinguish between the proof for the Goldbach conjecture, and the postulated existence of a constructive mapping. I could (hypothetically) prove the conjecture using non-constructive means without offering a means to actually tell you what two primes some arbitrary number is the sum of. Nevertheless, the proof implies it exists.

    Obviously the possibility of a sound Goldbach proof does not allow us to infer the existence of the function in question; so the argument fails by counter-example, and the likely culprit is premise one. It is possible that the right kind of distinction can be drawn between the two examples, but it seems to me that the crucial similarity is that the arguments hypothesized to be possible would be necessary if true. That’s the problem, and it shows us that my first worry is correct.

    A quick question, though, as I look over your post: why is
    (13) ~ (Ex)Ax
    and not
    (13*) ~ (Ex)Ax

    Intuitively it seems to me that 13* is the better version, since we would want to claim that it is not the case that it is possible there exists a sound ontological argument, again because putting the scope within the implies that the argument is contingent-as-in-dice, which can’t be right…


    • Dear Nicholas McGinnis,

      Thanks for the comment. I’m posting my comment from Facebook (with some minor additions) to mark the beginnings of an answer:

      If I understand you correctly, you’re saying that my (1) must really be based on conceivability rather than possibility, and that this is further obscured by my (13), which makes it seem as though I am genuinely treating the ontological argument as a contingency. But, as you point out, if there is a sound ontological argument, then it would seem to be sound in all possible worlds (given that all of the premises are analytic or a priori in nature).

      If so, then I cannot rely on presumption to grant the logical possibility of the argument, as it would violate my own restriction on presuming possibility, i.e. that if the contradictory is granted as possible, there is no implication that there would be a world where both contradictions obtain. I think that is the basic thrust of your points, but correct me if I am wrong.

      Mathematical examples prove analogous in this case, since it is generally agreed that if there were an argument for the Goldbach conjecture, or some other unproven theorem, such an argument would contain only necessarily true premises.

      My initial motivation for the argument was that a sound modal ontological argument may very well be contingent, i.e. there are possible worlds where it is sound and possible worlds were it is not sound though perhaps valid. And that there being possible worlds where the argument is not sound would not be worlds where it is implied that God doesn’t exist. I have a couple of reasons for thinking this 1) it is arguable that not all a priori propositions are logically necessary (some examples can be found in the works of Kripke) 2) not all ontological arguments strive to contain only analytic statements (see Rescher, N., 1959, “The Ontological Proof Revisited”, Australasian Journal of Philosophy, 37: 138–48). But in light of your concerns, I would like to pause and think about it some more. Perhaps this would defang the argument, but I need to look into what kind of epistemic warrant can be given to justify the idea that a sound modal ontological argument would be sound in all possible worlds. I suspect that no warrant can be provided. Briefly here is why:

      1. Any epistemic warrant for justifying your claim “if there is a sound modal ontological argument, it is necessarily a sound ontological argument” would either come from counterfactual reasoning about sound modal ontological arguments and their attributes in worlds where they obtain, or from a proof that the antecedent is impossible (that the conditional is a counterpossible).

      2. If you provide me epistemic warrant based on counterfactual information, then by S5 you have proved that a sound modal ontological argument actually exists.

      3. But you admit that you have no reason to think a sound modal ontological argument exists in this world.

      4. If (3) is so, your epistemic warrant cannot be based on counterfactual reasoning.


      5. Your epistemic warrant must originate from counterpossible reasoning, that the conditional is true because the antecedent is impossible.


      6. If your epistemic warrant for claiming that my (1) is false is really based upon the assumption that there is no sound modal ontological argument, then unless you have evidence for this, you’ve implicitly begged the question against my argument.

      In other words,

      7. Either have proof that there actually is a sound ontological argument, or you have proof that it is impossible that there be a sound modal ontological argument–a feat that Oppy has not admitted to.

      Since I don’t think you have epistemic warrant of either variety, I think you cannot actually prove that sound modal ontological arguments cannot be contingently sound. In fact, I suspect that defining God in relevant and meaningful ways to any ontological argument will depend upon certain contingencies in God’s will. Perhaps modal ontological arguments are only sound in worlds where God’s will leads to the relevant kinds of actions and effects where God can be relevantly defined such that a proof is sound. These are conjectures at this point, I admit.

      As an aside, it is somewhat ironic that a potential problem with my proof is that I may need to provide reason to think a sound ontological argument would not be necessarily sound. Typically, advocates of the ontological argument busy themselves trying to prove something is necessary, not contingent!

      As for (13*), the reason why I didn’t formulate it that way is because it would contradict (1). And while it is still consistent with the conclusion that God exists, it would prove to be a defeater for my argument (giving me little motivation to want to include it within the argument!).


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