This is somewhat more technical than our usual posts here at Secular Outpost. However, we have always thought of SO as a serious site for serious thinkers, and not for the usual invective that pollutes too much of the Internet. So, here is my take on the possible worlds version of the ontological argument.
Possible Worlds Version:
1) “God” =df “a perfect being.”
Premise: Definition of “God.”
2) If a perfect being exists, then that being exists in every possible world.
Premise: Perfection cannot exist contingently, i.e. only in some possible worlds but not in others.
3) If a perfect being exists, then that being necessarily exists.
Conclusion: From 2; definition of “necessarily exists.” A necessary being is one that exists in every possible world.
4) If it is possible that a perfect being exists, then that being possibly necessarily exists.
Conclusion: From 3; principle of modal logic. If p ⊨ q, then ◊p → ◊q.
5) If it is possible that a perfect being exists, then that being necessarily exists.
Conclusion: From 4; principle of modal logic: ◊□p ⊨ □p (If something is possibly necessary then it is necessary.).
6) It is possible that a perfect being exists.
7) A perfect being necessarily exists.
Conclusion: From 5 and 6 by Modus Ponens.
8) God necessarily exists.
Conclusion: From 1 and 7; definition of “God.”
9) God exists.
Conclusion: From 8; principle of modal logic. □p ⊨ p (necessarily p entails p.)
The crucial premise here is #6. It sounds innocuous enough. Surely it would seem closed-minded to deny that it is even possible that a perfect being exists. However, if God is defined as a perfect being, and perfection cannot exist contingently, then if God exists, God exists necessarily. In symbols, (∃x) Gx → □(∃ x) Gx. Further, if it is possible that God exists, then it is possible that God exists necessarily: ◊(∃x) Gx → ◊□(∃x) Gx. However, by an accepted modal principle, if something is possibly necessary, then it is necessary. Therefore, if God possibly exists, then he necessarily exists: ◊(∃x) Gx → □(∃x) Gx. So, to concede that God even possibly exists is already automatically to concede that God necessarily exists! Anyone who disputes that conclusion will therefore want to resist conceding premise 6.
What, though, besides not wanting to accept the conclusion, would motivate us to reject premise 6? Consider the following proposition G:
G: There exist one or more possible worlds containing gratuitous evil.
“Gratuitous evil” is evil that is so bad that no all-powerful, all-good being would permit it. An all-good being would prevent it and an all-powerful being could prevent it. Therefore, any possible world containing an all-good and all-powerful being will contain no gratuitous evil. Conversely, any possible world containing gratuitous evil will not contain—and necessarily will not contain—an all-good, all-powerful being. Put another way, if there are possible worlds that contain gratuitous evil, then, necessarily, no perfect being exists in that world, since a perfect being, by definition, is all-powerful and all good. Since God is defined as a perfect being, then, necessarily, God does not exist in any possible world containing gratuitous evil.If you accept Proposition G, then you hold that there exist possible worlds (one or more) containing gratuitous evil. In such worlds, God necessarily does not exist. If God necessarily does not exist in such worlds, then it is not even possible that God exists in such worlds. But if there are possible worlds in which God necessarily does not exist, then it is not even possible that God exists in every possible world. Therefore, whoever accepts Proposition G, will reject premise 6, and so will reasonably reject the possible worlds version of the ontological argument.
Yet acceptance of Proposition G has even worse consequences for those who define God as existing in every possible world. If, by definition, if God exists, then he exists in every possible world, and if, necessarily, there are worlds in which God does not exist (i.e. he cannot exist in those containing gratuitous evil), then God does not exist! So, anyone who accepts Proposition G, and who also accepts the definition of “God” given in the above version of the ontological argument, may easily construct an argument disproving the existence of God!
So, then, the matter comes down to which of the following propositions you accept:
It is possible that a perfect being exists.
It is possible that gratuitous evil exists.
If you accept the first of these propositions, then the above argument proves the existence of God. If you accept the second of these propositions, then you do not accept the sixth premise of the above argument, and so you do not have to accept its conclusion. Further, if you accept the second of these propositions and you accept the definition of “God” as a being who, if he exists, must exist in every possible world, then you can disprove the existence of God so defined.
I regard it as eminently reasonable to accept the second of the above propositions. Prima facie it seems easy to imagine worlds in which the balance of evil over good is so great that no perfect being would permit the existence of such a world. For instance, it seems possible that a world might exist in which all sentient creatures, innocent or guilty, suffer eternal torment in the afterlife. In such a world, there is no salvation, and the innocent all suffer eternally along with the guilty. If such a nightmare world is possible, then it seems that no morally perfect and all-powerful being would exist in such a world. Hence, this would be a possible world in which God does not exist.