Lib, I’ll try to make my objections explicit. I have at least three objections, and I’ve been playing mix and match in successive posts. I’ll present the obvious non-modal objection first, followed by my modal objection as clearly as I can state it.
Non-modal objection
Tisthammer’s first assumption is G->G. Things have already gone terribly wrong. It’s like watching a chess game where the first move is a4.
I would ask that you accept <>~G, that it is possible God does not exist. If one is not willing to admit (in their proof of God’s existence) that it’s possible he doesn’t exist, then they are clearly engaging in question-begging. Now, in all systems of modal logic, <>~G = ~G. If it is possible God does not exist, then it is not necessary that he exists. Now, using G->G, ~G, and plain old propositional modus tollens, we immediately conclude ~G. God does not exist.
The only way G->G does not lead immediately to God’s non-existence is if you insist that ~G is false. That is, by the excluded middle, you insist that G is true. So G->G compels you to accept the axiom “God necessarily exists” if you don’t want to be able to (trivially) prove that he doesn’t.
Clearly, the argument is still valid. The question-begging makes is suspicious, and unlikely to convince a non-believer.
Modal objection
There is a way out of the conundrum above. You can recognize that when Tisthammer claims G->G and when I claim <>~G, we are using two completely different notions of accessible possible world. Of course, this is exactly Tisthammer’s modal flaw: he fluidly varies his notion of accessible possible world to suit his needs.
The modal sentence G->G doesn’t mean “God has the greatest possible existence”. It’s really just a bit of syntax that doesn’t mean anything without a semantics. To give a semantics, we present a set of possible worlds, and an accessibility relation on possible worlds. Then, G->G means that if God exists in a world, He exists in all worlds accessible to that world. The semantics is important, because it determines the valid modal axioms.
Tisthammer’s flaw is that he never specifies the sort of accessible possible worlds he is talking about. There are several possibilities, all resulting in different valid modal axioms (or axiom schemas). Modal logic can be used to talk about what is necessary, what is known, what is believed, what ought to be, what hold locally, what holds now and in the future, what is provable in Peano arithmetic, what is true after the computer program terminates, and many others. Since Tisthammer doesn’t specify a system of modal logic, we’ll have to try to infer it.
First, he has G->G. If God exists in a world, then he exists in all worlds accessible to that world. Though this is purported to be a statement about what we mean by God’s existence, it also determines the kind of accessibility relation we can have on possible worlds. Assume two types of worlds, those with God and those without (either set can be empty). If we ever find ourself in a God-world, the only accessible worlds are other God-worlds. Once we get to a God-world, there is never an accessible non-God-world.
Next, he has <>G. God is possible in all worlds. That is, whether we’re in a God-world or a non-God-world, there is always an accessible God-world. Together, these two imply that if we’re in a God-world, we can only ever “get to” other God-worlds. If we’re in a non-God-world, we can always get to at least one God-world. This asymmetry bothers me a bit.
It’s not damning to Tisthammer’s argument, though. It is quite possible that the nature of God induces this asymmetry. God is always possible (“innocent until proven guilty”, according to Tisthammer), and his existence spreads out to all accessible possible worlds (perfect existence). The conclusion is that we must have an asymmetric accessibility relation.
Now, in step 5. of his proof, Tisthammer uses Becker’s postulate (that modal status is necessary), applied to ~G (which is <>~G). This is what I called a “dirty trick” before. We use accessibility to go from here (whether there is a God or not) to a world with God, then back to here; concluding God is here. Becker’s postulate can only be applied to <>A if the accessibility relation is symmetric. All this after carefully convincing us that the accessibility relation was asymmetric (otherwise we would not have blithely accepted <>G based on Tisthammer’s weak “innocent until proven guilty” argument).
This is my objection. His two assumptions seem to require an asymmetric accessibility relation, his use of Becker’s postulate requires a symmetric accessibility relation. Without Becker, no proof. The only way to insist on his axioms and insist on a symmetric transition relation is if he requires all possible worlds to contain God. That is, as in my first objection, the only way he can make the proof work is if he insists on the assumption that God exists.
Again, it’s obviously a valid proof, but Tisthammer fails to make clear that his proof only works if all possible worlds already contain God. So, if God exists then God exists. QED.