Status of Modal Logic
The first obvious issue is that modal logic is a formal system of axioms, and as such it tells us nothing about the real world unless we can establish that key features of the real world are correctly captured by the formal system. For example, Euclidean geometry is also a formal system, but we can only apply it to the real world in certain highly limited cases and even then the application is not precise.
Since modal logic is about possible alternative worlds, it seems unlikely that it could ever be adequately established to conform closely enough to reality to be a reliable guide to anything.
Definition of “God”
The post presents the premises of the argument in a misleading order; first discussing the possibility of God and then introducing properties to attach to the possible-God. For now, we need to skip the first premise and look at those properties.
All modal ontological arguments for God rely on introducing a clause of the form “if God exists in some possible world, then he exists in every possible world (i.e. he exists necessarily)”. Plantinga's argument does that in two steps, but the details don't really matter; what does matter is that the necessity operator is being smuggled in as part of a property in the definition of God.
This means that when the premise “it is possible that God exists” shows up, the term “God” is hiding a necessity operator inside it. So the premise we're really being asked to accept is, “it is possible that (an entity whose possibility implies its necessity) exists”. This form makes it clear that the entire argument is contained in this premise.
It is possible that God doesn't exist.
The above statement, taken as a premise of the argument, must be just as plausible as the “it is possible that God exists” premise. The only way to argue that the premise is implausible is to claim that God is necessary, which would be begging the question.
But if we substitute this premise, then:
- It is possible that God doesn't exist.
- If God exists in some possible world, then God exists in all possible worlds.
- (from 1, with ⋄X → ¬◻¬X) It is false that God exists in all possible worlds.
- (modus tollens on 2+3) God does not exist in any possible world.
- (from 4, with ¬⋄X → ◻¬X → ¬X) God does not exist in the actual world.
Equivocation on ‘possible’
In modal logic, ‘possible’ means no more and no less than “exists in at least one possible world”. Proponents of modal arguments invariably ignore this, and try and substitute an alternative meaning such as “not known to be logically impossible” or “conceivable”.
The problem here of course is that we can conceive of impossible things, and we might not know that something is impossible while still not knowing that it is possible.