Monday, July 20, 2015

Plantinga's Modal Ontological Argument

This is part of my series on debugging the ontological argument.

Finally, after explaining modal logic basics, I'm going to talk about Plantinga's Modal Ontological Argument (MOA).1  This argument takes as its centerpiece the following premise: $$G \Rightarrow \square G\tag{1}$$ G is the statement "God exists".  This premise is actually quite clever, because it's hardly an assumption at all, but a definition of God.  In order for an object to be God, then the same object must exist in all accessible worlds.2

This definition of God has the funny property that it depends on our choice of frame.  Even if you would call an object God in one frame, you might not be able to call it God in another frame.  As I explained before, frames are simply constructions and it seems odd that whether we call something God should depend on something so arbitrary.  Nonetheless, this difficulty can be overcome if we simply specify a frame as part of God's definition.  I'm not sure if this is what Plantinga does, but I'll propose a reasonable frame.

In this frame, let's say that world w is accessible from u if w is a possible past or future of u, or some combination (such as the possible past of a possible future).  God is taken to be some object which can never start existing, nor stop existing.  Thus if God exists, then God exists in all accessible worlds.  This justifies the premise \ref{ref1}.

Since premise 1 can be justified as a definition, we must look to the rest of the MOA for problems.

Statement of the Modal Ontological Argument

If you understand S5 modal logic well enough, the MOA is actually quite obvious.  If God is possible, then God exists in a world which is accessible from here.  And therefore God is also here.  More formally:
$$G \Rightarrow \square G\label{ref1}\tag{1}$$ $$\Diamond G\label{ref2}\tag{2}$$ $$\therefore G\label{ref3}\tag{3}$$
For those interested, I also include the formal justification for conclusion \ref{ref3}.  In parentheses, I cite the premises and S5 axioms in use.
$$\Diamond \lnot G \Rightarrow \lnot G~(\ref{ref1})\label{3a}\tag{3a}$$ $$\square( \Diamond \lnot G \Rightarrow \lnot G)~(\mathrm{N~and~\ref{3a}})\label{3b}\tag{3b}$$ $$\square\Diamond \lnot G \Rightarrow \square \lnot G~(\mathrm{K~and~\ref{3b}})\label{3c}\tag{3c}$$ $$\Diamond \lnot G \Rightarrow \square \Diamond \lnot G~(\mathrm{s5})\label{3d}\tag{3d}$$ $$\Diamond \lnot G \Rightarrow \square \lnot G~(\mathrm{\ref{3c}~and~\ref{3d}})\label{3e}\tag{3e}$$ $$\Diamond G \Rightarrow \square G~(\mathrm{\ref{3e}})\label{3f}\tag{3f}$$ $$\square G~(\mathrm{\ref{3f}~and~\ref{ref2}})\label{3g}\tag{3g}$$ $$G~(\mathrm{T~and~\ref{3g}})\label{3h}\tag{3h}$$
 Yes, this is logically valid.  All that remains is to justify premise \ref{ref2}, the statement that God is possible.

Is God possible?

Not many people put stock in ontological arguments, and indeed, neither does Alvin Plantinga, even though he formalized the MOA.  Instead, Plantinga merely argues that it is reasonable to think that God is possible, and therefore it is reasonable to believe that God exists.3

The problem is that possibility can mean oh so many different things.  Even if we confine ourselves to Kripke semantics, the meaning of possibility depends on our precise choice of frame.  As for the proof, it requires a very particular meaning of possibility.  Even if it is reasonable to believe God is possible, is it reasonable in the sense required by the proof?

For example, if you take the definition of accessibility that I used in the introduction, then the MOA can be phrased like so: "God existed some time in the past or future.  Since God by nature cannot begin to exist or stop existing, then God also exists now."  This isn't very compelling, because saying that God existed at some point in the past or future is a pretty major assumption.  I am no more likely to believe the assumption than I am to believe the conclusion.

Or suppose we had a broader accessibility relation.  Suppose that world u is accessible from world w if and only if the two worlds obey the same physical laws.  In this case, it's not clear to me how premise \ref{ref1} follows from a natural definition of God.  But let's just take that definition and be careful about the verification of any object as God.  The MOA can be phrased like so: "God exists in some possible world with the same physical laws as ours.  By definition, God must exist in all worlds with those physical laws.  Therefore he exists in our world."

This isn't very compelling because it raises the question, how do we verify that a particular object is God?  In order to check the object against the definition of God, we have to check all accessible worlds to see whether the same object exists in those.  Maybe that's not so bad if we chose a frame where there aren't so many possible worlds, but there is one world we need to check no matter what: our own world.  But by the time that you've verified God exists in our world, what need is there for the MOA?  The MOA is in this interpretation "useless".

While Plantinga argues that the premise of the MOA is "reasonable", we can see that this has no value.  It does not matter whether we believe in gods or not; because the MOA is "useless", it should not advance our belief in God even a little bit.


1. I'm really considering the simplest possible form of the modal ontological argument, skipping some steps for aesthetic reasons.  Plantinga's full argument can be found here.

2. For the moment we'll just forget about the problem of how to say that two objects in two separate possible worlds are the "same".  We'll also forget about whether existence is a predicate.  These are serious problems, but purely technical ones that will be solved later.

3. One thing you may not have known about Plantinga is that he has written an awful lot about his "reformed epistemology", and I'm sure he has something very particular in mind when he says the premise is "reasonable."  I am not familiar with Plantinga's reformed epistemology and won't address it.