Making sense of existence

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

In a previous post, I explained why existence is a meaningless predicate in First-Order Predicate Logic (FOPL).  If E(x) is a predicate that means "x exists", then E(x) is true over its entire domain, since the domain consists of exactly those things that exist.

On the other hand, even singing socks can think of things that don't exist.  What gives?

To some extent, we can understand "things that don't exist" with quantifiers.  For example, the fact that there are no monks singing chants in tight leather pants can be stated as follows:
$$\lnot \exists x F(x)\tag{1}\label{ref1}$$
where F(x) means "x is a monk singing chants in tight leather pants".  However, the logical language seems rather constrained compared to normal language.

This raises the question, is there some logical system that provides better ways to talk about things that don't exist?  Indeed, I will suggest two ways.

Second-Order Predicate Logic

Second-Order Predicate Logic (SOPL) is distinct from FOPL because we can now apply quantifiers to predicates.  For example, we can now have statements like:
$$\exists T ~\forall x ~T(x)\tag{2}\label{ref2}$$
This means "There exists a predicate T which applies to all objects."

Furthermore, we also have second-order predicates.  A first-order predicate takes a variable and outputs a proposition.  A second-order predicate takes a first-order predicate, and outputs a proposition.  In particular, let us define the second-order predicate S:
$$\forall F~ [S(F) \Leftrightarrow \exists x F(x) ]\tag{3}\label{ref3}$$
Basically, the predicate S(F) means "There exists some x for which F(x) is true."  S, in other words, is the existence predicate.   But it's not a first-order predicate, it's a second-order predicate.

We can now attempt to translate the DOA into SOPL:
$$\forall x ~[G(x) \Leftrightarrow P(x)]\tag{4a}\label{4a}$$ $$\forall x~ [P(x) \Rightarrow S(=_x)\tag{4b}]\label{4b}$$ $$\exists x~ G(x)\tag{4c}\label{4c}$$
Here, $=_x$(y) is the predicate that means "x is identical to y".  So \ref{4b} means "For all x such that P(x), there exists an object that is identical to x."

Unfortunately, \ref{4b} is tautological,¹ and the conclusion, \ref{4c}, does not follow, and the proof is invalid.

The conceptual ontological argument

Another solution to the problem is to consider a larger domain.  Previously, the domain of the predicates was the set of all things that exist.  What if we expanded the domain to include some things that don't exist?  We can now define the predicate E(x) to mean "x is part of the old domain", or "x exists in the real world".  In our new expanded domain, E(x) is no longer true for every object.

This leads to a new version of the ontological argument, which I'll call the Conceptual Ontological Argument (COA):
$$\forall x ~[G(x) \Leftrightarrow P(x)]\tag{5a}\label{5a}$$ $$\forall x~ [P(x) \Rightarrow E(x)\tag{5b}]\label{5b}$$ $$\exists x~ [ G(x)\wedge E(x)]\tag{5c}\label{5c}$$
This proof is almost enough to prove that God exists.  We need just one more premise:
$$\exists x G(x)\tag{6}\label{ref6}$$
Premise \ref{ref6} does not assume that God exists, since recall that we are using an expanded domain of objects.  For the moment, let's call the expanded domain "foo".  So the premise of the argument is merely that God is foo.  But what does that mean?

There are in fact some constraints on what things are foo.  For instance, you cannot have a foo object which is both red and not red.  And maybe there are other unknown constraints.

One philosophical interpretation of "foo" is that objects which are foo are "conceivable".  Of course, the meaning of conceivability is extremely contentious among philosophers.  For example, in a survey of philosophers, they found that 59% of philosophers believe p-zombies² are conceivable, 16% believed them inconceivable, and 25% said something else.  And if you look up conceivability, you find modern philosophical articles explaining all different kinds of conceivability.  This is a problem I cannot hope to solve!

Nonetheless, interpreting "foo" as "conceivable" provides some framework, however ambiguous, to understand the COA.  With this interpretation, the argument seems to say, if we can merely conceive of God, then God exists.  Isn't that interesting?

The objection I raise is, suppose we conceive of some object x.  x has all the properties we typically ascribe to God.  He created the world, he talked to a few prophets, he sacrificed his son, etc.  Last but not least, in our imagined version of the real world, this x exists in all his glorious ontological power!

It seems reasonable to conclude that the object that we are conceiving is in fact God.  But let's actually check whether G(x) is true.  If G(x) were true, then x would have all the perfections, and one of the perfections is that x exists in the real world (and not merely that x exists in our imagined version of the real world).  To check if this is true, we have to look at the real world and see if x exists.  Until we verify the existence of x, we can only say that x is very god-like, not that x is God in the sense of G(x).³

We're used to conceiving things, and the things we conceive of have exactly the predicates we conceive them to have.  But it is not true that given any predicate, we can automatically conceive of an object with that predicate.  In particular, it seems that we don't really have control over the predicates related to existence.

Here is another predicate H which may or may not be conceivable:
$$\forall x~[H(x) \Leftrightarrow \exists y [F(y) \wedge E(y)] ]\tag{7}\label{ref7}$$
Statement \ref{ref7} defines H(x) to mean "There exists in the real world a monk singing chants in tight leather pants."  This is an odd predicate, since H(x) just means the same thing no matter what x is. H is a completely sensible predicate, and has no contradictions.  But is H conceivable?  We cannot know, until we demonstrate the existence of the monk.

Similarly, in order to show that a god exists, we first need to show that a god exists.

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1. In fact, \ref{4b} is equivalent to statement (3b') from the previous post.  It seems that going from FOPL to SOPL really didn't help at all.

2. A p-zombie is a philosophical zombie, a being which behaves like us but which does not have conscious experience.  It's not remotely relevant to ontological arguments.

3. One possible resolution is to simply reject the equivalence of G and God.  But the ontological arguments are only valid if we use the predicate G, so this amounts to rejecting the ontological argument.