Is existence a predicate?

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

We will first consider a simple definitional ontological argument (henceforth referred to as the DOA).  I will address it in much greater detail than is strictly necessary, but part of the point is to build tools to understand more advanced ontological arguments later in the series.  The DOA is as follows:
$$\text{God is defined to be a being which has every perfection.}\tag{1a}\label{1a}$$ $$\text{Existence is a perfection.}\tag{1b}\label{1b}$$ $$\text{Hence God exists by definition.}\tag{1c}\label{1c}$$
This argument was famously refuted by 18th century philosopher Emmanuel Kant in his book Critique of Pure Reason.  Actually, my understanding is that Kant refutes it several times over with several distinct arguments.  Perhaps one of his most famous refutations is summed up in the statement, "Existence is not a predicate."

But what does it mean to be a predicate?  To understand what Kant meant, one would need to study Kant.  But here I will use a more modern concept of a predicate, taken from First-Order Predicate Logic (FOPL).

A predicate is a function that takes in a variable x, and outputs a proposition which may be true or false.¹  For example, consider the following predicates:

F(x) means "x is greater than 2."
L(x) means "x is alive."
G(x) means "x is a god."
P(x) means "x has all the perfections."

Note that the variable x can either represent a number, a person, or object, depending on the predicate being defined.  The domain of a predicate is the set of all things the predicate applies to.  In the case of the DOA, we are interested in the domain of all objects in the world

In this post, I will explain why existence is not a predicate, and the DOA is nonsensical.  But in a later post I will also explain why existence may be a predicate after all, so keep that in mind.

Why existence is not a predicate

What we would like is some sort of predicate E(x) that means "x exists".  The problem is, that the domain under consideration consists of all objects that exist.  In other words, E(x) is true for every x in its domain!  Extensionally,² the predicate E is equivalent to another predicate:

T(x) means "x is x"

Since E(x) is just tautologically true, it doesn't capture our meaning when we talk about existence.  And that's why existence is not a predicate.

The story doesn't end there though.  FOPL offers another natural way to think of existence: existence is a quantifier.  A quantifier specifies how many variables x there are for which a statement is true.  The two quantifiers we'll consider here are $\exists$ and $\forall$, which are defined as follows:

$\exists$x F(x) means "There exists at least one x in the domain for which F(x) is true."
$\forall$x F(x) means "For all x in the domain, F(x) is true."

There are lots of logical rules governing the allowable inferences with quantifiers, but they're mostly intuitive and details won't be necessary.

A note on definitions

To translate the DOA to logic, we first need a sense of what it means to define something, or to say that something is true by definition.  I'm used to defining things all the time in mathematics, but the trouble is you don't really "define" predicates in formal logic.  The predicates simply are.  What you can do is describe some or all of the properties of a named predicate.  For example, we could "define" G(x) with the statement:
$$\forall x~[G(x) \Leftrightarrow P(x)]\tag{2}\label{2}$$
This tells us that "by definition" G(x) implies P(x), and P(x) implies G(x).  If we replace G(x) and P(x) with plain English, then we could say, for example, that if x is a god, then x has all the perfections, by definition.

An important question: why is it that we can simply posit statement \ref{2} as true?  Maybe it's not true that G(x) means P(x)!  Certainly, if we're given plain English interpretations of G(x) and P(x), we may reject statement \ref{2} as simply untrue.

But in this series, I will begin all logical analysis by only considering the logical statements by themselves.  We are allowed to posit statement \ref{2} as true because there are some predicates G(x) and P(x) for which it is true.  Later on, if and when we're satisfied with the logic, we'll consider which English interpretations might fit the symbols.

On a side note, there are some definitions which are good, like statement \ref{2}, but there are also bad definitions.  Some definitions simply cannot be posited under any circumstance.  In general, distinguishing between good and bad definitions is hard.  Luckily, we'll only be dealing with good definitions until much later in this series.

Translating the Definitional Ontological Argument

We now have the tools to translate the DOA stated in \ref{1a}-\ref{1c} into logical statements.  Here is one translation, using existence as a predicate:
$$\forall x~[G(x) \Leftrightarrow P(x)]\tag{3a}\label{3a}$$ $$\forall x~ [P(x) \Rightarrow E(x)]\tag{3b}\label{3b}$$ $$\forall x~ [G(x) \Rightarrow E(x)]\tag{3c}\label{3c}$$
\ref{3a} and \ref{3b} are "good" definitions and can simply be posited.  The inference to \ref{3c} is valid.  However, it does not prove what we want to prove.  It just shows that if x is a god, then x exists.  So what?  We want to find a god, not just talk about its hypothetical properties once we find it.  The fact that E(x) is tautological for all x makes the conclusion even more worthless.  The conclusion we want is:
$$\exists x~ G(x)\tag{3c'}\label{4}$$
Unfortunately, \ref{4} simply does not follow from \ref{3a} and \ref{3b}.  The proof has fallen flat.

So here is an alternate translation of \ref{1b}, interpreting existence as a quantifier:
$$\forall x~ [P(x) \Rightarrow \exists y ~~y=x]\tag{3b'}\label{5}$$
This means "For all x such that P(x), there exists some object y which is identical to x."  This makes sense, but just as tautological as \ref{3b}.  It does not help us reach the desired conclusion.  You can try a few other translations, but I assert that none will work.  The DOA has gone nowhere fast.

In the subsequent posts we'll consider ways in which existence might be interpreted as a meaningful predicate after all.

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1. It's also possible to define predicates which are functions of two variables x and y, or even more variables.  For example L(x,y) could mean "x loves y".  I don't think this is ever relevant to ontological arguments.

2. Here I'm using the extension/intension distinction in philosophy.  The extensional meaning of a predicate is simply the complete set of objects that the predicate applies to.  The intensional meaning of a predicate is its conceptual content.  For example, "x was the president of the United States in 2015" and "x is Barrack Obama" are extensionally equivalent, but intensionally distinct.