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Allow me to present the argument:

(1) By definition God is that being which possesses every perfection.

(2) Existence is a perfection.

(3) Therefore, God exists.

Whether you agree is not important, I’m sorry. I’m trying to understand an objection to this argument. The following was written by Daniel Sampaio on a video titled "Objections to the Definitional Ontological Argument" published by Carneades:

The real problem with this argument seems to be much deeper than the objections you [Carneades] have pointed out. There is a great difference between existence as a property of an object, and existence as a property of a predicate. The argument shows that any entity which is a God has the property of being an existent entity; however, it does not show that there exists such an entity as a God. In other words, the first-order predicate of existence applies to every entity which is a God; however, the second-order predicate of existence must not apply to the property "… is a God".

Question #1: What does he mean by saying there is a difference between existence as a property of an object and existence as a property of a predicate?

Question #2: Also, what in the world is a second-order predicate?

Any explanation would be greatly appreciated.

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    I don't really get what this person is saying (and frankly I don't think this is my fault), but a second order predicate is a predicate whose argument can be another predicate. Put another way, given a domain of discourse, a second order predicate distinguishes a set of subsets from the domain of discourse. Where a predicate just distinguishes a subset. At least thats what I remember from logic class 6 years ago. – Tim kinsella May 27 '16 at 6:32
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    @Timkinsella I agree. This quote is just plain hard to follow. There at least to me to be much simpler ways to express the Kantian objection to the ontological argument (of which this appears to be a variant) than this pretzel. – virmaior May 27 '16 at 6:42
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    One big problem (thought not a problem for asking this question) is that the argument (1)-(3) at the top is not the ontological argument as Anselm (or many others) presents it. But it also is not not the ontological argument. It's the argument represented in a different ontology and logic than the context where it's made... – virmaior May 27 '16 at 6:42
  • Agree, although I never really thought about the fact that when you formalize the ontological argument, you have existence as a predicate, which is definitely pretty peculiar, if only because it collides with the quantifier. So maybe there's something there. – Tim kinsella May 27 '16 at 6:49
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    @Timkinsella right, "Consider the set of all things that do not exist..." Enter Raymond Smullyan and his pet demon on a leash. – user9166 May 27 '16 at 8:32
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Predicates

Predicates express properties.

First order predicates express properties of objects. For example, tall expresses the property of being tall, which is property of objects. So tall is a first order predicate (e.g. "John is tall").

Second order predicates express properties of properties. For example, a positive quality expresses the property of being a positive quality, which is a property of properties. So a positive quality is a second order predicate (e.g. "Courage is a positive quality").

The argument

An important thing to note here is that the guy you quoted takes the ontological argument to be talking about god as a predicate rather than as a name of an entity.

Thus he takes the conclusion of the argument to be the following:

∀x(G(x) → E(x))

That is, anything which is a god (G) has the property of existing (E). This formulation has existence (E) as a first order predicate, applying to objects. But then, he says, it doesn't apply to the predicate G, so it doesn't actually show that there is any x which is a G.

For the argument to show this, he says, it would have to use existence as a second order predicate, applying to the property expressed by G. Maybe something like this:

E(G)

Which can be taken to say "There is a thing which is god".

Finally, you can read about the problems of existence as a first order predicate in the SEP entry on Existence.


More on predicates

planet is a first order predicate, e.g. "Jupiter is a planet". Here planet applies to the object Jupiter. Now has 8 members is a second order predicate, since it applies to the predicate planet, e.g. "the predicate planet has 8 members", or "there are 8 planets".

Another example: prime number is a first order predicate, applying to numbers, e.g. "4 is not a prime number", while infinite is a second order predicate, e.g. "there are infinitely many prime numbers".

Similarly, you might say that exists is a second order predicate. So, for example, saying that unicorns do not exist is saying that the predicate unicorn is not instantiated (i.e. that there are no objects to which the predicate unicorn applies).

So a first order predicate says something about an object, while a second order predicate says something about a first order predicate.

  • I'm so sorry, but could you please provide some more examples of second order predicates? – London Jennings May 28 '16 at 8:51
  • Sure. See my edit, I've added more examples. – Eliran May 28 '16 at 8:52
  • Okay, so this guy is talking about “god” as a predicate, and second-order predicates apply to predicates. With this in mind, let me see if I understand: Since we defined it as we did in (1), once we say that such-and-such is a god, it must have existence. But it remains to be proven if there even is an object to which the property “is a god” applies. – London Jennings May 28 '16 at 9:26
  • @LondonJennings Yes, exactly. :) – Eliran May 28 '16 at 9:53
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What Sampaio is saying is that you can't get from 1 & 2 to 3.

1 and 2 allow you to say that any entity that is a god exists (first order predicate). But you can't then take the leap that such an entity exists because 1 and 2 don't establish that i.e. you can't go from saying that if there is an entity that's a god then it exists to a god exists (second order predicate).

If you follow the predicates correctly you end up with a conclusion 3 that's something like "if there is a God then God exists", which isn't the most Earth shattering statement.

  • But the mistake in logic has led to a lot of shattering, that is for sure! – user16869 May 29 '16 at 18:14
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Let's start from the end, and use yet a different approach to ontology.

A second order predicate is a predicate about predicates. If you think of a predicate as being identified by the set of things about which it is true, a second-order predicate is a set of predicates with given properties -- a collection of sets of things gathered according to their characteristics, where those characteristics satisfy a given rule.

If being square is a predicate, being true of all squares is a second-order predicate. We can model it as the set of all sets containing all squares, each of which also contains all the other things that have some single aspect in common with all those squares. For instance, the set of all rectangles qualifies, as does the set of all figures that can be described with a single dimension, since every square has each of these properties.

Presumably each perfection is a predicate, a set of perfect exemplars of some criterion. So we have the set of things that are God(s), and your claim is that this set lies within every set of things which are in some way perfect. This set of Gods then lies within the intersection of all these collections of things that are perfect in each given way.

You can state that this set exists, but how can it be non-empty? God would have to both be perfectly square and perfectly round, as these are both ways of being perfect. So you need rules about what kinds of predicates can conceivably apply to God, or we have lost already.

So what are those rules? Without establishing what 'perfections' to omit from consideration, your proof is incomplete. The set of Gods exists, but may still be empty. You can start relaxing the constraints, but there is no guarantee you will ever get to non-empty set without omitting some given, relevant 'perfection'.

Thus, the proof remains incomplete and does not establish what it claims.


There is a more direct, less mathematical, approach to this proof. One can follow one of the paths that leads to the Hindu neti-neti 'not this/not that' approach to God.

In what way is existence a perfection, more than nonexistence? Does the perfect circle actually exist? Can you produce one? How about an infinite straight line? Or a perfect romantic couple? Or a perfect meal?

It seems, to observation, that what most perfect things have in common is that they can only be partially attained. They do not fully exist in reality. To the extent they exist, it is in some place or manner beyond or outside reality.

So if perfections themselves tend to have the quality of not existing or lying outside reality, how can one deduce that existence in reality has anything whatsoever in common with perfection? One should more likely presume that nonexistence or lying outside reality, the thing we find in common in most perfect things, is the proper corresponding perfect state.

This observation casts your second premise into enough doubt that one should not pursue it further.

  • "The searing curve of beauty is a thought too bright to tell of, in fire and desire. Every meaning is a shell of polynomial perfection, never factored, not equated, in fluctuating fancy for perfection uncreated. Its crystallizing concept, agonizing extrospection, in perfect affirmation has denied its own perfection. Its perfectness imperfectly fulfills its own condition. Reality as realized refuses recognition." - Marilyn Hacker – user16869 May 29 '16 at 18:27

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