A proof is a proof. — Thorongil

What's a proof then? A proof to me is undeniable evidence that something is the case. Such cannot be given in the case of God.

I can prove the Theorem of Pythagoras within a certain system undeniably. But I cannot undeniably prove the existence of God.

"For "Ga" to be a wff, doesn't a have to be an object in your domain of discourse?"

Vulcan rotates.
Pegasus flies.
Santa wears a red suit.
Etc. are false WFF where the subject terms do not exist.

"What sense can be made of asserting "Ga" if you don't already know that a exists?"

Pegasus exists, is a sensible wff that is false.

There are no true propositions that have non-referring names or non-referring descriptions as their subject.

If that's your definition of proof, why are you then not a skeptic?

Vulcan rotates.
Pegasus flies.
Santa wears a red suit.
Etc. are false WFF where the subject terms do not exist. — Owen

Not helpful.

Russell would take all of these as false, but not as predicating anything of Vulcan or Pegasus or Santa.

How do you predicate, truly or falsely, of an object that does not exist in your domain of discourse?

Pegasus exists, is a sensible wff that is false. — Owen

And its logical form is what?

There are no true propositions that have non-referring names or non-referring descriptions as their subject. — Owen

But on your view, "Pegasus does not exist" should be true, shouldn't it?

Actually, I suppose the sentence should be

'Exists is not a predicate'

since 'exists' is grammatically a predicate and 'existence' is not.

It seems odd to say that a grammatical predicate is not a predicate. We must mean a different sort of predicate. I think we mean a logical predicate, so that the statement, at the expense of making it longer, becomes:

'Exists is a grammatical predicate but not a logical predicate.'

The way I make sense of this is to observe that when we say 'Santa Claus does not exist' what we actually mean is 'No being that ever lived had the key properties ascribed to Santa Claus'. Then in this sentence, the logical predicate is IsSantaClausLike, which means 'has all the key properties ascribed to the legendary Santa Claus' and it is formally rendered as something like:

for all x, for all t, (Alive(x,t) --> NOT IsSantaClausLike(x))

Yeah that's Russell's solution, to take the name as an abbreviated description.

I think we also want a way to talk about fiction (hypothesis, supposition, etc) "in world." So there are two answers, say, to "Does Santa fly in a helicopter?" One is "No, because he doesn't exist," but another is "No, it's a sleigh pulled by flying reindeer." Both have their use.

A distinction between reality and fiction isn't really a good one, unless parsed out in a very subtle way. Telling a story about what happened to you yesterday makes no more or less use of abstraction and representation than fiction does. You ate one orange for breakfast in no way delivers an actual existing entity to me, nor is it clear that an abstract entity like "one" is ever a material thing, yet science is far more based in math than material entities. In representations of general principles than particular physical things, nor is it even remotely clear that we come close to ever being able to represent a real entity to ourselves in all its fullness, particularity and complexity in thought.

When you tell people things, the actual physical exactitude of representation is entirely irrelevant, but the significance and effect of your experiences, and the events that transpired is what is remembered, and sought. None of which is any less real or true when represented with words, stones, stick figures, or bunny rabbits.

All that you do when you make that distinction is tell me that you're confusing some representations for the true true, and holding them as fundamentally different than others. You're a fanboy of a particular kind of story, is all.

@Joseph, @fishfry, technically these are well-formed propositions with mixed quantifications:

• ∀x∈S ∃y∈S φ(x,y)
• ∃x∈S ∀y∈S φ(x,y)

I'm not sure anyone wrote up a concise grammar for their syntax in general, but that'd be interesting to take a look at (feel free to point me in their direction if you know any).

In computing there are analogies to be found in comprehensions.

[...] if it's problematic, it's logic that has a problem, not existence. If existence declares that particles are waves or whatever quantum weirdness you care to mention, logic will just have get it's act together about it. — unenlightened

Yep.

All language presupposes thought/belief. Thought/belief consists in/of correlation. Correlation presupposes the existence of it's own content. Thus, since logic is language, logic itself presupposes existence.

That's the crux of the pseudo problem expressed by "existence is not a predicate".

I think we also want a way to talk about fiction (hypothesis, supposition, etc) "in world." So there are two answers, say, to "Does Santa fly in a helicopter?" One is "No, because he doesn't exist," but another is "No, it's a sleigh pulled by flying reindeer." Both have their use. — Srap Tasmaner

I think it might be helpful to distinguish the issue of whether existence is a predicate, with the issue of how philosophers handle fiction. That latter in itself is a huge topic.

If I say, "Ahab is the cabin boy of the Pequod," that statement is false. Yet Ahab does not exist, he's a fictional character. And someone could write fan fiction in which Ahab is the cabin boy, or the whale, or Gregory Peck. ("He tasks me. He heaps me!")

Now analyzing the truth values of statements about fictional characters is a deep business, but it's not really the same question as whether existence is a predicate.

After all, "Ahab is captain of the Pequod" is a shorthand for: "In Melville's novel Moby Dick, Ahab is captain of the Pequod." That statement is true, and there is no ambiguity or confusion. So perhaps one way out of the fiction dilemma is to fully qualify all statements. "Kirk is captain of the Enterprise in the original Star Trek tv show."

But all of this discussion of the truth values of statements about fictional entities, are red herrings (IMO) in the discussion of whether existence is a predicate.

But all of this discussion of the truth values of statements about fictional entities, are red herrings (IMO) in the discussion of whether existence is a predicate. — fishfry

Every discussion about existence ends up being a discussion about negative existentials!

But I agree. And I stand by what I said earlier:

Existence is not a predicate because it is something else, namely a quantifier. It's just a matter of getting it in the right logical bucket. — Srap Tasmaner

Frege makes the point that "The king's coach was pulled by four horses" has a very different logical structure from "The king's coach was pulled by black horses." That's the right place to start, in my opinion.

If that's your definition of proof, why are you then not a skeptic? — Thorongil

Sorry, I was never warned of this message, and just accidentally saw it now.

Because I don't think we need undeniable evidence (proof) of God to believe. Rather I, along with Pascal, take it that there is enough light for those who want to believe to believe, and enough darkness for those who don't want to believe, not to believe. Christianity does talk, in the end, about a Hidden God.

The problem with statements with the word existence in them is that they appear to be about individuals such as God or Santa Claus. But really the word existence is about the concept, i.e., it's saying that the concept either has an instance in reality or it doesn't. To say that, for example, "God exists," is to say that the concept has an instance in reality, which would mean the statement is true.

If someone says, "Hobbits exist," in order for the statement to be meaningful it would have to be about something, but what could the statement be about? It can't be about Hobbits since there are none. Thus it's about the concept of Hobbits. Thus, existence isn't something individuals possess - instead it's a way of talking about concepts of individuals.

If someone says, "Hobbits exist," in order for the statement to be meaningful it would have to be about something, but what could the statement be about? It can't be about Hobbits since there are none. Thus it's about the concept of Hobbits. Thus, existence isn't something individuals possess - instead it's a way of talking about concepts of individuals. — Sam26

Something seems awry here. The concept of Hobbits exists. Hobbits don't exist (at least not outside of the fictional media depicting them).

Under your theory, how could we coherently express the proposition that some X does not exist? If we can talk sensibly about X and whether or not it exists, I think it's safe to say that the concept of X exists (and is at least partially understood by the interlocutors). However, it would then be a contradiction in terms to state that X does not exist. Therefore, anything and everything which can be subsumed under some sensible concept can be said to exist!

However, it would then be a contradiction in terms to state that X does not exist. — Arkady

I think Sam was saying that talk of existence is really talk of whether a concept is instantiated. I don't see how saying that a concept isn't instantiated is a contradiction.

You get a contradiction if you treat existence as a predicate, because you can only predicate of objects in your domain of discourse.

Let me try again Arkady. For example, if I say, "Hobbits do not exist," for my statement to be meaningful, and in particular true, it would have to be about something. However, it can't be about hobbits, as I said above, since there are none; and if it were about hobbits, it would be about nothing. Thus, what the statement is about, is the concept of hobbits, not the subject of hobbits. The statement is saying that the concept of hobbits has no instances or individuals of which it is true. Therefore, existence is not something individuals possess; it is simply a way of expressing something about the concept.

Another important point, is that we must be able to explain the meaning of a proposition, including the subject, apart from knowing whether they're true or false. We also know that statements about hobbits are meaningful apart from knowing whether they are true or false. How is this possible? It possible because we understand the concept, and the only thing we know exists is the concept, not the subject. It can only make sense if the statement is about the concept, and not about the subject.

Moreover, we can coherently talk about the proposition that some X exists, or does not exist, because we are asking whether or not the concept X has an instance in reality. There is no inherent contradiction in the argument. Other philosophers who believed this were Kant and Russell, one being a theist, and the latter an atheist.

I think Sam was saying that talk of existence is really talk of whether a concept is instantiated. — Srap Tasmaner

The statement is saying that the concept of hobbits has no instances or individuals of which it is true. — Sam26

And what do you mean when you say that the concept is not instantiated, has no instances? (I deliberately emphasized the verb "to be" in these phrases.) Well, it means that there are no such things, that they do not exist. Oh, wait...

Fictional things don't exist, but fictions do. — unenlightened

That's roughly along the lines of how I use the words.
Say, Superman exists, but just isn't real.

But, exists is not a primary predicate. — Owen

Right.

(Y) I'll go with that.

In brief, something like:

• existence is not a logical predicate (∃ is not just another φ)
• existence can be used as a linguistic predicate

In brief, something like:

existence is not a logical predicate (∃ is not just another φ)
existence can be used as a linguistic predicate — jorndoe

Yes, absolutely. But this

Say, Superman exists, but just isn't real. — jorndoe

is confusing for everyone.

And what do you mean when you say that the concept is not instantiated, has no instances? (I deliberately emphasized the verb "to be" in these phrases.) Well, it means that there are no such things, that they do not exist. Oh, wait... — SophistiCat

If your point is that this is not an explanation of existence -- the ineffable there-ness of stuff -- I don't think it was intended to be. Neither, really, is "To be is to be the value of a bound variable," which comes to the same thing. Or saying the copula is sortal-hungry, which also comes to the same thing.

But it does clarify the logical form of existence statements and cut off the weirdness of saying "Superman is an object that has the property of not existing." If instead you say, "The concept of Superman has the property of having no instances," at least you haven't tried to predicate of an object that in the same breath you say doesn't exist. (And people get hinky about this and say things like, before it was built the Empire State Building did exist "as an idea".)

(And you don't even need to predicate of concepts: "Nothing has the property of being Superman" is a reasonable natural language equivalent to "¬∃xFx".)

You know all that though, so what was your point?

ADDED: That sounds belligerent, but was meant to sound puzzled.

Let me try again Arkady. For example, if I say, "Hobbits do not exist," for my statement to be meaningful, and in particular true, it would have to be about something. However, it can't be about hobbits, as I said above, since there are none; and if it were about hobbits, it would be about nothing. Thus, what the statement is about, is the concept of hobbits, not the subject of hobbits. The statement is saying that the concept of hobbits has no instances or individuals of which it is true. Therefore, existence is not something individuals possess; it is simply a way of expressing something about the concept.

Another important point, is that we must be able to explain the meaning of a proposition, including the subject, apart from knowing whether they're true or false. We also know that statements about hobbits are meaningful apart from knowing whether they are true or false. How is this possible? It possible because we understand the concept, and the only thing we know exists is the concept, not the subject. It can only make sense if the statement is about the concept, and not about the subject.

Moreover, we can coherently talk about the proposition that some X exists, or does not exist, because we are asking whether or not the concept X has an instance in reality. There is no inherent contradiction in the argument. Other philosophers who believed this were Kant and Russell, one being a theist, and the latter an atheist. — Sam26

I think I have a better idea of what you're driving at. A couple of questions/points: what does it then mean to say that a concept exists? That the concept of the concept is instantiated (presumably in one or more minds, or at least in one or more products of minds such as novels, etc., or wherever it is that concepts inhere)?

Secondly, in yoking existence to concepts, it would seem to imply that nothing exists which is unconceptualized. If the existence of X simply means that the concept of X is realized in at least one concrete instance, then the notion of existence is meaningless without reference to concepts. However, concepts require minds, do they not? Did that mean that nothing "existed" prior to the advent of minds which were capable of realizing concepts?

I wouldn't say that concepts exist in minds, and I'm not necessarily suggesting that you're saying this, but only pointing this out as a point of clarification. This seems similar to the idea that words point to objects, and in this case the concept somehow exists in the mind, as something we can point to. The concepts, as you seem to suggest, find their use in language. Thus, their existence is demonstrated in how we use the concepts in propositions for example. The existence of a concept is quite a separate issue from the issue of whether the concept has an instance in reality. So obviously there is the concept hobbit, but that is a separate issue in terms of whether hobbits actually exist, or have been instantiated. Concepts can inhere as part of their use in a language, and that is demonstrated in a variety of ways, such as, referring to concepts that get their life in a fictional work, in which case the concepts have no existence (hobbits for example), apart from the use of the concept in that work of fiction.

The existence of X is not dependent upon the existence of the concept. Thus, things can exist apart from us referring to them as part of a language. However, we want to know whether it is true or false, to say that hobbits exists in reality, as opposed to being part of a fictional work.

I'm not simply yoking existence to concepts, but yoking existence to whether or not that concept has an instance in reality. This doesn't mean that existence is dependent upon the concepts, but talking about existence is dependent upon the concepts. So obviously things could exist apart from minds.

[...] is confusing for everyone. — Srap Tasmaner

Hm. Seems reasonably clear to me.

For something to be real, it have to exist.
For something to exist, it doesn't have to be real. Like Superman or other imaginary things.

Do you have a simpler understanding or use of the words?

What do you mean by saying that Superman exists but isn't real?

I just don't want to say that Superman exists one way in fiction and another way outside it. The fiction exists in our world. Superman doesn't, not even "as" something.

If you want to say he exists here "as an idea", then as an idea of what? As an idea of an object, namely himself. So Superman is the idea of Superman, and that's an infinite regress.

We need a nice way to talk about fictional objects, but this is headed in the wrong direction, I'd say.

Existence can be modelled syntactically:
P(x) iff ∃x(x=x)
P(x) is true if and only if x exists. (x exists iff it equals with itself)

Semantical existence can't be modelled imo. We need an ontology as a naive metatheory for that.

If your point is that this is not an explanation of existence -- the ineffable there-ness of stuff -- I don't think it was intended to be. — Srap Tasmaner

No, you are right, I think I got carried away.

Existence can be modelled syntactically:
P(x) iff ∃x(x=x)
P(x) is true if and only x exists. (x exists iff it equals with itself) — Meta

So x exists iff there exists x (∃x) such that it is equal to itself? The last bit seems unnecessary, but otherwise you have a perfect tautology.

The last part is necessary for the formula to be well-formed.
And yes, it is a tautology, but it expresses existence. Any other tautologies would be sufficent.
If the predicate expressing existence isn't a tautology then it does not characterize existence.

So I still don't understand, what was the point of the exercise? To use the existence quantifier in a WFF?

I think the point could be to show that if we want to define a predicate P(x) in a formal language that means x exists then our only option is to make P a tautology.

I wouldn't say that concepts exist in minds, and I'm not necessarily suggesting that you're saying this, but only pointing this out as a point of clarification. — Sam26

I actually do believe that concepts inhere in minds (or at least their products), and all else that follows from that: I'm not sure where else they would inhere.