Existence is a property. One simple reason can be drawn from science. Entities are theorized, predicted and then sought for. Imagine a hypohetical particle x. We look for its existence. In that sense ''existence'' is a property and finding x we assign the property ''existence'' to x

How would you formalise the title?

There does not exist a predicate (x) in language (S) which has the meaning, 'exists' (φ).

I don't know, but there is a slight aroma of cutting off the branch one is standing on here... is it a rule that denies its own expression?

@TheMadFool, how to demarcate fictional and real entities?
If you suppose that a fictional entity exists, then what would it take for it to be real?
It seems like fictional (not real) entities exist as their hypotheses (or definitions) alone.

@unenlightened, according to those pesky objectivists "existence exists", though I think Russell argued otherwise.
Does existence then not exist...? Maybe we should have a poll on that one as well.
How about this one? "There exists an apple in the bowl of apples, so that apple does not exist." :)

There are cases where "existence is a predicate" comes through as nonsense, which makes me think that (generally) "existence is not a predicate".
As far as I can tell, there isn't anything further that "to exist" can reduce to; existence is ground if you will.

Another angle:
We commonly characterize things by predication, quiddity — what those things are.
These are different propositions from merely existing though — that something exists.

By the way, is the King of France bald or not?
Russell and Whitehead:

Did you answer my question, but I'm too stupid to understand it? Or was the question not clear enough? Can 'existence is not a predicate' be formalised without using existence as a predicate?

I'm not the big logician, but I know that existence is tricksy. It can be readily proved that there is no greatest prime number. One supposes that one exists, and then shows there is a bigger one. But then that feels different from the question of whether or not there exists hair on someone's head. I feel like we can put restrictions of this sort on language for certain purposes, but for other purposes I might define existence as that which is immune to argument.

I am not a logician either, I think thought indicates being, but it cannot formally confirm any being beyond itself since our only access to being is though thought.

I hope you don't get upset at me, but why should we care about answering this question? What is the ramification of it that makes it an interesting problem philosophically speaking, apart from appeals to authority (other philosophers have dealt with it), and the trivial reason that it just happens to be a question we dreamt up (cause I'm sure we don't pursue every question we dream up)?

The principle of sufficient reason.

The principle of sufficient reason. — Cavacava

How come? As far as I see, the only reason why existence being a predicate or not was ever philosophically interesting was because people were interested in the ontological argument.

Existence is a property. One simple reason can be drawn from science. Entities are theorized, predicted and then sought for. Imagine a hypohetical particle x. We look for its existence. In that sense ''existence'' is a property and finding x we assign the property ''existence'' to x — TheMadFool

In my opinion, concepts are concepts and the things they signify are the things they signify. For example, a circle can be taken purely as a concept. A circle as a concept is not round, it doesn't have a radius, etc. - it's just a concept. However, that which the concept of the circle signifies does have a radius, is round, and can exist or not exist. We get confused because we don't have two different words to distinguish between the circle, and the concept of the circle so we equivocate. It's one thing for a concept to exist, and it's another for that which it signifies to exist.

Existence isn't a predicate, because to exist is to stand out, be distinct, but saying that something exists doesn't introduce a new distinction to the equation. Saying that something exists doesn't tell you anything new about it.

Saying that something doesn't exist is more like saying that it cannot be found, it doesn't strip it of any particular quality, and I don't consider this a proper use of language anyway. If something truly didn't exist, then there would be nothing to talk about not existing.

In thinking that existence is a predicate, we immediately wish to predicate anything said to exist or not exist, and this leads to confusion in my view. It exists, therefore it has the qualities of being material, or occupying space and time. Or it doesn't exist, this must mean that it is lacks those and such qualities -- but then can it really be said that we're in possession of a such a powerful predicate, so that we know that universal quality for existence, and know that it must be present in every single thing that exists in all realms and modes, and possible worlds? All scales and possibilities?

Predicating something with "existence" simply imports our metaphysical presuppositions in my view, rather than actually zones in the very quality that grants something existence. My view is both cool because something that truly doesn't exist isn't there at all in the first place to not exist, and resting on distinction it's far more phenomenological, which I take to be both more grounded in experience, and humble in its reach.

Existence is a predicate because this statement makes complete sense:

∀x∈S[¬∃x]

The universal quantifier does not require existence. This statement says that for every member of S, it doesn't exist. Since there are no members of S, we can assign any attribute we want.

The only case where the above statement makes sense of course is when S is the empty set. If it had members, it would be false. And it works for the empty set because, again, the universal quantifier does not require existence.

∀x∈S[¬∃x] — Joseph

I do not believe the existential (or universal) quantifiers may stand alone. Rather there must be a unary predicate P so that you can write ∃xP(x).

If you have a reference illustrating a use of a standalone quantifier like ∃x I would appreciate a link or specific reference.

An expression such as ¬∃x is meaningless, as is ∀x∈S[¬∃x]. That's just not a well-formed formula in predicate logic.

If the general form of the existential quantifier is ∃xP(x) then in your example you are saying that ¬∃x may be a predicate. So you are assuming the thing you are trying to prove. In fact ¬∃x is not a predicate.

How about if you wanted to state that a set contains at least one member? The only sensible way to write it is ∃x∈S. That is a case of a standalone quantifier. That isn't to say anything about it being a predicate, but it could be extended for that case.

Next, how would you state that a set is empty in predicate logic?

The principal of sufficient reason requires reasons why the laws that govern the universe are as they are and are not otherwise and even if such a reason is found it must account for this reason ad infinitum...to "a reason not conditioned by any other reason, and only the ontological argument is capable of uncovering". (Quentin Meillassoux AF pg.33).
At least one absolute being is absolutely necessity for the laws, of universe to be necessary if not, everything could be otherwise. Kant's refutation of the ontological argument ended dogmatic metaphysics.

How about if you wanted to state that a set contains at least one member? — Joseph

∃x∈S (x ∈ S)

That would be the formal way, since x ∈ S is a predicate.

The only sensible way to write it is ∃x∈S. — Joseph

Technically that is a convenient abuse of notation. Everyone writes it that way, everyone knows what it means. But it's not actually correct if one is being precise.

That is a case of a standalone quantifier. — Joseph

No. Besides, it has a completely different form than your earlier example of writing ∃x without any predicate at all. You are giving two completely different examples. ∃x by itself is is meaningless. ∃x∈S is technically wrong but everyone understands that it's a shorthand for ∃x(x∈S).

Next, how would you state that a set is empty in predicate logic? — Joseph

¬∃x(x∈S).

Hmmm. To me science makes more sense the other way around, where you have some idea what your domain is -- and you might get that wrong and have to change it -- and the question is either "What in my domain has the properties my effect needs it to?" or "Does anything in my domain have such-and-such a property?"

The formalism was developed for mathematics, where you always specify the domain. Science has no choice but to follow suit. What are you going to do? look everywhere? at everything?

[retracted]

There does not exist a predicate (x) in language (S) which has the meaning, 'exists' (φ). — unenlightened

I think that may be the wrong approach. It's a little like saying, "There does not exist a predicate (x) in language (S) which has the meaning, 'and' (φ)."

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.

Well, what does it mean to exist? If, for example, to exist is to have a location in space, and if having a location in space is a predicate then existence is a predicate. Or for something abstract like numbers, we might say that to exist is to have a use in some system of mathematics, and if having a use in some system of mathematics is a predicate then existence is a predicate.

But under this way of looking at it, none of that would be an acceptable account of existence. To exist, logically speaking, is generally just to be the subject of a predicate.

To exist, logically speaking, is generally just to be the subject of a predicate. — jamalrob

But "is a fiction" is a predicate, yet fictional things don't exist.

I wonder if perhaps there's conflation between the existential quantifier in logic and "existence" as an ordinary language term.

Apologies @unenlightened, skim-reading is poor reading, my bad. The two propositions, "existence is not a predicate" and ∃x∈S [ φx ], are sufficiently different that I couldn't formulate the former as an example of the latter.

• ∃ ∉ Φ, where Φ is all φs (no unrestricted comprehension please)

Is "to be a predicate" a predicate? I suppose so, but haven't figured out whether that's problematic.

why should we care about answering this question? — Agustino

Philosophy and forums? In my own case, also looking for (in)consistencies among different things, e.g. ∃ and quiddity (old post).

Well, what does it mean to exist? — Michael

In this case, any such meanings (including hypothetical) already exemplifies existence. Seems in some ways, existence is auto-presupposed.


Anyway, using existence as a predicate can sometimes lead to nonsense, and other times make sense. So, ∃ is not just another φ, except sometimes it is?

Is "to be a predicate" a predicate? I suppose so, but haven't figured out whether that's problematic. — jorndoe

I haven't figured it out either, but 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. It might just be the usual problem of ordinary language being its own meta-language.

But "is a fiction" is a predicate, yet fictional things don't exist. — Michael

Fictional things don't exist, but fictions do.

To exist, logically speaking, is generally just to be the subject of a predicate. — jamalrob

This is where the problem is then. If, from the perspective of what is logical, to exist is simply to be the subject of a predicate, then logic isn't consistent with what we normal mean when we say "exists". This is why logic and epistemology must be based in a good ontology, not vise versa. If we turn this around, and try to base an ontology in what logic makes of existence, we are headed into problems.

Of course 'exists' is a predicate.
'a exists', has 'a' as its subject and 'exists' as its predicate.

But, exists is not a primary predicate.
(Ga & (a exists)) <-> Ga, ie. (a exists) does not add information to Ga.

If we define (a exists) as ∃F(Fa), and
(G exists) as ∃xGx.

1. |-. Ga -> ∃F(Fa). |-. Ga -> (a exists).
2. |-. Ga -> ∃x(Gx). |-. Ga -> (G exists).
3. |-. Ga -> ∃F∃x(Fx). |-. Ga -> ((Ga) exists).

(x exists) <-> ∃F(Fx).
(x exists) <-> ∃y(x=y).
(x exists) <-> x=x.
(x exists) <-> ((~F)x <-> ~(Fx)).

why should we care about answering this question? — Agustino

For one thing, classical theism is in jeopardy. To claim that God can be proven or believed to exist, despite not knowing his essence, becomes a nonsensical distinction if existence isn't a predicate.

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

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

For one thing, classical theism is in jeopardy. To claim that God can be proven or believed to exist, despite not knowing his essence, becomes a nonsensical distinction if existence isn't a predicate. — Thorongil

The whole conundrum seems a bit nonsensical to me. Nothing can be "proven" to exist. You can't even "prove" other minds exist. Classical theism does not claim to "prove" God in the sense of showing that it is irrational and illogical not to believe in God. Rather classical theism tries to give a defence of the faith which means to give very compelling reasons for believing in God, not a deductive and bullet proof argument. The premises of Aquinas' arguments can be denied for example. They are certainly sensible propositions that many people would be inclined to acknowledge, but one can still do the mental gymnastics required to deny them.

Philosophy and forums? — jorndoe

That's nothing but an argument from authority. I asked you to use your head and give me an actual reason. Philosophers of the past may have thought it is important to debate whether existence is a property or not (because they were interested in the ontological argument) but maybe they were bothering to address unimportant and sterile matters to begin with (and later philosophers like Russell merely picked up on such sterility without questioning it). Maybe they were asking the wrong questions, and discussing dead ends. So just because philosophers have thought it important to discuss it, doesn't mean it really is important. So again, why is it important? Why should I care about resolving this issue? How will it change anything?

I'm not seeing a difference between "proving God to exist" and "giving very compelling reasons for believing in God." The latter statement just reads as a more wordy restatement of the former. The question of whether the proof is inductive or deductive seems irrelevant. A proof is a proof.