Like when we define the truth of a formula of the form ∃n F(n) we say:
∃n F(n) is true iff there exists a natural number x such that F(x) is true
In that definition does the 'there exists' part mean the set theoretic provability of the existence or some kind of platonic metaphysical existence or some other kind of existence. — Meta

If the question is whether you are committed to interpreting ∃ metaphysically, maybe platonically, then the answer is no. "∃n F(n)" says only that F is true of something in the domain of discourse.

Of course you can go further and take a position on what the objects in the domain of discourse are, but just using quantifiers doesn't commit you to any particular view.

Thank you for answering!
Yes that's it! I Think we have to give up mathematics at some point and enter the realm of metaphysics. Because in the definition of the truth of an existential sentence of the form above we use the term "there exists". So existence is defined by some kind of "meta existence" and hence the rigour of Mathematics disappears at one point. We can interpret that differently (biologically, intuitively etc.) of course.

Edit: Basically platonism comes into the picture because in my opinion in practice most of the mathematicians think zfc provability means truth. But in theory that can not be the case of course. And I dont know a precisw definition. Precise in the most rigorous sense.

At the end we have to interpret existence as something like "it is trivial" based on our previous experience with reality. (for example we know what "there exists a dog in the room at the moment" means)

You needn't let that rigor disappear. The American philosopher W. V. Quine famously said, "To be is to be the value of a bound variable." He took "there exists" to mean exactly what it seems to, and argued that if somewhere in, say, a theory of physics, you have an expression like "∃x F(x)" then your theory is committed to the existence, real-world actual existence, of something that is F.

(Apologies if you know all this. If not, the place to start is "On What There Is" collected in Ontological Relativity and Other Essays.)

Thank you for the recommendation. I Will read it. I have only seen a bit from the mathematical side. I need to discover the philosophical side yet.

Edit: Jeah we basically agree in almost everything. Maybe I expressed myself not the best possible way. I really think that "mathematical" or "logical" rigour has to be given up while we define existence. Because either:
1. Existence is defined by itself. That does not meet the requirements of logic. Or
2. Existence ia defined by another type of existence. In this case we either need infinite previous definitions or there is an existence for which we need another method to define. (for example by using experience from the real word)

I do not say that is worse or less scientific than pure mathematics. But I do say that pure mathematics has to end somewhere.

It is also possible to construe quantifiers substitutionally. This means you take "∃n F(n)" to mean "There is an expression we can put in place of 'n' such that 'F(n)' is true." So in a way, the quantifiers range over linguistic expressions rather than objects. There's something quite natural about this approach. You explain "∀x F(x)" as meaning "F(x) is always true," and so on. I'm still not clear on the advantages and disadvantages of the two approaches, but taking variables as standing for objects is by far the more popular approach.

If you're just starting to do philosophy, trying to define existence is not the place to start.

One problem with construing the quantifiers substitutionally is that you will need a denumerable language if you're working with the natural numbers, and a uncountable language if you're working with the reals. But this seems awkward if you're trying to avoid existence assumptions.

That is not to say that one can't sidestep the problem from the op. One can work inside primitive recursive arithmetic as a metatheory and define ZFC inside it. Using ZFC, one can then formulate a truth-theory for the natural numbers in terms of provability inside ZFC. So everything is (ultimately) syntactic, a kind of formalism, if you will.

My main goal is not to define existence but to find a logic without the anomalies mentioned above. Or at least to formalize my problem (about mathematical existence)

He took "there exists" to mean exactly what it seems to, and argued that if somewhere in, say, a theory of physics, you have an expression like "∃x F(x)" then your theory is committed to the existence, real-world actual existence, of something that is F. — Srap Tasmaner

While it may seem compatible with natural language, the difficulty here is ascertaining the validity of the underlying quantification F as there are modal differences between names and descriptions.

Thanks. I was hoping you'd stop by.

ascertaining the validity of the underlying quantification — TimeLine

I don't know what this means.

Are you talking about Quine specifically or about the usual semantics for classical logic?

Kripke' view that there is a difference between name and descriptions, which challenges the idea that a name that can be replaced by a description and be 'bound by a variable' so without a remainder. But if names do refer to objects, the real-world actual existence of F does not make sense of non-existing objects, such as "the king of France is bald".

If truth in arithmetic means provability in ZFC then it is false that every PA formula is either true or false. Thats odd.

What kind of "truth" concept is used in the Gödel and Tarski theorems? Is it ZFC provability?

If truth in arithmetic means provability in ZFC then it is false that every PA formula is either true or false. Thats odd.

What kind of "truth" concept is used in the Gödel and Tarski theorems? — Meta

The first sentence is incorrect. ZFC can formulate a truth predicate for PA in such a way that a formula from PA is true iff ZFC proves that the natural numbers satisfy it. We can also prove that ZFC proves that for this predicate, a formula from PA is either true or false. This does not violate either Gödel's or Tarski's theorems because ZFC is strictly stronger than PA (in fact, it is much stronger), whereas the theorem only applies for theories weaker than PA (including PA itself).

The truth concepts used by Gödel and Tarski in their theorems is the usual model-theoretic one, i.e. a formula from PA is true iff it satisfied in the standard natural numbers.

Kripke' view that there is a difference between name and descriptions, which challenges the idea that a name that can be replaced by a description and be 'bound by a variable' so without a remainder. But if names do refer to objects, the real-world actual existence of F does not make sense of non-existing objects, such as "the king of France is bald". — TimeLine

But this has nothing to do with the original suggestion, which involves just names and not descriptions. It also does not mention names being "bound by a variable" (whatever that means). Here's the idea. Suppose I have a language, L, and an intended model, say M. Then, instead of saying that a formula such as "exists x such that Fx" being true when there is an object from M which satisfies Fx, we can instead introduce a name for each object in M and say that the formula is true iff there is a name a such that Fa is true. This is called the "method of diagrams" and is one way to avoid talking about satisfaction. Kripke himself proved that this idea can be deployed successfully to entirely avoid objectual quantifiers (see his paper "Is there a problem with substitutional quantification?". where he shows that, given a language L for which truth has already been defined, we can extend L by introducing substitutional quantifiers and this extension is well-defined).

EDIT: Of course, the mere fact that we can define substitutional quantifiers in terms of objectual quantifiers and vice-versa already shows that a facile reading of Quine's maxim is at least problematic...

Of course, the mere fact that we can define substitutional quantifiers in terms of objectual quantifiers and vice-versa already shows that a facile reading of Quine's maxim is at least problematic... — Nagase

The conversation between Barcan, Quine, and Kripke (and who else?) is interesting reading on this point.

One other point, just on Quine, is that he not only believed he could happily get along without modal logic, he also believed classical logic had no use for singular terms at all.

And now I'm done speaking for Quine. I still think his work is a pretty good place to start doing philosophy, especially if you come from logic and mathematics.

This is called the "method of diagrams" and is one way to avoid talking about satisfaction. Kripke himself proved that this idea can be deployed successfully to entirely avoid objectual quantifiers (see his paper "Is there a problem with substitutional quantification?". where he shows that, given a language L for which truth has already been defined, we can extend L by introducing substitutional quantifiers and this extension is well-defined). — Nagase

I don't disagree with Kripke, rather I was merely pointing out the flaw in Quine' interpretation of the objectual quantifier viz., the range of variable values. I will try and read that paper, despite being completely overworked at the moment.

Aw i didnt know that. Thanks. I dont know where my idea came from.

I accept Nagase's answer on arithmetic truth. If the mathematics we do is formalized within PRA then the problem of the truth of existence in PA is solved (or not so obvious anymore).

However I was reading a book today about logic and I faced the same problem again.
What if I want to define existential truth in ZFC or in a more powerful system? ZFC provability is not enough anymore as the set of provable sentences of ZFC form a real subset of the set of true sentences of ZFC. How do we define existential truth of ZFC? I think that we must use a metaphysical existence concept.

However I was reading a book today about logic and I faced the same problem again.
What if I want to define existential truth in ZFC or in a more powerful system? ZFC provability is not enough anymore as the set of provable sentences of ZFC form a real subset of the set of true sentences of ZFC. How do we define existential truth of ZFC? I think that we must use a metaphysical existence concept. — Meta

There are a couple of options, here. One way is to employ a weaker meta-theory, say primitive recursive arithmetic, and simply talk about what ZFC can prove. In fact, many mathematicians and logicians (such as Feferman) consider questions which are independent from ZFC, such as the continuum hypothesis, to be simply meaningless. So, while we can't formulate a truth predicate for ZFC, we can bite the bullet and say that the extra power of such a predicate would be meaningless anyway.

Another one would be to exploit something that is a bit bizarre about ZFC. Since ZFC can also code its own syntax, you can use ZFC as a meta-theory for ZFC. Since ZFC is able to formulate the satisfaction relation, you could try to use that to define a local truth predicate for ZFC. In fact, we know that, for every finite set of formulas from ZFC (remember, ZFC has infinitely many axioms), ZFC can prove that they are in a sense true (these are called "reflection principles"). So you could formulate local truth predicates and use these as a proxy for the global truth predicate.

Thanks Nagase your answer is top notch as always.
I dont wholly understand this thing about local truth predicates but sounds interesting. I Will read about it.

At the moment I cant imagine how would we define truth using the reflection principles.

I have been doing a lot of other stuffs lately so I couldn't learn enough about these reflection principles. I will read more about them for sure.
But for my conclusion of the thread:
Defining arithmetic truth is a hard task (I would say impossible).
Either we are platonists and believe that truth is given by the facts which are true in the existing abstract metaphysical world. In this case we can't deductively describe every aspect of truth.
Or we are formalists and we think something is true if we can prove it somehow (for example from ZFC).
In this case truth is easy to define and easier to describe but this concept does not meet some basic requirements of the concept of truth (e.g. completeness)
So the problem of defining truth without using truth is like the problem of life after death or the question what is beyond the universe. Impossible to answer.