@Banno, here's the response that you wanted from me (well perhaps it's not the one that you actually
wanted, but this is the best I got on this, mate. That doesn't mean that I'm right, so, grain of salt and all that standard nonsense.) Please excuse my style, of multiple quotations. I see it as a dialogue, actually. So, no disrespect from me, at least not intentional. Alright, here are my answers to your quiz, mate:
Pegasus is an individual in the domain we are discussing. — Banno
Agreed.
So not a predicate. — Banno
Agreed. But only for the sake of argument. I've been experimenting with the possibility of treating the term "Pegasus" as a predicate, but in a different manner than Frege, Russell, and Quine. Just an anecdote, mate. Nothing substantial.
We can write ∃(x)(x=a) were "a" is a constant that refers to Pegasus. It says very little. — Banno
Right, but it leads to the problem of literally saying that "Pegasus exists", if by that you mean that there is an "x", such that "x" is identical to some individual constant "p", such that "p" stands for "Pegasus". As in: ∃(x)(x=p). To me, all that means is that there is an "x", such that "x" is identical to "Pegasus". That's all it means to me. It has no ontological import as far as I'm concerned. It doesn't literally say "Pegasus exists in the real world, as a living horse that has the wings of a bird." It doesn't even say "Pegasus exists". That's not what existence is, in the context of Ontology. At least not how Bunge understands Ontology. And here I take his side. And he's not alone here.
Graham Priest, for example, might argue something along those lines as well, I believe. Not that such appeals to authority mean anything, what I'm saying in my last sentences is a fallacy, granted. But I'm just saying, mate. It's possible to make a valid, sound case for it.
Since it is true that Bellerophon rode Pegasus to Mount Helicon, there is something that was ridden to Mount Helicon, by existential introduction. — Banno
Of course. But you see, that's what I'm arguing here: semantics. That rule is fine. It's legit,
innit. All I'm saying is that it shouldn't be called "existential" introduction. It has nothing to do with the concept of existence, which is something that concerns Ontology, not Logic, and certainly not Mathematics. That's all I'm saying, mate. And some people sometimes make it seem like I'm saying something brutal or whatnot. You know what I'd call it? The "particularizing rule of introduction", or simply "particularizing introduction".
Something like "Pegasus exists in the context of Greek mythology, but it does not exist in the actual world" says little more than that Pegasus is an individual in the domain of Greek Myth, but perhaps not in the domain of chairs and rocks. Do you see a problem with such a simple and direct approach? — Banno
You're 100% correct in your interpretation of that statement. And yes, I do indeed see a problem with Bunge's simple and direct approach here. I'm just going on guts, instinct and intuition here, but I think that we should distrust individual constants for some reason. Quine distrusted them. And he was a smart man. I don't care if my argument here is a fallacy. I'm thinking this from the perspective of sound common sense now. Which is not to say that I'm right, but my suspicions aren't unfounded.
Note the dropping of the words "conceptually" and "really". They do not appear to be doing anything. — Banno
See, here's where I'm on the fence. I go back and forth on this one. Sometimes I think they do nothing. Sometimes it seems to me that they perform different functions, which, by "existential introduction", as you call it, there would be at least two "things", "x" and "y" such that they are performing different functions. I know this sounds cryptic, I can try to clarify it, if that sounds like something that might add anything positive to this Thread.
If needed, we could well put Pegasus and Mount Helicon into the same domain, and add a predicate something like "real", and say that Mount Helicon is real, but Pegasus is not real. But that has no implications for Pegasus' existence, as set out. It remains that Pegasus exists, but this amounts to little more than that Pegasus is one of the things about which we can talk - it is an item in the domain. — Banno
My only suggestion on this, is that you should be able to say, in first-order language, that Pegasus exists (is an item of) the domain of Greek mythology, to use your vocabulary, and that at the same time it does not exist (it is not an item of) the domain of Aztec mythology. In other words, you need to be able to say that Pegasus is neither in the domain of Reality nor Aztec mythology, etc. Bunge's approach allows you to say exactly that, since his existence predicate is a two-place predicate. But, as I've told you, I'm leaning towards Quine's approach here: like Quine, I simply don't "trust" individual constants like Bunge does. Not that such manner of speaking demonstrates anything at this level of the conversation, mind you.
What I've said here will be misunderstood and augmented by others, but to my eye it dissolves the issue of the OP. Infinitesimals exist, since they can be the subject of a quantification. Pegasus exists, since it can be the subject of a quantification. But neither are the sort of thing you might run into in the street. — Banno
Well I don't know if I would phrase it like that, but for the purposes of the OP, yes, I think you are correct: Pegasus exists if and only if Infinitesimals exist in the exact same sense. Now, if that sense is being "the subject of a quantification", that's where you and me personally begin to disagree. But that is not to say that you have not answered the question in the OP: in my eyes, you have.
And what is going on here is a clarification of what we mean by saying that something exists, made by looking at how a formal language can deal coherently with the problem. — Banno
Exactly, but it can't. No formal language can deal coherently with the problem of the meaning of existence. The concept of existence is not a concept of a formal language. It's a concept of ontology. And ontology is not a formal language. Now, there are some very smart people out there, who work in a place called "The Ontology Room", and they will tell you that there is such a thing as "Ontologese", which is a formal language, comparable to Portuguese as far as the poetics go. Those people, I believe, are wrong. And I take my cue here from someone smarter than me. Whatever the case may be, Ontologese is not a formal language, and there cannot be such language. You either get a formal language (like first-order logic) or a language with ontological import built into it from the get-go (i.e., "Ontologese"), you can't have both. You can't have your cake and eat it too, I would say. You have to choose one or the other, sadly.