If the people gathered here would allow me to "set the tone", so to speak, as if I was inserting a quarter into the jukebox of a Honky-Tonk, then I would share the following quote from Bunge:

Up until the mid 1960s whoever wished to engage in mysticism or freewheeling, intellectual deceit or antiintellectualism had to do so outside the hallowed groves of academe. For nearly two centuries before that time the university had been an institution of higher learning, where people cultivated the intellect, engaged in rational discussion, searched for the truth, applied it, or taught it to the best of their abilities. To be sure once in a while a traitor to one of these values was discovered, but he was promptly ostracized. — Mario Bunge

Mario Bunge,"In Praise of Intolerance to Charlatanism in Academia", page 96.

Others on TPF know the Tractatus a lot better than I do, but I think he meant something more than merely "not truth apt" or "not confirmable." I think it's closer to "incoherent" or "illusory." And he wasn't just thinking of ethics and religion, but also of certain supposedly bedrock metaphysical truths. In any case, what I meant by "inexpressible" was more like "unsayable save by metaphor and indirection."
— J

I am no expert either, but I understood that in the Tractatus Wittgenstein was concerned to make a distinction between what can be propositionally claimed and what cannot. I think that for him a coherent proposition just is a proposition which is truth-apt. — Janus

I wonder whether you have a response to this, or have you lost interest.

The date Bunge gives there seems imprecise, since much of the philosophy he is higly critical of, such as existentialism and phenomenology, considerably predated 1960.

↪Arcane Sandwich
The date Bunge gives there seems imprecise, since much of the philosophy he is higly critical of, such as existentialism and phenomenology, considerably predated 1960. — Janus

Right. That's why he says that before the 60's, you would have been ostracized if you were an existentialist or a phenomenologist. And rightly so. That's his entire point.

Both Husserl and Heidegger held respectable posts at universities. Not to mention Hegel, who I have no doubt Bunge would have criticized for indulging in philosophical confabulations.

Both Husserl and Heidegger held respectable posts at universities. Not to mention Hegel, who I have no doubt Bunge would have criticized for indulging in philosophical confabulations. — Janus

Bunge hated Hegel, and he hated him publicly as well as privately. That's no secret to anyone. He said that Hegel was a charlatan. He said it in lectures, in books, in press conferences, in the context of a coffee with friends, etc. Do I myself think that Hegel was a charlatan? Not necessarily. I don't agree with Bunge on everything.

As for Husserl and Heidegger, Bunge is speaking from the Analytic tradition. He defends people the likes of Rudolf Carnap, for example. Not Karl Popper, mind you. He thought that Popper was a fraud and a charlatan.

I am no expert either, but I understood that in the Tractatus Wittgenstein was concerned to make a distinction between what can be propositionally claimed and what cannot. I think that for him a coherent proposition just is a proposition which is truth-apt.
— Janus

↪J

I wonder whether you have a response to this — Janus

Thanks for checking -- I was sort of assuming you were right. It makes it easier to get a grip on what Witt meant, anyway. Talking about objects being "expressible" doesn't seem on target. I think Witt wanted to say something about what can and can't be said, sensibly. I'm not sure whether, for him, "sensibly" means "using truth-apt propositions," but it seems plausible. And there's the whole self-reflexive question about demonstration versus expression -- when Witt says that certain things can't be said, does he go on to show this or give it propositional expression?

EDIT - I meant, "give propositional expression to the impossibility of something being said."

Talking about objects being "expressible" doesn't seem on target. — J

Fair enough, point taken. Let me try something else instead. Quine famously said that the very reason why Pegasus does not exist is because there is no object or creature in the world that "Pegasizes". In the 50's, someone wrote a paper, asking Quine if President Truman exists because "something Trumanizes", there is an object or creature in the world (i.e., Truman himself) who "Trumanizes".

Those types of words, "Pegasizes", "Trumanizes". What do they mean? What do they express?

Quine meant, I suppose, that ¬∃x P(x), where P is the predicate corresponding to "Pegasus". Is your question about whether predicates need to be understood as verbs, a la "Pegasize"? Or is it the larger question of whether ∃x itself is a type of predication?

Quine meant, I suppose, that ¬∃x P(x), where P is the predicate corresponding to "Pegasus". — J

That is indeed what he meant. But that is also what Frege meant, and what Russell meant, though each of them had different reasons for it. What are Quine's reasons for even translating this discussion into common parlance when he speaks of "Pegasizing"?

Or is it the larger question of whether ∃x itself is a type of predication? — J

It has to do with this. The existential quantifier, ∃, does not have ontological import. Quine is averse to it because he thinks that it does have ontological import. But he's just plain wrong. Deluded, even. Frege and Russell had the same problem. If the universal quantifier, "∀", has no ontological import, there is no reason to believe that the existential quantifier has ontological import either, because you can switch these symbols under certain conditions, so what would you make of that, in ontological terms? Nothing, there is no ontology to symbols such as ∃ and ∀. They are not types of predication, they are types of quantification.

Free Logic is not the only option. You can keep classical logic while tracing a distinction (as Bunge does) between real existence and conceptual existence. — Arcane Sandwich

I haven't followed Bung, and you provide no reference, so I've no clear idea what he might be saying, but that sounds like a variation on free logic.

That mathematics is "pleasing" looks to be besides the point.

I haven't followed Bung, and you provide no reference, so I've no clear idea what he might be saying, but that sounds like a variation on free logic. — Banno

Let me craft an example for you, then. Consider the following statement: "Pegasus conceptually exists in the context of Greek mythology, but it does not really exist in the actual world."

That statement (I've argued this on paper, in an article that I published in a Bungean journal), is troublesome for someone the likes of Frege, Russell, and Quine. All of them would treat the term "Pegasus" as a predicate, no matter what differences they might later have (i.e., regarding the very concept of "Pegasizing").

Bunge's existence predicate manages to symbolize that statement in a very neat way. Like so:

∃x((x= p) ∧ E(gp) ∧ ¬E(rp)

That formula should be parsed like so: For some particular x, it is identical to Pegasus, and it exists conceptually in the context of Greek mythology, and it does not exist really in the actual world.

Notice that the existence predicate, "E", is a two-place predicate. It binds an individual constant, such as "p" (which stands for Pegasus), and it binds it to a context (i.e., Greek mythology, the actual world, etc.)

I would have been happier with an example that was a long way form philosophy - to look at how these terms are used in their natural environment, as it were, rather then in the lab. The sort of method advocated more by Austin than Wittgenstein, but captured in "Don't think, look" at PI §66. A way to check that our language hasn't taken a vacation.

I played with your idea of a high C for a while. The supposition was that the high C could be understood - perhaps C6, two ledger lines above the treble clef; but unachievable by a certain musician. We have a clear idea of what that is, even if it cannot be sung by our soprano. Is there a note that no soprano might sing? Apparently the highest achieved is F#8, truly awesome. But again, we know what it would mean for someone to say, sing C9, even if this is never achieved. These are tied to our use of this sort of language.

But is there a highest C? Here we start to leave firm ground. Is there a C higher than any other C? A first approach might be to claim that since for any given C there is a higher C, one octave above, that there can not be a highest "high C". In theses terms the term "Highest high C" doesn't have any referent, and very little sense.

But taking a step further, a high C must travel through a medium, and at frequencies above around 10 Ghz, the separation of air molecules is such that sound fails ton be transmitted. There will presumably then be a particular high C, far beyond anything we might hear, that is the highest C that can be transmitted via air. Higher frequencies might be achieved in other mediums.

Hopefully this digression shows that the context sets limits on the terms being used. Until we have a clear idea of what we mean by "high C" we don't have an answer to questions about what is the highest high C. Similarly, perhaps until we have a clear idea of what sorts of things are ineffable, we don't have a clear answer to the issues being discussed around ineffability. Trouble is, we don't have a way of saying what it is that is ineffable without the danger of thereby contradicting ourselves.

Put in more Wittgensteinian terms, we don't have a clear language game around ineffability. So we end up making one up. And that is fraught with the potential for contradiction.

Anyway, a bit off topic, I supose. Thanks for the reply.

Quine is averse to it because he thinks that it does have ontological import. But he's just plain wrong. Deluded, even. Frege and Russell had the same problem. — Arcane Sandwich

Well, I'm glad we've got that straightened out! :smile:

You might be interested in a recent thread on quantifier variance that tackles this question from number of perspectives.

Well, I'm glad we've got that straightened out! :smile: — J

It's a very small step, though. I'm not sure that it brings us any closer to reaching a general agreement on the status of Mathematical Platonism. It's just a tough debate to have, no matter if you've read all of the literature (well, all of the relevant documents, anyway).

EDIT: But thanks for the link to the Thread on quantifier variance, I'm a fan of Eli Hirsch, but more for his contributions to the Metaphysics of Ordinary Objects (i.e., his concepts of "incars" and "outcars". I don't believe in such things myself, but it's a fascinating debate nonetheless).

Thanks.

By way of continuing the example, here's how I might parse the same sort of thing.

Pegasus is an individual in the domain we are discussing. So not a predicate. We can write ∃(x)(x=a) were "a" is a constant that refers to Pegasus. It says very little.

Since it is true that Bellerophon rode Pegasus to Mount Helicon, there is something that was ridden to Mount Helicon, by existential 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?

Note the dropping of the words "conceptually" and "really". They do not appear to be doing anything.

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.

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.

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.

The existential quantifier, ∃, does not have ontological import. Quine is averse to it because he thinks that it does have ontological import. But he's just plain wrong. — Arcane Sandwich

That doesn't chime with my understanding. Did you mean ∃! ? But that's not a quantifier.

Quine certainly used quantification, to the extent that questions of existence and reality are for Quine to be answered using quantification. While I think this a but too tight, he's not wrong.

Did you mean ∃! ? But that's not a quantifier. — Banno

No, I meant the good old, classic ∃ from first-order logic. Let me show you what the problem is, with the notion that this quantifier has ontological import:

(1) ∀x(x = x) - Principle of Identity.
(2) p = p. From (1), by universal elimination.
(3) ∃x(x = p). From (2), by existential introduction.

Now, what does that mean? It means this:

(1) Everything is identical to itself.
(2) So, Pegasus is identical to Pegasus.
(3) So, Pegasus exists.

You arrive at the odd result that Pegasus exists. So what's the solution? To treat Pegasus like a predicate? Not at all. You can treat it as an individual constant, as I showed in my last example, while also claiming that it exists only in a conceptual sense, and in a specific context (Greek mythology instead of Aztec mythology, for example), and you can also say that it doesn't exist really, in the actual world. And you can say all of that at once, in the same breath, and with classic first-order logic.

Quine certainly used quantification, to the extent that questions of existence and reality are for Quine to be answered using quantification. — Banno

And I humbly think he's wrong. Better intellectuals than me have argued this point, I take my cue from them, but I don't simply take their word as one does in a fallacy of authority. I can see the actual reason why he's wrong. Then again, I should be humble, so perhaps I'm wrong.

Do you think that the analysis you offered here is much different from what I offered ?

Becasue I don't think it is.

I must have misunderstood you. You appeared to be saying that Quine had a problem with quantification. He didn't, he had a problem with individual constants, replacing them entirely with quantified variables.

The solution I offered makes use of them, contra Quine, and in accord with Kripke's answer to the sort of problem you presented. Pegasus does exist, which is to say no more than that he is an individual in the domain of discourse.

You appeared to be saying that Quine had a problem with quantification. He didn't, he had a problem with individual constants, replacing them entirely with quantified variables. — Banno

Quine didn't have a problem with quantification. If that's how what I said came across, then I apologize for the confusion, I did not intend it like that. He did have a problem with individual constants, and I have a bit of a problem with them myself, actually. But Bunge had no problem at all with individual constants, as I hope to have shown.

If I have not responded to this , it's because I'm thinking about what you said, and I want to take my time. You're not exactly asking me a triviality like "What time of the day / night is it over there in Argentina?"

Oh, no rush; indeed no obligation. Respond, or not, at your leisure. Go to bed!

Thanks for clarifying re Quine.

@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.

"Pegasus exists (is an item of) the domain of Greek mythology" looks to be a round about way to say that Pegasus is a myth. But we can do that without using any notion of existence. I just did. We can add that if something is a Greek Myth, then it is not an Aztec myth, and conclude that Pegasus is not an Aztec myth. All well and good and done without introducing two-placed existential predicates.

Greek myth(Pegasus)
For all x, Greek Myth(x) ≢ Aztec myth(x)
Hence
~ Aztec myth(Pegasus)

So I'm still not seeing the need for Bung's approach.

I could not follow that last paragraph, my apologies.

Greek myth(Pegasus)
For all x, Greek Myth(x) ≢ Aztec myth(x)
Hence
~ Aztec myth(Pegasus) — Banno

But here's the problem. Let's try to translate that to first-order language. You can't. You literally can't. Why not? Well, the closest you could get is the following:

First premise: G(p)
Second premise ∀x((Gx) ≢ (Ax))
Conclusion: ¬A(p)

The second premise is where the problem is at. You can state that premise in higher logics, but not in first-order logic. You can't declare that there's no identity between the variable "x" as the subject of a predication (Greek Myth), and that very same variable "x" as the subject of another predication (Aztec myth). It just makes no sense in the context of first-order logic, it's an error at the level of syntax. It's not a well-formed formula. And I specifically said the following:

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. — Arcane Sandwich

But we're just squabbling over details at this point. Fascinating conversation, I don't mind it, in fact I love it, but for the purpose of the OP, I already conceded and acknowledged that you have effectively solved the problem: Infinitesimals exist if and only if Pegaus exists in the exact same sense. So, you're right. Now, what that sense might be, is where our disagreement is to be found. But that's insubstantial for the purpose of this Thread and its main objective.

∀x((Gx) ≢ (Ax)) — Arcane Sandwich

is just
U(x)~((Gx⊃Ax) & (Ax⊃Gx))

Looks fine.

Cheers. Good chat.

U(x)~((Gx⊃Ax) & (Ax⊃Gx))

Looks fine. — Banno

I don't think it does. It seems to have a missing operator. Take a look:

https://www.umsu.de/trees/#U(x)~3((Gx~5Ax)~1(Ax~5Gx))

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. — Arcane Sandwich

I’m going to assume you meant “the meaning of ‛existence’” as in “what the term means,” as opposed to “the meaning of existence” in the more existential, what-is-my-life about? sense. If that’s right, can you explain how “existence” could be anything other than a concept of a formal language? The question connects, surprisingly, with my convo with @Banno about inexpressibility, which I’m about to try to continue.

I’m going to assume you meant “the meaning of ‛existence’” as in “what the term means,” as opposed to “the meaning of existence” in the more existential, what-is-my-life about? sense. If that’s right, can you explain how “existence” could be anything other than a concept of a formal language? — J

Sure. Existence is an ontological, or metaphysical concept, if you will. It's what analytic philosophers working in the field known as "Metaphysics of ordinary objects" study and discuss. In doing so, they use formal languages, most notably second-order predicate logic, and I'm averse to it for personal reasons (which are theoretical in nature), which is why I prefer to translate anything that is said in second-order language into first-order language. But that's beside the point. The point is that existence itself, not the concept, but existence itself, is a physical "thing", if you will. And in being a physical "thing", it cannot be formal. Now, there are some metaphysicians who dispute that last claim that I made, but they do it for metaphysical reasons. They will argue that forms are really out there in the world, that they're not just "in your mind" or "in your formal language". So it's a metaphysical debate. It's the same debate, more or less, about the literal existence of quantities in the objects themselves, which is the same debate regarding Natural numbers: do they exist in Nature, in some sense? Are they "out there", like apples and trees are? That's the actual "existence debate", as opposed to the debate that people have regarding the concept of "does the existential quantifier have ontological import or not?"

Not sure if anything that I said there was of any help. I can quote some books by smarter people than me, if not.

existence itself, is a physical "thing", if you will. And in being a physical "thing", it cannot be formal. — Arcane Sandwich

Thanks, I see where you're coming from now. I think equating "existence" with "physical 'thingness'," no matter how many scare-quotes we use, is debatable, though not for the reasons you suggest. I don't know whether forms or concepts are really "out there," but I'm pretty sure that the term "existence" only takes on meaning when given the sort of contexts you and @Banno are discussing. But what about Existence?!, we of course want to know. Yes, well . . . that takes us out of the Philosophy Room entirely.

Thanks, I see where you're coming from now. I think equating "existence" with "physical 'thingness'," no matter how many scare-quotes we use, is debatable, though not for the reasons you suggest. I don't know whether forms or concepts are really "out there," but I'm pretty sure that the term "existence" only takes on meaning when given the sort of contexts you and Banno are discussing. But what about Existence?!, we of course want to know. Yes, well . . . that takes us out of the Philosophy Room entirely. — J

Does it? It just takes us out of the Math & Logic Room. We're in the Ontology Room when we discuss the topic of existence from an ontological POV instead of the limited POV of formal languages in general (as in, both math and logic). Ontology, simply put, is done in ordinary language. Literally. You are of course within your epistemic rights to utilize formal languages as tools, just as the professional physicist uses math and logic merely as formal tools.

Again, the question of what is expressible in ordinary language. Let me see what I come up with for @Banno.