My understanding of ∃ is that it means exactly you originally quoted it as: "there exists some ....".

So a statement that ∃m(m is a man and m is Greek) means that there definitely does exist at least one instance where there is a man, and that man is Greek.

That is indeed different to saying "some men are Greek", because this statement doesn't imply anything about the existence of men at all.

I think what you're trying to say is that "some men are Greek" is more accurately represented as:
* given M = set of men, if cardinalogy(M) > 0 then ∃ m ∈ M: such that m is Greek.

More succinctly, what I'm trying to say is that the translation from "some men are Greek" to the use of ∃ is the problem here. It's not that the definition of ∃ needs changing.

Another possible source of disagreement and quandary is "value of a variable", which equivocates badly between word and object, as "numerical value" equivocates (but shouldn't) between numeral and number.

I was about to continue: "That settled by consulting a dictionary..." but no such luck. So yeah, very likely source.

If someone is talking about the empty domain they're either doing it wrong, or they're doing something in the neighbourhood of mathematical logic.

Still, that settled by following Quine's clear preference...

Both quantification functions, ∃ and ∀, only specify how many values of the variable they quantify make the statement that follows true, — Pfhorrest

Yes, i.e. they specify how many (actual, existent) things in the domain of discourse the predicate or open sentence is true of. So no call for the "only".

and the statement doesn't necessarily have to be asserting the existence of anything, — Pfhorrest

Do you mean in something like the way talking about numbers (or fictional characters) leaves it open whether they actually exist?

Sure, but that way is to talk as if they do actually exist. So ∃ still specifies that at least one (actual, existent) thing (number or unicorn) satisfies the predicate.

A fine observation. The word "exists" in the recommended translation of Ex as "there exists..." is not to be taken lightly. The word "exists" has a metaphysical meaning that the standard translation of Ex as "there exists..." fails to do justice to. For instance, while things like concepts and ideas "exist", they don't do so in the same way an elephant does. Take the following two sentences:

1. Some ideas are good idea: Ex(Ix & Gx). I hope the symbolism is self-explanatory

2. Some cows are brown: Ex(Cx & Bx)

Logic has failed to distinguish these two different flavors of existence.

That said, consider the following two statements:

1. Unicorns don't exist: ~Ex(Ux)

2. A unicorn is a horse that is white and has a horn: Ax(Ux -> (Wx & Hx))

As you might have already noticed, and it just dawned on me, statement 2, as per standard interpretation, doesn't make an existential claim, so no issues there. Statement 1 is also not problematic.

However take the following hypothetical sentence in some imagined children's book:

3. Some unicorn ate my sandwich: Ex(Ux & Ax)

Statement 3 makes an existential claim i.e. unlike statement 2, statement 3 asserts that unicorns exist but that's not true and there's no other way to translate statement 3 in predicate logic. Clearly, Ex translated as "there exists..." is an issue.

This is an example of the deep corruption inherent within modern logical systems. The requirement, to indicate that a set is not an empty set, comes about from the acceptance of the possibility of the empty set. The concept of "the empty set" is actually self-contradictory, and therefore ought to be banished as logically impossible. Then there would be no need for the phrase "there exists some m...", (which is actually a very misleading and deceptive piece of sophistry), because the question of whether the thing described exists or not would be irrelevant, as should be the case in deductive logic.

The word "exists" has a metaphysical meaning — TheMadFool

Matter of opinion :wink:

Statement 3 makes an existential claim i.e. unlike statement 2, statement 3 asserts that unicorns exist but that's not true — TheMadFool

Fiction generally isn't.

Clearly, ∃x translated as "there exists..." is an issue. — TheMadFool

Why?

Well, you'll have me repeat myself but for my own sake and yours, hopefully. here are two statements:

1. Real. Some dogs are good: Ex(Dx & Gx). Existential claim about dogs - there is at least ONE dog. TRUE

2. Fictional: Some unicorns have owners: Ex(Ux & Ox). Existential claim about unicorns - there is at least ONE unicorn. FALSE

Ex, interpreted as "there exists..." and "there is at least ONE..." clearly can't tell the difference between reality and fiction. But, the million dollar question is, Does the existential quantifier, Ex, need to make a distinction between fact and fiction?

But, the million dollar question is, Does the existential quantifier, ∃, need to make a distinction between fact and fiction? — TheMadFool

No.

No. — bongo fury

Yeah but with the caveat that fact and fiction, these terms understood in the conventional sense, don't overlap in a given argument, right? If I write "some Dodos are brown" it's logical equivalent is Ex(Dx & Bx) but we know Dodos are extinct and the logical translated of that is ~Ex(Dx)

1. Ex(Dx & Bx)........................premise
2. ~Ex(Dx)...............................premise
3. Ax~(Dx)...............................2, QN
4. De & Be..............................1, EI
5. ~De.....................................3, UI
6 De........................................4 Simp
7. De & ~De..........................5, 6 Conj [Contradiction]

What gives?

There's no issue with premise 2, it's true that Dodos are extinct. All lines 3 through 7 are valid equivalence or inference rules. Ergo, the problem must be with 1. Ex(Dx & Bx) - it's making an existential claim - the way it's defined, it has to - and we've translated "some Dodo is brown" in the approved way. However, we've arrived at a contradiction.

If I write "This Dodo is brown" it's logical equivalent is ∃x(Dx & Bx) but we know Dodos are extinct and the logical translation of that is ~∃x(Dx) — TheMadFool

Right... were you unsure whether these would turn out to be compatible or not?

Both quantification functions, ∃ and ∀, only specify how many values of the variable they quantify make the statement that follows true, and the statement doesn't necessarily have to be asserting the existence of anything, so saying that there exists some thing goes beyond what this function really does. — Pfhorrest

This reading is inconsistent with how ∃ is actually used in mathematical texts, at least the ones I am familiar with (which would be math textbooks mostly).

Interesting how a maths-logic term gets non-mathematicians/logicians all knotted up.

Meinongian quantifiers.

More fun reading on the existential import of quantifiers.

Meinongian quantifiers — Srap Tasmaner

The substance and conclusion of which appears to be pretty much "nothing to see here". As in, no answer to Quine.

"This defense leaves logic intact and also meets the objection, which is not a logical objection, but merely a reservation about the representation of natural language." https://plato.stanford.edu/entries/square/

And, same source,
"Ackrill’s translation contains something a bit unexpected: Aristotle’s articulation of the O form is not the familiar ‘Some S is not P’ or one of its variants; it is rather ‘Not every S is P’. With this wording, Aristotle’s doctrine automatically escapes the modern criticism.... On this view affirmatives have existential import, and negatives do not—a point that became elevated to a general principle in late medieval times The ancients thus did not see the incoherence of the square as formulated by Aristotle because there was no incoherence to see."

Nuance aside, the rules matter, and for right application you have to know what game you're playing. The one playing chess who on advancing his pawn to the eighth rank says, "King me!" isn't a chess player and isn't playing chess.

Right... were you unsure whether these would turn out to be compatible or not? — bongo fury

Well, as it turns out, if the logical equivalent of "some Dodos are brown" is Ex(Dx & Bx) then, it leads to a contradiction when I use it with "no Dodos exist", the approved translation of which is ~Ex(Dx).

Perhaps, one way out of this predicament is to restrict the domain of discourse temporally. The statement "some Dodos are were brown" doesn't look like it can be translated as Ex(Dx & Bx) and that would prevent the contradiction from arising.

However, this still doesn't solve the earlier problem:

Some unicorns have owners = Ex(Ux & Ox). Ex(Ux & Ox) makes an existential claim and means that there exists at least one unicorn. That's clearly false - unicorns don't exist. The only option here, like before, is to ensure there's on overlap between fact and fiction in the argument containing such sentences.

For example, if there's a book that contains real and mythical/fictional creatures and has in it the statements, "some dogs are brown" and "some unicorns have owners" then both sentences would have to use the existential quantifier as so: Ex(Dx & Bx), and Ex(Ux & Ox). That unicorns don't exist is known and that dogs exist is also known but the officially approved [logical] translations of these sentences make it look like unicorns exist in exactly the same sense as dogs exist. This, at best, is a cause for confusion, at worst, is a sign that there's a something seriously amiss in logic.

There's more to it though. The "for all", universal quantifier never makes an existential assertion. If I say "all dogs are mammals", it translate as: IF something is a dog then, it's a mammal. That there are such things as dogs is not part of a universal statement. If this is the case then why does the particular statement, "some A are B", Ex(Ax & Bx) have to be translated as "there exists something that is an A and a B"?

Well, as it turns out, if the logical equivalent of "some Dodos are brown" is Ex(Dx & Bx) then, — TheMadFool

I argue you have a translation problem, at least in part created by your "if." Boiled, peeled, reduced, it amounts to saying, if something that isn't is, then it doesn't make sense because it isn't. I think we all share the experience one time or another of turning out into this Holtzwege; the trick is not to get lost in it, and then to recognize them without having to traverse them.

@TheMadFool One thing at a time please.

The "for all", universal quantifier never makes an existential assertion. — TheMadFool

Not so.

why does the particular statement, "some A are B", ∃x(Ax & Bx) have to be translated as "there exists something that is an A and a B"? — TheMadFool

It doesn't. ~(∀x(~(Ax & Bx)))

Hence the square of opposition stuff.

But the configuration of prefixes '~∀x~' figures so prominently in subsequent developments that it is convenient to adopt a condensed notation for it; the customary one is '∃x', which we may read 'there is something that'. — Quine, Mathematical Logic

If all you're interested in is truth values, then maybe "A dog is barking" can be rendered into philo-English as "There is something which is a dog and is barking".

But does anyone think that, in saying "A dog is barking", you are asserting the existence of dogs? You're assuming or presupposing there are dogs, and so far as that goes you are committed to the existence of dogs, in Quine's sense. As above with truth values, if what you're looking for are the ontological commitments of a theory, the translation does what you want.

But the existence of dogs isn't even your assumption; it's background knowledge. Not only you but everyone you know is aware of the existence of dogs. In particular, whoever you're saying "A dog is barking" to is one of those people who already knows that dogs exist.

If the usual translation is taken as an explication -- what we're "really saying" or something -- then at least half of what people tell us everyday is stuff we already know, and that they know we already know.

(Math doesn't suffer from this weirdness because the domain is always specified. It's not like when you conclude that there is a point within this interval such that ..., you are asserting the existence of points, whatever that would even mean.)

But does anyone think that, in saying "A dog is barking", you are asserting the existence of dogs? — Srap Tasmaner

Yes, me, exactly in the sense of,

You're assuming or presupposing there are dogs, and so far as that goes you are committed to the existence of dogs, in Quine's sense. — Srap Tasmaner

(Math doesn't suffer from this weirdness — Srap Tasmaner

How is it weird?

It's not like when you conclude that there is a point within this interval such that ..., you are asserting the existence of points, whatever that would even mean.) — Srap Tasmaner

I think it's exactly like that, and we end up here on TPF discussing what it might mean.

Not so. — bongo fury

If I recall correctly, the modern interpretation of universal statements don't make an existential claim for some reason I forgot. Aristotelian universal statements do make existential claims.

In the 19th century, George Boole argued for requiring existential import on both terms in particular claims (I and O), but allowing all terms of universal claims (A and E) to lack existential import. — Wikipedia

Any ideas why?

It doesn't. ~(∀x(~(Ax & Bx))) — bongo fury

An existential statement is one which expresses the existence of at least one object (in a particular universe of discourse) which has a particular property. That is, a statement of the form: ∃x:P(x) — Google

I argue you have a translation problem, at least in part created by your "if." Boiled, peeled, reduced, it amounts to saying, if something that isn't is, then it doesn't make sense because it isn't. I think we all share the experience one time or another of turning out into this Holtzwege; the trick is not to get lost in it, and then to recognize them without having to traverse them. — tim wood

What's wrong with my "if"? :chin:

If you tell me that a dog is barking, are you also telling me there are such things as mammals?

Of course, all manner of things are implied.

If I recall correctly, the modern interpretation of universal statements don't make an existential claim for some reason I forgot. Aristotelian universal statements do make existential claims. — TheMadFool

That's right, although in everyday day speech universal statements still tend to carry existential import: from 'Everyone on the ship got sick' you may conclude 'Some people on the ship got sick'.

You can see in the SEP article how this leads to trouble with empty terms, but Parsons also makes the intriguing point there that weakenings, deriving a "some" from an "all", were not traditionally of much interest, much as empty terms were ignored. Indeed, what is the point of concluding that some people got sick if you know everyone did?

Still the modern version preserves our ability to say that if everyone on the ship got sick and so-and-so was on the ship then they got sick, which is all math needs. It saddles us with all the Martians on the ship having gotten sick too, though, but in fairness that's not just an issue with universal quantification but with the material conditional.

That's right, although in everyday day speech universal statements still tend to carry existential import: from 'Everyone on the ship got sick' you may conclude 'Some people on the ship got sick'.

You can see in the SEP article how this leads to trouble with empty terms, but Parsons also makes the intriguing point there that weakenings, deriving a "some" from an "all", were not traditionally of much interest, much as empty terms were ignored. Indeed, what is the point of concluding that some people got sick if you know everyone did?

Still the modern version preserves our ability to say that if everyone on the ship got sick and so-and-so was on the ship then they got sick, which is all math needs. It saddles us with all the Martians on the ship having gotten sick too, though, but in fairness that's not just an issue with universal quantification but with the material conditional. — Srap Tasmaner

:up: I think the reason the modern interpretation of universal statements lack existential import is basically because of empty terms which seems to fits right in with what I've been saying all along, to wit, Ex should also be neutral on the matter of existence like its companion Ax.

Thanks.

So by the time we get to asserting all of modern science every time you ask for the salt, you'll still be fine, because holism, right?

But also because you don't mean the same thing I do by "assert".

Ex should also be neutral on the matter of existence like its companion Ax. — TheMadFool

Whoa! No. That is not the conclusion you should draw.

Yes, i.e. they specify how many (actual, existent) things in the domain of discourse the predicate or open sentence is true of. So no call for the "only". — bongo fury

The “only” is because not every proposition is in the business of saying what does or doesn’t exist. “There ought to be some apples in this box” doesn’t say that there exist some apples with the property of oughting-to-be-in-this-box; perhaps the reason why no apples are in the box is because no apples exist. We can nevertheless make sense of saying some ought to exist, in this box.

Do you mean in something like the way talking about numbers (or fictional characters) leaves it open whether they actually exist? — bongo fury

Yes, that is another case. Consider geometry. We can in one sense say that, given the geometric definition of a rectangle, there exist no rectangles: all the “rectangular” things that actually exist are imperfect approximations of rectangles, not actual rectangles. But nevertheless there are true statements about some rectangles meeting certain criteria, like having all equal lengths of their sides, even though no such rectangles actually exist in the sense of “existence“ we were just using before.

(Unless in some platonic sense, but that’s exactly the kind of assumption I think we need to avoid making just by doing math at all, even though I’m not here arguing against platonism, just that it’s not necessarily entailed by doing any math).