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

What follows then?

I’m not talking about the empty set issue or anything like that. I fully support the standard modern relations between “some”, “all”, and “none”. It is perfect correct in my view to take “some rectangles have equal length legs” as equivalent to “it is not the case that all rectangles have different length legs” or “it is not the case that no rectangles have equal length legs”.

I’m more going on about how “all rectangles have different length legs” fleshes out to “if something is a rectangle then it has different length legs”, and we can affirm or deny that conditional statement without asserting the existence, in any ordinary sense, of any rectangles at all: a disagreement about that conditional is a disagreement about what would count as a rectangle if any such things existed, not about what kind of things exist.

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). — SophistiCat

Can you elaborate?

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? — Srap Tasmaner

Obviously you're being sarcastic, but again I have to be grateful for being at least half understood. :smile:

you don't mean the same thing I do by "assert". — Srap Tasmaner

I'd be thrilled if anyone in this thread were prepared to dissolve statements, assertions, beliefs, propositions and truths into one colour — bongo fury

Which is a bit extreme :snicker:

Interesting read. Unlike the Meinongian, or the Quinean interpretation of him at least, I don’t support the use of an existential predicate. Rather, I think only some sentences are in the business of describing reality in the first place, while others instead prescribe, and still others only discuss relationships between ideas without saying either that the world is or that it ought to be any way. In any of these kinds of sentences we can find use for quantifying over variables used in them, whether that quantity be “all” or only “some”.

I had another thread already about a logic for clarifying what kind of sentence we mean to assert, here:

https://thephilosophyforum.com/discussion/9066/logical-mood-functions-and-non-bivalent-logics

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

It’s only the very end of this that I have any objection to: reading the DeMorgan dual of universal quantification as asserting that there is (or exists) something. This reading works if, but ONLY if, it occurs in a sentence that is already talking about what does or doesn’t exist. If a sentence is in the business of doing something other than describing, then that reading brings in unnecessary ontological commitments.

I’m more going on about how “all rectangles have different length legs” fleshes out to “if something is a rectangle then it has different length legs”, and we can affirm or deny that conditional statement without asserting the existence, in any ordinary sense, of any rectangles at all: a disagreement about that conditional is a disagreement about what would count as a rectangle if any such things existed, not about what kind of things exist. — Pfhorrest

Some thoughts of a non-logician that may have been covered already by those more expert...

Doesn't it just depend on context, determined by the domain? Shouldn't the domain always be defined, thus making it clear how "exist" is meant to be understood, which is not necessarily "in any ordinary sense"? Although I'm not sure what counts as ordinary for you: do mathematical objects exist ordinarily?

Or, one could say that with different domains of discourse, different ordinary language interpretations of the quantifier will seem more or less appropriate, among "there exists", "for some", etc. Incidentally, "for some" seems to be pretty common.

Are you worried that an interpretation along the lines of "there exists a rectangle that...", implies the existence of rectangles, thereby introducing ontological commitments in your philosophy of mathematics? I'm prepared to be told that your worry is more subtle than that, and that I'm missing the point.

No sarcasm -- it's just that, I used to be pretty well-versed in the position I take you to espouse (there's a lot of Quine and Goodman on my bookshelf), not so much anymore and not enough to have the discussion it deserves. But if you give me a raincheck, we'll do this sometime.

:ok: :up:

I'm a little confused now, but it's probably my own fault!

I put on my "speaking for the received view" hat to address a couple of your questions, and if I'm still wearing that hat then absolutely the existential quantifier has existential import, and the universal quantifier doesn't -- it's just a kind of souped-up conditional.

If you want me to put on a "reforming logic" hat, I don't have one of those.

I do have a "logic is swell for math and generally ham-fisted dealing with ordinary language" hat and I'm almost always wearing that one, enough that I forget to take it off even when I meant to, which might have happened in this thread, I'm not sure.

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). — SophistiCat

Can you elaborate? — Pfhorrest

The way ∃ would typically be used would be to say things like "∃x (x∈R, f(x) = 0)", that is to say, "equation f(x) = 0 has a real solution." The way you would have it, that formula would say "equation f(x) = 0 may or may not have a real solution," which is trivially true. What would be the point of such an operator?

All existential operator does is assert existence. If you remove that, you have nothing left.

I'm a little confused now, but it's probably my own fault!

I put on my "speaking for the received view" hat to address a couple of your questions, and if I'm still wearing that hat then absolutely the existential quantifier has existential import, and the universal quantifier doesn't -- it's just a kind of souped-up conditional.

If you want me to put on a "reforming logic" hat, I don't have one of those.

I do have a "logic is swell for math and generally ham-fisted dealing with ordinary language" hat and I'm almost always wearing that one, enough that I forget to take it off even when I meant to, which might have happened in this thread, I'm not sure. — Srap Tasmaner

No problem. :up:

In itself, nothng. "If" is a tool, but as a tool it needs be used correctly and not abused. Ex.: if circles had corners, then we could square them. Because we can square them, there are square circles.

Yours is, if an expression of math/logic is "logically equivalent" to a natural language proposition about Dodos, then....

And it isn't. Part of the difficulty is the grammar. It should be, if...were, then. Using "is" instead of the correct "were" adds to the confusion.

Yours is, if an expression of math/logic is "logically equivalent" to a natural language proposition about Dodos, then....

And it isn't. — tim wood

Ok. Thanks for spotting the grammatical error but that doesn't have anything to do with the fact that "some aliens have legs" is expressed in logic as Ex(Ax & Lx) where Ax = x is an alien and Lx = x has legs. That being the case, the existential quantifier is forcing us to commit to something we should have an option not to viz. that aliens exist.

That use is not contrary to what I’m saying at all. In fact it’s a great illustration of the alternative reading of the “existential” operator I’m suggesting: instead of “there exists some x such that [formula involving x] is true”, I suggest “for some value of x, [formula involving x] is true”. The formula involving x may or may not be asserting the existence of anything. If it is, then saying some x satisfies it does assert the existence of something. But if it’s not, then it doesn’t.

for some value of x, [formula involving x] is true — Pfhorrest

That's exactly the "substitutional interpretation" of quantifiers.

You forget the twin criteria of validity and truth. In short, whichever way you wriggle, you're always a few cards short of a full deck. And your example suffices. Whether or not aliens exist is a matter of speculation that presumably someday will be resolved to a fact. No symbolic or grammatical manipulation will alter that. As with unicorns, in translating to natural language and back, the existential quantifier is always an assumption they exist for the purpose of the argument.

I'd be thrilled if anyone in this thread were prepared to dissolve statements, assertions, beliefs, propositions and truths into one colour... — bongo fury

Mental correlations drawn between different things. But that's another topic in it's entirety, and I'm unprepared to add anything more to this one. Seem there are enough knowledgable folk hereabouts already, and I'm not one of them to begin with.

:wink:

:grin: :up:

Yep, and that’s what I’m advocating.

You forget the twin criteria of validity and truth. In short, whichever way you wriggle, you're always a few cards short of a full deck. And your example suffices. Whether or not aliens exist is a matter of speculation that presumably someday will be resolved to a fact. No symbolic or grammatical manipulation will alter that. As with unicorns, in translating to natural language and back, the existential quantifier is always an assumption they exist for the purpose of the argument. — tim wood

Yes, "always an assumption they exist" but that's the issue here. The existential quantifier forces us to make an ontological commitment while ordinary language doesn't.

If I say "some unicorns have owners" people don't immediately reach the conclusion that there are such things as unicorns. They would, quite naturally, think that I maybe discussing a hypothetical.

Likewise, if I say "some aliens have legs" people would, again, think on those very same lines viz. I'm entertaining a hypothesis.


This flexibility, the possibility that what is being said could be hypothetical or fiction, is absent in the logical translations of the above statements. Ex(Ux & Ox) and Ex(Ax & Lx) can only be interpreted in one way - that unicorns and aliens actually exist.

that’s what I’m advocating — Pfhorrest

I was just thrown because you hadn't said anything suggesting this is where you were headed -- nothing about changing what kind of variables we quantify over, for instance.

There's some equivalence of course, but I don't think anyone is going to convince mathematicians to quantify over expressions instead of objects.

Still, I do often find myself thinking it's an attractive option for at least some cases in natural language.

And of course you trade whatever is a pain-in-the-ass about existence for whatever is a pain-in-the-ass about truth.

This flexibility, the possibility that what is being said could be hypothetical or fiction, is absent in the logical translations of the above statements. Ex(Ux & Ox) and Ex(Ax & Lx) can only be interpreted in one way - that unicorns and aliens actually exist. — TheMadFool

*sigh* All right. Ex(Ax & TMFx). TMF, of course, is you. I guess I just proved you're an A. After all, that's the logic. And any variations n this theme, yes?

Ex(Ux & Ox) and Ex(Ax & Lx) can only be interpreted in one way - that unicorns and aliens actually exist. — TheMadFool

Please make this explicit. How do you demonstrate that U or A is unicorn or alien?

There's some equivalence of course, but I don't think anyone is going to convince mathematicians to quantify over expressions instead of objects. — Srap Tasmaner

I'm not asking mathematicians to change anything at all. I'm just suggesting we interpret the ontological import of the things they write differently.

And of course you trade whatever is a pain-in-the-ass about existence for whatever is a pain-in-the-ass about truth. — Srap Tasmaner

Yes, but that's fine with me, because that's where I think the important discussion need to be had: are all true statements true in virtue of the (non)existence of something, or can there be true statements of kinds that aren't even trying to describe what does(n't) exist?

I’m not talking about the empty set issue or anything like that. I fully support the standard modern relations between “some”, “all”, and “none”. It is perfect correct in my view to take “some rectangles have equal length legs” as equivalent to “it is not the case that all rectangles have different length legs” or “it is not the case that no rectangles have equal length legs”. — Pfhorrest

The point is that you need to qualify these terms "some", "all", and "none", as you do in your example, with "rectangles". And, it makes sense to say "some rectangles", and "all rectangles", but it makes no sense to say "none", or "no rectangles". This is because "rectangle" requires a definition, and once defined, it is an object whose existence cannot be negated with "none". By defining rectangle you say "this is a rectangle". What sense could it make to turn around and say there are none of these things which I have just shown you? Such a claim could only be supported by showing the definition as self-contradicting.

I’m more going on about how “all rectangles have different length legs” fleshes out to “if something is a rectangle then it has different length legs”, and we can affirm or deny that conditional statement without asserting the existence, in any ordinary sense, of any rectangles at all: a disagreement about that conditional is a disagreement about what would count as a rectangle if any such things existed, not about what kind of things exist. — Pfhorrest

The problem here, is that if a rectangle is any sort of object at all, it is a mathematical object. So it exists by having an acceptable formula, or definition. So when you say "all rectangles have different length legs", you give existence to "rectangle", in this way. Therefore you cannot deny the existence of rectangles, as you desire, because you've already necessitate the existence of rectangles through your description of them.

Many hours ago we had a weird exchange, which left me with a vague feeling that I hadn't answered a question or that there was something I meant to come back to. (I've had kind of a confusing day.)

I'm a little confused now, but it's probably my own fault! — Srap Tasmaner

So I've come back thinking I'm now in a frame of mind to figure out what was bothering me.

We were talking about why the O form (All As are Bs) doesn't carry existential import, I linked the SEP article about the square again, and then in follow-up you said something that struck me as way wrong, though I wasn't really tuned in just then:

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

And eventually I posted the above and also this:

absolutely the existential quantifier has existential import, and the universal quantifier doesn't -- it's just a kind of souped-up conditional. — Srap Tasmaner

Which, I mean, wtf?

I can see how it happened. You had switched from talking about "universal statements" -- like All As are Bs -- to universal quantification, like ∀xFx, and I only half realized it. You can see that in the "conditional" comment there, in which I'm clearly still thinking about the O form even while I'm typing "universal quantifier"! Didn't this confuse the shit out of you?

So, for the record, these are nothing alike. With modern unary quantification, such as ∃xFx and ∀xFx, you don't have the same question of who has existential import and who doesn't. Variables like x range over a domain of discourse (giddily unspecified in natural language), a bunch of objects that you have already stipulated to "exist" (in whatever sense); all you're doing is figuring out which of them satisfy which predicates.

Since ∃ and ∀ can readily be defined in terms of each other, either they both commit you to the existence of, let's say, things that are F, or neither does. Quine more or less started this particular way of talking, and he says they do. If nothing satisfies a predicate F, you can say, 'There's nothing that's F' or 'There are no Fs,' etc.

tl;dr: 'Everything is a unicorn' and 'Something is a unicorn' both commit you to there being unicorns. 'Nothing is a unicorn' doesn't. 'Something is not a unicorn' (equivalently, 'Not everything is a unicorn') doesn't, but be careful with this one.

That use is not contrary to what I’m saying at all. In fact it’s a great illustration of the alternative reading of the “existential” operator I’m suggesting: instead of “there exists some x such that [formula involving x] is true”, I suggest “for some value of x, [formula involving x] is true”. — Pfhorrest

I fail to see the difference. We are expressing a commitment to the existence of something from the variable's domain. So in what sense are we not making an existential commitment?

Reading your other comments, it seems like in my example ∃x∈R ( f(x) = 0 ) you would want to say that if there were such things as reals (and all the other things that are tacitly assumed by the usual interpretation of that formula), then some real would satisfy the formula f(x) = 0. Is that all? Are you just concerned about (not) making metaphysical commitments when we write formulas?

Are you just concerned about (not) making metaphysical commitments when we write formulas? — SophistiCat

Yes.

My logic is a bit rusty so bear with me. You've made a distinction between:

1. Universal Statements: All F are G = Ax(Fx -> Gx)

and

2. Universal quantification: Ax(Fx) = Everything is an F

Then you mentioned O statements = Particular Negative statements:

3. Particular negative: Some F are not G = Ex(Fx & ~Gx)

I want to add:

4. Particular affirmative: Some F are G = Ex(Fx & Gx)

I'm not clear why you want to bring in 3. O statements because I clearly didn't involve them in my discussion. Perhaps their relevance stems from the fact that O statements also make existential claims.

Coming to 2. universal quantification/Ax(Fx), I'm actually not sure whether universal quantification has existential import or not. If I were to bet though I'd say, yes, they do.

Universal statements like "All F are G" = Ax(Fx -> Gx), under the modern reading, aren't supposed to be existential claims as the "if...then..." translation [Ax(Fx -> Gx)] clearly demonstrates. It's a hypothetical.

I gave it some thought last night and have come to the conclusion that the existential quantifier Ex uses the word "exist" in the metaphysical sense i.e. it's infused with ontological meaning. Just as a primer I call your attention to the reason why there is a modern interpretation/version of Aristotle's square of the opposition. You, if I recall correctly, gave me a big hint on that score.

Consider the category of vampires, an empty set (hopefully :grin: ). I could make the statement, X = "all vampires are bloodsuckers" = Ax(Vx -> Bx). Is X true/false? There are no vampires, at least to the extent we're aware, and so the statement X is false.

Now take the statement Y = "some vampires are not bloodsuckers". Statement Y, when translated as Ex(Vx & ~Bx). This too is false as vampires don't exist.

But then this leads to a problem as universal statements [all vampires are bloodsuckers] and particular negatives [some vampires are not bloodsuckers] are supposed to be contradictory and under this interpretation, the interpretation that universal quantifiers have existential import both Ax(Vx -> Bx) and Ex(Vx & ~Bx) are false.

In other words, we lose the important relationship of contradiction between universal statements (all vampires are bloodsuckers) and particular negative statements (some vampires are not bloodsuckers).

We need to devise a method by which universal statements like "all vampires are blood suckers" and particular negative statements like "some vampires are not bloodsuckers" have opposite truth values so that we can continue to have the contradictory relationship between them.

It seems the best option is to remove the existential from universal statements like "all F are G". One way of doing that is to translate them as hypotheticals with "if...then..." The statement X = "all vampires are bloodsuckers" becomes "IF there are vampires THEN they are bloosuckers". Looked at this way, universal statements like "all vampires are bloosuckers", because they're translated as hypotheticals (if...then...) can be assigned the truth value TRUE even when the subject term, here vampires, is empty. Retaining the existential import of the corresponding particular negative, "some vampires are not bloodsuckers" we assign the truth value FALSE to "some vampires are not bloodsuckers" because there are no vampires, we're able to ensure that the contradictory relationship between universal statements like "all vampires are bloodsuckers" and their corresponding particular negatives like "some vampires are not bloodsuckers" is intact. :chin:

*sigh* All right. Ex(Ax & TMFx). TMF, of course, is you. I guess I just proved you're an A. After all, that's the logic. And any variations n this theme, yes? — tim wood

I think I get it now. Yes, the statement, "some unicorns have owners" gets translated into predicate logic as Ex(Ux & Ox) but Ex(Ux & Ox) is false.

1. Ax(~Ux).........................unicorns don't exist
2. Ex(Ux & Ox).................assume for reductio ad absurdum
3. Ue & Oe.......................2 EI
4. ~Ue..............................1 UI
5. Ue................................3 Simp
6. Ue & ~Ue...................4, 5 Conj
7. ~Ex(Ux & Ox).............2 to 6 reductio ad absurdum

Since Ex(Ux & Ox) is false, even if the translation involves the existential quantifier Ex, there's no issue.

I forgot all about the possibility that Ex(Ux & Ox) could be false and assumed, mistakenly, that it had to be true. Were that the case then, it would've been a problem but since it isn't it's all ok.

Basically Ex(Ux & Ox) is definitely making an existential claim BUT that claim is false.

Please make this explicit. How do you demonstrate that U or A is unicorn or alien? — tim wood

I don't think I have to answer this question anymore.

This is as far as I got. Did I get it? I might want to resume this discussion if you don't mind. Thank you.