The meaning of the existential quantifier
In the usual type of predicate logic, a statement like "some men are Greek" would be written as ?m(m is a man and m is Greek), and read as "there exists some m such than m is a man and m is Greek".
I think that that manner of reading the ? symbol aloud is problematic, because I think it implies unnecessary assumptions or at least raises unnecessary questions about the existence of things in a more robust sense than this logical function strictly implies. I think a much better reading of the ? function is simply "for some..." (just like ? is read as "for all..."), rather than "there exists some...".
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. ? merely says that some value of the variable satisfies the following formula, just like ? merely says that any value of that variable satisfies the formula.
I think that that manner of reading the ? symbol aloud is problematic, because I think it implies unnecessary assumptions or at least raises unnecessary questions about the existence of things in a more robust sense than this logical function strictly implies. I think a much better reading of the ? function is simply "for some..." (just like ? is read as "for all..."), rather than "there exists some...".
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. ? merely says that some value of the variable satisfies the following formula, just like ? merely says that any value of that variable satisfies the formula.
Comments (89)
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.
I was about to continue: "That settled by consulting a dictionary..." but no such luck. So yeah, very likely source.
Quoting 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".
Quoting 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.
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.
Matter of opinion :wink:
Quoting TheMadFool
Fiction generally isn't.
Quoting 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?
No.
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.
Right... were you unsure whether these would turn out to be compatible or not?
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).
Meinongian quantifiers.
More fun reading on the existential import of quantifiers.
The substance and conclusion of which appears to be pretty much "nothing to see here". As in, no answer to Quine.
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 [s]are[/s] 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"?
Quoting TheMadFool
Not so.
Quoting TheMadFool
It doesn't. ~(?x(~(Ax & Bx)))
Hence the square of opposition stuff.
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.)
Yes, me, exactly in the sense of,
Quoting Srap Tasmaner
Quoting Srap Tasmaner
How is it weird?
Quoting Srap Tasmaner
I think it's exactly like that, and we end up here on TPF discussing what it might mean.
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.
[quote=Wikipedia]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.[/quote]
Any ideas why?
Quoting bongo fury
[quote=Google]
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)[/quote]
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?
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.
: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".
Whoa! No. That is not the conclusion you should draw.
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.
Quoting 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).
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.
Can you elaborate?
Obviously you're being sarcastic, but again I have to be grateful for being at least half understood. :smile:
Quoting Srap Tasmaner
Quoting bongo fury
Which is a bit extreme :snicker:
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
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.
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.
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.
Quoting 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.
No problem. :up:
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's exactly the "substitutional interpretation" of quantifiers.
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:
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.
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.
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.
Quoting 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?
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.
Quoting 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.)
Quoting 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:
Quoting TheMadFool
And eventually I posted the above and also this:
Quoting 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.
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?
Yes.
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:
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.
Quoting 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.
Quoting bongo fury
Quoting Srap Tasmaner
Although of course it does commit you to there being non-unicorns :wink:
Par for the course. Obviously I meant A.
:up:
No, it's vacuously true. The suggestion you make toward the end of your post:
Quoting TheMadFool
That's already what we do. You can do the proof yourself:
1. (x)(~Vx)
2. ~(x)(Vx ? Bx)
3. (?x)~(Vx ? Bx) ...... 2
4. ~(Va ? Ba) ............ 3
5. Va ........................... 4
6. ~Va ......................... 1
Quoting Pfhorrest
I don't think that's an issue above and beyond the old realist/non-realist divide. Non-realists simply mean something different than realists when they say "there exists x such that..." - or so they say. I am not even convinced that there is a substantive difference between these positions.
What makes me uncomfortable about the predicate calculus is that sortals aren't really like attributes, and sortals are the natural way to talk about what exists. 'To be' is substantive hungry: if you say 'X is', the question is, 'Is a what?'
We now have coyotes where I live, but we didn't when I was a kid. On first hearing them howl at night, I might remark, 'Listen to those dogs howling.' If someone else tells me, 'Those aren't dogs; they're coyotes,' they're not telling me I assigned the wrong predicate to an object, they're telling me I picked the wrong sortal. A coyote might or might not be howling -- that's a predicate; but could a coyote be a dog, or a block of cheese, or a representative democracy? And we recognize this in our grammar: 'is ...' is not the same as 'is a ...' We'll never say that a coyote is a howling, though it might be a-howlin'.
I think this is why I'm inclined to bring up the old syllogistic and the square of opposition when talk turns to existence. It feels like there's room there to make the distinction I want.
If I say, 'Some dogs over there are howling,' I'm attributing "howling" to some of the individuals in the domain "the dogs over there". But suppose they're coyotes, not dogs. There may be dogs over there and they're not howling, and that would be one kind of error; but the main error seems to be picking the wrong collection of individuals to consider attributing "howling" to. That looks to me like a very different kind of mistake.
But what if there aren't any dogs over there, howling or otherwise? I've implied there are. When corrected, I might say I thought the coyotes were dogs.
It's plain enough what I mean, but the plain language of that sentence is ludicrous. I'll only add that I couldn't have said this before being informed that there were coyotes over there, and if I had known that beforehand I wouldn't have been tempted to say that the coyotes over there are dogs, and they're howling.
I.e., that you assigned the right predicate to the wrong things? Apparently so:
Quoting Srap Tasmaner
In which case I get:
Quoting Srap Tasmaner
... As per @jamalrob's comment and probably others.
But "specifying a domain" is flagging up a likely rupture of your individual discourse from the wider "web" :wink:, e.g. your specified domain might be fictional, or (per the OP?) hypothetical, or for other reasons resist identification with any more widely recognised domain. Mending ruptures is difficult, and hence (maybe):
Quoting Srap Tasmaner
But then, isn't mending or patching together and reconciling domains what science is about? In which case I am surprised if,
Quoting Srap Tasmaner
... is the quantifiers requiring us to make our selections in terms of predicates, define our sorts in terms of attributes. Which it does, so that the selection can be from a maximally inclusive domain. Maybe?
I'm not seeing what you're seeing, so maybe you can fill me in. I don't know why sortals would be especially problematic. It's still just public language, public conceptual apparatus, picking out individuals in the way a speech community does. "Dogs". "Those dogs over there." "Some of those dogs over there." What struck you as uniquely problematic about this, more problematic than what we do with predicates?
Anyway it seems natural to me that insofar as trouble arises, the parties to a conversation will negotiate through it, as my dogs and coyotes example runs right into. (I think David Lewis talks about this in Scorekeeping, which I ought to reread.) I'm just splitting the negotiation into (a) what are we talking about? and (b) what are we saying about it?
Does that seem terribly unnatural to you?
What things, or what kinds of thing?
I'm honestly not sure how to answer. I leaned on the word "about" there but I often find analysis of "about" kinda slippery.
What I have in mind is pretty minimal, just approaching quantification in the restricted way math does.
So the analysis of
would not be
but
How do I describe that? If I want to say I'm talking only "about" the dogs that are over there and barking -- there's nothing left to say about them! Yuck.
Honestly it feels like I want to push "over there" back into the subject, that what I'm talking about is all those dogs over there and what I'm saying is that some of them are barking.
That might work, and in a sense it's okay if our sortal isn't a natural kind, but just an ad hoc count noun.
Shrug. The ad-hoc sortal thing is appealing, but we lose some of the other stuff we might want to say, even though the analysis of the sentence feels rightish. For instance, if something is a dog, it's necessarily a dog, but if it's over there it's only per accidens over there. So natural kinds.
Does it? Do you mean something involving x?R as alluded to previously? So the drift from
Quoting SophistiCat
to
Quoting SophistiCat
isn't a mere abbreviation, and ? a mere binary predicate? (In math?)
As you like, it's just that math never uses quantifiers that range over even all mathematical objects, much less everything in this and all possible worlds.
And just as a side effect, it's clearer that what you're asserting is that one of the reals is such that f(x)=0; you're certainly not asserting that the set of reals is non-empty -- you know it is, or you wouldn't be saying things like f(x)=0 anyway.
I'm not all that concerned about the metaphysics, but I am interested in finding the least misleading way to analyse ordinary language. "There is something that is a dog and is barking " is not it.
So thanks, because what I needed was to think of searching "range of quantification", and it turns out that @SophistiCat's drift, which I recognised as a (kind of a) thing, was into what's called "bounded quantification". Thing is, though, it is a mere abbreviation:
Quoting Wikipedia
It's an abbreviation that supports your point of view, that the range of quantification is restricted, but it doesn't resist undoing (as shown here), so that ? is a binary predicate and ? ranges over the whole domain of discourse - which in a mathematical discourse is presumably all mathematical objects, no?
Also supporting your point of view:
Quoting Wikipedia
So yes, you can (informally at least) use "existence" to highlight more and more predicates (sub-domains?) applying to the specific entities you are about to assign a predicate.
And then I think I might get your whole agenda here: you want to separate the process of identifying and setting up a target (which is to be the subject for a forthcoming predication) from the actual predication, the 'firing' of the predicate at that target. Anything like that?
Exactly like that!
Actually I think I'm probably just recreating work Strawson did years ago ...
Quoting Pfhorrest
I think I've been running together a few different sorts of concerns.
I want to say that what we predicate of is not a generic unknown object but a more or less specific sort of object, an object taken as some kind of thing, or taken to be some kind of thing. In a sense, I am taking back the incredibly useful step of abstracting; that is, when we say that x*x = -1 has no real solution, but does have a complex solution, there is a step of generalizing what numbers are, abstracting away certain properties and leaving a smaller defining set, thus enlarging the set of potential solutions. (Probably not how complex numbers come about, but how they are eventually understood algebraically, I think.)
I've been thinking there are modal claims that we might want to make about sortals and predicates that are obscured by this phantom abstraction, i.e., the presumption that we predicate of a generic object. But I think that turns out to be at least a little wrong, and it shows in my posts. Yes, to be a coyote is necessarily not to be a dog. But also to be over there is necessarily not to be over here, and to be howling is necessarily not to be silent. On the other hand, none exclude the others: you can be a silent coyote over here, or a howling dog over there, and all the other variations.
But there is still a difference right? If you're a coyote, you're always a coyote, wherever you go and whatever you do. If you're howling, that's a temporary state. When a coyote dies, there is one less coyote in the world, though others are probably added. When something stops howling, does a howling thing blink out of existence? There is one less howling thing in the world, okay, but wouldn't we rather just say that fewer of the things in the world are howling? And same for the converse: it's not that a thing that is howling springs into existence; one of the things already here begins howling. If that thing is a coyote, it was already a coyote, and doesn't begin being a coyote at the same time it begins howling.
I also want to say that by avoiding premature abstraction, we get to save it for when we need it, as a step we take to solve a problem. Once there's doubt about the source of the howling, we might find it useful to speak in more abstract terms like "the source of the howling".
In a sense, one could say that when a coyote dies, that matter stops doing whatever it was that constituted being a coyote. The matter still exists, it just stopped doing something. To be is to do.
Yeah that's worth considering, but it looks like a category mistake to me.
Last post resorted to heavy use of tenses, and maybe it turns out this is the most obvious difficulty in "applying" classical logic to everyday speech, a difficulty not faced in mathematics.
Goodman famously connects natural kinds to tense through projectibility, but I haven't looked at that in a long time.
I'm torn now between wanting to find a way to distinguish two types of "predicates", say by sortals being tenseless (the difference between being a coyote and howling); or just distinguishing the role in a sentence or an assertion. Maybe I get what I want just by saying this part of the sentence determines the subject -- these predicates constitute a sortal, what we're talking about -- and this other part is what we say about the subject.
Btw though:
Is the aforementioned problem (and other ordinary talk) posable as a finite one, and would it help?
Also:
:chin:
If the plan is just to cash out talk that relies on "universals", broadly construed, into talk that doesn't, because it just uses names to refer to individuals, then my mistake of thinking it's dogs howling and not coyotes seems to just drop out. If that's not a big enough problem -- I have clearly made a mistake we want to be able to point at -- how am I supposed to name the individuals in order to make this translation? If not in practice, then in principle I should be able to do so. If I were to try -- "let's go see!" -- I can land right in a de dicto/de re swamp, of a sort Quine talks about somewhere with propositional attitudes: I am looking for a dog that is howling.
(( Right now reading Sellars's "Grammar and Existence: A Preface to Ontology" -- if I understand it, I may report back. ))
When we talk about concrete stuff (be it mindly or physical), we often use ? to mean the existence of a piece of information, which is equivalent to the truth of a proposition. See that comment of mine for more info.
(Where “some-of(a,b,c,...)” just means “a or b or c or ...”).
I actually construct “for-some()” explicitly from a more general “for()” function with an embedded “some-of()” function; and I similarly construct “for-all()” out of that same “for()” function with an embedded “all-of()”, which in turn is my version of conjunction.
(Where “all-of(a,b,c,...)” just means “a and b and c and ...”).
There’s just one problem with regarding all-quantification and there-is-quantification as mere infinitary AND and OR operators, respectively. I’ll quote myself but make certain important words italic:
Quoting Tristan L
Also, note that for every property E, “?x:E(x)” means the disjunction of all propositions of the shape E(x), and “?x:E(x)” means the conjunction of all propositions of the shape E(x), where “x” is a variable that varies over all things.
We see that the very definitions of the two quantifiers need allness, and the latter cannot be reduced to AND alone. Moreover, there has to be at least one existing thing, for otherwise, the variable couldn’t vary over anything, and there wouldn’t even be a variable to vary (for it is a thing, too). So, we can’t reduce allness and existence to anything else, but we only need to invoke them once (in the definition of the quantifiers or the definition of families indexed by all things), after which we need only deal with infinite AND (or infinite OR; the two can be defined in terms of each other and negation).
That’s at least my take on this thing.