Eeek, this ended up really really long—sorry about that! Good luck to anyone trying to make it through
zompist wrote: ↑Mon Mar 22, 2021 10:46 pm
As (I hope) a minor point, I'm not sure why you're saying your analysis is not an analysis. If you want to call it an "intuition", fine; to me that just means an analysis you're not very committed to. Which is fine!
I feel uncomfortable calling the semantics I sketched for zero number an analysis because it's not an analysis of any actual data out in the world—it's more of a description of an abstract formal object. And then I showed that that kind of abstract formal object has perhaps undesirable qualities.
I am fine with calling the link I draw between the undesirable qualities of that formal object and the nonexistence of zero number in natural language an analysis.
zompist wrote: ↑Mon Mar 22, 2021 10:46 pm
You present an argument that a statement about "cat-ZERO" is a statement about the empty set. You see that this would not be "useful", but my point is that when you come up with an absurdity like this, you should step back and see what's wrong with your analysis. You seem to conclude that languages avoid the problem by not having grammatical zero number. But it's unclear to me why you think grammatical number, but not numerals and quantifiers, run into an intractable problem.
I think grammatical number runs into the problem because the standard kinds of semantics for grammatical number, when naturally generalized to a hypothetical zero number, leads to this kind of "absurdity". Generalizations of a very basic semantics for quantifiers don't lead to this same kind of problem.
(I think maybe a miscommunication is happening because I'm making an assumption here that I haven't been explicit about: the assumption being that the kinds of things in natural language that linguists would analyze as instantiating the category of "grammatical number" form some kind of semantic natural class. Perhaps this is a nontrivial assumption.)
zompist wrote: ↑Mon Mar 22, 2021 10:46 pm
So you think grammatical number semantically scopes over the whole proposition? So something like "the cats slept" means something like "the cardinality of the set of cats that slept is greater than one"? That's a very unusual semantics for grammatical number; I don't think I've ever seen anyone propose anything like that. All semantics for grammatical number I've seen have it scope inside the noun phrase.
But we don't have an example of negative grammatical number to look at! Negatives are weird-- McCawley, to give an old example, wanted to always handle them at the S level. We don't need to look above the NP
if we don't look at negatives, but extrapolating from that to a hypothetical negative grammatical number is unwarranted.
You're right that in principle we could give a hypothetical zero number a drastically different semantics than singular or plural or dual. But if zero number was drastically different from singular or plural or dual etc., then it would be highly unlikely that a linguist describing a language would end up describing that "zero number" as a grammatical number (rather than, e.g. an affixed negative quantifier).
zompist wrote: ↑Mon Mar 22, 2021 10:46 pm
But really, I made that comment because you were talking about cat-ZERO as Ø, which you recognized leads to problems, and I was trying to present an alternative. It's my intuition, but I don't think the hypothetical "Cat-ZERO slept here" is a statement about the set of no cats. It's a statement about the set of cats-that-slept-here. And when you think of it that way, there's no problems with the cardinality of the set being any number.
I...don't understand how you're having intuitions about the semantic properties of nonexistent abstract objects like "zero number". What I did in my post was stipulate a semantics for zero number, one that was a natural extension of the semantics of grammatical number, and explore some of the properties of that semantics. I don't know how you could do anything else besides that, when talking about hypothetical formal objects.
zompist wrote: ↑Mon Mar 22, 2021 10:46 pm
BTW, this goes for other numbers too. If I say "One cat slept here and one cat slept there", I'm still not talking about a set of one cat, and in fact pragmatically the two sets must be different. The relevant sets are still "the cats who slept (here/there)".
Again, not sure why you're assuming that the semantics of numerals necessarily tells us anything about the semantics of grammatical number.
For what it's worth, it's perfectly possible to give a denotation to
one that is purely intersective (i.e. it just tells you that the particular set of cats that you're talking about has cardinality one), and get the right truth conditions for that sentence. Indeed, this is a common analysis of the semantics of numerals (Bylinina and Nouwen (2018) discuss this in section 3.1.1, and also see section 3.1.2). This kind of semantics does require you to have unbound variables in the syntax that can then get bound by quantificational operators elsewhere in the sentence (perhaps null quantificational operators).
So the NP
one cat would denote the following property:
λx.|x|=1 & cat(x) (i.e. the property that's true of a singularity which is a cat). And you can combine that (conjunctively) with the property denoted by
slept here to get you
λx.|x|=1 & cat(x) & slept-here(x) to get you the property that's true of singularities which are cats which slept here. And then you need to bind the
x variable to get you a well-formed formula
∃x[|x|=1 & cat(x) & slept-here(x)]. You do the same thing with
one cat slept there, and then you conjoin those to formulas to get you
∃x[|x|=1 & cat(x) & slept-here(x)] & ∃x[|x|=1 & cat(x) & slept-there(x)].
That's the same result you'd get as if you gave
one cat an existential quantifier denotation (e.g.
λP.∃x[|x|=1 & cat(x) & P(x)])—you're just getting at that result via a different compositional route. And these two different compositional routes make different predictions: the existential quantifier approach should always give you existential force for the NP, whereas the property approach would in principle allow you to get different quantificational forces, depending on if there's another kind of quantifier floating around in the sentence that could bind the free variable in
one cat. And example (18) from Bylinina and Nouwen (2018) is supposed to show that the free variable in
three men can be bound by a generic quantifier.
zompist wrote: ↑Mon Mar 22, 2021 10:46 pm
(It seems to me that you're confusing the semantics of grammatical number (singular, dual, plural, etc.) with the semantics of actual numbers (one, two, three, etc.)?
I wasn't clear about it, but I think these things are related, and
till proven otherwise I assume that cat-ZERO is semantically equivalent to "zero cats" or "no cats".
I'm not sure why you're making that assumption. Is it supposed to hold for all grammatical numbers? i.e. is singular supposed to be semantically equivalent to the numeral
one? Or is it a special assumption you're making just for our hypothetical zero number?
For what it's worth, I did mention that you could imagine giving your hypothetical zero number a negative quantifier denotation in my original reply. But then you'd have to think about whether that's really grammatical number, or just an affixed negative quantifier.
zompist wrote: ↑Mon Mar 22, 2021 10:46 pm
Is the idea that you think it should be possible for a grammatical number marker to have a quantificational semantics? It would then have a radically different distribution than other grammatical number markers: for instance, you can say "one cat-SG", but if you had a hypothetical zero number with a quantificational semantics, you wouldn't be able to say "zero cat-ZERO", presumably for a similar reason to why you can't say "zero no cat" or "no zero cat". And that differing distribution might lead a linguist studying this hypothetical language with a "zero number" to analyze it not as grammatical number, but as an affixed negative quantifier.
I don't follow this at all, sorry. I don't see that you can make blanket statements about a hypothetical language, especially based on English syntax. Also, syntax is not semantics; I'm certainly not claiming that grammatical number is a quantifier.
You write here that you don't see how I can make blanket statements about a hypoethetical language; but it seems to me that you're making "blanket statements" about a hypothetical language (e.g. assuming, somehow "intuitively", certain properties about zero number)? Am I misunderstanding something?
I wasn't super explicit in what I wrote in that paragraph, so I apologize for that. Essentially, the very basic kind of semantics people give for quantifiers (e.g. for "some",
λP.λQ.∃x[P(x) & Q(x)]) is such that you cannot compose a quantifier with a quantified expression—you get a type clash. Quantifiers are looking for something with a
property/predicate denotation, not a quantifier denotation. I'm not sure if this is understandable without some basic background in a compositional extensional semantics (e.g. as in
Heim and Kratzer 1998). If this is too technical, just trust me that it won't work out.
Also, a clarificatory note: when I say "quantifier" (both in this reply, and in my previous replies), I mean an expression that has the semantic denotation of a quantifier. I'm not referring to a syntactic class of items.
When you write "I don't see that you can make blanket statements...based on English syntax", it seems like your assumption is that you cannot have a quantifier with a numeral together for syntactic reasons in English. That might be true. However, it might also be possible that that's due to semantics (either just semantics, or semantics in addition to syntax).
Sometimes it's hard to tell which is the right kind of analysis to take (for instance, you can explain the badness of
a furniture by saying that it's syntactic—
a needs to combine with things that have some kind of [+count] feature—or you could say that it's semantic—give
a a denotation that can't combine with mass nouns). But it's certainly true that at least
some quantifiers are well-formed with numerals, like "all three cats", or even "all zero cats". (The possibility of that is also probably another point in favor of treating numerals not as quantificational, but as modificational/intersective; I hadn't realized that.)
My English example notwithstanding (it's actually not really crucial to the main point), if you give zero number the same kind of denotation you'd give to the negative quantifier
no (at least under a very basic semantics 101 version of quantification, like in Heim and Kratzer), then it's impossible to compose another quantifier on top of
NP-ZERO/
no NP. There'd be a type clash.
zompist wrote: ↑Mon Mar 22, 2021 10:46 pm
If you're interested in differences between zero and negative quantifier no, I'd recommend checking out the aforementioned Bylinina and Nouwen (2018)—they provide several arguments that they must be distinguished, and eventually argue against a negative quantifier analysis of zero.
OK, I just read most of this. Very interesting stuff. I'm interested in the differences between "no" and "zero" that they've found, but these may well be more historical than semantic.
Not sure what you mean by "historical [rather] than semantic". Speakers don't have access to the history of their language, so their synchronic grammars must be telling them
something about the properties of
no and
zero, such that they show different properties. Of course, those properties might not be
semantic properties, but rather
syntactic ones. Bylinina and Nouwen (2018) argue that you can derive all the differences they provide from
semantic differences between
zero and
no, since their proposed analysis gives them different semantics. You might imagine an alternative theory which derives those differences from
syntactic differences between
zero and
no (is this what you had in mind?). But I'm not sure a syntactic account would be able to derive the facts about tag questions and negative polarity items.
zompist wrote: ↑Mon Mar 22, 2021 10:46 pm
Grammatical number doesn't work syntactically much like numerals-- or quantifiers. But I don't see that the semantics are interestingly different.
One obvious difference is that in some languages you get singular marking with all numbers, like Turkish:
iki çocuk ‘two boy.SG’. If singular number meant something like "one", then that would be a semantic contradiction. But it's not. Similarly for "zero cats"—you might expect that to be a semantic contradiction, but again it's not.
Of course, this is only an issue if you assume that singular and plural marking has a uniform semantics in every syntactic context. Maybe you don't like that assumption. Then for Turkish you could say that singular marking systematically becomes semantically vacuous when there's a numeral floating around, and for English you'd have to say something even funnier: plural marking systematically becomes semantically vacuous with all numerals except
one (to account for
zero cats,
0.1 liters, etc.).
There are other differences. If you imagine that
chip-s is semantically identical to
more than one chip, then you'd predict that the following questions should be identical:
(1) Did you eat chips?
(2) Did you eat more than one chip?
Now imagine a scenario where you've only eaten one chip, and someone asks you either (1) or (2). For (1), you could answer "yes", but for (2), you couldn't. This seems to suggest that there's a semantic difference between plural marking and the complex numeral expression
more than one. (Facts like these are also the reason why some people want to include atoms/singularities in the denotation of plural nouns, which I briefly alluded to in my original reply.)
zompist wrote: ↑Mon Mar 22, 2021 10:46 pm
Bylinina and Nouwen do suggest one semantic difference, sec. 3, but I'm not convinced by their example. Compare:
No men lifted the piano.
No men can lift the piano.
This seems exactly parallel to "three", in that the first sentence is existential and the second is about capability.
So the point of their original example (18) is that
three men can be read nonspecifically, and it gets a reading with generic quantification, something like "generally, groups that contain three man have the capability to lift the piano" (note how this is different than the existential quantifier reading "there exist three men that have the capability to lift the piano (together)"). This ambiguity is supposed to show that the free variable in
three men can be either bound by a (covert) existential or a (covert) generic quantifier.
It doesn't seem like a comparable ambiguity exists for your second example. It might be instructive to replace
no with
zero:
(3) No men can lift the piano.
(4) Zero men can lift the piano.
(3) only has a quantificational reading: there doesn't exist any men that can lift the piano. In other words, the piano is too heavy for anyone to lift. (4) has both a quantificational reading, which seems identical to (3), in which the piano is really heavy, but it also has a (somewhat fantastical) generic reading, something like "generally, groups that contain zero men have the capability to lift the piano", and in that reading the piano is so light that it literally lifts itself, or it can levitate, or something like that. Another (semantic!) difference between
no and
zero!
KathTheDragon wrote: ↑Mon Mar 22, 2021 11:28 pm
It might be worth pointing out that "cat-zero" only makes sense with an indefinite reading, where IMO Zompist is correct about what it would mean. Trying to impose a definite reading on it is nonsensical, which I think is what Priscianic discovered.
Again, I don't see how you can make any conclusions about what
cat-ZERO would mean, unless you have already defined a semantics for zero number. (Since it's hypothetical, we don't have access to any native speakers to tell us what makes sense or not, whether a given sentence is felicitous in a given context or not, etc.)
Under the semantics that I sketched in my original reply, where
cat-ZERO denotes the empty set, existentially quantifying over the empty set (which is presumably what you mean when you say "an indefinite reading", right?), will always result in something trivially true. i.e. if
cat-ZERO denotes the empty set, and
a/some existentially quantifies over the set denoted by its sister,
a/some cat-ZERO slept would mean something like "there exists an element of the empty set such that that element slept", which is trivially true.
