zero as grammatical number?

[Emily]

had a thought today during breakfast: are there any (natural) languages that treat zero as a grammatical number, the same as other grammatical numbers in that language? so for example, if the language expresses plural ("dogs") with an inflectional suffix on nouns, zero ("no dogs") would also be an inflectional noun suffix. are there any languages where this happens?

[Zju]

Long shot, but German maybe? As analytical inflection at least:
kein Buch, ein Buch, mehrere Bücher

[Emily]

the inflection in german isn't the "kein/ein/mehrere" word, it's the ⸚er. so what i'm asking is if there's a language that handles it like:

1 Buch
2 Bücher
0 Büchkein (or whatever)

(doesn't have to be a suffix of course)

[Richard W]

There's data related to this at https://unicode-org.github.io/cldr-stag ... rules.html. There are no examples of grammatical number (I wouldn't expect it), but Latvian uses the genitive plural after zero and Welsh uses the plural, whereas after other numbers it uses the singular. There's something going on with Arabic, but I can't work out what.

[Vijay]

There's something going on with Arabic, but I can't work out what.

What's confusing you about Arabic here? From what I can tell of the table, it's showing how you would say "zero books," "one boy is present," "two boys are present," and "three (or more) boys are present," all of which are different. I take from this that it's not possible in Arabic to say "zero boys are present" instead of "there are no boys."

[priscianic]

Corbett (2000) Number doesn't provide any examples of languages with a "zero number". And I think the general consensus is that such languages do not exist.

I think it's worth noting that the standard kinds of semantics for grammatical number (e.g. Link 1983, Schwarzschild 1996, Winter 2001, Harbour 2011, 2014, among many others), if you generalized them to a "zero number", would make zero number essentially semantically completely vacuous.

Here's the general kind of intuition behind the formal semantic analysis of grammatical number. A singular predicate, like cat-SG, denotes a set of "singularities" of cats—i.e. individual cats. So if our model contained three cats, Felix, Whiskers, and Socks, (f, w, and s, respectively) cat-SG would denote the set {f,w,s}. A plural predicate, like cat-s, would denote a set that contained pluralities of cats. This can either be implemented as mereological sums (as in Link 1983), or as sets (as in Winter 2001; see also Schwarzschild 1996 for discussion of these two options). Let's just do sets for our purposes here. So cat-s would denote the following set of sets: {{f,w},{f,s},{s,w},{f,w,s}}. Here, we have a set containing all the possible subsets of catkind with cardinality >1. Usually, people also include singleton sets in here, for various reasons. But that would take us too far afield; see Sauerland (2003), Sauerland, Andersson, and Yatsushiro (2005) for discussion. You might also wonder why cat-SG denotes a set of cats, whereas cat-s denotes a set of sets of cats. Why shouldn't cat-SG denote a set of sets of cats, but only the singleton sets of cats: {{f},{w},{s}}? You could do that too. It doesn't really matter for the purposes of this reply though.

You can then extend this idea to duals by saying that cat-DU would denote a set containing sets of cats with cardinality 2—i.e. the set {{f,w},{f,s},{w,s}}. And so on.

So, what would a predicate like cat-ZERO then denote, extending this kind of analysis? It would presumably denote a set of sets of cats with cardinality zero. In other words, it would denote a set containing the empty set Ø as its only member: {Ø}. Strikingly, dog-ZERO would denote the same set, as would twig-ZERO, garden-ZERO, thought-ZERO, or any other zero-number predicate. If a language had this kind of zero number, then all zero-number-marked nouns would be synonymous. That doesn't seem very communicatively useful.

When would a sentence like the cat-ZERO slept be true? First, let's consider when a sentence like the cat-s slept is true. It's true when all the cats (that are currently relevant in the discourse) are sleeping. So in our toy model with Felix, Whiskers, and Socks, the cat-s slept is true when Felix slept, Whiskers slept, and Socks slept, and false otherwise. In other words, the sentence is true when slept is true of the maximal element in the denotation of cat-s (the maximal element in {{f,w},{f,s},{s,w},{f,w,s}}, which is {f,w,s})—i.e. when slept is true of Felix, Whiskers, and Socks (true of all the members in the set {f,w,s}—this is a characteristic property of distributive predicates).This is known as a "maximality" theory of definite descriptions, and it also generalizes to singular definite descriptions to good effect, capturing the uniqueness/familiarity properties of expressions like the cat. This goes back to Sharvy (1980), so I refer you there for more discussion. Again, a digression.

So back to the original question: when would a sentence like the cat-ZERO slept be true? It would be true if slept is true of the maximal element in the denotation of cat-ZERO (the maximal element in {Ø}). The maximal element is just the null set. So our sentence is true if slept is true of all the members of the set Ø. Since there are no members of Ø, given that it's the empty set, this means that the cat-ZERO slept is always vacuously true. Again, this doesn't seem very communicatively useful. Quantification over cat-ZERO would also be trivial: all/some/no cat-ZERO slept would all similarly be vacuously true. Particularly relevant would be no cat-ZERO slept; that would be true if there doesn't exist an element in the denotation of cat-ZERO that slept. In other words, if there doesn't exist an element in Ø that slept. Again, vacuously true.

Of course, all this shows is that "zero number", with a semantics parallel to the semantics of other grammatical numbers (singular, dual, plural, etc.), would be communicatively useless, and it doesn't exist in natural language for presumably exactly that reason. Of course, if you give "zero number" a different kind of semantics (e.g. maybe the semantics of a negative quantifier, like English no or German kein), then it would be communicatively useful. But it would then arguably be a different thing entirely: a negative quantifier that's suffixed/cliticized to the noun, rather than a grammatical number marker.

(You might imagine that what I sketched above as a semantics of zero number is actually more or less an accurate semantics for the word zero (see in particular Bylinina and Nouwen 2018, who argue for something like this for zero), and that zero does seem to be somewhat useful. That may be true. (The alternative view is that zero, at least in non-mathematical contexts, is synonymous with the negative quantifier no, which crucially has a very different semantics than the stuff I sketched out above.) But note that, in natural discourse, zero NOUN is rarely used—only really when you're counting things, or in math problems, etc. And the concept of zero is also a relatively recent innovation. But if you want to have a zero number, and you want it to have a semantics akin to zero, it might be worth pondering when people actually use zero NOUN. It also might be worth thinking about the difference between two NOUN versus NOUN-DU, and thinking about how that would extend to zero NOUN and NOUN-ZERO.)

[Pabappa]

i could imagine a language taking a word like "catlessness" which would be in a mass-noun class and then evolving towards that as the form you want ... e.g. "(a/some) catlessness in the room" for "no cats in the room". Looks clumsy in English but in a language that grammaticalizes this, it would be roughly equivalent to the normal English method of getting the idea across.

Thus, zero would not have its own number category ..... it would be a mass noun ..... but this is pretty close to having its own category, since you would normally not need to mark countable things like cats with a mass noun agreement marker.

In theory, you could then do something like what French did with personne and have the mass noun marker function on its own to become a true zero-number marker.

[zompist]

So back to the original question: when would a sentence like the cat-ZERO slept be true? It would be true if slept is true of the maximal element in the denotation of cat-ZERO (the maximal element in {Ø}). The maximal element is just the null set. So our sentence is true if slept is true of all the members of the set Ø. Since there are no members of Ø, given that it's the empty set, this means that the cat-ZERO slept is always vacuously true. Again, this doesn't seem very communicatively useful. Quantification over cat-ZERO would also be trivial: all/some/no cat-ZERO slept would all similarly be vacuously true. Particularly relevant would be no cat-ZERO slept; that would be true if there doesn't exist an element in the denotation of cat-ZERO that slept. In other words, if there doesn't exist an element in Ø that slept. Again, vacuously true.

I appreciate the thorough reply, but this part of the analysis more or less convinces me that the analysis is wrong. That is, "zero cats" certainly does not make an assertion about the null set Ø.

Surely the set in question is not "cats" but "cats who have slept (here)". Then the cardinality of this set corresponds to numbers-- zero, one, two, many, etc. all work fine.

Of course, all this shows is that "zero number", with a semantics parallel to the semantics of other grammatical numbers (singular, dual, plural, etc.), would be communicatively useless, and it doesn't exist in natural language for presumably exactly that reason. Of course, if you give "zero number" a different kind of semantics (e.g. maybe the semantics of a negative quantifier, like English no or German kein), then it would be communicatively useful. But it would then arguably be a different thing entirely: a negative quantifier that's suffixed/cliticized to the noun, rather than a grammatical number marker.

I'm not sure why you wouldn't just use quantifiers. (Or to put it another way, I don't see that "zero cats" differs from "no cats".) The quantifiers (no, one, some, all) are easy to handle in predicate calculus. "Many" is not so easy, because it expresses a value on the cardinality ("N is large") that depends on the type of set.

The Finnish abessive ("without a house") seems relevant. It's a case rather than number, but it certainly shows that basic morphology can refer to a lack rather than the presence of something.

[priscianic]

Sorry, I feel like I misunderstood something? I'm not really understanding where your reply is coming from—some explanation of my confusion below. (Maybe I was misleading/confusing in my post?)

So back to the original question: when would a sentence like the cat-ZERO slept be true? It would be true if slept is true of the maximal element in the denotation of cat-ZERO (the maximal element in {Ø}). The maximal element is just the null set. So our sentence is true if slept is true of all the members of the set Ø. Since there are no members of Ø, given that it's the empty set, this means that the cat-ZERO slept is always vacuously true. Again, this doesn't seem very communicatively useful. Quantification over cat-ZERO would also be trivial: all/some/no cat-ZERO slept would all similarly be vacuously true. Particularly relevant would be no cat-ZERO slept; that would be true if there doesn't exist an element in the denotation of cat-ZERO that slept. In other words, if there doesn't exist an element in Ø that slept. Again, vacuously true.

I appreciate the thorough reply, but this part of the analysis more or less convinces me that the analysis is wrong. That is, "zero cats" certainly does not make an assertion about the null set Ø.

Not sure what you mean by "the analysis"...I'm just pointing out an intuition behind a natural extension of the semantics of grammatical number to a hypothetical "zero number", and showing that that leads to undesirable results (as you seem to note when you say "'zero cats' certainly does not make an assertion about the null set Ø", though I'm unsure where you're getting "zero cats" from, as my post was about a hypothetical zero number, rather than the lexical item zero). What I wrote isn't really an analysis of anything (e.g. it's not an analysis of "zero cats"); if anything, it's a hypothetical analysis of a hypothetical zero number that doesn't actually exist in any natural language.

Surely the set in question is not "cats" but "cats who have slept (here)". Then the cardinality of this set corresponds to numbers-- zero, one, two, many, etc. all work fine.

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.

(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.)? Or you thought that I was writing about the semantics of numerals, rather than the semantics of grammatical number? I feel like there's a miscommunication here.)

Of course, all this shows is that "zero number", with a semantics parallel to the semantics of other grammatical numbers (singular, dual, plural, etc.), would be communicatively useless, and it doesn't exist in natural language for presumably exactly that reason. Of course, if you give "zero number" a different kind of semantics (e.g. maybe the semantics of a negative quantifier, like English no or German kein), then it would be communicatively useful. But it would then arguably be a different thing entirely: a negative quantifier that's suffixed/cliticized to the noun, rather than a grammatical number marker.

I'm not sure why you wouldn't just use quantifiers. (Or to put it another way, I don't see that "zero cats" differs from "no cats".) The quantifiers (no, one, some, all) are easy to handle in predicate calculus. "Many" is not so easy, because it expresses a value on the cardinality ("N is large") that depends on the type of set.

I think what most conlangers have in mind when they say "zero number" is a negative quantifier.

I'm not sure I understand why you say "I'm not sure why you wouldn't just use quantifiers". I'm not trying to make any suggestions about what people should or shouldn't do, or give an analysis of any natural language phenomenon—just pointing out a potential reason why natural languages don't have zero number. 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'm also not sure I understand why you're talking about "zero cats"—my post was about a hypothetical "zero number" (distinct from the lexical item zero, which I never made a comment on except briefly in the end of the reply, when I referred to Bylinina and Nouwen's (2018) analysis of zero.).

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.

[dɮ the phoneme]

I appreciate the thorough reply, but this part of the analysis more or less convinces me that the analysis is wrong. That is, "zero cats" certainly does not make an assertion about the null set Ø.

Actually, I think it does, at least in the context priscianic is discussing. Certainly if I say "zero cats slept here", I'm saying something equivalent to "there are no cats who slept here", i.e. ¬∃(a cat who slept here). But if I say "yesterday I saw zero cats, and the zero cats I saw slept here" then indeed I appear to be making a claim about elements of the empty set (and priscianic is right that this makes it vacuously true). I believe it is this latter type of usage, however strained it may sound in English, to which priscianic is referring. Certainly in a language with a zero-number, this sort of phrase would be easier to construct

[priscianic]

But if I say "yesterday I saw zero cats, and the zero cats I saw slept here" then indeed I appear to be making a claim about elements of the empty set (and priscianic is right that this makes it vacuously true). I believe it is this latter type of usage, however strained it may sound in English, to which priscianic is referring.

(Just to be clear, I was not referring to any kind of usage of any kind of expression in any natural language in my reply. I was just exploring a pure hypothetical—a zero grammatical number with the same kind of semantics that people give to grammatical numbers like singular, dual, and plural—and showing how that leads to undesirable results.

For what it's worth, I don't find your English example particularly felicitous, and I'm not sure what it would actually mean.)

[zompist]

Sorry, I feel like I misunderstood something? I'm not really understanding where your reply is coming from—some explanation of my confusion below. (Maybe I was misleading/confusing in my post?)

I was writing very quickly before dinner, so I was undoubtedly unclear myself!

So back to the original question: when would a sentence like the cat-ZERO slept be true? It would be true if slept is true of the maximal element in the denotation of cat-ZERO (the maximal element in {Ø}). The maximal element is just the null set. So our sentence is true if slept is true of all the members of the set Ø. Since there are no members of Ø, given that it's the empty set, this means that the cat-ZERO slept is always vacuously true. Again, this doesn't seem very communicatively useful. Quantification over cat-ZERO would also be trivial: all/some/no cat-ZERO slept would all similarly be vacuously true. Particularly relevant would be no cat-ZERO slept; that would be true if there doesn't exist an element in the denotation of cat-ZERO that slept. In other words, if there doesn't exist an element in Ø that slept. Again, vacuously true.

I appreciate the thorough reply, but this part of the analysis more or less convinces me that the analysis is wrong. That is, "zero cats" certainly does not make an assertion about the null set Ø.

Not sure what you mean by "the analysis"...I'm just pointing out an intuition behind a natural extension of the semantics of grammatical number to a hypothetical "zero number", and showing that that leads to undesirable results (as you seem to note when you say "'zero cats' certainly does not make an assertion about the null set Ø", though I'm unsure where you're getting "zero cats" from, as my post was about a hypothetical zero number, rather than the lexical item zero). What I wrote isn't really an analysis of anything (e.g. it's not an analysis of "zero cats"); if anything, it's a hypothetical analysis of a hypothetical zero number that doesn't actually exist in any natural language.

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!

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.

Surely the set in question is not "cats" but "cats who have slept (here)". Then the cardinality of this set corresponds to numbers-- zero, one, two, many, etc. all work fine.

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.

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.

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)".

(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".

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.

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.

Zero goes back farther than they say-- the Babylonians had a zero by the 1st millennium BCE. It's often said, but erroneously, that it was used only as a separator (e.g. in 603); it is attested in other positions (e.g. 036, 630).

But linguistically, "zero" is a relative newcomer and, not surprisingly, it gets shoehorned into existing syntactic systems. In English, for some reason, we say "zero cats", not "zero cat". In French, if I'm not mistaken, you say "zéro chat". The obvious way to make "zero" work is by analogy with other numerals.

Grammatical number doesn't work syntactically much like numerals-- or quantifiers. But I don't see that the semantics are interestingly different.

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.

[zompist]

Another approach to answering Green's original question is to look at other alternatives to singular/plural.

The obvious first step is dual, and this is highly limited according to Wikipedia: Indo-European, Semitic, Inuit, Sami, Austronesian (pronouns only), Khoisan.

Trial and paucal are even more limited-- see https://en.wikipedia.org/wiki/Grammatical_number#Trial

So I think it's possible that fancy plural systems just don't happen a lot, or decay quickly.

Again, Finnish (and also Quechua) have an abessive case, which seems like it's in the ballpark.

[KathTheDragon]

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.

[priscianic]

Eeek, this ended up really really long—sorry about that! Good luck to anyone trying to make it through

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.

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.)

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).

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.

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.

(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.

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.

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.

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.)

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!

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.

[Richard W]

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.

Or that 'zero' is acquired later during language learning. I learnt to count from 'one'; it surprised me that my daughter was taught to count from 'zero'. The CLDR does caution that it may be better to treat zero as a special case and just have a totally different string for messages in that case.

[zompist]

(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.)

I think we've disagreed about natural classes before. More on what we're disagreeing about below.

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).

Now now, an adjective is not an argument. What's "drastically different" about generalizing from what we know about grammatical number, and what we know about negation? Negation leads to a lot of complications, so it's not saying much to say that the analysis would be different.

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.

I don't see a difference here; we are both speculating, hopefully by extrapolating from concepts that do exist. If speculation is wrong, I would point out that you are the one who claims to have proven something by speculating about nonexistent objects.

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.

I think we're running into problems here because (in this discussion at least) you're a splitter and I'm a lumper. You're focused on how grammatical number, numerals, and quantifiers differ; I'm focused on how they're the same. And this is mostly a matter of focus, not ultimate truth. Of course I recognize that these things differ... but languages are full of weird little differences, and that doesn't prevent us from abstracting upwards. To me it's obvious that these things are related, and I find it bizarre that you expect that each category tells us nothing about the others.

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?

Try to keep our different orientations in mind here. Do I think singular is the same as "one"? No, I expect there to be quirky differences, like everything else in language. Do I think they are completely unrelated? No, I think their prototypical meanings both involve 1*. I think we've disagreed before about prototypes and your natural classes, so if the disagreement is there we're going to have to just let it rest.

* Or maybe "not plural", in a singular/plural system. Maybe singular is unmarked. I'm curious what you'd think of the earlier discussion Brad, Richard W, and I were having about alignments and null marking, starting here: viewtopic.php?p=41619#p41619

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.

Of course speakers have access to non-semantic information: they hear how their language is used. There is no semantic reason why you decline "four" in Sanskrit but not in Latin; speakers just followed what they were taught. At some point "zero" was a new word, and people had to decide how to use it. Centuries later, we may just follow their lead.

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.

I don't know Turkish, but my first guess would be that number marking is considered unnecessary when an explicit number is given. That fits with the singular being the default or unmarked form. But in general, I expect languages to show weird variation. E.g. English has "zero cats", perhaps with the idea of "it's not one"; French has "zéro chat", perhaps with the idea of "it's not more than one".

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.)

Neat example! I would point out, though, that "chips" in (1) can be taken as either a plural, or as a sort of mass noun, and thus (1) can be answered either yes or no. The example doesn't work so well with things firmly on the count-noun side. "Did you explore planets?" is much harder to construe as a mass noun, so it'd be hard to answer "yes" if you only explored Earth today.

(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!

Also neat, but I'm not sure I agree. Consider a dialog:

A: That piano is heavy. It'd take three men to lift it.
B: No, no, it lifts itself. You don't need three men, or any men. No men can lift the piano.

There may be a difference in that "zero" is more likely to be used to contradict another number. ("Chomsky hasn't written one readable book, it's zero.")

[Richard W]

Try to keep our different orientations in mind here. Do I think singular is the same as "one"? No, I expect there to be quirky differences, like everything else in language. Do I think they are completely unrelated? No, I think their prototypical meanings both involve 1*. I think we've disagreed before about prototypes and your natural classes, so if the disagreement is there we're going to have to just let it rest.

* Or maybe "not plural", in a singular/plural system. Maybe singular is unmarked. I'm curious what you'd think of the earlier discussion Brad, Richard W, and I were having about alignments and null marking, starting here: viewtopic.php?p=41619#p41619

English

She has three children and four dogs. He has one child and two cats. Megan has no children. Robert has one cat.

translates, at least according to Google Translate, to

Mae ganddi dri o blant a phedwar ci. Mae ganddo un plentyn a dwy gath. Nid oes gan Megan blant. Mae gan Robert un gath.

The words for 'cat' and 'dog' have unmarked singulars - cath and ci, and marked plurals - cathod and cŵn. Conversely, Welsh has for 'child' has an unmarked plural plant but a marked singular ('singulative') plentyn. Thus, with positive integers, Welsh uses the singular for 'cat' and 'dog', but for numbers of children greater than one it seems to use the periphrastic o blant, literally 'of children'. Other words with singulative seem not to need the preposition 'o', but generally take the 'plural', rather than the singulative, for numbers greater than one.

[Ephraim]

I seemed to recall that Wikipedia had an article about the ”Nullar number” at one point, and indeed there once was an article for ”Nullar” which was apparently merged into the article for ”Grammatical number” in 2006:
https://en.wikipedia.org/w/index.php?ti ... d=39870330

The mention of a nullar number then seems to have been removed altogether around 2007. In any case, the original article made the following unsourced claim:
”However, in some other languages, e.g. Latvian, there is another form as well as singular and plural, the nullar, used for zero objects. So in Latvian, the singular is used for one object, the plural for more than one object, the nullar for zero objects.”

The idea that Latvian has a ”nullar form” may stem from the use of the genitive plural with the numeral zero that Richard W mentioned above.

Wiktionary still has an article for ”nullar”, that makes the following claim (no sources are mentioned):
”It is generally believed that no natural language uses true nullar number except when the noun is omitted; thus no noun ever takes on a nullar form.”
https://en.wiktionary.org/wiki/nullar

I don’t know where Wikipedia and Wiktionary picked up the term ”nullar number” but from what I could find through a Google search, the term seems to be mostly restricted to a conlanging context (or a few places that appears to have used Wikipedia as a source).

Quite a few conlangs actually have a ”nullar number”. So it may be interesting to look at how conlangers have interpreted the concept.

——

But in any case, I do think what KathTheDragon mentioned above about ”nullar” marking on nouns mostly making sense in an indefinite context is pretty important. Of course, this will depend on what you think nullar marking actually means (and how you interpret definiteness). But if you want to think about what a nullar grammatical number could potentially mean in a language that treated it mostly the same way as singular, dual and plural marking, it may be interesting to think about the following questions:
- When would you use nullar marking with definite noun phrases?
- Can you have nullar-marked topics?
- When would you use a nullar anaphoric pronoun?

[zompist]

But if you want to think about what a nullar grammatical number could potentially mean in a language that treated it mostly the same way as singular, dual and plural marking, it may be interesting to think about the following questions:
- When would you use nullar marking with definite noun phrases?
- Can you have nullar-marked topics?
- When would you use a nullar anaphoric pronoun?

I can think of contexts for these. I've indicated where the zero number would go with a subscript z.

1. "I told you not to drink my craft beers! I had three bottles, and now there's bottle_z!"
2. "Coin_z, that's what I have in my pocket."
3. "We're discussing languages with zero grammatical number, but there are none_z. They_z are intriguing though!"

Now, I would be fine with not allowing definiteness on zero-marked constituents. But when I think about using them, I come up with things that used to exist, or should exist, and somehow don't. A salient absence, you could say.