2.1 Distributional differences

If “no” and “zero” are equivalent generalised quantifiers, then we would expect them to have a similar distribution. It turns out, however, that in many respects, “zero” behaves distributionally like a numeral, not like a quantifier. A first indication of this is that, like other numerals, it allows NP ellipsis. In contrast, “no” disallows such ellipsis. Only the full DP “none” can be anaphoric to cars in (3).

According to our native speaker informants, measure nouns like “litre” and “metre” combine with numerals, and also with “zero”. Yet, they are infelicitous with “no”:

Ratio expressions like “DP per N” only allow a very specific class of DPs, especially numeral ones: “they sold { a hundred/*all } tickets per week”. Our informants tell us that, here too, “zero” pairs with numerals and not with “no”. For instance:

Another environment that is suitable for numerals is multiplicatives, as in (6).

In terms of distribution, then, “zero” appears to behave like a normal numeral. Although we take this to be suggestive of a non-quantificational nature to “zero”, it could of course in principle be that, although “zero” and “no” are semantically equivalent, they differ starkly in their syntactic requirements. That is, in order to show that a quantificational analysis is on the wrong track, we will have to show that “zero” and “no” differ in important semantic respects. As we will show next, they do.

2.2 Polarity and NPI licensing

“Zero” and “no” are obviously negative expressions. They differ, however, with respect to the nature of their negativity. In subject position, negative quantifiers trigger positive tag questions, just like sentential negation does, as shown in the following contrasts.

“Zero”, however, pairs with positive quantifiers like “most”, not with “no”, as was previously noticed by De Clercq (2011), who gives (11) as an example.²

This datum suggests that, in subject position, “zero” contributes a different semantics of negation than “no”. This is confirmed by the inability of “zero” to license NPIs. In (13), we show this for the strong NPI “in years”.

The judgements reported in the literature regarding weak NPIs are not totally consistent. However, what is consistently reported is that there is a clear contrast between “no” and “zero” in licensing weak NPIs. For instance, Gajewski (2011) reports the contrast between (14) and (15), giving (15) a question mark.

Zeijlstra (2007) thinks the contrast is clearer and gives (16), a totally parallel example, a star:

If both “no” and “zero” are equivalent negative quantifiers, it is hard to see how their licensing of tag questions and NPIs can be so different.

In summary, the negative force of “zero” is weaker than that of “no” in the sense that it does not license sentence negation phenomena like positive tag questions and NPIs.³ We take this fact to be sufficient to dismiss an analysis of “zero” as a negative quantifier. In the remainder of this article, we will show that these properties follow once we assume that the semantics of “zero” is parallel to that of (other) numerals.

3.1 A standard account

We start by discussing two prominent non-quantificational approaches of numeral semantics.

3.2 Numerals and plurality

Because in these views the cardinality information provided by a numeral is thought to be the property of an entity (as in #x = n), the semantics of numerals is intrinsically related to semantic plurality. Clearly, singular entities cannot have the property “thirteen”, only plural entities, i.e. so-called pluralities, can. Because our argument will be that “zero” is important to semantic plurality, it is worthwhile to be a bit more specific about the assumptions on semantic plurality.⁶

Pluralities are made of atoms. In particular, if a and b are two atomic entities, then there exists a plural entity a ⊔ b, the plurality that consists of nothing but a and b, or, the sum of a and b. In general, for any set of entities X, there exists an entity ⊔X whose parts are the elements of X as well as their parts, while nothing else is part of that individual. So, ⊔{john, mary} is john ⊔ mary and ⊔{john ⊔ mary, sue, ann} is john ⊔ mary ⊔ sue ⊔ ann.

Following Link (1983), we write * for the operator that allows us to map any set to the set of all pluralities that can be built from the elements of this set.

This operation is illustrated in Figure 1. The arcs between the nodes correspond to inclusion, ⊏, when read from bottom to top. So, a ⊏ a ⊔ b, and a ⊔ b ⊏ a ⊔ b ⊔ c.⁷

Figure 1

The set of pluralities *{a, b, c, d}.

The operation * is essential to explain how predicates that are incompatible with group action can nevertheless combine with plural arguments. For instance, breathing is a property of atoms only and so, one would expect that its extension only contains atoms, this in contrast to collective predicates like “to meet”. However, if “to breathe” is true only of atoms, how come we can truthfully state that “John and Mary are breathing”? The answer is that in this case the predicate is pluralised using *. If B is the set of breathing atoms, and if that set includes both John and Mary, then *B will include john ⊔ mary. “John and Mary are breathing” is true if and only if the plurality john ⊔ mary is indeed a member of *B.

Once we have plural individuals in this way, we can also express numerical properties. In Figure 1, there are four layers. The bottom layer is the layer of atoms, entities of cardinality 1. The layer above that has the pluralities of cardinality 2, and so forth.

The definition we provided for * is the one commonly assumed in the literature. It explicitly excludes the sum of the empty set. Formally, this makes the resulting structure a so-called semi-lattice.

The sum of the empty set is an individual that has no proper parts. It is the bottom element in a full lattice.⁸ We will write this element as ⊥ and, so, ⊔∅ = ⊥. Correspondingly, we could have suggested a way of forming pluralities from a set Z that includes the sum of the empty subset of Z. We will write this operation as ^×:

This operation yields the full lattice in Figure 2, which is exactly the same as the semi-lattice in Figure 1 except that it has this extra element, which turns out to be a proper part of any other entity in the structure. Crucially for our argument, while the semi-lattice in Figure 1 only allows us to express cardinalities of 1 or more, the full lattice of Figure 2 contains an entity of cardinality 0 as well.

Figure 2

The set of pluralities *{a, b, c, d}.

We are not the first to contemplate adding the bottom element to the domain of entities. Landman (2011) and Buccola & Spector (2016) have entertained and discussed a similar move. For most scholars, however, the choice between a semi-lattice domain and one in the shape of a full lattice is purely cosmetic: since the bottom element has no obvious use, it is easier to do without it. As Landman (1991) explains: “In a full [lattice] we have this 0 element. In all the important definitions, we want to apply our concepts to singular or plural individuals, excluding 0. This means that we have to add exclusion clauses (x ≠ 0) to all of them. Assuming from the start that 0 is not there, will make the definitions simpler and more readable.” (Landman 1991: 302).

A straightforward illustration of the consequences of including ⊥ comes from bare plurals. Intuitively, a sentence like (31) should receive an analysis along the lines of (32).

This simply says that the text is not without typos. But what would happen if we applied ^× instead of *, as in (33)?

It turns out that this is a tautology. The reason for this is that for any predicate P, it is true that ^×P(⊥). This is because the empty set is a subset of the extension of P no matter what that extension is. Consequently, ⊥, the sum of ∅, is a member of the extension of ^×P, no matter what P denotes. It follows that there will always be an x such that ^×typo(x) ∧ ^×in-the-text(x), just take x = ⊥.

3.3 Numeral “zero”

If we want to maintain that “zero” is a numeral, a significant dilemma emerges. On the one hand, if we maintain the dominant choice in the literature of adopting *, we have no hope of doing justice to “zero”, since it expresses 0 cardinality, a concept that is not defined in the semi-lattice obtained by applying *. On the other hand, if we go against the grain and adopt ^×, then it turns out that sentences with “zero” inherit the problem we just observed for bare plurals: we wrongly predict them to be tautological (and, as we will see below, other problematic cases of the same kind can be found).

We illustrate this dilemma using the degree denoting account of numerals. On such an account, we would interpret “zero” as in (34).

In combination with MANY, it yields the determiner meaning in (35):

Given this, “zero students passed the test” is assigned the truth-conditions in (36).

This is a contradiction. The reason is that the following two requirements made by (36) clash. First of all, the only values for x that could satisfy pluralized predicates like *student(x) and *pass-the-test(x) are values from a semi-lattice. Second, the only value for x that could satisfy #x = 0 is ⊥. Since ⊥ is never part of the semi-lattices formed by the *-operation, these two requirements are irreconcilable.

This is, of course, an unwelcome result. We can very well imagine what it is like for it to be true that zero students passed. Using a full lattice, moreover, the result is not much better.

No matter what the extensions of student and pass-the-test is, the full lattices formed by applying ^× to them are always going to contain ⊥. So, (37) is always true: just take x = ⊥.

Note that the modificational view, on which “zero” would denote the property of having 0 cardinality (λx.#x = 0), gives exactly the same result – it would also assign (36) or (37) to “zero students passed the test”.⁹

While in section 2 we presented arguments that “zero” is not a quantifier, we now have seen that giving “zero” a numeral semantics creates significant problems. As we will show next, however, once we take exactly readings of numerals into account and derive them from at least readings, things start falling into place.

4.1 Tautological semantics and exhaustification

Recall that on an at least reading account of numerals (both on the modificational and the degree-denoting account), (38) can be given the semantics in (39).

As we explained, as it stands this is a trivial statement: it is true irrespective of how many students passed, since any property ^×P holds of the bottom element. This means it is an at least meaning, compatible with there being a y s.t. #y > 0 & ^×student(y) & ^×pass-the-test(y). And, so, in other words, (39) says that zero or more students passed the test.

Like with other numerals, however, an exactly meaning can be derived from the at least meaning by exhaustification. Here, and in what follows, we will assume that this meaning comes about via a silent operator EXH, defined along the lines of (Chierchia 2004; Chierchia et al. 2013; Fox & Spector 2018).¹⁰

Since at sentence-level “zero” means “zero or more”, other numerals offer stronger statements (“one or more”, “two or more” etc.). Exhaustification denies all such stronger statements (the meaning component added by exhaustification is underlined):

This now states that there are zero or more students that passed the test, but that there are not more than 0. In other words, the number of passing students is exactly 0: everyone failed.

Unlike other numerals, “zero” invokes exhaustification obligatorily. This is for purely pragmatic reasons. Quite simply, statements with “zero” are semantically defective without exhaustification. On our syntactic view on exhaustification, interpretation of a sentence with “zero” always yields two meanings, a defective one derived without EXH and a non-defective one with EXH. That latter meaning will always be the only one to surface (provided it is not itself defective for independent reasons).

In contrast to other scalar implicatures, the “not more than 0” component of exhaustified “zero” does not disappear in embedded contexts. While (41) strongly implicates that John does not take both sugar and milk, i.e. that he takes just milk or just sugar, (42) does not mean that nobody takes just sugar or just milk. Instead, it means the stronger nobody takes either. This is standard behaviour for scalar terms. Weak scalar terms, like disjunction, trigger implicatures to make them more informative. Yet, in certain embedded positions, like in the scope of negation, for instance, their weak semantics ends up being very strong because of the scale reversal introduced by the higher operator. Weak scalar terms are strong in downward entailing contexts and this is why they fail to trigger implicatures in such contexts.

The example in (43) differs from (42) in that the exactly implicature is still in place, even though “zero” is in the scope of a downward entailing operator. This is simply because in contrast to other scalar terms, the semantics of “zero” is not more informative in downward entailing contexts. The negation of a tautology is just as defective as the tautology itself.

Crucially, having exhaustification rescue the defective semantics of “zero” is not the same as assigning “zero” lexical semantics that amounts to “exactly zero”. As we will see in the next section, detaching the “exactly” (EXH) component from the word “zero” plays an essential role in our analysis of the polarity of “zero”.

4.2 “Zero” and negative polarity items

As we observed in section 2, “zero” appears not to be able to license negative polarity items, in contrast to “no”. As we will explain in this section, adopting an at least semantics for “zero” and other numerals provides a way to approach this contrast. Under the view we adopted above, “zero” differs semantically from “no” in that its negative effect comes about via exhaustification. As we will show, this difference can indeed account for the differences in NPI licensing. Let us first introduce our assumptions.

Traditionally, NPIs are thought to be licensed in downward entailing (DE) environments (Ladusaw 1979). We follow Gajewski (2011) in assuming that an NPI is licensed when the environment it occurs in is DE. Assuming an exhaustification operator as we have done above, this licensing condition has two parts:

These two conditions are needed to account for differences between weak NPIs (for instance, “any”) and strong NPIs (for instance, “either”, “in years”).

To illustrate, consider the case of “few”. “Few” licenses weak NPIs, as in (46), but not strong ones, (47).

“Few” is downward entailing in its second argument. If it is true that “few” students read City of Glass, then it will also be true that “few” students read City of Glass twice. For any statement “few A B” there is a stronger alternative statement “no A B”, though. This means that the exhaustified meaning of “few A B” will make it false that “no A B” and thus true that “some A B”. In other words, after exhaustification “few” is no longer downward entailing. If nobody read City of Glass, then “few students read City of Glass” is true on the non-exhaustified and false on the exhaustified reading. In terms of Gajewski’s licensing conditions this means the following (assuming no polarity reversal intervenes between “few” and the NPI):

Given the distinction in licensing conditions between weak and strong NPIs stated in (45), this setup now correctly predicts that “few” licenses weak, but not strong NPIs.

If we apply this approach to “zero”, we are not immediately going to be successful. Given our at least + EXH semantics, “zero” creates the following entailment patterns. (Again we are assuming nothing affecting polarity intervenes between “zero” and the NPI.)

“Zero” is clearly downward entailing after exhaustification. If exactly 0 students read City of Glass, then exactly 0 students read it twice. The non-exhaustified version is a bit trickier. On our approach, “zero” has a defective semantics. Any statement of the form “zero A B” is – without exhaustification – a tautology. In other words, “zero A B” is a tautology, but “zero A C” is also a tautology, irrespective of our choices of A, B, and C. Let ⊤ be the tautological proposition, the proposition that is always true. Obviously, ⊤ entails ⊤. Now choose B and C such that the extension of C is a proper subset of that of B. Since “zero A B” and “zero A C” both express ⊤, they entail each other. Since the extension of C is contained in that of B, “zero” must be both downward entailing and upward entailing.

Given the situation in (49), we wrongly predict “zero” to license both kinds of NPIs. It is well-known, however, that the requirement of being in a downward entailing environment is at times too lax. In quite a few places in the literature, it is suggested that not only should NPI environments be downward entailing, they should also not be upward entailing (Progovac 1993; Lahiri 1998; Gajewski & Hsieh 2014; Barker 2017). One illustration of this comes from singular definite descriptions. Since definite descriptions are presuppositional, the relevant notion of entailment we need is so-called Strawson entailment (Von Fintel 1999). A premise Strawson-entails a conclusion, if the premise together with the presuppositions of the conclusion entail the conclusion. Given such a definition, the restrictor argument of a singular definite description is Strawson DE, as illustrated in (50).

Given the above setup, it would now be predicted that definite descriptions license (at least weak) NPIs. They do not:

However, the restrictor argument is not just downward but also upward entailing (Lahiri 1998).

Given such observations, we can revise our earlier statement of NPI licensing conditions, replacing downward entailingness by non-trivial downward entailingness (NTDE), i.e. being downward, and not upward entailing.

If we now return to “zero”, we see the following:

In other words, the case of “zero” is exactly the opposite of the case of “few”. Whereas the latter met condition 1 but not condition 2, “zero” meets condition 2 but not 1. Since both kinds of NPIs are subject to condition 1, we now correctly predict that “zero” licenses no NPIs of any kind.

Our account of the lack of NPI licensing by “zero” is dependent on our assumption that numerals come with an at least semantics, which is turned into an exactly meaning by the additional operation of exhaustification. If we had assumed that the exactly meaning is basic, then we would have had no account of the NPI licensing contrast between “zero” and “no”, since “zero” is non-trivially downward entailing on that reading.

In other words, two crucial ingredients have allowed us to explain the polarity profiles of “zero”: (i) the at least semantics of numerals; (ii) the inclusion of ⊥ in the domain of entities. The first of these is relatively uncontroversial. The second of these is more controversial than the first. For that reason, we now turn to the consequences of that second assumption.¹¹

6.1 De-Fregean “zero”

On the Kennedy (2015) view, the sentence “Zero students passed the test” would yield (72) or (73).

In order to see what these formulas express, we first need to understand which sets are described by λi.∃x[#x = i ∧ *student(x) ∧ *pass-the-test(x)] and λi.∃x[#x = i ∧ ^×student(x) ∧ ^×pass-the-test(x)]. In any world with at least some students passing the test, the former lambda term will describe the set {1, …, n}, where n is the number of students passing the test. The latter lambda term, the scope of the max operator in (73), adds 0 to that set: {0, …, n}. In both cases, the maximum of the set is clearly not 0, and so both (72) and (73) correctly predict that “Zero students passed the test” is false whenever some students passed.

In a world with no students passing the test, things are different. The lambda term in (72) now yields the empty set. Since there is no student who passed the test, there is no x that satisfies the *-pluralised predicates, and so there is no number that corresponds to the cardinality of such an x. However, the lambda term in (73) does not describe the empty set in a world in which no students pass the test. This is because there is an x satisfying the ^×-pluralised predicates, namely x = ⊥, and so there is exactly one number in this set, namely 0. In other words, in such a world, (72) and (73) correspond to the propositions max(∅) = 0 and max({0}) = 0, respectively.

Clearly, then, (73) yields the correct truth-conditions. What about (72)? Under quite standard assumptions, we would have to assume that this semantic analysis fails. The reason is that max(X) is normally defined as returning the unique element in X such that no other element exceeds it on the relevant scale. Taking the maximum from the empty set, as we would need to do for (72) is simply undefined.

However, one could alter the definition of max by stipulating that the maximum of the empty set is the bottom element of the scale used by the operator, 0 in this case.¹⁴ If we do, then (72) and (73) become equivalent, since now max(∅) = max({0}) = 0.

Importantly, the option of stipulating that max(∅) = 0 opens up a route to talk about zero cardinalities without assuming the existence of ⊥. The fact that (72) and (73) are now equivalent means that we have found a semantics of numeral “zero” that does not depend on extending our ontology with the bottom element. Does this mean that a de-Fregean analysis is a contender for a full, yet ontologically light theory of numeral “zero”?

To answer this, we will look at two cases: the scopal ambiguity (or lack thereof) of sentences with “zero” and the polarity data discussed above.

6.2 “Zero” and scope

If, as our own proposal has it, the exactly reading of “zero” comes about via exhaustification, then we could expect to see the effects of the exhaustification operator engaging in scope relations with other operators. Take (74), for example.

On our analysis, we expect this to have two possible logical forms, (75a) and (75b).

These correspond to two truth-conditionally different readings: (76a) says that the company is not allowed to fire anyone, while (76b) says that the company does not need to fire anyone (this kind of reading is often referred to as the “split scope” reading for reasons beyond our immediate concern, although see Jacobs 1980; Rullmann 1995; De Swart 2000; Penka & Zeijlstra 2005; Abels & Martí 2010; Penka 2011). Both readings are attested, so our proposal seems to make a valuable prediction here.

The de-Fregean analysis is no different, however. Since on that account “zero” is a degree quantifier, it can take scope at different sites. As with our own proposal, again two logical forms are possible for (74):

The meanings expressed by (76a) and (76b) correspond exactly to those of (75a) and (75b), respectively. It turns out then that it is going to be very hard to distinguish between the de-Fregean account and ours on the basis of scope matters: the readings generated by the scope flexibility of the exhaustivity operator are exactly the same as those created by the different QR landing sites that degree quantifier “zero” may inhabit.

The situation is different when it comes to sentences with non-modal quantifiers.¹⁵ In a context with a nominal quantifier, such as “every student”, our proposal predicts two logical forms to be available, quite like with the modal quantifier:

The latter LF (77b) amounts to a weak, “split-scope”, reading according to which it is not the case that every student read one book or more – it should be compatible with the situation where one of the students read some non-zero amount of books. Contrary to the prediction our analysis makes, the sentence in (77) does not have this reading – as shown by the unavailability of a continuation like in (78):

The lack of the reading corresponding to the LF in (77b) thus posits a challenge for our proposal. Kennedy’s (2015) analysis, to the contrary, predicts the unavailability of this reading. Two candidate LFs for sentence (77) under the de-Fregean theory would be the following – depending on the landing site of QRed degree quantifier “zero”:

But only one of these logical forms is predicted to be viable – the one to be filtered out is (79b), corresponding to the unattested “split scope” reading. (79b) is ill-formed according to what is known as the Heim-Kennedy generalization (Kennedy 1997; Heim 2000), which states that nominal quantifiers can never intervene between a degree quantifier and its trace:

It is easy to see that (79b) realizes the prohibited scheme (80): it involves a degree quantifier (“zero”) QRed over a nominal quantifier (“every student”). Therefore, in an analysis in which “zero” is a degree quantifier and scope ambiguities are derived via QR of “zero” to different positions, the asymmetry between modal and nominal quantifiers can be traced back to (80). Kennedy (2015) is one such theory, while our proposal is not.

As it stands, things are as follows. We assumed a lower-bounded semantics for numerals, which allowed us to explain the lack of NPI licensing for “zero” in terms of its indirectly derived negative meaning, namely via exhaustification. That same mechanism now poses a problem for our theory, since we have not provided sufficient constraints on exhaustification to prevent over-generation. We will have nothing to say on how to solve this issue, but instead we will point out a dilemma. While the at least semantics for “zero” over-generates readings, it provides a neat explanation for the polarity data. The de-Fregean, doubly bounded, theory of numerals, on the other hand, has a salient remedy against over-generation via the Heim-Kennedy generalization.¹⁶ As we will explain next, the degree quantifier approach to “zero” has little to no hope to provide an explanation for the NPI data.

6.3 Polarity revisited

As we observed in section 2.2, “zero” doesn’t seem to be grammatically negative in the same way as, for example, negative indefinites are. Unlike “no”, “zero” in subject position doesn’t license positive tag questions, doesn’t trigger negative inversion and – the case we focus on – doesn’t license NPIs (we repeat examples (13) and (14)):

In section 4.2 we argued that adopting an at least semantics for numerals, including “zero”, allows us to account for this lack of NPI licensing.

Assuming that the structure of sentences with “zero” involves EXH attached to the structure corresponding to the at least reading, we formulated NPI licensing conditions for this configuration (following Gajewski 2011):

All NPIs are subject to condition 1 – domain β has to be non-trivially downward-entailing. Environments with “zero” before exhaustification don’t satisfy this condition – they are both upward- and downward-entailing. Thus, NPIs are not predicted to be licensed – as desired.

The difference between exactly and at least analyses of numerals is that the LFs generated by the former crucially lack the structural point corresponding to the split between domains α and β in (83). A doubly bounded semantics will not guarantee the presence of a β-environment that violates Condition 1. This means that exactly analyses of “zero” will always predict NPI licensing. This prediction is independent of further assumptions (the exact definition of maximality or presence or absence of the bottom element).

The degree quantifier analysis (Kennedy 2015) is an exactly analysis. This fact is fatal for capturing the NPI facts – what gets the data right is the lack of a specific configuration in the language inventory – namely, the lack of configuration in which numerals (including “zero”) have an exactly semantics lexically. As long as this option is available, NPIs are incorrectly predicted to be licensed, whether there is an additional at least + EXH option or not. The polarity data are deeply problematic for the de-Fregean analysis.

There is no clear way for the de-Fregean analysis to account for the NPI data. The only potential way to get numeral “zero” to satisfy the conditions above is to detach maximality from the numeral after all and treat it as a kind of exhaustification operator. The numeral itself would be a degree quantifier with an at least semantics.

This raises a number of concerns. First and foremost, deriving at least readings as basic via at least degree quantification would undermine the very idea behind degree quantifier semantics for numerals. If numerals are at least degree quantifiers (and there is no other way to derive at least readings), the exclusively exactly readings of numerals in non-existential contexts will be left unaccounted for.¹⁷ An additional mechanism deriving in situ readings would be needed for these cases – this would be the at least type-shift. But if there are two distinct mechanisms for deriving at least readings, a number of new questions arise as to how to constrain each of these derivations both in terms of interaction with their syntactic and semantic environment and, potentially, in terms of their competition with each other, to avoid over-generation. Finally, whether NPI licensing conditions would end up directly applicable to this hypothesised new structure without further stipulation also depends on the details of the implementation, which is beyond the scope of this paper.

Summing up, polarity facts, as we argue in this section, are fundamentally problematic for a degree quantifier analysis of “zero” – but fall out naturally under our analysis.