As has been pointed out by Israel (1996), Iatridou & Zeijlstra (2010, 2013) and Homer (2015), universal modals that take wide scope with respect to sentential negation, like English must, should or ought to, should be analysed as PPIs.

Iatridou & Zeijlstra (2010, 2013) and Homer (2015) analyse these modals as PPIs, as only these modal auxiliaries outscope negation. Other modals auxiliaries, existential modals and other universal modals alike, in principle take scope under negation:

In fact, in the domain of deontic modals, all existential modals scope under sentential negation.6 Iatridou & Zeijlstra (2010, 2013) argue that modals auxiliaries are base-generated in a position below negation, where they should also be interpreted, regardless of whether they raise to a higher position at the surface structure or not. This then explains that the modals in (15) and (16) take scope below negation. Only if an interpretation under the scope of negation would lead to ungrammaticality, Iatridou & Zeijlstra argue, may the modal be interpreted in its surface position or even raise to a higher position above negation. Elements that are scopally banned from negation (and other DE contexts) are by definition PPIs. Assuming that modals that outscope negation are PPIs thus ensures that these, but no other, modals take scope above negation if they appear in a negative clause.

Empirical evidence for the PPI-status of universal modals that outscope negation comes from two facts. First, other polarity-sensitive verbs/auxiliaries have been attested in the domain of modals as well. English need (the need that does not select to-infinitivals) as well as its German and Dutch translations (brauchen and hoeven, respectively) are all NPIs, as the following examples show:

Given that generally PPIs surface in domains where NPIs appear, it is likely to expect that modal PPIs can be attested as well.

Second, it is a well-known fact that PPIs may take scope under negation under special circumstances (cf. Szabolcsi 2004). For a detailed description and application of these tests to modal PPIs I refer to Iatridou & Zeijlstra (2013) and Homer (2015), but for the sake of illustration I address two tests already here, which will also prove to be relevant in the remainder of this article. The first test concerns intervention effects, where a scope-taking intervener disrupts the relation between an (anti-)licenser and a polarity item. If an intervener shows up between the PPI and the anti-licenser, this PPI is fine under the scope of negation, as shown in (18), taken from Iatridou & Zeijlstra (2013). Both sentences in (18) allow a reading where must takes scope below negation.

The second test (after Baker 1970) concerns the appearance of a PPI under two DE operators. PPIs that are banned from anti-additive contexts may appear under the scope of an anti-additive operator if that operator, in turn, is outscoped by another (non-anti-additive) DE operator. The data in (19), again taken from Iatridou & Zeijlstra (2013), show that this also applies to English must.

If modals like must, should and ought to are PPIs, the question naturally pops up as to why they are PPIs in the first place. One answer, in the light of the discussion, suggests itself: they are universal quantifiers that have the same properties that Chierchia (2006, 2013) attributes to existential quantifiers that are NPIs. If must, should and ought to are introducers of domain alternatives that require the presence of a covert exhaustifier, it follows immediately that they may not take scope under negation: that would give rise to a semantic contradiction again.

To see this, take (20), where the modal does not reconstruct below negation. Let’s assume that the modal base of (20a) consists of the worlds w₁, w₂, w₃. The semantics of She must leave (without the exhaustifier) is then as in (21a). Since no domain alternative is stronger than (21a), exhaustification applies vacuously and the meaning of (20b) is as in (21a).

Now suppose that must would reconstruct under negation. The logical form of (22a) would then be as in (22b). But now all relevant domain alternatives are stronger than the proposition [she e -n’t must_[uσ]-leave] and therefore (22b) has the semantics in (23). But (23) is a contradiction, as it states that not in all worlds w₁, w₂, w₃ she leaves, and at the same time that she leaves in all worlds w₁, w₂, w₃. This proposal thus correctly predicts that must cannot reconstruct below negation.

One caveat should be made, though. The toy model used in (21)–(23), contains only three possible worlds. In more realistic models, the number of worlds is of course much bigger (and can even be infinite). More importantly, the alternatives will not reduce down to singleton sets, as there is never enough information for a speaker/hearer to narrow down the domain of worlds into singleton sets. Note, though, that for this approach to yield the contradiction it is only necessary that the domain alternatives are partitioned and that the union of all domain alternatives is already the same set as the original domain of quantification. For instance, in (23) the contradiction would also be yielded without any singleton alternatives. In fact, if there is already a partition with two subdomains, with, for some proposition p, one subdomain consisting of all and only all p-worlds and the other subdomain consisting of all and only all not-p-worlds, the contradiction is yielded. Both subdomain alternatives would then be stronger, whereas the union of these subdomains of quantification is still identical to the original domain of quantification.

The fact that modal PPIs can be analysed along the lines sketched above does, however, not entail that they have to be analysed as such. It could very well be the case that the source of their PPI-hood lies elsewhere. However, there are good reasons to analyse them along the suggested lines.

First, Chierchia’s approach builds upon the idea, originally proposed by Kadmon & Landman (1993), that existential NPIs (and thus universal PPIs in their slipstream) are elements that must be strengthened. Kadmon & Landman argue that existential NPIs are domain wideners that stretch their domain of quantification beyond its contextual restrictions. They argue, for instance, that (24b) is stronger than (24a):

And even though the judgements are subtle, it seems that similar effects are attestable in the case of must and have to. Although both sentences in (25) seem slightly degraded, native speakers prefer (25b) over (25a), suggesting that (26a) is stronger than (26b).

However, the idea that all NPIs are domain wideners has been criticised (see, e.g., Krifka 1995, among others). For this reason, Chierchia replaced it by the introduction of domain alternatives, where a licensed NPI (and thus also a non-anti-licensed PPI) should not necessarily exhibit strengthening effects, as strengthening by the covert exhaustifier may take place vacuously. Hence, even though the data in (25) do not run against the suggested analysis, they cannot be taken to be conclusive.

Second, an alternative way to evaluate the suggested analysis is by comparing it to other competing analyses. One potential alternative that comes to mind, though, is that certain modals like must or should are in some sense performative and would therefore require a higher LF position than negation. However, such an analysis, especially for must, runs at odds with the fact that in various cases the modal PPI may indeed take scope below negation and is thus allowed in an LF-position below the position of negation (see (18)–(19)). Another alternative would be Giannakidou & Mari’s (2016) approach to PPI-hood of epistemic modal adverbs, which takes a positive speaker’s perspective to be the source of the PPI-hood of speaker-oriented modal adverbs such as unfortunately or probably. Giannakidou & Mari’s approach particularly focuses on modal adverbs, but less so on modal auxiliaries. More importantly, their approach only applies to non-universal epistemic modal PPIs, whereas the reported modal PPIs here are all deontic modals (cf. Iatridou & Zeijlstra 2013; Homer 2015).8

Third, the strongest piece of evidence, however, comes from the hitherto unreported observation that PPIs under this approach should be able to “self-intervene” (though see Iatridou & Zeijlstra 2013: 557, footnote 40). The argument is quite complex, but takes at heart the possibility that the exhaustifier that the PPI induces may appear in a scopal position between the PPI and its anti-licenser and therefore allows the PPI to take (indirect) scope under it. In the following section this argument will be spelled out in detail.

Under Chierchia’s approach, every NPI that is exhaustified gives rise to a logical contradiction unless some DE operator intervenes between the exhaustifier and the NPI:

All other scopal configurations of EXH, DE and NPI than (27) give rise to either a feature-checking violation (when the uninterpretable feature [uσ] remains unchecked) or give rise to a logical contradiction. However, in the domain of PPIs, things are different. Whereas (28) is a scopal configuration that yields ungrammaticality, other scopal configurations between EXH, a DE operator and a PPI are fine:

One such configuration that is fine is the one where the DE operator appears under the scope of the PPI (as in (29)), which we saw was the scopal configuration that emerges when the modal does not reconstruct (in the case of She must not leave).

But another licit scopal configuration that has not been discussed so far is the one in (30).

Nothing in (30) violates any rule of grammar. The PPI’s uninterpretable feature [uσ] has been checked by higher EXH; application of EXH over its (propositional) complement does not give rise to any contradiction (as no DE operator is embedded in the complement of EXH); and, finally, since exhaustification applies vacuously, nothing forbids the DE operator to take scope over its own complement that contains the vacuously exhaustified PPI. Since no further EXH is included, (30) can underlie acceptable sentences.

To see this, take (31a) again, repeated from (11), but now with the logical form in (30)/(31b):

Now, the exhaustifier in (31b) applies vacuously. The reason is that I read pevery book is first exhaustified, before negation applies. But then exhaustification applies vacuously: of all the propositions in (32), (32a) is the strongest.

Hence, the meaning of [EXH [I read [pevery book]]] is the same as the meaning of [I read [pevery book]] (both mean ∀x.[x∈{a,b,c} → read(I, x)]), which can subsequently be negated without any problem (yielding ¬∀x.[x∈{a,b,c} → read(I, x)]). Hence, a universal quantifier PPI can actually take scope below negation, provided the logical form is one where negation does not take scope in between the (higher) EXH and the (lower) PPI.

The only difference between unacceptable (28) and acceptable (30) concerns the position of the covert exhaustifier. The question thus arises as to what determines the position of a covert operator in a sentence. In this we follow Zeijlstra (2008a; b, 2012) who for the inclusion of abstract negative operators argues that the locus of the abstract operator must be in a position c-commanding the highest overt marker of it at surface structure. For Zeijlstra, neg-words in negative concord languages (where multiple negatively marked elements give rise to one semantic negation only) carry an uninterpretable feature [uNEG] that, much alike NPIs/PPIs under the approach pursued here, must be c-commanded by a (possibly) covert operator that carries the matching interpretable feature [iNEG]. Under the postulation that the covert negative operator is in a position higher than any of the neg-words (or other elements carrying an uninterpretable feature [uNEG]), it follows that all neg-words take scope below negation, a desirable result under the relatively well-established assumption that neg-words are existentials or indefinites. An illustrating example for Italian is given below:

One might wonder what the nature of the principle is that states that covert operators must be assumed to be present in a position c-commanding the highest overt marker of them. There are two ways to think about that. First, the mechanism can be derived from Chomsky’s Merge-over-Move constraint. Inserting an abstract operator below an element that marks this operator (i.e. that stands in an Agree relation with it), requires an additional step of movement. To see this, take (33) again. If Op¬ were inserted below highest nessuno, nessuno should have been base-generated in a position even lower than Op¬ to have its feature checked under c-command, or Op¬ should have been raised to a higher position:9

Since a competing derivation, (33b), lacks these extra movement steps, (34a–b) can be ruled out under Merge-over-Move.

Alternatively, one can also conceive the principle that posits the abstract operator in a position above the highest overt agreeing marker as an extra-grammatical constraint on grammatical structures. Under such a conception, all derivations that contain an abstract operator in some position where it can check off the relevant uninterpretable features and that do not give rise to semantic anomalies are grammatically fine, but are not parsable. It is the parser that includes the abstract operator the moment that the parser detects its marker; so the only sentences that are both parsable and grammatically correct are the ones where the operator is higher than its highest agreeing marker. Such extra-grammatical constraints are not unfamiliar in grammatical theory. For instance, Ackema & Neeleman (2002), Abels & Neeleman (2012), and Biberauer et al. (2014) allude to similar mechanisms, albeit in different ways, to account for various left-right asymmetries in grammar, such as the ban on rightward movement.

As for now, I remain agnostic about how exactly EXH-inclusion is constrained, but, in full analogy to the inclusion of abstract negative operators, I take the following to hold:

We are now in a position to determine which scopal configurations are fine for exhaustified PPIs and which ones are not. The question at stake is when a scope configuration like (30), repeated in (36) below, is licit. After all, when (30)/(36) would reflect a possible LF, it shows that PPIs may indeed appear under the scope of their anti-licenser, provided that the exhaustifier they induce acts as an intervener. The answer to this question is now straightforward: if a PPI is in a position lower than its anti-licenser, EXH may c-command the PPI under the DE operator, provided that the complement of EXH is propositional. Then, the PPI can take scope under its anti-licenser:

If, however, the PPI c-commands its anti-licenser at surface structure, the exhaustifier must be in a position c-commanding both the PPI and the DE anti-licenser:

Note that both (36) and (37) are acceptable scope construals. However, a consequence of this is that any sentence where the exhaustifier that checks off the uninterpretable feature [uσ] of a PPI that c-commands a DE operator at surface structure would forbid the PPI to reconstruct below the DE operator, even in cases where non-PPIs would be allowed to reconstruct. The reason is that, then, this PPI would end up in the illicit scope configuration (38) that yields a logical contradiction.

So, a particular prediction that I take the application of Chierchia’s (2006, 2013) mechanism to make is that PPIs, at least those PPIs that are PPIs due to the presence of [uσ], are elements that (i) may appear and take scope under DE operators at surface structure; and (ii) when they appear above a DE operator at surface structure, may not reconstruct below it.

Now strikingly, all modal PPIs in English appear above the negative marker:

Also, several other languages discussed in Iatridou & Zeijlstra (2013) place PPI modals in a position above the negation, such as (Northern/Western) Dutch moeten (‘must’) or German sollen (‘should’):

The other languages involving modal PPIs discussed in Iatridou & Zeijlstra (2013) do not place the modals in a position below negation, but rather have them form one morpho-phonological unit, suggesting that these complex expressions occupy the same position in the syntactic structure. Examples here come from Greek and Spanish.

Both Greek dhen and Spanish no are clitic-like elements (hosted in Neg°) that head-adjoin from a lower position into the modal head (cf. Zanuttini 1997, 2001; Giannakidou 1998; Zeijlstra 2004; Merchant 2006 for a cross-linguistic survey of the morpho-syntactic status of such negative markers). Arguably, no abstract operator can intervene in such complex words and the PPI may only appear at LF in between the negation and the DE operator.

Hence, all modal PPIs attested so far appear either above negation or form a morpho-phonological complex with it. This is already strongly in line with the proposed analysis. But even more evidence can be provided. Dutch is a language that exhibits V-to-C movement in main clauses, but not in embedded clauses. Modal auxiliaries, being verbal in nature, thus appear above negation in main clauses, but below negation in subordinate clauses. Consequently, Dutch moeten (‘must’) in negative main clauses can only yield the scopal configuration EXH > MUST > DE and behaves PPI-like:

However, in embedded clauses, two scopal orderings should be fine: one where EXH intervenes between the negation and the modal, and one where EXH takes widest scope:

Example (45) is thus predicted to be ambiguous, unlike (44), a prediction that is indeed born out:

The fact that the Dutch pattern shows exactly the predicted behaviour can be taken as a piece of evidence in favour of the proposal that takes PPI modals to be universal quantifiers that introduce domain alternatives and that require covert exhaustification. But it gives rise to a new question as well: why have universal quantifier PPIs only been attested in the domain of modal auxiliaries and never in the domain of quantifiers over individuals?

Nothing in this proposal on the PPI status of modals hinges on the fact that they are quantifiers over possible worlds. In fact, the approach to NPI-hood has even been developed for quantifiers over individuals. Hence, nothing in the approach should forbid the existence of universal quantifiers over individuals that are PPIs (for the same reason). However, no such PPI has been attested so far.

In this section, I argue that the reason why only universal PPIs have been attested among quantifiers over possible worlds and not among quantifiers over individuals, again, lies in the fact that universal quantifier PPIs may actually scope under negation, and therefore do not appear to be PPIs. To see this, take again the scopal ordering of a universal quantifier with a feature [uσ], negation and the covert exhaustifier that gives rise to the logical contradiction, the ordering in (48):

If negation intervenes between the exhaustifier and the universal, a contradiction arises. But, as argued above, nothing requires that a universal quantifier with a feature [uσ] (henceforward ∀_[uσ]) has its exhaustifier scope higher than the negation: the feature [uσ] only requires that the exhaustifier c-commands the ∀_[uσ] and therefore has scope over it, but does not require that it has immediate scope. An alternative underlying syntactic configuration for such a universal quantifier carrying a feature [uσ] is (49):

But (49) does not give rise to a logical contradiction. As shown in (31)–(32), the proposition I read every_[uσ] book, denoting ∀x.[x∈{a,b,c} → read(I, x)], could be exhaustified (a vacuous operation, since it is already stronger than any of its alternatives) before it gets negated. The denotation of (49) is then just (50), where the exhaustifier intervenes.

Universal PPIs (or to be more precise: universal quantifiers that obligatorily introduce domain alternatives that must be exhaustified) are fine in negative/DE contexts as long as the exhaustifier takes scope in between the anti-licenser and the universal quantifier itself. Universal quantifier PPIs may appear under negation without being unacceptable and therefore are unrecognizable as such.

How do we know, then, if such PPIs that are universal quantifier over individuals exist in the first place? As discussed before, we can only do so on the basis of examples where a universal quantifier negation appears above a morphologically independent negative marker at surface structure. In that case the surface scope order would be EXH > ∀ > NEG. Under such a configuration, the universal quantifier that is equipped with a feature [uσ] cannot reconstruct below negation (as this would give rise to a logical contradiction), but a universal quantifier that is lacking [uσ] would be able to reconstruct below negation.

Interestingly, variation between universal quantifiers that may and that may not reconstruct under negation when appearing above it at surface structure has indeed been attested (and never been properly explained). In the remainder of this section, I argue that the only distinction between such quantifiers is the presence or the absence of a feature [uσ] on ∀. Following this line of reasoning, it can actually be established that English everybody is not a PPI, but that Dutch iedereen (‘everybody’) is a PPI, to the best of my knowledge a novel observation.

In English and most other languages (cf. Zeijlstra 2004 for an overview), for almost all speakers a universal quantifier that precedes negation may reconstruct under negation. Sentence (51) is ambiguous between a wide-scope reading of the universal quantifier with respect to the negation (‘nobody left’) and a narrow-scope reading (‘not everybody left’), the latter being the most salient reading. This shows that English everybody cannot carry a feature [uσ].

However, for a small number of languages this is not the case. For most speakers of Dutch (and several Northern German varieties), this reconstructed reading is not available (cf. Zeijlstra 2004; Abels & Martí 2010). The same holds for (Levantine) Arabic and Japanese.

This observation has never received a satisfactory explanation, but directly follows once universal quantifiers in Dutch, Northern German, Levantine Arabic and Japanese are taken to be PPIs. Focusing here on the Dutch example, if iedereen is a PPI, it must be c-commanded by EXH at surface structure, and reconstructing it below negation would result in the contradictory reading EXH > NEG > ∀, thus providing a simple solution for this hitherto unsolved problem. Moreover, if iedereen is a PPI, the prediction that our analysis makes with respect to universal quantifiers, namely that universal quantifiers equipped with a feature [uσ] should also be attested (since nothing principled rules them out), is also confirmed.

However, how can we independently investigate whether Dutch iedereen is indeed a PPI? As shown by Szabolcsi (2004) and briefly discussed earlier, PPI-hood can be diagnosed in four different ways. First, PPIs should be fine under metalinguistic negation. This is indeed the case for Dutch iedereen, which may take scope under metalinguistic negation:

Second, PPIs can take scope under clause-external negation. Again, this applies to iedereen as well.

Third, PPIs can scope under negation if a proper intervener scopes between the PPI and its anti-licenser. In a way, we already saw that this is the case for those PPIs that appear under the surface scope of negation (since EXH then acts as an intervener), but more examples of intervention effects can be attested (see (18)). Example (57) can be true in a situation where it is not always the case that everybody leaves the room. Note that this reading is facilitated by adding extra stress on altijd (‘always’).

Finally, Szabolcsi (2004), following Baker (1970), shows that PPIs can be rescued under two anti-licensers (with the highest one being a non-anti-additive anti-licenser). Again, this is the case for Dutch iedereen. Take (58). The most salient reading of this sentence is the one where the speaker is surprised that some people left (i.e. that not everybody stayed). Again this reading is only possible if iedereen is allowed to reconstruct under negation.

So, Dutch iedereen, when preceding negation, exhibits all the diagnostics of PPI-hood, thus allowing us to safely conclude that it is indeed a PPI in the classical sense (i.e. in the same way that anybody and ever are NPIs). The fact that it may appear under the scope of surface negation when negation precedes it immediately follows as this PPI introduces an exhaustifier, which in turn may acts as an intervener between the PPI and its anti-licenser.10,11

So far, we have not distinguished between PPIs of different strength. As Iatridou & Zeijlstra (2013) point out, various modal PPIs are subject to different strength requirements. Focusing on English, at least the following two PPIs can be distinguished: strong PPIs (banned from all DE contexts) and weak PPIs (banned from anti-additive contexts only). English should is a strong PPI. For instance, it cannot take scope below few, at most N or only, whereas English must, which is a weak PPI, can:

The distinction between strong and weak PPIs is reminiscent of the distinction between strong and weak NPIs. For instance, English in years is a strong NPI, since it can be licensed by anti-additive contexts only:

By contrast, English ever is fine in all DE contexts:

As noted by Collins & Postal (2014), in English strong NPIs are also strict NPIs, NPIs that must be licensed within a local syntactic domain, such as a finite clause or an island. Licensing such strong/strict NPIs across such locality boundaries, like adjunct islands, is not possible:

Again, weak NPI licensing is not subject to such syntactic locality constraints and may apply across locality boundaries:

Gajewski (2011) and Chierchia (2013) take the weak-strong distinction to lie in the distinction whether the exhaustifier looks only at the semantics of the NPI licenser, or also at its pragmatics (both the presupposition and the implicatures). Weak NPIs trigger EXH to look at the semantics of the licenser only; strong NPIs also trigger EXH to look the licenser’s pragmatics.

Let’s illustrate this for few N. Few students passed the exam asserts that not many students passed the exam and implies that at least some students passed it. That the latter meaning contribution is an implicature and not part of the assertion is evidenced by the following example:

If few meant ‘not many, but some’ there would be no reason why the department would face budget cuts in this case. This means that if one only looks at the semantics of few N, it is DE and it can shield an NPI from yielding a contradiction when embedded under an exhaustifier. However, the joint meaning contribution (the semantics and pragmatics) of few N is no longer DE. If few but still some students wear a shirt, it is not entailed that few but still some students wear a red shirt. Hence, when an exhaustifier takes the implicatures of the licenser into consideration, few N is no longer DE and cannot shield an NPI from yielding a contradiction when embedded under an exhaustifier either. Since anti-additive contexts do not give rise to such non-DE implicatures, such pragmatically active exhaustifiers can still shield NPIs under anti-additive contexts. This, then, may explain the difference between strong and weak NPIs.

This solution, however, faces two serious problems. First, it is unclear how it can be a property of an NPI that its exhaustifier must (not) consider the pragmatics of its licenser. And second, it is unclear how the locality conditions fit in. The latter is actually a problem for understanding weak NPI licensing in the first place, for how can it be possible that in (69), repeated below as (71), ever’s uninterpretable feature [uσ] can be checked off by EXH if this checking relation violates a syntactic locality constraint?

In order to circumvent these problems, I suggest, slightly speculatively, to make the following amendments to Chierchia’s theory, based on these NPI facts. Later on, I demonstrate that they make very precise and correct predictions in the domain of PPI-hood.

Let’s assume that there are two ways in general to trigger the presence of an exhaustifier: one way is by syntactic agreement (as has been shown in this article at various places); another one would be the result of a pragmatic mechanism that states that if there have been introduced some (domain) alternatives in the sentence and they have not been applied to by any operator that applies to alternatives, as a last resort, the entire clause is exhaustified. Implicitly or explicitly, such mechanisms have been suggested in the literature before (cf. Krifka 1995). This amounts to the following picture:

Now, the strict vs. non-strict distinction among NPIs may naturally follow: both types of NPIs obligatorily introduce all scalar and domain alternatives, but strict NPIs must syntactically agree with the exhaustifier; non-strict NPIs are subject to pragmatic exhaustification. But if strict NPIs are exhaustified by a different exhaustifier than non-strict NPIs, nothing forbids that these two exhaustifiers may have different semantic properties. This opens up ways to understand the weak-strong distinction and its correlation with the strict-non-strict NPI/PPI distinction. Since weak NPIs are subject to pragmatic exhaustification, it seems plausible to assume that the pragmatic exhaustifier has the same properties that we assigned thus far to EXH. In this sense, EXH functions like a classical Neo-Gricean operation. However, if syntactic exhaustification always considers the implicatures of the licenser as well it makes rather sense to assume that this is a different type of exhaustifier, call it EXH+, which also considers the implicatures generated in its entire complement.

As sketchy as the above may be, it provides a solution for the two problems addressed before. First, it predicts that all weak NPIs, unlike strong NPIs, can be licensed by a licenser outside their syntactic domain. Second, it is now possible to syntactically encode the difference between weak and strong NPIs by postulating that all NPIs obligatorily introduce domain alternatives (which is what renders them NPIs), but that only strong NPIs carry an uninterpretable feature [uσ] that triggers the presence of EXH+ that they agree with; weak NPIs are simply subject to the pragmatic exhaustification requirement (given that they obligatorily introduce domain alternatives). Note that an additional advantage is that what underlies NPI-hood under the exhaustification approach boils down to only one requirement (obligatory introduction of domain alternatives) and not two (obligatory introduction of domain alternatives and the presence of a [uσ]-feature on the NPI).

This solution also makes predictions for the domain of PPI-hood. If the presence or absence of an uninterpretable feature [uσ] is what distinguishes weak from strong existential NPIs, this should also be behind the distinction between strong and weak universal PPIs, again in the reverse configuration (as weak NPIs and strong PPIs have a distribution described in terms of downward entailment, whereas strong NPIs and weak PPIs have a distribution described in terms of anti-additivity): whatever makes a weak NPI a weak NPI, makes a strong PPI a strong PPI, and whatever makes a strong NPI a strong NPI, makes a weak PPI a weak PPI. Both weak and strong PPIs introduce domain alternatives, but only weak PPIs carry a feature [uσ] (and trigger EXH+ rather than EXH).

Now, the presence of EXH+ no longer causes non-anti-additive DE operators to give rise to a contradiction when they scope over universal PPIs. As we saw, a plain DE operator like few N is no longer DE when its implicatures are taken into consideration; and if DE-ness is what is behind the semantic contradiction that renders anti-licensed PPIs unacceptable, few under EXH+ can no longer give rise to this contradiction when it scopes over a PPI.

So now we may assume that English must carries an uninterpretable feature [uσ], whereas English should does not do so and is subject to a pragmatic exhaustification constraint. This has, however, serious repercussions for the analysis we presented so far. If only weak PPIs (the ones like must with an uninterpretable feature [σ]) syntactically trigger the presence of EXH+, and strong PPIs like should do not, then the mechanism that is responsible for postulating the covert exhaustifier EXH+ in a position between the anti-licenser and the PPI is only available for weak PPIs like must and not for strong PPIs like should. Strong PPIs trigger the presence of EXH, which can only be present at clausal/CP level. Hence, we predict that only weak PPIs are fine under higher clause-mate anti-licensers and strong PPIs are not. The relevant configurations are below:

Now, for English, given the fixed position of modal auxiliaries in the clause, this may be hard to test, but other languages exhibit constructions that can be used to evaluate these predictions. As said before, Dutch exhibits V-to-C movement in main clauses only. We already established that whereas Dutch moeten behaves like a typical PPI in main clauses, it can easily appear under the scope of negation in embedded clauses. Dutch moeten is also a weak PPI:

This is in line with our predictions. But Dutch also exhibits a universal modal PPI meaning ‘should’ (of the form would must). This modal, however, is a strong PPI. The same applies to German sollen (‘should’).

Now, the final prediction is that in an embedded clause (both Dutch and German exhibit V-to-C movement in main clauses only), these stronger PPIs, being pragmatically exhaustified only, still must outscope negation. The pragmatic exhaustifier EXH can only be introduced at clausal/CP level. And, indeed they do:

Hence, the evidence presented so far shows that when applied to the domain of PPI modals the suggested amendment to Chierchia’s theory that was independently necessary given the different syntactic and semantic behavior of weak and strong NPIs again makes correct predictions for the treatment of universal PPIs. Finally, note that this predicts that Dutch iedereen must also be a weak PPI (and not a strong PPI), as otherwise it would also behave PPI-like when appearing in object position.

To conclude, universal quantifier PPIs do exist, both in the domain of quantifiers over individuals and in the domain of quantifiers over possible worlds, as is predicted by Chierchia’s (2006, 2013) approach to NPI-hood, which in turn is based on Kadmon & Landman (1993), Krifka (1995) and Lahiri (1998). Since the exhaustifier that is induced by these PPIs can act as an intervener between the PPI and its anti-licenser (at least for weak PPIs), universal quantifier PPIs may appear in disguise and take scope below their anti-licenser. Their PPI-like behavior only becomes visible once they morpho-syntactically precede their anti-licenser. In addition, I provide a possible analysis for why this mechanism is restricted to weak PPIs only, based on some independently necessary amendments of Cherchia’s theory of NPIs.