This paper analyzes the syntax and compositional semantics of scalar modifiers of quantifier phrases in expressions like

The universal quantifier phrase headed by

(1)

a.

b.

c.

Almost everybody was there.

Absolutely everybody was there.

Nowhere near everybody was there.

Other expressions that fall in the same class as

Such expressions can be used with other kinds of quantifier phrases as well:

(2)

a.

b.

c.

Almost all the students were there.

Absolutely all the students were there.

Nowhere near all the students were there.

(3)

a.

b.

c.

Almost no students were there.

Absolutely no students were there.

?Nowhere near no students were there.

In this paper, I present a compositional semantics for such scalar modifiers, on analogy with the cross-categorial semantics of negation presented in

Section 2 presents the framework for the syntax and semantics of negation in

I follow

(4)

If X has a semantic type ending in <…, t>, then

NEG takes X with semantic value: λP_{1}….λP_{n} […]

And returns Y with semantic value: λP_{1}…λP_{n} ¬[…]

For propositional variables p (no predicate abstraction), the negation is simply ¬p. On this view, negation can combine with constituents of various different types, parallel to the analysis of conjunction given in Partee and Rooth (

Consider the following example:

(5)

Not everybody was there.

_{DP} everybody]] (see also

(6)

Not every day do you hit three home runs in a postseason elimination game at Dodger Stadium.

(

Following

In the framework of (4), negation in (5) has the following semantic value in (7):

(7)

⟦NEG⟧ = λQ.λP.¬[Q(P)]

In other words, NEG takes a generalized quantifier argument of type <<e,t>,t> and yields a generalized quantifier of type <<e,t>,t>.

On analogy with the proposals about the syntax and semantics of negation in

(8)

⟦almost⟧ = λQ.λP.∃X[near(X,Q) ∧ X(P)]

In both cases,

(9)

a.

⟦almost⟧ (⟦everybody⟧) =

b.

λQ.λP.∃X[near(X,Q) ∧ X(P)] (λR.∀x[person(x)

c.

λP.∃X[near(X, λR.∀x[person(x)

As (9c) shows, ⟦almost everybody⟧ takes an argument of type <e,t> (a predicate) and yields a truth value (of type <t>), and so ⟦almost everybody⟧ is a generalized quantifier of type <<e,t>, t>.

An analysis closely related to the one given in (8), also in the spirit of

(10)

⟦almost⟧ = λQ.ιX[X < Q ∧ distance(X,Q) = n]

According to this definition,

In the next section I will discuss scales of generalized quantifiers and the definition of

Following (

(11)

a.

b.

some

not every

<

<

half

less than half

<

<

every

no

(positive)

(negative)

All generalized quantifiers on a scale have the same NP restriction. In other words, each restriction (the denotation of NP) gives rise to two scales: a negative scale and a positive scale. The positive scale is defined by the smallest number of elements (satisfying the restriction) in each member set of a generalized quantifier. For example, for the DP [some boy], each member set of the generalized quantifier has one or more boys in it. The negative scale is defined by the greatest number of elements in each member set of the generalized quantifier.

The scales in (11) are model theoretic objects (each defined as a total ordering of generalized quantifiers), not conventionalized scales of lexical items (contra

For example, consider a quantifier phrase like

From the standpoint of the scales in (11), negation and scalar modifiers have complementary functions. The function of negation is to move from one scale to another (e.g., when

In (8), _{S}, where S is the contextually appropriate scale (with similar changes to (10)). But I leave out the scale subscript for brevity’s sake.

What counts as

(12)

Almost everybody showed up.

(12) has the following truth conditions:

(13)

ƎX[near(X, ⟦everybody⟧) ∧ X(⟦showed up⟧)]

Translating into English: There is a generalized quantifier X that is near to ⟦everybody⟧ on the positive generalized quantifier scale such that X applied to ⟦showed up⟧ is true.

In this case, the generalized quantifier X can be taken as 95% of the people, so that (13) entails (14):

(14)

95% of the people showed up.

A property of the analysis in (13) is that (15a) does not entail (15b).

(15)

a.

b.

c.

Almost everybody was there.

Not everybody was there.

Everybody was there.

The reason why (15b) is not an entailment of (15a) is that (15a) and (15c) can be true at the same time (see (16) and (17) below for supporting evidence). Adding a negative entailment to the semantics of

Rather, I claim that (15b) is a scalar conversational implicature of (15a) (see

To show that

(16)

a.

b.

c.

d.

e.

As long as you finish nearly/almost all of your vegetables, you can watch TV.

Just about/almost everybody should be present, before we can vote.

We should start as soon as almost everybody is here; in fact, everybody is here, so let’s start.

Not only did almost everybody show up, (in fact) everybody did.

If you want to pass the exam, you have to answer almost all questions correctly.

In (16a), if the child finishes eating all of the vegetables, they will not be stopped from watching TV. In (16b), if everybody is present, the vote will not thereby become invalid. (16c) and (16d) were suggested to me by Jack Hoeksema and Larry Horn respectively. (16e) is from Nouwen (

Similarly, there is a clear difference between the following two sentences (personal communication, Paul Postal):

(17)

a.

b.

Almost everybody showed up. In fact, everybody did.

Not quite everybody showed up. *In fact, everybody did.

In context, (17a) is acceptable (e.g., “Let’s start the meeting….”). But in any context, (17b) is a contradiction (indicated by the * before the second clause). The scalar modifier

The sentences in (16) all involve

Penka (

My account has no such syntax/semantics discrepancy. The scalar modifiers are sisters of DP and compose semantically with their sisters. This is achieved by giving

Scalar modifiers also modify negative quantifier phrases (see

(18)

Almost nobody showed up.

(18) will have the following truth conditions:

(19)

ƎX[near(X, ⟦nobody⟧) ∧ X(⟦showed up⟧)]

Only quantifiers X near the right edge of the scale in (11b) will count as near to

Now consider (20):

(20)

Almost half of the people were there.

This means a little less than half of the people were there, not that a little more than half of the people were there. So, the following condition on the meta-language relation

(21)

near(X,Q) only if X is less than Q (on the contextual scale S)

A reviewer makes the interesting proposal that in the case of

Examples where

(22)

This means there are almost fewer than 1 million veterans remaining of the 16 million who served our nation in World War II.

(

(23)

At this point, we have fewer than 40 days of walking and almost fewer than 600 miles to go,

(

In these examples the generalized quantifier X which is near Q is less than Q on the negative scale in (11b).

The semantic value in (8) involves existential quantification over generalized quantifiers. There are other scalar modifiers that are more transparently quantificational. Consider negative scalar modifiers, such as

(24)

Nowhere near everybody showed up.

(25)

¬ƎX[near(X, ⟦everybody⟧) ∧ X(⟦showed up⟧)]

The question is what counts as near in (25). The use of the phrase

I will now show how to derive (25) compositionally given the analysis of scalar modifiers in this paper. I assume that

(26)

[_{DP} [NEG SOME] [_{NP} where [_{PP} near everybody]]]

In other words,

And I assume the following semantic values:

(27)

a.

b.

c.

d.

e.

⟦NEG⟧

⟦SOME⟧

⟦where⟧

⟦near⟧

⟦everybody⟧

=

=

=

=

=

λX.λQ.λP.¬X(Q)(P)

λQ.λP.ƎX[Q(X) ∧ X(P)]

λQ.positive-scale(Q)

λQ.λX.near(X,Q)

λP.∀x[person(x)

NEG in (27a) is the semantic value of negation, which directly modifies SOME (as opposed to negation in (5) which modifies the entire universal quantifier phrase, the distinction is not relevant to this paper). The first argument of SOME is a predicate of generalized quantifiers and the second argument is a predicate of individuals.

The only semantic value that needs comment is (27b). Compare (27b) to the regular semantic value of

(28)

a.

b.

⟦some⟧

⟦SOME⟧

=

=

λQ.λP.Ǝx[Q(x) ∧ P(x)]

λQ.λP.ƎX[Q(X) ∧ X(P)]

The reason for the difference is that English

The compositional calculation is shown below:

(29)

a.

b.

c.

d.

e.

⟦near⟧(⟦everybody⟧)

⟦where⟧(⟦near everybody⟧

⟦NEG SOME⟧

⟦(26)⟧

⟦(24)⟧

=

=

=

=

=

λX.near(X, λP.∀x[person(x)

λQ.[positive-scale(Q) ∧ near(Q, λP.∀x[person(x)

λQ.λP.¬ƎX[Q(X) ∧ X(P)]

λP.¬ƎX[positive-scale(X) ∧ near(X, λP.∀x[person(x)

¬ƎX[positive-scale(X) ∧ near(X, λP.∀x[person(x)

Alongside of

(30)

a.

b.

I saw nowhere near every student.

I didn’t see anywhere near every student.

Following

(31)

I NEG_{1} see [[<NEG_{1}> SOME] [where near every student]]

In this structure, <…> stands for an unpronounced occurrence. In this case NEG_{1} raises from a position modifying SOME to a post-auxiliary (post T) position. Examples like (30b) arise when NEG_{1} raises from the DP object, and SOME is spelled out as

The analysis in section 5 of

The instructions were the following: “This is an informal judgment survey about scalar modifiers of quantifier phrases. Please rate each sentence from 1 (unacceptable) to 5 (acceptable).” The survey was sent out to the individual people by e-mail, and the responses were received by e-mail.

In

Survey of acceptability judgments of scalar modifiers.

The survey in

(32)

a.

b.

Nowhere near everybody will be at the party.

*Somewhere near everybody will be at the party.

This contrast is surprising. One reason that the contrast is surprising is that I showed in section 5 that (32a) has a compositional interpretation, so we might expect such a compositional interpretation to extend to (32b). Another reason the contrast in (32) is surprising is that in the domain of physical location, both of the following are possible:

(33)

a.

b.

I parked nowhere near here.

I parked somewhere near here.

I do not at the present time have any convincing analysis of the contrast in (32). But I note that a similar contrast applies in several other cases (personal communication, Richard Kayne and Paul Postal respectively):

(34)

a.

b.

John is nowhere near as smart as Bill.

*John is somewhere near as smart as Bill.

(35)

a.

b.

He is nowhere near a .300 hitter.

*He is somewhere near a .300 hitter.

The unacceptability of

(36)

a.

b.

*Somewhere near everybody was at the party.

*Somewhere around everybody was the party.

The survey shows that the following sentences involving

(37)

a.

b.

c.

*Nowhere around everybody will be at the party.

*Nowhere around half of the people will be at the party.

*Nowhere around nobody will be there.

These facts suggest that

(38)

*I didn’t see around 10 rabbits.

In fact, (38) is only unacceptable on a low scope interpretation: “It is not the case that there were around 10 rabbits that I saw.” On a high scope interpretation, it is much better: “There were around 10 rabbits that I didn’t see.” These facts are consistent with

In the survey, the following sentences involving

(39)

a.

b.

c.

Nowhere near half of the people will be at the party.

Somewhere near half of the people will be at the party.

Somewhere around half of the people will be at the party.

These facts generally support the analysis proposed here. For example, (39c) could be given a compositional analysis similar to the one given in (29) above.

(40)

a.

b.

Somewhere around half of the people showed up.

ƎX[around(X, ⟦half of the people⟧) ∧ X(⟦showed up⟧)]

The difference between the metalanguage

The following sentence involving a negative scalar modifier modifying a negative quantifier was rejected by the majority of respondents, but is predicted to be acceptable on my theory of scalar modifiers:

(41)

*Nowhere near nobody showed up.

The truth conditions should be:

(42)

¬ƎX[near(X, ⟦nobody⟧) ∧ X(⟦showed up⟧)]

Suppose in this case that 20% or fewer counted as near. Then these truth conditions would be consistent with 21% or more of the people showing up.

Only three people rated the double negative in (41) to be 3 (on a scale of acceptability). However, one respondent commented on the importance of prosody. I suggest the problem with (41) is the presence of double negation. Even straightforward sentences like the following are not easy to judge out of the blue, because they involve double negation:

(43)

Nobody saw nobody.

Intended: “Everybody saw somebody.”

Just like double negation is improved by context and prosody, the double negation in (41) seems to be improved by context and prosody. Consider the following dialogue between person A and person B (based on a suggestion by Larry Horn):

(44)

A:

B:

Nobody at all showed up at my party.

Nowhere NEAR nobody showed up, the party was packed.

Consider now

(45)

Absolutely everybody showed up.

I propose that

(46)

⟦absolutely⟧ = λQ.Q

However, this semantic value does not account for the following contrast:

(47)

a.

b.

c.

d.

absolutely everybody was there.

absolutely nobody was there.

*absolutely half of the people were there.

*absolutely 75% of the people were there.

Such contrasts suggest that

(48)

An endpoint quantifier of S (a scale of generalized quantifiers) occupies the greatest position on the scale.

Given this definition, the semantic value of

(49)

a.

b.

⟦absolutely⟧

⟦absolutely⟧

=

=

λQ:Q is the endpoint of S.Q

λQ:∀X[X ε S

Paul Postal suggests that there is also an element of certainty of judgment in the semantics of

(49) predicts that combining

(50)

Almost absolutely everybody showed up.

In (50),

While the combination in (50) sounds a bit strained, there are actually a fair number of Google hits on the internet:

(51)

Almost absolutely everyone has their own website, be it a corporate website or a private web page.

(

The reverse combination is much worse:

(52)

*Absolutely almost everybody showed up.

The unacceptability of (52) follows from the fact that

The analysis in (49) also predicts that

(53)

Strange as it may seem, not absolutely everybody is hunting for rare Pokémon on their phones right now

(

(54)

But not absolutely everybody who voted Labour in Canterbury has mental health problems.

(

Once again, these sentences are OK because

Universal quantifier phrases modified by

(55)

a.

b.

Everybody who has ever been to France smokes.

Almost everybody who has ever been to France smokes.

However, the restriction of the modified universal quantifier phrase does not allow downward entailing inferences. For example, (56a) does not entail (56b):

(56)

a.

b.

Almost everybody who owns a car is happy.

Almost everybody who owns a Jaguar is happy.

If the number of Jaguar owners is small, it might be that there are hardly any happy Jaguar-owners, but still almost all the car owners are happy. The reason for this shift, on the theory presented in section 2, is that the scale of generalized quantifiers is calculated with respect to the restriction. The restriction in (56a) is the set of people who own a car, and the restriction in (56b) is the set of people who own a Jaguar (a much smaller set). Almost everybody who owns a car might be two billion people, but almost everybody who owns a Jaguar might be two thousand people. Even if every Jaguar owner was unhappy, it would not make much of dent in the number of happy car owners.

If the restriction of the modified quantifier phrase in (56) is not a downward entailing context, then why are NPIs licensed in (55b)? I propose that the answer can be found in the proposed structure of the modified quantifier phrases (see section 2 for proposal):

(57)

[almost [everybody who has ever been to France]]

Consider (57) from the point of view of the following condition (for discussion see

(58)

“A negative-polarity item is acceptable only if it is interpreted in the scope of a downward-entailing expression.” (from Ladusaw 2002: 467)

Crucially, this formulation makes no reference to whether the overall sentential context of the NPI is downward entailing or not. Therefore, since the restriction of a universal quantifier phrase is a downward entailing context, the negative polarity item is licensed (for related discussion, see

As discussed in Collins (

(59)

a.

b.

c.

Every professor who has ever been to France smokes.

*Some professors who have ever been to France smoke.

Not every professor who has ever been to France smokes.

Collins (

The distribution of NPIs in the restriction of modified quantifier phrases (modified by either negation or

As opposed to universal and negative quantifier phrases, existential quantifier phrases do not admit scalar modifiers (see

(60)

a.

b.

c.

*Almost some boy was there.

*Absolutely some boy was there.

*Nowhere near some boy was there.

(61)

a.

b.

c.

*Almost some boys were there.

*Absolutely some boys were there.

*Nowhere near some boys were there.

(62)

a.

b.

c.

*Almost some of the boys were there.

*Absolutely some of the boys were there.

*Nowhere near some of the boys were there.

(63)

a.

b.

c.

*Almost a boy was there.

*Absolutely a boy was there.

*Nowhere near a boy was there.

(64)

a.

b.

c.

*Almost boys were there.

*Absolutely boys were there.

*Nowhere near boys were there.

(65)

a.

b.

c.

*Almost several boys were there.

*Absolutely several boys were there.

*Nowhere near several boys were there.

One possibility is that these examples are all unacceptable because indefinites are not quantificational (rather, indefinite DPs would be of type <e,t> denoting predicates). As predicates, they could not be modified by a scalar modifier defined for generalized quantifiers. However, I propose a different account based on the assumption that indefinite DPs denote generalized quantifiers.

Since

As for the (a) sentences, consider the following contrast:

(66)

a.

b.

c.

*Almost one boy was there.

?Almost two boys were there.

Almost ten boys were there.

(66c) is fine and would be true if eight or nine boys were there. (66b) seems acceptable, but odd because it implies that one boy was there (so one could have said

In other words, scalar modification cannot yield a value that jumps between the two scales in (11), but only moves around on a single scale (see Hitzeman (

I do not attempt to account for the unacceptability of

The condition in (21) allows the following examples:

(67)

a.

b.

c.

d.

Almost half of the people were there.

Almost 75% of the people were there.

Nowhere near half of the people were there.

Nowhere near 75% of the people were there.

In none of these cases does the quantifier phrase denote the lowest point on the relevant scale, and so it can be modified by

Consider now the lowest element of the negative scale

(68)

a.

b.

c.

*Almost not every student was there.

*Absolutely not every student was there.

*Nowhere near not every student was there.

Once again, the unacceptability of (68b) follows from the fact that

According to the condition in (21), near(X,Q) entails that X is less than Q on the negative scale. But there is no X less than the denotation of

Free choice

(69)

a.

b.

c.

Almost anybody should be able to do that.

Absolutely anybody should be able to do that.

*Nowhere near anybody should be able to do that.

(69b) strongly suggests that free choice

I have analyzed the syntax and semantics of scalar modifiers of quantifier phrases. The syntax is based on the structure [almost DP], analogous to the structure of negated quantifier phrases [not DP] in the framework of

A future topic will be to extend this kind of analysis to other modified elements. As noted by Morzycki (

(70)

a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

The cup is almost full.

The cup is absolutely full.

The cup is nowhere near full.

Dinner is almost ready.

John has almost finished his homework.

He almost died.

Paris is near Versailles.

He is almost there.

He is almost always late.

He is almost never late.

He is absolutely never late.

(adjective)

(adjective)

(adjective)

(adjective)

(VP)

(VP)

(Locative)

(Locative)

(Adverb)

(Adverb)

(Adverb)

Morzycki proposes that one can account for cross-categorial

I thank Richard Kayne, Larry Horn and Paul Postal for discussions of the issues in this paper. I also thank the respondents of my acceptability judgments survey. I am grateful to the editor Johan Rooryck and three reviewers who provided me with much useful feedback.

The author has no competing interests to declare.