Getting the most out of most

Joseph Lochlann Smith

September 21, 2012

Contents

1 Generalised Quantifier Theory 31.1 Limitations of GQT . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Three critiques of generalised quantifiers 72.1 Most and more than half . . . . . . . . . . . . . . . . . . . . . 72.2 At least three and three . . . . . . . . . . . . . . . . . . . . . . 92.3 At least three and more than two . . . . . . . . . . . . . . . . 11

3 The syntactic distribution of at most 143.1 Comparison with adverbs . . . . . . . . . . . . . . . . . . . . . 153.2 Comparison with only . . . . . . . . . . . . . . . . . . . . . . 163.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 The proposal: from most to at most 184.1 Intensional most . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Ranking propositions . . . . . . . . . . . . . . . . . . . . . . . 194.3 Refining the account . . . . . . . . . . . . . . . . . . . . . . . 214.4 At best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Semantic or pragmatic modality? 24

6 Conclusion 26

References 27

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Introduction

Generalised Quantifier Theory (GQT) is one of the great success stories offormal semantics. In recent years, though, it has come under increasingcriticism. Much of this criticism is directed at the perceived coarsenessof GQT. Because its aims and successes lie in characterising generalproperties of large classes of disparate quantificational expressions, it glossesover subtle differences of meaning and ignores internal syntactic structure.

What much subsequent research has shown is that the two failings areintimately linked. Differences in meaning between expressions equated byGQT often find their explanation in the internal structure which GQT hasignored. Conversely, more detailed exploration of the structures subsumedunder the Det category of GQT has often led to the discovery of differencesin their behaviour.

In this paper I will attempt to bring together two separate strands ofresearch in this vein. The first is Hackls (2009) analysis of the differencesbetween most and more than half. This is work in the explicitly compositionaltradition of Heim and Kratzer (1998) and its critique of GQT is centred onthe latters insensitivity to the internal structure of multi-word and evensingle-word quantifiers.

The second is work on inadequacies of GQT with respect to the expres-sions at most and at least, which I will henceforth (following Geurts andNouwen (2007)) refer to as superlative scalar modifiers or simply superlativemodifiers. The specific puzzles under consideration vary from paper to pa-per, but all the work from Geurts and Nouwen (2007) through Buring (2008)to Cummins and Katsos (2010) has built on the formal foundations of Krifka(1999).

My goal is to address the question speculatively posed at the end ofGeurts and Nouwen (2007): what exactly is the relationship between com-parative and superlative modifiers, on the one hand, and comparative andsuperlative morphology, on the other? I will concern myself only with thesuperlative side of things, and my specific aim will be to derive the lexicalentry for at most hypothesised by Geurts and Nouwen from Hackls lexicalentry for the superlative morpheme -est. In so doing, I hope to explain therelationship between at most and the closely-related expression at best, whichis an issue overlooked by all authors that I am aware of. This will, I hope,pave the way for exploration of superlative scalar modifiers as a wider classof expressions than previously assumed, and ultimately the formulation ofsome cross-linguistic generalisations about them.

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1 Generalised Quantifier Theory

The mathematical notion of generalised quantifiers dates back to Mostowski(1957) but their most famous application to linguistic issues is Barwise andCoopers classic 1981 paper.1 Their key insight was that there are quantifi-cational expressions in natural language whose meaning cannot be expressedin first-order predicate logic. More than this, the fact that first-order logicis inadequate is not simply because its inventory of quantifiers ( and ) istoo limited but because of fundamental limitations in its formulation.

The key witness to this fact is the word most. Barwise and Cooperproved that no matter how we enrich its inventory of quantifiers, first-orderlogic cannot express the proportional relation denoted by most. Comparethe following (where in (1-d), {,,,} denotes any combination of thelogical operators contained in the set):

(1) a. JeveryK(A)(B) = 1 iff x[A(x) B(x)]b. JsomeK(A)(B) = 1 iff x[A(x) B(x)]c. JnoK(A)(B) = 1 iff x[A(x) B(x)]d. *JmostK(A)(B) = 1 iff Mx[A(x) {,,,} B(x)]

Unlike for every, some, and no, which can be expressed using the existingresources of predicate logic, there is no first-order quantifier Mx which willallow us to represent the meaning of most.

The solution to the issue is to employ Mostowskis generalisation of thenotion of a first-order quantifier, such that relations like and are infact specific instances of the more general class of relations between sets ofentities. On this view, we can give a uniform treatment to the quantifierslisted above, as well as to many more that we find in natural language.

(2) a. JeveryK(A)(B) = 1 iff A Bb. JsomeK(A)(B) = 1 iff A B 6= c. JnoK(A)(B) = 1 iff A B = d. JmostK(A)(B) = 1 iff |A B| > |AB|

We must be careful to distinguish early on between various conflicting uses ofthe terms quantifier and determiner. A quantifier in the GQT sense is notan individual word like every or most, which are known as quantificationaldeterminers. A generalised quantifier is the denotation of the DP formed

1It was not, however, the first: they were employed in Montague (1974). Montaguesaims, though, did not coincide with those of Chomskyan linguistics to the extent thatBarwise & Coopers did, as Montague was concerned with delimiting the class of allpossible natural and artificial languages, as opposed to just natural languages.

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by the combination of a quantificational determiner with an NP, for instanceevery boy. (Note that in the syntactic theory of the early 80s, every boy wasconsidered an NP, not a DP).

(3) . . . semantically . . . more than half is not acting like a quantifier,but like a determiner. It combines with a set expression to producea quantifier. On this view, the structure of the quantifier may berepresented as below:

Quantifier

Set expressionDeterminer

. . . we can see that the structure of the logical quantifier correspondsin a precise way to the English noun phrase (NP) as represented in:

NP

Noun

people

Det

most

(Barwise and Cooper 1981: 162)

The term determiner is, in Barwise and Coopers usage, potentially am-biguous between a semantic sense (i.e. the denotation of a word like every)corresponding to Determiner in the first tree diagram above, and a syntacticone, corresponding to Det in the second tree. Here, I will follow Szabolcsi(2010) in referring to the former as semantic determiners or determinerdenotations and the latter simply as determiners.

This clarified, we can now say that the relations between sets that consti-tute the denotations in (2) are not quantifier denotations but the denotationsof quantificational determiners. To understand the denotations of generalisedquantifiers, lets restate our definitions in (2) using lambda abstraction.2

(4) a. JeveryK = AB[A B]b. JsomeK = AB[A B 6= ]c. JnoK = AB[A B = ]d. JmostK = AB[#(A B) > #(AB)]

The arguments A and B are predicates, i.e. sets of individuals or their char-

2For clarity in combination with the square brackets I use the notation #(P ) as asubstitute for |P | here and elsewhere.

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acteristic functions, depending on our perspective. Lets combine a predicatewith our determiner denotations in (4) via functional application to producegeneralised quantifers:

(5) a. Jevery boyK = AB[A B](JboyK) = B[JboyK A]b. Jsome boyK = AB[A B 6= ](JboyK) = B[JboyK B 6= ]c. Jno boyK = AB[A B = ](JboyK) = B[JboyK B = ]d. Jmost boysK = AB[#(A B) > #(AB)](JboyK)

= B[#(JboyK B) > #(JboyKB)]The generalised quantifiers thus formed are second-order predicates, that is tosay, they denote (the characteristic functions of) sets of sets. The denotationof (5-a) is the set of all properties possessed by every boy, which is the samething as the set of all sets which contain every boy.

Generalised quantifiers have proven to be an immensely powerful tool forinvestigating the deep semantic structure of natural language quantification.They have been particularly useful at uncovering potentially universal se-mantic properties of determiners, such as conservativity and extension (see,for instance, Keenan and Stavi (1986)), and exploring interesting implica-tions for the learnability and processing of quantificational expressions. Butthey are not without their limitations.

1.1 Limitations of GQT

There are two respects in which GQT can be considered coarse. The firstis syntactic, the second semantic.

The syntactic coarseness is the result of the rather blunt way in whichBarwise and Cooper identify the Det in the lowermost tree diagram of (3).Essentially, what they choose to identify as the determiner is the result ofremoving the noun from the noun phrase. In the phrase at least three boys,boys is obviously the noun, and therefore the determiner must (accordingto Barwise and Cooper) be at least three. Once a determiner has been soidentified, its internal structure is entirely ignored. This extends even toexamples like not more than two people, where the determiner is identifiedas not more than two and the matter is left at that. For instance, considerhow the following syntactically and morphologically diverse expressions areindistinguishable in GQT:

(6) a. JoneK(A)(B) = 1 iff |A B| 1b. Jat least oneK(A)(B) = 1 iff |A B| 1c. Jless than twoK(A)(B) = 1 iff |A B| 1

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The semantic coarseness of GQT, on the other hand, has a number ofaspects. GQT is a strictly truth-conditional theory, and as such has nothingto say about implicatures which may come packaged with the meaning of anexpression. This is, of course, hardly surprising in view of the aims of formalsemantics in the late 70s and early 80s, as Szabolsci (2010, p. 76) points out.Nonetheless, it leaves the GQT lexical entries unable to account for certainaspects of the semantic behaviour of quantifiers, and puts GQT increasinglyout of step with modern developments in the field.

GQT is also strictly extensional. This may seem entirely justified for atheory of quantification. However, we will see that some of the expressionslabelled as quantificational determiners by GQT give rise to modal meanings(though whether these modalities arise via semantic or pragmatics means isthe subject of ongoing debate).

A number of authors (Hackl (2009), Pietroski et al. (2009), Lidz et al.(2011)) have cited examples such as (7) as evidence of GQTs perceivedinsensitivity to the form in which truth conditions are stated.

(7) a. JnoK(A)(B) = 1 iff A B = b. JnoK(A)(B) = 1 iff |A B| = 0c. JnoK(A)(B) = 1 iff |A B| < 1

They argue that certain statements of truth conditions more accurately con-vey the intermediate representations that language users make use of in com-puting truth values, citing experimental data relating to the verification pro-cedures employed by subjects in evaluating statements.

While it seems right that there is more to meaning than truth conditions,I dont see how this can possibly be a criticism of GQT without being acriticism of the whole of truth-conditional semantics. After all, the fact thattruth conditions can be stated in multiple forms is true of any truth con-ditional lexical entry in formal semantics. One can, for instance, substitutean expression of the form x for one of the form preserving truth,and two lexical entries so differentiated cannot be considered substantivelydifferent. One may be preferred for reasons for readability, or to make the au-thors intentions clearer, but such issues cannot be confused with substantivedifference in meaning.

As I see it, if one wants to seriously pursue the study of verification pro-cedures, one needs a new (explicitly procedural) representational frameworkin addition to truth-conditional semantics. Otherwise, one is simply imbuingnotation with intuited meaning over and above its intended use.

These thorny issues aside, the main point is that much recent work hasstarted from a standpoint of dissatisfaction with GQT and attempted to find

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ways to differentiate, both syntactically and semantically, expressions whichare equated by GQT. I will, therefore, introduce each critique of GQT withreference to the truth-conditional equivalences it objects to, but implyingnothing more than that the meaning thus expressed does not fully accountfor the semantic behaviour of the expressions in question.

2 Three critiques of generalised quantifiers

2.1 Most and more than half

In his 2009 paper, Hackl notes that most and more than half are indistin-guishable in GQT. Both of the below formulations of their truth conditionsare equivalent and interchangeable.

(8) a. JmostK(A)(B) = 1 iff |A B| > |AB|b. Jmore than half K(A)(B) = 1 iff |A B| > |AB|

(9) a. JmostK(A)(B) = 1 iff |A B| > 12|A|

b. Jmore than half K(A)(B) = 1 iff |A B| > 12|A|

Hackl provides persuasive evidence of systematic differences between the twoexpressions which can be attributed to their compositional structure butwhich are a mystery from the GQT perspective.

The main evidence is a puzzling asymmetry between most and fewest.While in English more than half has a polar opposite less than half (whosemeanings are straightforwardly related as in (10)), the relationship betweenmost and fewest is more complex. (Similar facts hold for German).

(10) a. Jmore than half K(A)(B) = 1 iff |A B| > 12|A|

b. Jless than half K(A)(B) = 1 iff |A B| < 12|A|

Most has two readings, associated with slightly different surface syntax inEnglish (in German only one surface pattern is attested and is ambiguousbetween the two readings).

(11) a. John climbed most of the mountains. (PROPORTIONAL)b. John climbed the most mountains. (RELATIVE)

For the statement in (11-a), which gives rise to what Hackl calls the pro-portional reading, most has a meaning equivalent to more than half. Thestatement in (11-b), by contrast, means that John climbed more mountainsthan anyone else in a contextually-determined comparison set.

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(12) a. *John climbed fewest of the mountains. (PROPORTIONAL)b. John climbed the fewest mountains. (RELATIVE)

In (12-a), we see that the proportional reading (less than half the mountains)is unavailable for fewest, though the reading on which John climbed fewermountains than anyone else is available. In German the surface syntax ofthe two is identical but fewest still lacks the proportional reading.

Hackl demonstrates that the explanation for these facts lies in the word-internal compositional structure of most and fewest, specifically the presenceof the superlative morpheme in both. As noted by Szabolcsi (1986) and Heim(1999), superlatives exhibit a very similar ambiguity to most.

(13) John climbed the tallest mountain.

(13) can mean that John climbed a taller mountain than anyone else, or (inthe absence of a context which limits the comparison class) that he climbedMount Everest. Following Heim (1999), Hackl claims that the superlativemorpheme -est moves at LF for interpretability, and thus attributes the am-biguity in (13) and the two forms in (11) to differences in the LF scope of-est.

(14) JmanyK(d)(A) = x.[A(x) |x| d](15) For all C of type e, t, D of type d, e, t and x of type e,JestK(C)(D)(x) is defined only if

x C y[y C]When defined,JestK(C)(D)(x) = 1 iffy C[y 6= x max{d : D(d)(x = 1} > maxd: D(d)(y)=1. ]

(16) a. [John climbed [the [-est C]i [di-many mountains]]]b. [John [-est C]i [climbed [the di-many mountains]]]

Hackl assumes that if [-est C] stays inside the DP, the comparison class C isidentical to the NP sister of the degree function. In (16-a) this makes it theset of pluralities of mountains. If [-est C] moves into the matrix clause, thecomparison class is a set of salient individuals including the subject. In (16-b)this makes it the set of individuals who climbed pluralities of mountains.

Many-est comes out with a proportional reading because of the assump-tion that the pluralities in the comparison class C have no overlapping atomicparts. We can see why this meaning comes out if we consider an analogywith seats in Parliament. A party which has an overall majority (i.e. morethan half the seats) is guaranteed to be in government, because the party

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with more seats than any other forms a government. If one party has morethan half the seats, it is impossible for any other party, even if they controlall the remaining seats, to have more seats. Similarly, Hackls lexical entrymakes Jmany-estK(C)(x) true of x if x has more atomic members than anydistinct (i.e. non-overlapping) plurality in C. If x contains more than halfof all the atomic members under consideration, this condition is guaranteed.

Hackls achievement is to show how a careful consideration of the struc-ture that GQT leaves unanalysed can provide insightful explanations of oth-erwise puzzling phenomena. Deriving the semantics of most from the seman-tics of the superlative morpheme is a triumph for a compositional approach,and one which I hope to extend to at most.

2.2 At least three and three

Krifka takes exception to the following equivalence in GQT:

(17) a. Jat least threeK(A)(B) = 1 iff |A B| 3b. JthreeK(A)(B) = 1 iff |A B| 3

He claims that the indistinguishable truth conditions fail to explain why threegives rise to a scalar implicature that stronger alternatives on the Horn-scale are false, while at least three does not. Krifka argues that part ofthe reason for this failure of GQT is that it gets the syntax of modifiednumeral expressions wrong. Barwise and Cooper analyse at least three asa determiner which combines with a noun phrase to give the structure [atleast three [boys]]. Krifka disputes this, maintaining that [at least] is anindependent constituent which combines with [three [boys]] to give [[at least][three [boys]]].

The distributional data are discussed in more detail in section 3, butsuffice to say for now that the separability of [at least] from the numericexpressions it sometimes attaches to seems indisputable in light of exampleslike the following:

(18) a. At least John drank three beers.b. You should at least call her.c. Mary is at most an associate professor.d. Three people came, at most.

The above examples demonstrate not only the syntactic deficiency of theGQT analysis of superlative scalar modifiers but some of its correspondingsemantic blind spots, for nothing in the GQT truth conditions can explainhow at most is able to modify an associate professor, or at least modifies the

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VP call her.Krifkas solution to the puzzle crucially relies on another facet of superla-

tive scalar modifiers which GQT overlooks, namely their ability to associatewith focus in a way which affects truth conditions.3 Consider:

(19) a. At most three boys left.b. At most three boys left.

Sentence (19-a) could still be true if, say, three boys and three girls left.Sentence (19-b) is ambiguous between a reading where the focus is specificallyon the noun boys, and one where focus projects to the entire noun phrasethree boys. If we assume the latter, the sentence would be false (given anappropriate context) if three boys and three girls left.

As Krifkas formal account of focus will be essential for my own account ofat least, it is worth considering in some detail. Krifka follows Rooth (1985) inassuming that focussed expressions have the semantic function of introduc-ing context-determined alternatives which percolate upwards compositionallythrough the tree until they meet an operator which makes use of them. Hethen additionally assumes that number words and other expressions whichnaturally line up along a scale are special, in that they can introduce alterna-tives without being focussed, and that the set of alternatives they introducecomes with an ordering relation. Superlative scalar modifiers like at mostand at least depend upon this set of alternatives and its ordering relation fortheir meaning.

Lets consider a concrete example.

(20) At most John drank three beers.

According to Krifka, the number word three introduces a set of ordered alter-natives which percolate upwards compositionally according to the followingrule (this idea goes back to Hamblin (1973)):

(21) If J[]K = f(JK, JK), then J[]Ka ={f(X, Y ), f(X , Y ) | X,X JKa and Y, Y JKa}

Those expressions which do not introduce alternatives (i.e. that are neitherfocussed nor come with an ordering relation) are assumed for formal reasonsto have come with a set consisting of both their meaning proper (to allowcomposition with unordered alternatives) and an ordered pair formed withtheir meaning proper (to allow composition with ordered alternatives).

3This phenomenon has been explored in depth for adverbials such as only and even inwork such as Rooth (1985).

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Heres how the first few steps of such composition would play out for ourexample (20).

(22) a. JthreeK = Px[3(x) P (x)]JthreeKa = {Px[n(x)P (x)], Px[m(x)P (x)] | n n m}b. JbeersK = beersJbeersKa = {beers, beers,beers}c. Jthree beersK

= Px[3(x) P (x)](beers)= [3(x) beers]Jthree beersKa= {x[n(x) beer(x)], x[m(x) beer(x)] | n n m}

Since in (20) at most attaches to the S node of the syntactic tree, the alter-natives that serve as input to at most are those that percolate all the way tothe top, and are thus an ordered set of propositions:

(23) J[John [drank [ three [beers]]]]Ka ={x[n(x) beers(x) drank(john)(x)],x[m(x) beers(x) drank(john)(x)] | n n m}

In Krifkas analysis, at most and at least use the focus-induced alter-natives in such a way that leaves no alternatives for scalar implicatures tonegate. This is the explanation for the contrast between at least three andthree which GQT fails to capture.

While I will make use of Krifkas ideas on projection of ordered alterna-tives by focus, I will not use his specific analyses of at most and at least,mainly because they fail to account for the phenomena which are the topicof the next section.

2.3 At least three and more than two

While Krifka criticised the equivalence of at least n boys and n boys in GQT,he made no such distinction between at least n boys and more than n-1 boys,and his focus-sensitive definitions for at least and more than in effect equatethe two, just as GQT does (likewise for at most n and less than n+1 ):

(24) a. Jat least threeK(A)(B) = 1 iff |A B| 3b. Jmore than twoK(A)(B) = 1 iff |A B| 3

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(25) a. Jat least threeK(A)(B) = 1 iff |A B| > 2b. Jmore than twoK(A)(B) = 1 iff |A B| > 2

(26) a. Jat most twoK(A)(B) = 1 iff |A B| 2b. Jless than threeK(A)(B) = 1 iff |A B| 2

(27) a. Jat most twoK(A)(B) = 1 iff |A B| < 3b. Jless than threeK(A)(B) = 1 iff |A B| < 3

Geurts and Nouwen (2007) take exception to this equivalence of compara-tive and superlative scalar modifiers. Their key pieces of evidence are thefollowing.

Firstly, superlative modifiers allow a specific construal that is infelicitousfor comparative modifiers:

(28) a. I will invite at most two people, namely Jack and Jill.b. ?I will invite fewer than three people, namely Jack and Jill.

(29) a. I will invite at least two people, namely Jack and Jill.b. ?I will invite more than one person, namely Jack and Jill.

Secondly, there is a contrast in the accessibility of inferences to statementsinvolving superlative and comparative modifiers:

(30) a. Beryl had three sherries.b. Beryl had more than two sherries.c. ?Beryl had at least three sherries.

While (30-b) indisputably follows from (30-a), the inference from (30-a) to(30-c) is more questionable, a result which has been borne out in severalexperimental studies (Geurts (2007), Geurts, Katsos, Cummins, Moons, andNoordman (2010)).

Thirdly, in combination with modals superlative modifiers give rise toambiguities that comparative modifiers do not.

(31) a. You must have at least three beers.b. You must have more than two beers.

Example (31-a) has a preferred reading that you are required to drink aminimum of three beers. This is what Buring (2008) calls the authoritativereading. It also has another, less accessible reading that three beers is theminimum number that you are required to drink, but its possible you arerequired to drink four beers, or five beers, etc. This Buring calls the speakerinsecurity reading.

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Example (31-b) is, by contrast, not ambiguous in the same way. It cannothave the speaker insecurity reading, and means simply that you are requiredto drink a minimum of three beers.

Geurts and Nouwen solve these puzzles by ascribing a modal componentto the meanings of superlative modifiers. Here are their lexical entries:

(32) a. If is of type t, thenJat least K = [ B ]b. If is of type a, t, thenJat least K = X[(X) [ B (X)]]

(33) a. If is of type t, thenJat most K = [ B ]b. If is of type a, t, thenJat most K = X[(X) [ B (X)]]

Notice that, like Krifka, Geurts and Nouwen ackowledge the fact that su-perlative scalar modifies can combine semantically with entire propositions,a fact not just overlooked in GQT but essentially inexpressible within itsframework. They also make use of Krifkas notion of ordered alternatives:the B operator in the lexical entries above indicates an ordering relationbetween two entities so joined (namely that the former precedes the latter).

Geurts and Nouwens introduction of an epistemically modal componentto the meaning is, however, novel. On their view, a speaker uttering Johndrank at most three beers means that, as far as they know, it is possible thatJohn drank three beers, and not possible that he drank more than three. Onthe other hand, John drank at least three beers means it is certain that hedrank three, and possible he drank more than three.

The infelicity of the specific construal in (28) and (29) is therefore ascribedto the fact that comparative modifiers can only combine semantically withfirst-order predicates, whereas as we can see from presence of a variable a inthe type definitions in (32-b) and (33-b), superlative modifiers can combinewith higher-order predicates. This means that the operation of existentialclosure with Geurts and Nouwen assume must take different scope in the caseof superlative and comparative modifiers:

(34) a. [I will invite [at most [two people]]]b. [I will invite [fewer than [three people]]]

In (34-a) the closure operation, which results in a type shift from predicateto quantifier (first- to second-order predicate), can apply within the scope ofat most because at most is allowed to combine with higher-order predicates.In (34-b), however, it cannot, and because it does not scope directly over the

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numeral, the numeral cannot have a specific construal.The questionable inference patterns in (30) are straightforwardly ascribed

to the fact that the conclusion contains modal components which are notpresent in the premise.

The ambiguity in (31-a) when superlative modifiers combine with modalsis ascribed to the interaction of the two modalities according to a rule ofmodal concord, for which Geurts and Nouwen provide some independentevidence in Dutch. This rule causes stacked modals which agree in modalforce (i.e. both are necessity or both are possibility operators) to merge.This rule is posited to be optional but preferred (hence the inaccessibility ofthe speaker insecurity reading).

In summary, Geurts and Nouwen convincingly demonstrate that superla-tive modifiers exhibit modal behaviour which comparative modifiers do not.They assume that this modal behaviour is semantically encoded in the lexicalentries of the superlative modifiers, though they note the possibility that itarises pragmatically (a point which others have taken up and to which wewill later return).

3 The syntactic distribution of at most

Krifka correctly points out that superlative scalar modifiers have distribu-tional properties which are incompatible with their GQT analysis as deter-miners. Namely, they have a much freer distribution than syntactic deter-miners, and can in fact form constituents with them, something which isimpossible for true determiners.

(35) a. At least every girlb. At most some boyc. *Every some girld. *Some every boy

They can also attach to VPs:

(36) You must at least talk to him.

The evidence is thus compelling that superlative scalar modifiers are separa-ble from the numerals they sometimes attach to.4

Before providing an explicit compositional account of at most, however,we need to make this observation more precise. Exactly how free is the

4By contrast, Krifkas further claim that comparative scalar modifiers are also thusseparable is more dubious, as pointed out in Szabolcsi (2010), pp 165-166.

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distribution of at most? To what other expressions is it most similar in itssyntactic characteristics?

3.1 Comparison with adverbs

First, lets compare at most and a quantificational adverb, sometimes, ina sentence with a numerically modified DP. Here a habitual reading of thepresent tense must be assumed to make the examples felicitous (imagine acontext where we are describing what John does in the evenings after work).

(37) a. (At most/Sometimes) John drinks three beers.b. John (at most/sometimes) drinks three beers.c. John drinks (at most/*sometimes) three beers.d. John drinks three beers (at most/sometimes).

At most can appear everywhere that sometimes can, and in one position notavailable to sometimes, namely post-verbally as in (37-c).5 The same pat-tern appears if we consider a non-quantificational adverb straightforwardlyderived from an adjective, e.g. slowly.

(38) a. (At most/Slowly) John drank three beers.b. John (at most/slowly) drank three beers.c. John drank (at most/*slowly) three beers.d. John drank three beers (at most/slowly).

Why is at most, but not slowly or sometimes, allowed in (37-c) and (38-c)?A comparison with another focus-sensitive adverbial, only, can provide someinsight here.

(39) a. (At most/Only) John drank three beers.b. John (at most/only) drank three beers.c. John drank (at most/only) three beers.d. John drank three beers (at most/only).

Interestingly, only has, on the surface, the same distribution as at most : itcan also appear in the post-verbal position which is off-limits to the otheradverbial expressions we have considered. How can we account for this?

5The ungrammaticality of many adverbs in post-verbal position in English (as in (37-c)and (38-c)) is of course a well-documented phenomenon (see, for instance, Carnie (2012)).It forms part of the key evidence that in French (and many Romance languages) the verbraises from V to I while in (modern) English it does not.

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It it well-known in the literature6 that only is a special sort of adverbin that, unlike the manner or frequency adverbials which we tested above,it can form a constituent with DP. In fact, it can combine with a variety ofphrase types. Consider the following from Koopman et al. (2003):

(40) a. only John (only DP)b. only with John (only PP)c. only happy (only AP)d. only put pepper on the tomatoes (only VP)

The same possibilities are available to at most, with the caveat that we canimagine a context which places what follows at most on an ordered scale ofalternatives.

(41) a. at most John (at most DP)b. at most with John (at most PP)c. at most happy (at most AP)d. at most put pepper on the tomatoes (at most VP)

So at most has in common with only both focus-sensitivity and a distribu-tional freedom not shared by other adverbials, in particular the ability toform a constituent with DP. This explains the apparently post-verbal po-sition of at most in (39-c): it is not so much post-verbal as pre-nominal,forming a constituent [at most [three beers]].

3.2 Comparison with only

There is, however, a crucial difference between at most and only. Only cannotcombine with a whole sentence, while at most can. We need to look slightlybeyond the surface syntactic distribution to find supporting evidence for this.Consider the following (again from Koopman et al. (2003)):

(42) a. Only John drinks beerb. *Only John drinks beerc. *Only John drinks beer

Placing the focus on John is felicitous, but placing it later in the sentenceis not. (Or rather, it is not felicitous on a reading which associates it withonly and thus creates truth-conditional differences). This is because of thefollowing rule (as quoted in Koopman), which restricts the scope of focus-sensitive operators:

6My source is Koopman, Sportiche, and Stabler (2003)

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(43) The focus associated with only must be contained in a constituentsister to only

Clearly (42-b) and (42-c) are unacceptable because only can only form aconstituent DP with John, and therefore cannot associate with focus outsideof this DP.

The same restriction does not apply to at most.

(44) a. At most John drank three beers.b. At most John drank three beers.c. At most John drank three beers.d. At most John drank three beers.

All of these are felicitous, and all have slightly different truth-conditions(although we need a suitable context to bring out this fact). For instance,(44-a) would be false if, say, John and Peter both drank three beers, becausethe set of alternatives under comparison might be the following:

(45) {John drank three beers, John and Peter drank three beers, Johnand Peter and Mary drank three beers, . . . }

By contrast, (44-c) would be true if John and Peter both drank three beers,because the alternatives would then be:

(46) {John drank one beers, John drank two beers, John drank threebeers, John drank four beers, . . . }

3.3 Summary

In summary, at most has a wider distribution than comparative scalar mod-ifiers and many other adverbials.7 Its closest rival in this respect is only,but at most can in fact form a constituent with a superset of the syntacticphrases that only can. Specifically, unlike only it can combine syntacticallywith S and associate with an focussed subconstituent of S. What this meansfor the compositional semantic analysis of at most will be the topic of thenext section.

7Geurts & Nouwen do highlight certain cases where the distribution of superlativescalar modifiers appears more restricted than that of comparatives. However, as their ownexplanation indicates, this is a semantic infelicity to do with the interaction of negationand modals rather than a syntactic distributional restriction, and the weakness of thecorresponding judgments tallies with this.

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4 The proposal: from most to at most

My aim is to characterise the relationship between the degree quantifier mostand the superlative scalar modifier, at most, and in so doing provide a com-positional analysis of the related expression at best. In what follows I willrestrict my attention to cases where the superlative scalar modifier has at-tached to the S node and thus takes a propositional argument, but it shouldnot be too difficult to refine the account to include those cases where at mostcombines with a DP and thus takes a predicative argument.

4.1 Intensional most

The first element of my proposal is the idea that most and its suffixal coun-terpart -est each come in two closely-related flavours.8 The first of these isthe lexical entry from Hackl (2009) which we discussed above. Here I use avariant of Hackls formulation due to Gajewski (2010).9

(47) For all C of type e, t, D of type d, e, t and x of type e,JmostK(C)(D)(x) is defined only ifx C d[D(d)(x) = 1] y[y 6= x y C d[D(d)(y) = 1]]

When defined,JmostK(C)(D)(x) = 1 iffd[D(d)(x) = 1 y[y 6= x y C D(d)(y) = 0]]

Echoing the two variants of Geurts and Nouwens lexical entries in (32) and(33), I propose that most is in fact systematically ambiguous between form(47), which is a generalised degree quantifier over individuals, and an inten-sionalised form, mostint, which is a generalised degree quantifier over propo-sitions (that is, over functions from worlds to truth values). Thus whilemost is of type d, e, t, e, t, mostint is of type d, s, t, t, s, t, t.The truth conditions of mostint are otherwise identical to (47), but as weshall see, the shift in type allows mostint to interact compositionally with itssurroundings in ways which are impossible for most.

8Following Hackl (2009), I take it for granted that most is additionally ambiguousbetween its compositional form many-est and its primitive form. The latter is equivalentto -est, and is the only form under consideration here.

9It may appear that the presuppositions are substantively different from Hackls here,given that they require the members of the comparison class to meet D to some degree.However, as Gajewski points out, Hackls truth conditions for -est do reference the maximaldegree to which each individual in C has D, and he must therefore assume that everymember of C has D to some degree.

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(48) For all C of type s, t, t, D of type d, s, t, t and p of types, t,JmostintK(C)(D)(p) is defined only if

p C d[D(d)(p) = 1] q[q 6= p q C d[D(d)(q) = 1]]

When defined,JmostintK(C)(D)(p) = 1 iffd[D(d)(p) = 1 q[q 6= p q C D(d)(q) = 0]]

For mostint, the covert contextual argument C is a set of salient alternativepropositions, and D is (the characteristic function of) a set of propositionspossessing some gradable property to a given degree. What gradable propertycould a proposition possess?

4.2 Ranking propositions

Recall that Krifka (1999) claimed that certain expressions in certain contextsintroduce a partially ordered set of semantic alternatives into the universeof discourse. These are expressions which either come pre-packaged with anatural ordering (e.g. number words) or else have an ordering which is insome sense aided by context (e.g. John is at least a Texan creating a tax-onomic ordering on the alternatives to Texan, e.g. american tax texantax austinite). Such alternatives bubble up compositionally through thesentence until they meet a focus-sensitive operator which makes use of them.In the example under consideration, this will be at the syntactic level of theS node and semantic level of an ordered set of a complete propositions.

I propose that this ordered set in fact constitutes the covert contextualargument, C. Consider again

(49) At most John drank three beers.

As described in (23) and repeated below, the end result of the percolationof the ordered alternatives up the tree will be the following set C as anargument to at most :

(50) J[John [drank [ three [beers]]]]Ka ={x[n(x) beers(x) drank(john)(x)],x[m(x) beers(x) drank(john)(x)] | n n m}

Because the propositions which make up the comparison class are now em-bedded inside ordered pairs, we will need to replace every instance of C inour lexical entry for mostintwith C to get at them directly. Given that our

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intensional variant of most can only be used as part of at most and C isby assumption an ordered set in such cases, this will work for our purposes.However, in the long run it suggests a refinement of the formalisation maybe needed. Firstly, vindication for the existence of mostintwould come fromevidence for its presence in other contexts, perhaps outside of superlativescalar modifiers. Such contexts, if found, might not provide an ordered setfor C, and C would not be a valid operation. Secondly, the further diver-gence between mostintand most weakens the underlying argument that theyare essentially the same thing. Ultimately, I believe a solution will lie intweaking the formal machinery used.

This noted, I can use the availability of the ordered set C to define thefollowing function as a mapping from pairs of degrees and propositions totruth values:

(51) rank(C)(d)(p) = 1 iff |{q | q, p C}| = dThat is to say, rank(C)(d)(p) = 1 if there are d lower-ranked propositionsthan p in C.

Lets assume, as a nave initial sketch, that [at most] has the followingstructure (never mind at this point about the syntactic ramifications of a Phead selecting a DegP complement).

(52) PP

DegP

most

P

at

By our assumptions about semantic composition, atmust be of type d, s, t, t,and combine with mostint via Functional Application. Remember our goal isto move compositionally from the semantics for most found in Hackl (2009),Gajewski (2010), and Heim (1999) to the semantics for at most found inGeurts and Nouwen (2007). For now, lets place most of the burden for thison the denotation of at. Note that the possibility operator represents epis-temic possibility: the set of worlds implicitly under consideration are onlythose that are compatible with what the speaker knows.

(53) JatK = dd.ps, t.[rank(C)(d)(p) = 1 p]By Functional Application, then, we have the following.

(54) JmostintK(C)(JatK)(p) is defined only ifp C d[rank(C)(d)(p) = 1 p]

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q[q 6= p q C d[rank(C)(d)(q) = 1 q]]When defined,JmostintK(C)(JatK)(p) = 1 iffd[rank(C)(d)(p) = 1 p

q[q 6= p q C [rank(C)(d)(q) = 1 q]]The resulting complex expression [at most] still requires a propositional ar-gument. It presupposes that this proposition has a given rank in the setof semantic alternatives, and that it is possible (true in at least one worldcompatible with the speakers knowledge). It also presupposes that thereis at least a second possible ranked proposition under consideration. It as-serts that given the rank of the proposition argument, there is no seconddistinct proposition that is both higher-ranked and possible. That is to say,the speaker asserts both that p is possible, and that it is the highest-rankedproposition that is so. Thus the alternative possible proposition which sat-isfies the presupposition must be lower-ranked than p.

It is easy to see that, apart from the presuppositions, this amounts to thesame thing as Geurts and Nouwens definition in (33) for the variant of atmost which combines with a propositional argument, repeated below.

(55) If is of type t, thenJat most K = [ B ]4.3 Refining the account

In the previous sections we made the dubious syntactic assumption that atmost has the simple structure in (52). This led to a rather dubious conclusion,namely that the denotation of a particular variant of the word at carrieswhat seems to be a large amount of construction-specific baggage. This goesagainst the motivating principle of our compositional account, which is tofind nuggets of reusable meaning in apparently idiosyncratic expressions.

Fortunately, we can improve on our account. Consider:

(56) At the most John drank four beers.

If the intervenes between at and most then at cannot in this case be the sisterto most, and cannot supply an argument to most. A plausible structure mightbe

(57) [pp at [dp the most]]

However, if the is the sister to most, how does most combine with a predicate

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containing the notions of rank and possibility? Moreover, there does notseem to be much independent evidence for a determiner selecting a DegPcomplement.

Lets consider an alternative well-formed PP structure.

(58) [pp in [dp the [ [ most probable] [case]]]]

Here, a P head selects a DP complement. Within the DP, D selects an NP.The N within the NP is modified by an AP which consists of an AP itselfmodified by the DegP adjunct, [most]. All very orthodox. What if we assumethat our the structure of at the most and indeed at most is identical,but filled partly with covert elements?

To maximise the structural similarity between the various PPs we havelooked at, I propose that the determiner position can be filled with a covertdefinite determiner (for whose general existence there is a wealth of indepen-dent evidence see in particular Abney (1987)). The only difference betweenat most and at the most is in the overtness of the determiner.

(59) a. [at [ [ [most RANKED][POSSIBILITY]]]]b. [at [the [ [most RANKED][POSSIBILITY]]]]

Slightly more controversially, I assume the existence of a covert gradableadjective encoding nothing more than the rank function we defined above.10

(60) JRANKEDK(C)(d)(x) = 1 iff rank(C)(d)(x) = 1I also posit a covert abstract noun POSSIBILITY denoting a possible propo-sition.

(61) JPOSSIBILITYK = ps, t.pOf course, most is now uninterpretable as it does not have a sister of typed, s, t, t. It must move at LF in the manner described in Hackl (2009).(62) [at [most]i d [ [[di RANKED][POSSIBILITY]]]]

RANKED and POSSIBILITY combine compositionally just as any degreeadjective and noun would, by Predicate Modification. This is how we getthe component p rank(C)(d)(x) which we formerly stuffed awkwardlyinside the denotation of at. Composition up to the parent node of [most] isachieved exactly as in the account of Hackl and Heim. Because by this pointwe already have the correct denotation, we can now view at as semantically

10Echoing Kaynes practice, I write covert lexical items in all caps.

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vacuous in this construction. The denotation simply passes up to the topnode of the PP where it is a sister to S, and can receive its propositionalargument.

4.4 At best

Is there any independent evidence for the structure I have hypothesisedabove? We have seen a few examples of parallel PP structures which sug-gest that it is possible. Further evidence comes from the expression at best,hitherto overlooked by other authors on the topic of superlative modifiers.

At best is similar to at most in that it can combine with propositionsand similar to all superlatives in that it depends on a comparison class. Itis also focus-sensitive, as shown in (63-a), and thus depends on projectedalternatives.

(63) a. At best John drank three beersb. At best John drank three beers.c. At best John drank three beers.d. At best John drank three beers.

These have truth conditional differences. (a) would be false if there is apossible situation in which Bill drank three beers which is considered betterthan the one in which John did. (c) could be true in the same conditions,provided that the situation in which John drank three beers is better thanthe one which he drank an alternative number of beers.

For at best, I propose the surface structure in (64-a) and the LF in (64-b).

(64) a. [at [ [ [good-est][POSSIBILITY]]]]b. [at [-est]i d [ [[di-good][POSSIBILITY]]]]

Just as is the case for at most, at best appears in two surface syntactic forms,at best and at the best.

(65) a. [at [ [ [good-est][POSSIBILITY]]]]b. [at [the [ [good-est][POSSIBILITY]]]]

We still need the covert POSSIBILITY noun to make the semantics comeout right, however. I argue that attributing a degree of desirability to aproposition without further qualification normally entails the truth of theproposition, as in (66).

(66) Its good that hes still alive.

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Since use of at best always entails uncertainty about the truth of the propo-sition, I propose that the POSSIBILITY noun is a necessary element of thesemantic representation.

From a semantic perspective, the comparison of at best with at most isintriguing because it perhaps indicates a connection between the fact that atmost piggy-backs on pre-existing semantic scales and the fact that it lacksan overt adjective. The alternatives required by at best do not need to become with a pre-existing semantic ordering, because the adjective good doesnot require the comparison class C to be an ordered set.

The evidence for this comes from the fact that at best can seeminglyreverse what would be the default semantic ordering. Consider a contextwhere John is an alcoholic on the point of death from liver failure. He wentto the bar last night and we are speculating about how many beers he drank.(Also assume that we want John to live).

(67) a. At best John drank three beers.b. At most John drank three beers.

When uttering (67-a), the semantic alternatives are compared by their desir-ability. Given that we want John to drink as little as possible, the alternativesincrease in desirability as the number of beers decreases. With (67-b), nosuch option is available. The default number ordering (or quantity of alcohol,say, depending on focus) bubbles up through the tree and makes alternativesin which John drank more beers higher-ranked than those in which he drankfewer.

5 Semantic or pragmatic modality?

Since the publication of Geurts and Nouwen (2007) several authors havedirectly taken up the issue briefly discussed in its conclusion, namely whetherthe modal effects Geurts and Nouwen identify are encoded in the truth-conditional semantics or arise by pragmatic inference. This is not an issueI have space to explore in detail, but it merits a brief discussion becausethe pragmatic account has some advantages over the modal account I haveassumed herein.

Buring (2008) hypothesises that the truth-conditional component of atleast is simply disjunctive: at least three cars means three cars or morethan three cars. He demonstrates how the modal phenomena can be derivedpragmatically via a Gricean implicature of quantity, namely that when aspeaker utters a disjunction it is because they are not certain of the truth of

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either proposition.11 This account has the advantage of parsimony over thesemantically modal account of Geurts and Nouwen because it can also explainthe interaction of modals and superlative modifiers that forces Geurts andNouwen to invoke to the rather heavy machinery of modal concord. Buringclaims that there is no independent evidence for modal concord, althoughhe seems to have overlooked some examples from Dutch which are cited inGeurts and Nouwens paper.

Cummins and Katsos (2010) (ironically in continuation of their work inGeurts et al. (2010) which led to the opposite conclusion) provide experimen-tal evidence in support of a disjunctive view of superlative scalar modifiers.Some of this is compelling, but I note an instance where my compositionalaccount provides an explanation where neither the Burings nor Geurts andNouwens analysis does.

One of their experiments involved asking subjects to rate the acceptabilityof a second statement as a revision of a first. For example,

(68) Jean has at most three houses. Specifically, she has three houses.

Revisions from at most n to exactly n-1 and exactly n were found to besignificantly more acceptable than revisions from some to all. As the au-thors note, this is unexpected if both revisions are simply pragmaticallyself-contradictory. It is also unexpected on the semantically modal account,since on this view the superlative modifier encodes the possibility of exactequality, a possibility which is then either denied by the second sentence orstrengthened to a statement of fact.

However, recall that in my semantics for at most, the superlative mor-pheme introduces a presupposition that there exist at least two possiblepropositions in the comparison class. The first is the proposition denotedby the S node. The second is one of the focus alternatives. The existenceof a higher-ranked possible proposition is explicitly denied by the semantics,meaning that this second presupposed alternative must be lower ranked thanthe over proposition. It is conceivable that when subjects revise from at mostn to exactly n-1 they are simply fixing the rank of the second possibility. Thisis evidently less coherent than a straightforward entailment, but equally it ismore coherent than an implicature cancellation.

11However, for reasons which remain obscure to me (and which seem to have beenignored in Cummins and Katsos (2010)), he notes that extending the account to at mostwould require a crucial refinement of his ideas.

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6 Conclusion

In this paper I have examined a number of critiques of GQT and consid-ered how their insights might be connected. Specifically, I have attempted toaddress the link between superlative morphology and superlative scalar mod-ifers by connecting the work of Hackl on the former with the work of Geurtsand Nouwen on the latter. In the process I have explored in depth some ofthe syntactic properties of scalar modifiers and proposed that they contain anumber of covert elements which are crucial to their compositional analysis.I have also opened up the possibility that the class of scalar modifiers is widerthan assumed, including at best.

The pragmatic account of Buring does pose a strong challenge to thesemantically modal account which I assume here. As far as future prospectsfor my compositional account, I see two options. The first is to accept thatit is irrevocably tied to the semantic account. If the pragmatic account wereultimately vindicated, then presumably the compositional account would bediscarded. The alternative is to look for ways to modify it so as to derive thetruth-conditional disjunction of the pragmatic account, and let the modalityarise by implicature. In my view this is probably the wisest, as the pragmaticaccount appears to have a number of decisive advantages, the strongest beingits parsimony.

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