Degree pluralities : distributive, cumulative and ...archive.sfl.cnrs.fr/sites/sfl/IMG/pdf/dotlacil_nouwen_14Feb2014.pdf · Degree pluralities : distributive, cumulative and collective

Degree pluralities : distributive, cumulativeand collective readings of comparatives

Jakub Dotlacil (Groningen)

&

Rick Nouwen (Utrecht)

February 14, 2014, Paris

1 John lifted the box.

2 The boys lifted a box.

Collective reading Distributive reading

3 Ten boys lifted five boxes.

1 John lifted the box.2 The boys lifted a box.









Plural individuals

Scha 1981, Link 1983, Landman 1996,

Schwarzschild 1996, Sternefeld 1998,

Landman 2000, Beck & Sauerland 2000,

Winter 2002, Dotlacil 2010, Nouwen 2013

Plural events

Krifka 1989, Schein 1993, Landman 2000,

Kratzer 2003

Plural information states

Van den Berg 1996, Krifka 1996, Nouwen

2003, Nouwen 2007, Brasoveanu 2008

Plural propositions

Sharvit & Beck 2002

Plural times

Artstein 2003

Plural degrees

. . .

Plural individuals





Plural events


Kratzer 2003




Plural propositions

Sharvit & Beck 2002

Plural times

Artstein 2003

Plural degrees

. . .

Plural individuals





Plural events


Kratzer 2003




Plural propositions

Sharvit & Beck 2002

Plural times

Artstein 2003

Plural degrees

. . .

Weak claim:the domain of degrees contains degreepluralities

John, Peter and Bill are 20, 22 and 26 years old.

Strong claim:the comparative expresses a relation betweendegree pluralities

comparison can be distributive, cumulative and, potentially,collective.









1 A simple framework for plural degree semantics

2 Quantifiers in than-clauses

3 Cumulative comparison

4 Collective comparison





Distributivity & Cumulativity

Two boys carried two boxes.

distributive construal: there are two boys such that each ofthese boys carried two boxes

cumulative construal: there are two boys and two boxes suchthat each boy carried at least one of those boxes and each boxwas carried by at least one of these boys

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A semantic framework for distributivity & cumulativityLink, 1983, Landman, 1996, 2000, cf. Nouwen 2014

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a t b is the plural individual that has a and b as its parts

Predicate cumulation:

*X := the smallest set s.t. *X ⊇ X & ∀x , y ∈ *X : x t y ∈ *X

X = {a,b}, *X = {a,b ,a t b}

Relation cumulation:

**R := the smallest set s.t. **R ⊇ R &∀x , x′, y , y′[〈x , x′〉, 〈y , y′〉 ∈ **R → 〈x t y , x′ t y′〉 ∈ **R]

R = {〈a,b〉, 〈b , c〉}, **R = {〈a,b〉, 〈b , c〉, 〈a t b ,b t c〉}

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a t b is the plural individual that has a and b as its parts



X = {a,b}, *X = {a,b ,a t b}



R = {〈a,b〉, 〈b , c〉}, **R = {〈a,b〉, 〈b , c〉, 〈a t b ,b t c〉}

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1 Two boys *[carried two boxes] distributivity

If b1 carried two boxes and b2 carried two boxes,then b1 t b2 *[carried two boxes]

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2 Two boys **carried two boxes cumulativity

If b1 carried bx1 and b2 carried bx2,then b1 t b2 **carried bx1 t bx2

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3 Two boys carried two boxes collective

b1 and b2, as a group, carried bx1 and bx2

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Plural degree semantics

Novel assumptions:

— the structure of De = the structure of Dd— cumulation (i.e. * and **) may apply to relations of any type

1 John, Peter and Bill are 20, 22 and 26 years old.

**old(j t p t b,20 t 22 t 26)

2 150 t 170 **> 140 t 160170 **> 140 t 160150 **≯ 140 t 160

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The syntax of than clauses

John is taller than Mary is.than Opi Mary is ti tall

Evidence for Op movement in than-clauses (Chomsky, 1977)(i) *He found more marbles than Mary went to the store in order to buy t(ii) *What did Mary go to the store in order to buy t?

(Adjunct islands)

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The semantics of than clauses

John is taller than Mary is.Opi Mary is ti tall.

Adjectives are monotoneJtallK = λdλx .x ’s height ≥ d

The interval strategy von Stechow 1984, Heim 2000

Jthan Op Mary is t tallK = λd.Mary is d-tallJohn is taller than the maximum of Jthan WH Mary is t tallK

The negation strategy Seuren 1973, Klein 1981, van Rooij 2011

Jthan WH Mary is t tallK = λd.¬Mary is d-tallJohn is tall to some degree satisfying Jthan WH Mary is t tallK

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John is taller than Mary is.

Interval: the maximal degree towhich John is tall exceeds themaximal degree to which Mary istall

Negation: there is a degree towhich John is tall such that it is notthe case that Mary is that tall

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John is taller than every girl is.

Interval: the (maximal) degree towhich John is tall exceeds the(maximal) degree to which everygirl is tall

Negation: there is a degree towhich John is tall such that it is notthe case that every girl is tall tothat degree

λd.every girl is d-tall = (0,150] λd.¬every girl is d-tall = (150,∞)

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λd.every girl is d-tall = (0,150] λd.¬every girl is d-tall = (150,∞)18 / 44





λd.every girl is d-tall = (0,150] λd.¬every girl is d-tall = (150,∞)18 / 44


Notice:

λd.every girl is d-tall = (0,150]

λd.¬every girl is d-tall = (150,∞)

Jthan every girl is tallK = the shortest girl’s height

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Wide-scope behaviour for the universal quantifier (von Stechow):

John is taller than every girls is- For every girl: the maximal degree to which John is tall exceeds

the maximal degree to which the girl is tall

- For every girl: there is a degree to which John is tall such that itis not the case that this girl is so tall

Problem: than-clauses are islands (cf. Schwarzschild & Wilkinson)

*The room was painted precisely that shade of greenwhich Raffaelo Di Sanzio would have bitten off his righthand rather than use (Douglas Adams)

Solution: stipulate a scope pivot within the than-clause

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Negation as a scope pivot Larson, Gajewski, van Rooij, Schwarzschild

than every girl is{than WHj every girli NOT ti is tjtall

λd.every girl is not d-tall

= (190,∞)

= than the tallest girl is

Notice: the original(problematic) reading isavailable on a different scoping

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Beck’s interval strategy (Beck 2010)

Core idea: selection from sets of intervals I(Schwarzschild & Wilkinson 2002, Heim 2006)

max(min(I))

Example I’s (than clauses)λI.Mary’s height∈ IλI.every girl’s height∈ I

min picks the smallest interval from a set of intervalsmatrix-clause: John’s heightwhole construction: matrix-clause > than-clause

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Beck’s interval strategy: example

John is taller than Mary is

max(min(λI.Mary’s height∈ I))

= max([Mary’s height])

= Mary’s height

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Beck’s interval strategy: example

John is taller than every girl is

max(min(λI.every girl’s height∈ I))

= the height of the tallest girl

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A conservative reinterpretation of Beck’s selectionapproach

λd.every girl’s height v d =

. . .34 t 67 t 150 t 155 t 156 t 190,. . . ,150 t 155 t 190 t 1554,. . . ,150 t 155 t 190,. . .

the minimum = 150t155t190

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170 > 150 t 155 t 190 ??

170 **> 150 t 155 t 190

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John is taller than every girl (is)

Beck’s approachtake the smallest interval thatcontains the height of every girltake the maximum of thatinterval

John’s height > the maximumof [150,190]

Our approachtake the smallest plurality thatcontains the height of every girlthis plurality is in the **>relation to the matrix clause

John’s height > each of theatoms in 150 t 155 t 190

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John is taller than every girl (is)

Beck’s approachtake the smallest interval thatcontains the height of every girltake the maximum of thatinterval

John’s height > the maximumof [150,190]

Our approachtake the smallest plurality thatcontains the height of every girlthis plurality is in the **>relation to the matrix clause

John’s height > each of theatoms in 150 t 155 t 190

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Comparison of Beck’s and our approach

Beck’s approachmax(min(λI.every girl’sheight∈ I))min picks the smallest intervalfrom a set of intervals

Our approachmin(λd.every girl’s heightv d)min picks the smallest pluralityfrom a set of pluralitiesJohn’s height **> Jthan-clauseK

Immediate advantage: no excessive maximality

Instead, we deal with pluralities in the standard way

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Compositional derivation

Cumulativity on relation

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Compositional derivation

Pluralization of degree predicate

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Differentials

Standard semantics for differential:John is exactly 2 inches taller than Bill.⇔ John’s height − 2in = Bill’s height

John is exactly 2 inches taller than every girl (is).{ every girl is the same height

% Beck: John’s height − 2in = the height of the tallest girl

" John’s height − 2in **= the plurality of heights of the girls

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Interim summary

The comparative as a potentially cumulative relationbetween degree pluralitiesConservative reinterpretation of Beck’s selection approachSlightly simplifies the selection procedureEmpirical advantage for differentials

If we really want to argue for our framework then we needto show that some examples could not be interpretedwithout cumulation

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The relation between distributive and cumulativereadings

Fact: If one of the arguments of a relation is singular, then thedistributive and cumulative reading collapse into one.

John and Bill know Sue.John and Bill **know Sue. (cumulative analysis)

= John and Bill *[know Sue] (distributive analysis)

John is taller than every girl is.John’s height **> Jthan-clauseK (cumulative analysis)

= Jthan-clauseK *λd.John’s height > d (distributive analysis)

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Cumulative readings in phrasal comparativesScha & Stallard 1988, Schwarzschild 1996, Matushansky & Ruys 2006

(i) The frigates were faster than the carriers.

“Imagine [..] a context in which it is clear that these shipsare sent out in teams to different areas of the globe witheach team consisting of frigates and carriers. It may bethat one area calls for very fast action while another willtolerate a sluggish response. If that were the case, Iwould judge [(i)] true just in case the frigates in a givenarea were faster than the carriers of that area, regardlessof what speed relations obtained between ships ofdifferent areas.”Schwarzschild 1996, p. 89

F1 C1

F2 C2

F3 C3

〈the frigates, the carriers〉 ∈ **λx .λy .x is faster than y

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Cumulative readings in phrasal comparativesScha & Stallard 1988, Schwarzschild 1996, Matushansky & Ruys 2006

(i) The frigates were faster than the carriers.

“Imagine [..] a context in which it is clear that these shipsare sent out in teams to different areas of the globe witheach team consisting of frigates and carriers. It may bethat one area calls for very fast action while another willtolerate a sluggish response. If that were the case, Iwould judge [(i)] true just in case the frigates in a givenarea were faster than the carriers of that area, regardlessof what speed relations obtained between ships ofdifferent areas.”Schwarzschild 1996, p. 89

F1 C1

F2 C2

F3 C3

〈the frigates, the carriers〉 ∈ **λx .λy .x is faster than y

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Cumulative readings in clausal comparatives

The frigates were fasterthan the carriers were

How do we get a cumulativerelation?

% QR is clause-bounded!

**

∅were

ti

than

faster

were

λi

the carriers

the frigates

cf. Bale 2006

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Alternative (Winter 2000, cf. Beck & Sauerland 2000):cumulative readings are distributive readings (e.g. *)cumulative effects are due to dependent interpretation

The frigates eachi were faster than the carriersi were

Where the carriersi functionally depends on quantification overthe frigates.

Each student managed to hand the essay in on time

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So far in summary:

There are cumulative phrasal comparativesthese can be handled using the ** operator

There are cumulative clausal comparativesUsing ** is unavailable: QR over a clause boundaryAlternative: distributivity + dependency

Next: there exist cases where the dependency analysis isalso unavailable

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The state economies of Ireland, the Netherlands andAustralia all scored higher than they each did in themid-1980s. from www.oapen.org/download?type=document&docid=340206 (simplified)

A dependency analysis is out of the question, due to they each.

(i) The boys all think they won the race(ii) #The boys all think they each won the race(iii) The boys all think that they each failed the course







Similar data

(i) The students earned less than each of them hoped theywould

(ii) The Baltimore Ravens scored more points than each of theiropponents did(the actual scores: 24 - 9; 38 - 35; 28 - 13)

Cumulative readings in the comparative

— no dependency analysis possible- #The boys all think each of them won the race

— no relation as a target of ** available- QR is clause-bounded

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Similar data

(i) The students earned less than each of them hoped theywould

(ii) The Baltimore Ravens scored more points than each of theiropponents did(the actual scores: 24 - 9; 38 - 35; 28 - 13)

Cumulative readings in the comparative

— no dependency analysis possible- #The boys all think each of them won the race

— no relation as a target of ** available- QR is clause-bounded

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The negation approach / Beck’s approach

The state economies ofIreland, the Netherlandsand Australia all scoredhigher than they each didin the mid-1980s.

We want pairwisecomparison of theindividual scores; neitherthe negation approach,nor Beck’s could give usthis

They give us a singledegree of comparison

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Our analysis

These sentences are genuinely cumulativebut do not involve a cumulative 〈e, 〈e, t〉〉-relationbut rather a cumulative relation on degrees

The state economies of Ireland, theNetherlands and Australia all scoredhigher than they each did in themid-1980s.

6 t 8 t 9 **> 4 t 5 t 7

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Collective readings?

WOW! John was almost 4 seconds faster than every opponent,and a 9 second gap on Lance. Beck

John: 10m40sRui: 10m44sTony: 10m45sLance: 10m49sLars: 11m30s

" max(min(λD.∀x[opponent(x)→ x ’s time-4s ∈ D])) =10m40s% 10m40s − 4s **= (10m44st10m45st10m49st11m30s)

Collective reading: degree point representative of degree group

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Collective readings?

Ben was almost a year older than everyone else in class(because he had just missed the deadline for the previousschool year). (Beck)

- #For all x , Ben: Ben was almost a year older than x- #Ben was almost a year older than the next oldest in the

class- ?The others’ ages center around a point almost a year

younger than Ben.

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Summary and conclusion

There exists properly cumulative readings for comparativesproblematic to all existing accountsProposal: than-clauses with quantifiers receive theirparticular interpretation via a process of pluralisationThe maximality associated with than-clauses is aconsequence of plural semantics and the monotonicity ofscalesWhen the monotonicity is cancelled (differentials) pluralsemantics correctly derives the-same-height reading

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Degree pluralities : distributive, cumulative and ...archive.sfl.cnrs.fr/sites/sfl/IMG/pdf/dotlacil_nouwen_14Feb2014.pdf · Degree pluralities : distributive, cumulative and collective