Non-canonical comparatives:Syntax, semantics, & processing

ESSLLI 2018

Roumyana Pancheva, Alexis Wellwood

University of Southern California

August 15, 2018

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cardinality comparison

I Two generalizations about nominal comparatives

(1) a. I bought more co�ee than you did. volume, weight, *temperature

b. I bought more co�ees than you did. cardinality, *volume, *weight

Monotonicity (Schwarzschild 2002, 2006)

Comparatives with bare Ns show variable but constrained dimensionality, sensitive to

part-whole relations.

Number (Hackl 2001, Bale & Barner 2009)

Comparatives with plural NPs may only be compared by number.

There is a class of apparent counter-examples, the so-called ‘mass plurals’ (e.g., muds). See Acquaviva 2008,Schwarzschild 2012, Solt 2015, among others.

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Two generalizations about verbal comparatives

I The generalizations hold for VPs as well (Wellwood et al. 2012)

(2) a. as much co�ee volume, weight, *temperature

b. too many co�ees cardinality, *volume, *weight

(3) a. run on the track as much distance, duration, *speed

b. run to the track more/as many times cardinality, *distance, *duration

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Capturing the monotonicity constraintI Permissible values of the measure function µ encoded in many and much must be

S(chwarzschild)-monotonic.

S-monotonicity (Schwarzschild 2002, 2006)

∀x, y ∈ DP , if x ≺P y , then µ(x) < µ(y).

(4) Let Jco�eeK = {. . . , c, c′, c ⊕ c′, . . .} = DC (where c, c’, etc. are non-atomic)

Intuitively, for any x, y ∈ DC such that x ≺P y ,

a. volume(x) < volume(y)b. weight(x) < weight(y)c. temperature(x) 6< temperature(y)

(5) Let Jco�eesK = {. . . , c, c′, c ⊕ c′, . . .} = DC (where c, c’, etc. are atomic)

Intuitively, for any x, y ∈ DC such that x ≺P y ,cardinality(x) < cardinality(y)

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An ambiguity theory of the plurality-cardinality link

I These pa�erns may be explained in part by an ambiguity in the morphosyntax of more,such that it spells out much plus -er with mass nouns, but many plus -er with plural nouns.

(6) a. more1 co�ee ! much-er coffee

b. more2 co�ees ! many-er coffee-pl

I Correspondingly, much contributes a variable over measure functions in general (with

constraints), while many specifically contributes a cardinality function.

(7) a. JmuchK = λd.λrη. µ quantity(r) ≥ d type η stands for e or v

b. JmanyK = λd.λrη. µ cardinality(r) ≥ d

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Some problems for the ambiguity theory

I One immediate issue with the ambiguity theory is that many never surfaces in the verbal

domain, yet the cardinality-semantic plurality link still obtains for event comparison

(8) a. too much co�ee

b. too many co�ees

(9) a. run on the track too much

b. run to the track too much

I Similarly, the interpretation of object mass nouns is an issue, since they too do not surface

with many but allow comparison by cardinality

(10) too much tra�ic

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Some problems for the ambiguity theory

I Another issue is that an adequate analysis of more1 (i.e., much-er) must capture the fact

that such comparatives display variable dimensionality both across and within predicates.

(11) Variability across predicates

a. more1 co�ee volume, *temperature

b. more1 global warming temperature, *volume

(12) Variability within predicates

a. more1 co�ee volume, weight

b. run more1 distance, duration

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Li�le cross-linguistic support for the ambiguity theory

I If the distinction between primitive much and many was semantic, we would expect it to

appear more robustly cross-linguistically, but this is not the case

(13) Spanish (from Wellwood forthcoming)

a. mucha cerveza volume

b. muchas cervezas number

(14) Bulgarian

a. mnogo bira volume

b. mnogo biri number

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Li�le cross-linguistic support for the ambiguity theoryI Typologically more broadly, any way that a language has of indicating plurality marks a

shi� in interpretation from volume to number

language volume number di�erenceEnglish much soup many cookies ‘lexical’

Spanish mucha sopa muchas galletas agreement

Italian molta minestra molti bisco�i agreement

French beaucoup de soupe beaucoup de biscuits morphology

Macedonian mnogu supa mnogu kolaci morphology

Mandarin henduo tang henduo kuai quqi classifier

Bangla onek sup onek-gulo biskuT classifier

Table 1:Where many signals number with plural Ns in English, other languages combine a univocal

form with (broadly) plural marking. (Wellwood 2014)

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An alternative theoryI Wellwood 2014, 2015, forthcoming, building primarily on Schwarzschild 2002, 2006 and

Bale & Barner 2009, argues for an alternative, univocal account, in which the pa�erns of

constrained variability are captured by a new, stronger condition on the selection of

measure functions.

string morphosyntax semanticsambiguity more N muchµ-er n . . . σ(µ)(x) . . .

more Ns many-er n-pl . . . cardinality(xx) . . .univocality more N muchµ-er n . . . σ(µ)(x) . . .

more Ns muchµ-er n-pl . . . σ(µ)(xx) . . .

Table 2: Di�erences in the alignment between strings, morphosyntax, and semantics on the two

accounts.

By xx , yy , etc. in Table 2 and below, only visual clarity of talk of pluralities is intended - their nature isn’t at

issue.

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An alternative theory

I Wellwood’s proposal: the Number generalization is a special case of the Monotonicity

generalization, and arises due to the semantics of much.

I The solution has two pieces:

I Surfacing many : many spells out much in the context of a nominal plural.

I Restricting much: permissible µs that much encodes are invariant under structure-preserving

permutation. Claim: only number meets this and S-monotonicity for Jco�eesK and JfurnitureK.

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An alternative theory

I What unifies plurals and object mass nouns, such that they require number comparisons in

the comparative? Perhaps simply: their (ordered) domains have atomic minimal parts (Bale

& Barner 2009).

abc

ab ac bc

a b c

Figure 1: Hypothetical extension for furniture and co�ees in a context. Nodes represent pieces of

furniture / containers of co�ee and pluralities thereof, lines represent plural-part relations.

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An alternative theory

I The intuition: cardinality is the only permissible measure of such domains, because it

uniquely characterizes these domains. That is, measures by cardinality assign all of the

atomic minimal parts to 1, the 2-atom pluralities to 2, etc.

A function like weight, in contrast, can assign di�erent values to each atom, each 2-atom

plurality, etc. Only a mapping by cardinality represents the structure of an atomic join

semi-la�ice.

(Wellwood forthcoming)

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An alternative theory

I A new constraint, augmenting S-monotonicity, which says that permissible σ assignments

to µ must be A(utomorphism)-invariant—i.e., they assign the same value to all x in P as to

x’s image under any structure-preserving permutation.

A-invariance∀x ∈ DP , ∀h ∈ Aut(〈DP ,%P〉), µ(x) = µ(h(x))

(15) JmuchµKσ(d)(x) = σ(µ)(x) ≥ d

(Wellwood forthcoming)

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An alternative theory

I Automorphism: a bijective function that maps a set, say DP , to itself, and meets the

condition in (16).

In e�ect, an automorphism maps elements of an ordered set to those same elements, such

that all of the same ordering relations are preserved.

(16) ∀x, y ∈ DP , x %P y i� h(x) %P h(y)

(Wellwood forthcoming)

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An alternative theoryI An illustration:

Let DP = {a, b, c, ab, ac, bc, abc} (the inclusive set of pluralities whose minimal parts are

the individuals a, b, and c), and the ordering -P on this set has all of the properties that we

think the domains of plural nouns like toys or superordinate mass nouns like furniture have(i.e., they are atomic join semi-la�ices).

Then, h in (17) is an example of an automorphism on DP .

(17) Automorphism h in Aut(〈DP ,-P〉)a. h = [a 7→ b, b 7→ c, c 7→ a, ab 7→ bc, ac 7→ ab, bc 7→ ac, abc 7→ abc]b. range(h) = domain(h) [endomorphy]

c. there is a function g such that domain(g) = range(h) [bijectivity]

d. ¬∃x, y[x -P y ∧ h(x) 6-P h(y)] [order preservation]

(Wellwood forthcoming)

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An alternative theoryI An illustration:

Let DP = {a, b, c, ab, ac, bc, abc} (the inclusive set of pluralities whose minimal parts are

the individuals a, b, and c), and the mereological ordering -P on this set.

There are many functions h that are not automorphisms on DP ; (18) gives some examples,

along with reasons for their failure.

(18) Functions h not in Aut(〈DP ,-P〉)a. Any h = [a 7→ d, . . . ], since d 6∈ DP [not endomorphic]

b. Any h = [a 7→ b, c 7→ b, . . . ], since not invertible [not bijective]

c. Any h = [a 7→ c, ab 7→ a, . . . ], since a -P ab, but h(a) 6-P h(ab)[not order-preserving]

(Wellwood forthcoming)

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An alternative theoryI Distinguishing cardinality and weight measures

Since any automorphism h on atomic 〈DP ,%P〉 pairs singletons with singletons, doubletons

with doubletons, etc., then any plurality xx ∈ DP is such that cardinality(xx) =cardinality(h(xx)). Thus, measures by cardinality are A-invariant with respect to such a

domain.

However, measures by weight are not; a counter-example is given in (19).

(19) Let DP = {b, c, bc}, h an automorphism on DP such that h(b) = c, andweight : [b 7→ 120lbs, c 7→ 240lbs, . . .].Then, since

a. weight(h(b)) = weight(c),b. weight(h(b)) = 240lbs;

therefore,

c. weight(h(b)) 6= weight(b), because 120lbs 6= 240lbs.

(Wellwood forthcoming)

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An alternative theory

I Can A-invarience supplant S-monotonicity?

It seems that the answer is ‘no’. There are measure functions that fail to preserve the

structure of the domain for measurement, but which would satisfy A-invariance.

Consider a hypothetical such function, one, that maps everything to the number 1. This

function trivially satisfies A-invariance, since any x ∈ DP will be such that one(x) = 1, and

of course one(h(x)) = 1, etc. Such a function will not satisfy S-monotonicity, however,

since any case of some x, y ∈ DP such that x �P y , it is not the case that one(x) � one(y).

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An alternative theory

I Does A-invarience apply in the case of measurement of mass nouns?

It seems that the answer is ‘yes’.

Suppose that the extension of co�ee is a dense ordering of portions of co�ee by inclusion.

Any automorphism (hence, any h ∈ Aut(〈DC,%C〉)) will preserve this structure exactly.

It seems that just in the same way that cardinality can be said to represent essential

structure of plural part-of relations, volume or weight do the same for material part-of

relations.

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References I

Acquaviva, Paolo. 2008. Lexical plurals: A morphosemantic approach Oxford studies in theoretical

linguistics. Oxford, UK: Oxford University Press.

Bale, Alan & David Barner. 2009. The interpretation of functional heads: Using comparatives to

explore the mass/count distinction. Journal of Semantics 26(3). 217–252.

Hackl, Martin. 2001. Comparative quantifiers and plural predication. In K. Megerdoomian &

Leora Anne Bar-el (eds.), Proceedings of WCCFL XX, 234–247. Somerville, Massachuse�s:

Cascadilla Press.

Schwarzschild, Roger. 2002. The grammar of measurement. In B. Jackson (ed.), Proceedings ofSALT XII, 225–245. Cornell University, Ithaca, NY: CLC Publications.

Schwarzschild, Roger. 2006. The role of dimensions in the syntax of noun phrases. Syntax 9(1).67–110.

Schwarzschild, Roger. 2012. Neoneoneo Davidsonian Analysis of Nouns. Handout for the 2nd

Mid-Atlantic Colloquium for Studies in Meaning, University of Maryland.

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References II

Solt, Stephanie. 2015. Q-adjectives and the semantics of quantity. Journal of Semantics32(221-273).

Wellwood, Alexis. 2014. Measuring predicates: University of Maryland, College Park dissertation.

Wellwood, Alexis. 2015. On the semantics of comparison across categories. Linguistics andPhilosophy 38(1). 67–101.

Wellwood, Alexis. forthcoming. The semantics of more Studies in Semantics and Pragmatics.

Oxford UK: Oxford University Press.

Wellwood, Alexis, Valentine Hacquard & Roumyana Pancheva. 2012. Measuring and comparing

individuals and events. Journal of Semantics 29(2). 207–228.

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