Comparative vs. superlative quantifiers Superlative and comparative ...

Superlative vs. comparative Indefinites, numerals, and scales Comparative modifiers Superlative modifiers Experimental evidence 

Comparative vs. superlative quantifiers 

Bart Geurts 

(Nijmegen) 

Rick Nouwen 

(Frankfurt) 

Bart Geurts & Rick Nouwen: Comparative vs. superlative quantifiers 


Superlative and comparative quantifiers 

◮ Superlative quantifiers: at least/most 4 sheep 

◮ Comparative quantifiers: more/fewer than 5 camels 

◮ Attested examples: 

[1] We need a president who is fluent in at least one language. 

[2] Never eat more than you can lift. 

It is usually assumed that these quantifiers are interdefinable: 

◮ at least 7 bath ducks = more than 6 bath ducks 

◮ at most 7 bath ducks = fewer than 8 bath ducks 

This is a mistake. 

Bart Geurts & Rick Nouwen: Comparative vs. superlative quantifiers


Puzzle #1: Specificity 

NPs in superlative quantifiers may have specific readings: 

[1] I will invite at most 2 people, namely Jack and Jill. 

Jack and Jill will be invited. 

[2] I will invite fewer than 3 people, namely Jack and Jill. 

⇒ Jack and Jill will be invited. 

[3] ∗I will invite fewer than 2 people, namely Jack and Jill. 



Puzzle #2: Inference patterns 

Barney ate 3 apples. 

⇒ Barney ate fewer than 4 apples. 

Barney ate at most 3 apples. 

⇒ Barney ate more than 2 apples. 

Barney ate at least 3 apples. 



Puzzle #3: Distributional restrictions 

By and large, superlative modifiers occur more freely than their 

comparative counterparts: 

[1] I might survive a few days {at most/∗fewer than}. 

[2] It actually dates back to {at least/∗more than} the 14th century. 

[3] Men like me because I don’t wear a brassiere. Women like me 

because I don’t look like a girl who would steal a husband. 

{At least/∗More than} not for long. 

But in some ways superlatives are more restricted. 



Distributional restrictions (cont.) 

Downward entailing contexts 

[1] Bach wrote more/fewer than 27 fugues. 

Bach wrote at least/most 27 fugues. 

[2] Bach didn’t write more/fewer than 27 fugues. 

?Bach didn’t write at least/most 27 fugues. 

[3] No composer wrote more/fewer than 27 fugues. 

?No composer wrote at least/most 27 fugues. 

[4] Few composers wrote more/fewer than 27 fugues. 

?Few composers wrote at least/most 27 fugues. 



Distributional restrictions (cont.) 

Quantifiers 

[1] {All/Nearly all/Most/?Some/?About 7/∗None} (of the) children 

ate at least/most 2 sandwiches. 

[2] There were so many/few chocolates left that 

{(almost) everybody/most guests/?about 7 guests/∗no one} 

got at least/most 3. 

[3] Before going to bed, she 

{(nearly) always/usually/?often/∗occasionally/∗rarely/∗never} 

says at least/most 4 prayers. 



Puzzle #4: Missing readings 

[1] You may take fewer than 3 cards. 

a. You are allowed to take 1 or 2 cards (or more). 

b. You are allowed to take 1 or 2 cards, but no more. 

[2] You may take at most 2 cards. 

a. — 

b. You are allowed to take 1 or 2 cards, but no more. 



Outline of a proposal 

◮ The standard analysis of comparative quantifiers is correct: 

[1] Wilbur saw more than 4 films. 

∃x[#x > 4 ∧ film(x) ∧ see(w, x)] 

◮ Superlative quantifiers have a bi-modal meaning: 

[2] Wilbur saw at least 5 films. 

□∃x[5(x) ∧ film(x) ∧ see(w, x)] ∧ 

⋄∃x[#x > 5 ∧ film(x) ∧ see(w, x)] 

[3] Wilbur saw at most 5 films. 

⋄∃x[5(x) ∧ film(x) ∧ see(w, x)] ∧ 

¬⋄∃x[#x > 5 ∧ film(x) ∧ see(w, x)] 



Indefinites, numerals, and scales 



Indefinites and numerals 

◮ Nouns and number words denote properties: 

◮ ‖book‖ = λx[book(x)] 

◮ ‖three‖ = λx[#x = 3] = λx[3(x)] 

✐ Note that this is an ‘exact’ reading. 

◮ ‖three books‖ = λx[3(x) ∧ book(x)] 

◮ Existential closure turns a predicate into a quantifier: 

◮ ‖∅[three books]‖ = λP∃x[3(x) ∧ book(x) ∧ P(x)] 

◮ ‖Betty read [∅[three books]]‖ = ∃x[3(x) ∧ book(x) ∧ read(b, x)] 

✐ Note that this is an ‘at least’ reading. 

◮ Implicature: 

◮ ¬∃x[#x > 3 ∧ book(x) ∧ read(b, x)] 

◮ Specificity requires existential closure. 

◮ Betty has read three books. [May be specific.] 

◮ These are three books. [Non-specific only.] 



Digression on number terms 

How are we to account for the fact that number terms behave 

differently from bona fide scalar expressions? 

Some of the options: 

◮ Short-circuited implicature (whatever that may be). 

◮ Existential closure comes in several flavours, and the ‘exactly’ 

flavour is the default for expressions like ‘three’, ‘half of’, etc. 

◮ The lexical meaning of a number term is quantificational and 

‘exact’: type shifting yields first a predicative, and then a 

quantificational (‘at least’) sense. 

♥ Number terms are polysemous between predicative and 

quantificational senses, and both are exact, so the ‘at least’ 

reading must still be derived. 

This view is recommended by experts in 

Germany and the Netherlands. 



Scales Krifka (1999) 

◮ Focus-induced alternatives are ordered. 

◮ E.g., if ‘three’ is in focus, we get: 

. . . λx[3(x)] ⊲ λx[2(x)] ⊲ λx[1(x)] 

✐ Note: this is not an entailment scale. 

◮ This ordering is propagated up the derivation tree. 

Hence, the alternatives for ‘[three] F books’ are ordered thus: 

. . . λx[3(x) ∧ book(x)] ⊲ λx[2(x) ∧ book(x)] ⊲ λx[1(x) ∧ book(x)] 

◮ Scalar modifiers operate on such ordered alternatives (cf. ‘only’). 

◮ Example: When applied to ‖three‖, ‖more than‖ returns the 

union of all the properties that are ‘higher’ than λx[3(x)], i.e. 

λx[4(x) ∨ 5(x) ∨ . . .] = λx[#x > 3]. 



Comparative modifiers 



The grammar of scalar quantifiers 

◮ Scalar quantifiers are standardly parsed like this: 

[[more than three] books] 

◮ This is problematic for several reasons. E.g. 

[1] Betty read more than [three] F books: she read four. 

[2] Betty read more than [three books] F : she read fifteen essays 

as well. 

◮ Alternative analysis: 

[more than [three books]] 



The semantics of comparative modifiers 

◮ The argument of a comparative modifier is a first-order 

predicate: ‘more than happy’, ‘more than three books’, but 

‘∗more than Fred’. 

◮ This explains why ‘three books’ in ‘more than three books’ does 

not allow for a specific construal. 

◮ ‖more than α‖ = λx∃α ′ [α ′ ⊲ α ∧ α ′ (x)] 

[1] With λx[scalding(x)] ⊲ λx[hot(x)] ⊲ λx[warm(x)], 

‖more than warm‖ = λx[hot(x) ∨ scalding(x)] 

[2] With . . . λx[3(x)] ⊲ λx[2(x)] ⊲ λx[1(x)], 

‖more than [[three] F books]‖ = 

λx[[4(x) ∨ 5(x) ∨ . . .] ∧ book(x)] = λx[#x > 3 ∧ book(x)] 

So: ‖Betty read ∅[more than [[three] F books]]‖ = 

∃x[#x > 3 ∧ book(x) ∧ read(b, x)] 



‘Fewer than’ 

◮ ‖fewer than α‖ = λx∃α ′ [α ⊲ α ′ ∧ α ′ (x)] 

So: ‖Betty read ∅[fewer than [[three] F books]]‖ = 

∃x[#x < 3 ∧ book(x) ∧ read(b, x)] 

Unfortunately, this is too weak. 

[Actually, it’s a tautology.] 

◮ Implicature saves: 

¬∃x[#x ≥ 3 ∧ book(x) ∧ read(b, x)] 

◮ If this is an implicature it should be cancellable—which it is: 

[1] You may take fewer than three cards. 

[2] ∃x[#x < 3 ∧ card(x) ∧ take(y, x)] 

[3] ¬∃x[#x ≥ 3 ∧ card(x) ∧ take(y, x)] 



Superlative modifiers 



The semantics of superlative modifiers 

The argument of a superlative modifier is any predicate (it may be a 

quantifier, for example). 

◮ ‖at least α‖ = λX[□α(X) ∧ ∃α ′ [α ′ ⊲ α ∧ ⋄α ′ (X)]] 

◮ ‖at most α‖ = λX[⋄α(X) ∧ ¬∃α ′ [α ′ ⊲ α ∧ ⋄α ′ (X)]] 

[1] a. Betty read [at least ∅[[four] F books]]. 

b. □∃x[4(x) ∧ book(x) ∧ read(b, x)] ∧ 

⋄∃x[#x > 4 ∧ book(x) ∧ read(b, x)] 

[2] a. Betty read [at most ∅[[four] F books]]. 

b. ⋄∃x[4(x) ∧ book(x) ∧ read(b, x)] ∧ 

¬⋄∃x[#x > 4 ∧ book(x) ∧ read(b, x)] 

☞ In either case, the phrase ‘four books’ may be specific. 



Modal concord 

[1] Hij moet zeker in Parijs zijn. 

he must certainly in Paris be 

◮ Compositional reading: I suppose he has to be in Paris. 

◮ Concord reading: He (definitely) has to be in Paris. 

[2] Hij moet misschien in Parijs zijn. 

he must perhaps in Paris be 

◮ Compositional reading: Perhaps he has to be in Paris. 

◮ Concord reading: none 

[3] Ze zou misschien wel eens dronken kunnen zijn. 

she could maybe wel eens drunk can be 

◮ Compositional reading: ⋄. . .⋄(she is drunk) [not available] 

◮ Concord reading: ⋄(she is drunk) 



Modal concord with superlative quantifiers 

◮ The two conjuncts introduced by a superlative quantifier do not 

have the same status: 

↓ 

primary operator 

□∃x[n(x) ∧ φ(x)] 

}{{} 

∧ ⋄∃x[#x > n ∧ φ(x)] 

principal message 

◮ If a superlative quantifier combines with a modal expression 

whose force is the same as that of its primary operator, the two 

operators will fuse by default (modal concord). 

◮ Otherwise, we have a double modal (compositional reading). 

◮ The modal operators contributed by the quantifier are epistemic 

by default, hence will prefer to take wide scope in most cases. 



Examples 

[1] You may take at most three cakes. 

a. ∃x[3(x) ∧ cake(x) ∧ take(y, x)] ∧ 

¬∃x[#x > 3 ∧ cake(x) ∧ take(y, x)] 

b. ⋄∃x[3(x) ∧ cake(x) ∧ take(y, x)] ∧ 

¬⋄∃x[#x > 3 ∧ cake(x) ∧ take(y, x)] 

[2] You must take at least three cakes. 

a. ⊡∃x[3(x) ∧ cake(x) ∧ take(y, x)] ∧ 

∃x[#x > 3 ∧ cake(x) ∧ take(y, x)] 

b. □⊡∃x[3(x) ∧ cake(x) ∧ take(y, x)] ∧ 

⋄⊡∃x[#x > 3 ∧ cake(x) ∧ take(y, x)] 

[concord] 

[compositional] 

[concord] 




Examples (cont.) 

[1] You may take at least three cakes. 

a. — [no concord reading] 

b. □∃x[3(x) ∧ cake(x) ∧ take(y, x)] ∧ 

⋄∃x[#x > 3 ∧ cake(x) ∧ take(y, x)] 


[2] You must take at most three cakes. 

a. — [no concord reading] 

b. ⋄⊡∃x[3(x) ∧ cake(x) ∧ take(y, x)] ∧ 

¬⋄⊡∃x[#x > 3 ∧ cake(x) ∧ take(y, x)] [compositional] 



Modal modifiers 

If the modal analysis is correct, it makes sense that superlative 

modifiers sometimes alternate with overt modals: 

[1] Hij heeft voor zeker zes kranten gewerkt. 

he has for certainly six newspapers worked 

‘He has worked for at least six newspapers.’ 

[2] Hij heeft voor misschien drie kranten gewerkt. 

he has for maybe three newspapers worked 

‘He has worked for at most three newspapers.’ 



Distributional restrictions 

If the modal analysis is correct, there should be restrictions on the 

distribution of superlative quantifiers just as there are on the 

distribution of bona fide modals (and epistemic ones in particular): 

{Each / Most / ?About five / ∗None} of the guests . . . 

[1] . . . may have dispatched the butler. 

[2] . . . had at least 3 cocktails. 

[3] Nabokov might not have written ‘Pnin’. (Only wide scope for ⋄.) 

[4] ∗Nabokov didn’t write at least/most 3 novels. 



Experimental evidence 



Predictions 

If the proposed analysis is on the right track, then superlative 

quantifiers should differ from comparative ones in that: 

◮ they are more complex, 

◮ they are acquired later, and 

◮ they license fewer inferences. 



Evidence from acquisition Musolino (2004: Experiment 3) 

Materials: 

‘The puppet only likes to keep cards with 

{exactly/at least/at most/more than} two Ns on them.’ 

Results (% correct): exactly 2 100 

more than 2 88 

at least 2 50 

at most 2 54 



Pilot study on reasoning in adults (n = 6!) 

Fred had . . . 

[1] a. at least 3 beers ⇔ more than 2 beers 

b. at most 2 beers ⇔ fewer than 3 beers 

[2] a. 3 beers ⇒ more than 2 beers 

b. 3 beers at least 3 beers 

[3] a. 2 beers ⇒ fewer than 3 beers 

b. 2 beers at most 2 beers 

[4] a. 3 or 4 beers ⇒ at least 3 beers 

b. 1 or 2 beers ⇒ at most 2 beers 

[5] a. at least 3 beers ⇒ at least 2 beers 

b. at most 2 beers at most 3 beers 

c. fewer than 3 beers ⇒ fewer than 4 beers 



Summary 

Key assumptions: 

◮ Scalar modifiers are focus-sensitive and operate on scales. 

◮ Comparative modifiers are extensional and take first-order 

predicates as arguments. 

◮ Superlative quantifiers are modals and take any type of predicate 

as argument. 

These assumptions enable us to solve our four puzzles: 

Puzzle #1: Specificity 

Puzzle #2: Inference patterns 

Puzzle #3: Distributional restrictions 

Puzzle #4: Missing readings

Comparative vs. superlative quantifiers Superlative and comparative ...

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