CAS LX 502

12a. Plurals, mass, andmeasurement

Singular individuals

• The Last Juror is a book.• Cat’s Cradle is a book.• Semantics is a book.

• [book]M,g = x [ x F(book) ]• F(book) = {The Last Juror, Cat’s Cradle,

Semantics}

• F(The Last Juror) F(book)• F(Cat’s Cradle) F(book)• F(Semantics) F(book)

Plurals

• The Last Juror and Cat’s Cradle are books.• The Last Juror and Semantics are books.• Cat’s Cradle and Semantics are books.• The Last Juror, Cat’s Cradle, and Semantics

are books.

• So, something F(books).• The question is: what is in F(books)?

‘What is the denotation of books?’

Collections

• Intuitively, The Last Juror and Semantics is something like a collection of individuals (each a book).

• A natural thing to suppose is that F(books) contains collections of individuals—all of those collections such that each member is in F(book).

• F(books) = { {The Last Juror, Cat’s Cradle},{The Last Juror, Semantics},{Cat’s Cradle, Semantics},{The Last Juror, Cat’s Cradle, Semantics} }

• NP1 and NP2 creates a collection {[NP1]M,g, [NP2]M,g}

The structure of plurals

• A, B, C are the atoms; all sums (collections) of atoms are in the plural. (The atoms aren’t: The Last Juror isn’t books)

A+B+C

A+B A+C B+C

(A) (B) (C)F(book)

F(books)

Distributivity

• Great. So, The Last Juror and Cat’s Cradle comes out as denoting the collection {The Last Juror, Cat’s Cradle}.

• The Last Juror and Cat’s Cradle are boring.• Notice that agreement registers whether we’re looking

at a plural or singular noun phrase here.

• Under what conditions is this true?

Distributivity• Notice that the way we’ve been thinking of the definition

of boring from the formal fragment of English, we do not predict this:

• [boring]M,g = x [ x F(boring) ]

• So while it may (hypothetically) be the case that The Last Juror F(boring) and that Cat’s Cradle F(boring), we hadn’t considered the possibility that {The Last Juror, Cat’s Cradle} F(boring).

• Moreover, we probably don’t want that.

Distributivity

• It’s very predictable that a collection C like these is only boring if, for each member x in the collection, x is boring.

• The predicate distributes through the collection, applying to each member.

• Predictable things should be in the grammar.

Plural predicates• The standard way to think about this is to suppose that for

every predicate like boring there is a plural predicate, notated boring, that holds of collections:

can be viewed as a function from singular predicates to plural predicates:

• []M,g = P [ C [ x[ xC P(x) ] ] ]

• The idea is that the word boring can either correspond to boring or boring; the one you choose is based on whether it is an individual or a collection that it applies to.

This works fabulously.

• Loren and Bond are tall.

• Nemo and Dory are fish.

• It is not the case that Loren and Nemo are fish.

• The books are boring. The fish are hungry.• (what are the collections here?)

Er…

• Loren and Bond bought a house.

• Pavarotti and Bond lifted a piano.

• The men gathered.

• Loren and Pavarotti are similar.

• The meaning here is does not seem to be distributive.

Groups• A collection of individuals can be viewed either

distributively (as individuals) or collectively (as a group).

• The students gathered.• The students have blue eyes.

• We even have names for some groups: committee, jury, senate, …

• The jury gathered.• Note that these are morphologically singular even though

they are semantically plural in some sense.

Collections of groups

• The animals filled the barn to capacity.

• The pigs and the cows filled the barn to capacity.

• The pigs and the cows argue.

• The boys and the girls touched the ceiling.

• …There are many complex issues left to address with plurals and groups, but this gives you a taste…

Plurals and part-whole structure

• One of the main things to take away here is that plurals are structured.

• Additivity:• If C1 F(books), C2 F(books), then

C1 and C2 F(books)

• Books and books are books.

Mass nouns

• Certain kinds of nouns, mass nouns, also show this kind of part-whole structure (unlike count nouns)

• Soup and soup is soup.• Parts of soup are soup.• A chair and a chair is not a chair.• Parts of a chair are not a chair.

Distinguishing mass and count

• John bought four books.

• *Mary bought four barley(s).

• Each chair is brown.

• *Each furniture is brown.

• *John doesn’t have much chair(s).

• Mary doesn’t have much furniture.

Coercions

• It is possible to interpret nouns that are “naturally” mass as count, or interpret nouns that are “naturally” count as mass. Although you can do these coercions, you can feel that they are coercions.

• John bought three coffees.• There is chair all over the room.

Making mass nouns countable:How do you count soup?

• I have two gallons of soup.• I have two pounds of soap.• I have three pieces of furniture.

• By providing a measurement, we can begin to count the number of those measurements. These measurements can in some sense be natural or imposed.

• I found three grains of rice in my pocket.• I found three pocketfuls of rice in my closet.

Measurement

• Measurement can come in a couple of different kinds. Let’s explore two of them for a while:

• Pseudopartitive: One quarter inch of wire

• Compound: One quarter inch wire

• What do these refer to?

• How do we know that?

Measurement

• It seems that the pseudopartitive and compound forms of measurement have certain restrictions on how they measure:

• 80° water.• *80° of water.• *Two mile highway.• Two miles of highway.

• What’s the difference here?

Kinds of measurement• Consider measuring the wire. Say we measured a wire

and found it to be 10 inches. If we cut it in half, what’s the measurement of each bit?

• *5 inch wire. 5 inches of wire.• *5 pound sugar. 5 pounds of sugar.

• Consider measuring the water. Say we measured the water and found it to be 80°. If we split it in two, what is the measurement of each part?

• 80° water. *80° of water.• 25 mph wind. *25 mph of wind.

Metrics

• Some metrics are monotonic.

• Monotonic:For a a part of b, where a measures m, and b measures n, m < n.

• Length, volume, weight, depth are like this.• Temperature is not.

• Three miles of highway. Three inches of dirt. Three pounds of sugar. *80° of water. *25 mph of wind.

Hypothesis

• Pseudopartitive (measure of stuff) is possible only when the measure is monotonic.• 2 pounds of sugar. *30° of water.

• Compound (measure stuff) is possible only when the measure is not monotonic.• 30° water. *2 pound sugar.

Monotonicity and parts

• Pseudopartitive—monotonic.

• Twenty pounds of sugar.• Twenty pounds of apples.

• So, weight is monotonic. The pseudopartitive is allowed…

• Twenty pounds of turtle.

• So why does this lead to an unpleasant reading?

Parts and unpleasantness

• It seems that the requirement on the pseudopartitive (monotonicity) not only requires that parts measure less than wholes, but that there are parts.

• 20 pound turtle

• 20 pounds of turtle

Compounds

• Compounds are used when the requirement (monotonicity) is not met…

• …either because the measure is not monotonic• Some 80° water (*80° of water)• A 20 mph wind (*20 mph of wind)

• …or because the noun doesn’t have parts• A two-hour job (*2 hours of job)• A twenty-pound dog (20 pounds of dog)

8 inches of snow

• What does 8 inches of snow mean? What is being measured? Is it monotonic? • Delaware got more 2 inches more rain than Texas.

• What is 1/4 inch wire? What is being measured? Is it monotonic?

• What is 1/4 inch of wire? What is being measured? Is it monotonic?

So

• Moving beyond the simple conception of individuals, we find complex individuals.• Mass nouns• Plural nouns

• Language is sensitive to the abstract property of having parts (divisibility, additivity), e.g., in choosing between the pseudopartitive and compound measure forms.