CAS LX 502CAS LX 502SemanticsSemantics

5a. Plurals, mass, and5a. Plurals, mass, andmeasurementmeasurement

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Singular individualsSingular individuals The Last JurorThe Last Juror is a book. is a book. Cat’s Cradle Cat’s Cradle is a book.is a book. What is Meaning?What is Meaning? is a book. is a book.

[[bookbook]]MM = = xx [ [ x x F F((bookbook) ]) ] FF((bookbook) = {) = {The Last JurorThe Last Juror, , Cat’s CradleCat’s Cradle, , What is Meaning?What is Meaning?}}

FF((The Last JurorThe Last Juror) ) F F((bookbook)) FF((Cat’s CradleCat’s Cradle) ) F F((bookbook)) FF((What is Meaning?What is Meaning?) ) F F((bookbook))

PluralsPlurals The Last Juror The Last Juror and and Cat’s Cradle Cat’s Cradle are books.are books.

The Last Juror The Last Juror and and What is What is Meaning? Meaning? are books.are books.

Cat’s Cradle Cat’s Cradle and and What is Meaning? What is Meaning? are books.are books.

The Last JurorThe Last Juror, , Cat’s CradleCat’s Cradle, and , and What is Meaning? What is Meaning? are books.are books.

So, So, somethingsomething F F((booksbooks).). The question is: what is in The question is: what is in FF((booksbooks)?)?‘What is the denotation of ‘What is the denotation of booksbooks?’?’

CollectionsCollections Intuitively, Intuitively, The Last JurorThe Last Juror and and What is What is Meaning?Meaning? is something like a collection is something like a collection of individuals (each a book).of individuals (each a book). A natural thing to suppose is that A natural thing to suppose is that FF((booksbooks) ) contains collections of individuals—all of contains collections of individuals—all of those collections such that each member is in those collections such that each member is in FF((bookbook).).

FF((booksbooks) = ) = {{ { {The Last JurorThe Last Juror, , Cat’s Cat’s CradleCradle},},

{{The Last JurorThe Last Juror, , What is Meaning?What is Meaning?},},{{Cat’s CradleCat’s Cradle, , What is Meaning?What is Meaning?},},{{The Last JurorThe Last Juror, , Cat’s CradleCat’s Cradle, , What What

is Meaning?is Meaning?} } }}

NPNP11 and NP and NP22 creates a collection {[NP creates a collection {[NP11]]MM, , [NP[NP22]]MM}}

The structure of The structure of pluralsplurals

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

A+B+C

A+B A+C B+C

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

F(books)

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

The Last Juror The Last Juror and and Cat’s Cradle Cat’s Cradle are are boring.boring. Notice that Notice that agreementagreement registers whether we’re registers whether we’re looking at a plural or singular noun phrase looking at a plural or singular noun phrase here.here.

Under what conditions is this true?Under what conditions is this true?

DistributivityDistributivity Notice that the way we’ve been thinking Notice that the way we’ve been thinking of the definition of of the definition of boringboring from the from the formal fragment of English, we do not formal fragment of English, we do not predict this:predict this: [[boringboring]]MM = = xx [ [ x x F F((boringboring) ]) ]

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

Moreover, we probably Moreover, we probably don’tdon’t want that. want that.

DistributivityDistributivity

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

The predicate The predicate distributesdistributes through the through the collection, applying to each member.collection, applying to each member.

Predictable things should be in the Predictable things should be in the grammar.grammar.

Plural predicatesPlural predicates The standard way to think about this is to The standard way to think about this is to suppose that for every predicate like suppose that for every predicate like boringboring there is a there is a plural predicateplural predicate, notated , notated boringboring, that holds of collections:, that holds of collections: can be viewed as a function from singular can be viewed as a function from singular predicates to plural predicates:predicates to plural predicates:

[[]]MM = = PP [ [ CC [ [ xx[ [ xxCC PP((xx) ] ] ]) ] ] ]

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

This works fabulously.This works fabulously.

Loren and Bond are tall.Loren and Bond are tall. Nemo and Dory are fish.Nemo and Dory are fish. It is not the case that Loren It is not the case that Loren and Nemo are fish.and Nemo are fish.

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

Er…Er… Loren and Bond bought a house.Loren and Bond bought a house. Pavarotti and Bond lifted a Pavarotti and Bond lifted a piano.piano.

The men gathered.The men gathered. Loren and Pavarotti are similar.Loren and Pavarotti are similar.

The meaning here is does The meaning here is does notnot seem to be distributive.seem to be distributive.

GroupsGroups A collection of individuals can be viewed A collection of individuals can be viewed either either distributively distributively (as individuals) or (as individuals) or collectivelycollectively (as a group). (as a group). The students gathered.The students gathered. The students have blue eyes.The students have blue eyes.

We even have names for some groups: We even have names for some groups: committeecommittee, , juryjury, , senatesenate, …, … The jury gathered.The jury gathered.

Note that these are Note that these are morphologically morphologically singularsingular even though they are even though they are semanticallysemantically plural plural in some sense.in some sense.

Collections of groupsCollections of groups The animals filled the barn to capacity.The animals filled the barn to capacity. The pigs and the cows filled the barn to The pigs and the cows filled the barn to capacity.capacity.

The pigs and the cows argue.The pigs and the cows argue. The boys and the girls touched the The boys and the girls touched the ceiling.ceiling.

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

Plurals and part-whole Plurals and part-whole structurestructure

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

Additivity:Additivity: If CIf C11 F F((booksbooks), C), C22 F F((booksbooks), then), thenCC11 and C and C22 F F((booksbooks))

Books and books are books.Books and books are books.

Mass nounsMass nouns Certain kinds of nouns, Certain kinds of nouns, mass nounsmass nouns, , also show this kind of part also show this kind of part structure (unlike structure (unlike count nounscount nouns))

SoupSoup and and soupsoup is is soupsoup.. Parts of Parts of soupsoup are are soupsoup.. A chairA chair and and a chaira chair is not is not a chaira chair.. Parts of Parts of a chaira chair are not are not a chaira chair..

Distinguishing mass and Distinguishing mass and countcount

John bought four books.John bought four books. *Mary bought four barley(s).*Mary bought four barley(s). Each chair is brown.Each chair is brown. *Each furniture is brown.*Each furniture is brown. *John doesn’t have much *John doesn’t have much chair(s).chair(s).

Mary doesn’t have much Mary doesn’t have much furniture.furniture.

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

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

Making mass nouns Making mass nouns countable:countable:

How do you count soup?How do you count soup? I have two gallons of soup.I have two gallons of soup. I have two pounds of soap.I have two pounds of soap. I have three pieces of furniture.I have three pieces of furniture.

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

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

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

Pseudopartitive: Pseudopartitive: One quarter inch of One quarter inch of wirewire

Compound:Compound: One quarter inch wire One quarter inch wire

What do these refer to?What do these refer to? How do we know that?How do we know that?

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

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

What’s the difference here?What’s the difference here?

Kinds of measurementKinds of measurement Consider measuring the wire. Say we Consider measuring the wire. Say we measured a wire and found it to be measured a wire and found it to be 10 inches. If we cut it in half, 10 inches. If we cut it in half, what’s the measurement of each bit?what’s the measurement of each bit? *5 inch wire. 5 inches of wire.*5 inch wire. 5 inches of wire. *5 pound sugar. 5 pounds of sugar.*5 pound sugar. 5 pounds of sugar.

Consider measuring the water. Say we Consider measuring the water. Say we measured the water and found it to measured the water and found it to be 80°. If we split it in two, what be 80°. If we split it in two, what is the measurement of each part?is the measurement of each part? 80° water. *80° of water.80° water. *80° of water. 25 mph wind. *25 mph of wind.25 mph wind. *25 mph of wind.

MetricsMetrics Some metrics are Some metrics are monotonicmonotonic..

Monotonic:Monotonic:For For aa a part of a part of bb, where , where aa measures measures mm, and , and bb measures measures nn, , mm < < nn..

Length, volume, weight, depth are like Length, volume, weight, depth are like this.this.

Temperature is not.Temperature is not. Three miles of highway. Three inches of dirt. Three miles of highway. Three inches of dirt. Three pounds of sugar. *80° of water. *25 mph of Three pounds of sugar. *80° of water. *25 mph of wind.wind.

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

Compound (Compound (measure stuffmeasure stuff) is ) is possible only when the measure possible only when the measure is is notnot monotonic. monotonic. 30° water. *2 pound sugar.30° water. *2 pound sugar.

Monotonicity and partsMonotonicity and parts

Pseudopartitive—monotonic.Pseudopartitive—monotonic.

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

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

Twenty pounds of turtle.Twenty pounds of turtle. So why does this lead to an unpleasant So why does this lead to an unpleasant reading?reading?

Parts and Parts and unpleasantnessunpleasantness

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

20 pound turtle20 pound turtle 20 pounds of turtle20 pounds of turtle

CompoundsCompounds Compounds are used when the requirement Compounds are used when the requirement (monotonicity) is (monotonicity) is notnot met… met…

……either because the measure is not either because the measure is not monotonicmonotonic Some 80° waterSome 80° water (*80° of water)(*80° of water) A 20 mph windA 20 mph wind (*20 mph of wind)(*20 mph of wind)

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

8 inches of snow8 inches of snow What does What does 8 inches of snow8 inches of snow mean? What mean? What is being measured? Is it monotonic? is being measured? Is it monotonic? Delaware got more 2 inches more rain than Delaware got more 2 inches more rain than Texas.Texas.

What is What is 1/4 inch wire1/4 inch wire? What is being ? What is being measured? Is it monotonic?measured? Is it monotonic?

What is What is 1/4 inch of wire1/4 inch of wire? What is ? What is being measured? Is it monotonic?being measured? Is it monotonic?

SoSo Moving beyond the simple conception Moving beyond the simple conception of individuals, we find complex of individuals, we find complex individuals.individuals. Mass nounsMass nouns Plural nounsPlural nouns

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