sentence semantics

Sentence Semantics I

Boredom

• Classes• Meetings• No problems• Playing computer games• Free time

Blue

• Sky• Sea• Music• Soccer team shirt

Bear

• Forest• Zoo• Mountains• Hair

Bad

• Politicians• Global warming• Examinations• Stress

If X then Y

• If you phone me, I will speak to you• If you study hard, you will pass the test• If I get up early, I will be able to go to the zoo• If you put the air conditoner on, it will get

cooler• If you give me money, I will be happy

If X -> Y

• If you phone me, I will speak to you• If you study hard, you will pass the test• If I get up early, I will be able to go to the zoo• If you put the air conditoner on, it will get

cooler• If you give me money, I will be happy

If X -> Y

• If you study hard, then you will pass the test• X = Bob is a student• Y = Mary bought the book• If Bob is a student, Mary bought the book• ????• There doesn’t have to be a logical connection• You can say anything

If X & Y, then Z

• If you buy me the ticket and I have free time, I will go to the theater with you

• If we win and they lose, I will laugh at them• If a bear comes in the classroom and I have a gun,

I will shoot it• If I get the job and the job is well-paid, I will be

happy• If you drink the beer and you drink the whisky,

you will feel bad

If X & Y Z

• If you buy me the ticket and I have free time, I will go to the theater with you

• If we win and they lose, I will laugh at them• If a bear comes in the classroom and I have a gun,

I will shoot it• If I get the job and the job is well-paid, I will be

happy• If you drink the beer and you drink the whisky,

you will feel bad

If X & Y Z

• If Pochi is a dog and you finished your homework, then sushi is delicious.

• X = Pochi is a dog• Y = you finished your homework• Z = sushi is delicious

• ?????

If (X & Y) Z

• If Pochi is a dog and you finished your homework, then sushi is delicious.

• X = Pochi is a dog• Y = you finished your homework• Z = sushi is delicious

• ?????

Why do this?

• Seems completely pointless• Maybe it is!

Propositional Logic

• Crazy name• Crazy idea• It’s actually very simple• So don’t worry about it• Just going to have a quick tour

Words have amazing power

Incredibly complex

Mysterious

Magical

And don’t forget … we don’t know where meaning comes from!

So there’s a problem for Semantics

• We don’t know where meaning comes from• And meaning in words is incredibly

complicated• So just thinking about words, we get lost

Just individual words

Full of information, concepts, meaning

That we cannot really explain

And another problem

Sometimes just word is OK

But usually we speak in sentences

• And sometimes sentences are very long, and keep going on and on without giving very much useful information and you start to lose interest and ……

Even simple sentences …

… contain huge amounts of information

And we cannot …

… really explain …

… at all

I saw a brown bear

• Who is “I”?• Let’s say “I” is an individual called Jim• What does the meaning of “Jim” look like in

your mind?• What does “see” look like in your mind?• etc

Lots of things we cannot explain …

Really … not at all

So how do we begin to understand?

• First step• Simplify• Simplify a lot• So it sometimes seems stupid• And very very very very • Very• Boring

Boredom

Boredom is good!

Interesting is BAD!

Boredom feels secure

Don’t forget … it’s a mystery

Complicated and mysterious

So let’s simplify

• Any sentence = p• Or = q• Or = some other letter, like r for example• It’s not really important

Propositional logic: atomic statements

• Truth values• p = 1• Means: p is true

• Truth values• p = 0• Means: p is false

• p = T• p = F• That is also OK• It’s not really important

• The important thing is …• … talking about TRUTH values.• It’s a theory about true and false.• Any problem with that?• No?• OK.• Prepare to be bored!

Propositional logic: no structure at all

• p = a grizzly is a bear• q = a bear is a mammal• r = a grizzly is a mammal• If p is true• And q is true• Then r is true• p & q r

Sentences have NO structure!

• p = some sentence or other• q = some other sentence or other• r = some other, different, sentence or other• And so on

• Notice they’re supposed to be lower-case letters

• Let’s look at an example of this• A very simple example

• p (some sentence)• r (some sentence)• (if … then connective)

• p r

• What does this mean?• If p is true, then r is also true

• If you are human, then you are a mammal

• If … then connective

• p r• p = you are human• r = you are a mammal

• If p is true• Then r is also true

• p & q r• p = you are Japanese• q = You go to university• r = you can write Kanji

• That’s a commonsense example• But it doesn’t HAVE to be commonsense

• We can have crazy examples if we want

• p & q r• If p is true and q is true, then r is true• p = a salmon is a fish• q = a fish is human• r = a salmon is human

• p is true• q is false• r is false

• p & q r

• What does this mean?• Three sentences p, q, r• It means …• If p and q are both true,• Then r is also true

Propositions

• A fish is human• My teachers are turtles• John’s friend is flying• Her camera is transparent• The Little Prince is standing

Propositions

• Try to keep it simple• Passives are treated as the same as the active

form• John kicked the ball = the ball was kicked by

John

Propositions

• John hit Ben• Ben was hit by John

• Same proposition• call it p• or q• or r• etc

p = Ben is studying

p: Ben is studying

• Ben is studying = True• p = True• p = T• p = 1

p = Ben is studying

p: Ben is studying

• Ben is studying = False• p = False• p = F• p = 0

But maybe he’s studying?!

Maybe, but don’t worry too much

Keep things simple

Imagine simple little worlds

Where everything is either T or F

p = The Little Prince is standing (= T)

q = The Little Prince is gardening (=T)

r = The Little Prince is Playing soccer (= F)

Logical Constants or connectives

• ∧• &• And

• ∨• Or

• →• If … then

Connectives

• ¬• Not• ~• Not• -• Not

Connectives

⇒→If… thenp → qp = you are cleverq = you will studyIf you are clever, you will study

Connectives

⇒→If… thenp → qp = you are cleverq = you will pass the testIf you are clever, you will pass the test

Connectives

⇒→If… thenp → qp = you are studying logicq = you are boredIf you are studying logic, you are bored

Connectives

⇒→If… thenp → qp = you are studying logicq = you hate your teacherIf you are studying logic, you hate your teacher

Connectives

⇒→If… thenp → qp = Eri hates logicq = Eri studies logicIf Eri hates logic then she studies logic

Connectives

• ≡• ⇔• equivalence• if and only if (iff)• p = Jim is a man• q = Jim is an adult male human• p≡q• p q ⇔

Truth tables for p & q

p & q

• p = The weather is fine• q = Tom is sleeping

• Is p & q true?• If both p & q it is true• Not otherwise

Truth tables for p or q

p q

¬ p

p q⇔

Predicate Logic

• P v Q• P or Q

• - P• Not P

• If we know P or Q is true• And we know P is not true• Then Q must be true

• Reasoning

Propositional Logic – too simple

• p• q• No internal structure

John likes Mary

• p = John likes Mary• p• What happened to John and Mary?• And what happened to the verb?• What happened to the meaning?

Predicate Logic

Predicate Logic

Simple sentence (grammar)

Predicate Logic

• Predicate• For example …• Love• Love (x,y)

Predicate

• Love• Two arguments• Love (x, y)• x = john• x = mary• Love (john, mary)

Predicate

• Ken is crazy• What’s the predicate?• Crazy• How many arguments?• One• What is it?• Ken

Ken is crazy

• Predicate• Crazy• One argument• Crazy (x)• x = ?• x = ken• Crazy (ken)

Love

• Two-place predicate• John loves Mary• Loves (john, mary)

Crazy

• One-place predicate• Ken is crazy• Crazy (ken)

How about three-place predicate?

• Send• Ken sent Mary a letter• Send (ken, mary, letter)

Predicate is capital

• Love (x,y)

Arguments are small letters

• Love (john, mary)• Why?• Why not?• Who cares?

Japan is a country

Japan is a country

• Country (x)• x = japan• Country (japan)

Easy so far?

• Yes• But what is the point?• Good question• Don’t think about that!

And it helps with computer programming

Peter fell over

• Predicate?• Fall_over• How many arguments?• One – peter• Fall_over (peter)

Jim donated $100 to the city hospital

• Donated (x,…..n)• Donated (jim, $100, city_hospital)

Ben hates computers

• Hate (ben, computers)

Eri gave up

• Gave_up (eri)

Eri is a genius

• Genius (eri)

Quantifiers

The Little Prince is wearing a brown scarf

But the Little Prince is the only person in this world

Everyone is wearing a brown scarf

Imagine simple little worlds

What do quantifiers mean?

All – upside-down A

Think of sets

• Two sets• Set A • Set B

• Set A = the set of Linguists• Linguist (x)

• Set B = the set of crazy people• Crazy_person (x)

Who is in these sets?

• Linguist (x)• {evans, • imai, • ono}

Who is in the set of crazy people?

• Crazy_person (x)• {ken, • jim, • ben, • mary, • evans}

All linguists are crazy

• ∀x (Linguist (x) Crazy_person (x))• Is this true?

• Set A = the set of Linguists• Linguist (x)

• Set B = the set of crazy people• Crazy_person (x)

Who is in these sets?

• Linguist (x)• {evans, • imai, • ono}

Who is in the set of crazy people?

• Crazy_person (x)• {ken, • jim, • ben, • mary, • evans}

All linguists are crazy

• ∀x (Linguist (x) Crazy_person (x))• Is Untrue• Because two members of the set of linguists

are not in the set of crazy people.

Some linguists are crazy

• Is this True?

Some linguists are crazy

• ∃x (Linguist (x) & Crazy_person (x))• Backward E• Existential Quantifier• There is at least one individual x• x is a linguist• And x is crazy

Who is in the set of crazy people?

• Crazy_person (x)• {ken, • jim, • ben, • mary, • evans}

Evans is in the set of crazy people

• So this is true• ∃x (Linguist (x) & Crazy_person (x))• Backward E• Existential Quantifier• There is at least one individual x• x is a linguist• And x is crazy

Predicate

• Mary is a girl• Girl• Girl (mary)

Predicate

• Mary lives in Tsuru• Lives_in_Tsuru (mary)

Set of girls

• Girl (mary)• Girl (eri)• Girl (rie)

• Girl = {mary, eri, rie}

Set of people who live in Tsuru

• Live_in_tsuru (ben)• Live_in_tsuru (ken)• Live_in_tsuru (mary)

• Live in Tsuru = {ben, ken, mary}

A girl lives in Tsuru

• ∃x • (• Girl (x) • & Lives_in_tsuru (x)• )

• ∃x (Girl (x) & Lives_in_tsuru (x))

Set of girls

• Girl (mary)• Girl (eri)• Girl (rie)

• Girl = {mary, eri, rie}

Set of people who live in Tsuru

• Live_in_tsuru (ben)• Live_in_tsuru (ken)• Live_in_tsuru (mary)

• Live in Tsuru = {ben, ken, mary}

• ∃x (Girl (x) & Lives_in_tsuru (x))

• Is True!

A girl lives in Fujiyoshida

• ∃x • (• Girl (x) • & Lives_in_fujiyoshida (x)• )

• ∃x (Girl (x) & Lives_in_fujiyoshida (x))

Set of girls

• Girl (mary)• Girl (eri)• Girl (rie)

• Girl = {mary, eri, rie}

Set of people who live in Fujiyoshida

• Live_in_tsuru (ben)• Live_in_tsuru (len)• Live_in_tsuru (stan)

• Live in Tsuru = {ben, len, stan}

• ∃x (Girl (x) & Lives_in_fujiyoshida (x))

• Is False!

• But• ¬ x (Girl (x) & Lives_in_fujiyoshida (x))∃• Is true• ~ x (Girl (x) & Lives_in_fujiyoshida (x))∃• Is true• - x (Girl (x) & Lives_in_fujiyoshida (x))∃• Is true

Any problems with Predicate Logic?

• Yes• We are not always trying to say things that are

true

The sky is blue

We don’t say the sky is blue at night

Even though it’s true

• We sometimes say things that are not true• “My brain exploded”

• And do we really think in Logical Form?• ¬ x (Girl (x) & Lives_in_fujiyoshida (x))∃

• And if I say “A girl lives in Fujiyoshida” …• … is it really a statement about existence of an

individual?• Or am I more concerned with number• Or the fact that we’re talking about a girl

rather than a boy?• Or a girl rather than a woman?• Or something else related to CONTEXT?

New ideas about how we understand meaning

Strong evidence for IMAGES rather than CODE

Strong evidence for ACTION simulation

Mental Models as IMAGES

Many people say Logical Form cannot be real

But Logic is VERY important in Linguistics!

• The nurse kissed every child on his birthday.

• [The nurse] kissed [every child] on his birthday.

• Kissed (nurse, every_child)

• ∀x (Child (x) Kissed (nurse, x)) on x’s birthday

• The nurse kissed every child on his birthday

But Logic is VERY important in Linguistics

• What do you think?• Do these words really MOVE in the grammar?• Just because our theory of logical meaning

takes that form?• Or is it completely wrong?• If it is completely wrong …• … maybe you can think of something better.

• In what ways is Predicate Logic superior to Propositional Logic?

• It deals with internal structure

• Which part of a sentence does the Predicate correspond to most closely?

• The verb

• Give an example of a two-place predicate.• Eats • Eats (john, fish)

• Give an example of a three-place predicate.• Gives• Gives (john, mary, the banana)

• How could you represent “Every Linguist is crazy” in Predicate Logic?

• ∀x (Linguist (x) Crazy_person (x))

•

• How could you represent “A boy sent Mary $300” in Predicate Logic?

• ∃x (Boy (x) & Sent (x, mary, $300))

• What does an Upside-down A mean?• Every, All

• What does a backward E mean?• There is • Existence

• What does an upside-down A say about two sets?

• One is contained in the other

• What does a backward E say about two sets?• One intersects with the other

• Do you think this kind of code REALLY plays a part in our thinking?

• Yes• No• We think in pictures• We think in action-images• Or whatever you believe