Lecture 7: Structure and Meaning August 27,...

Lecture 7: Structure and Meaning

August 27, 2012

Lecture 7: Structure and Meaning Last Time

Review Questions‚ What does syntactic knowledge describe?

‚ How can we represent syntactic knowledge?

‚ How do we test syntactic knowledge?

‚ Phrase Structure Rules (p150):

§ S Ñ NP VP

§ NP Ñ Det N'

§ Det Ñ NP poss

§ NP Ñ N'

§ NP Ñ NP PP

§ N' Ñ Adj N'

§ N' Ñ N

§ VP Ñ V

§ VP Ñ V NP

§ VP Ñ V CP

§ VP Ñ Aux VP

§ VP Ñ VP PP

§ PP Ñ P NP

§ CP Ñ C S

§ Is this list of PSRs completely adequate for describing all

of English?

§ Why not?

2

Lecture 7: Structure and Meaning Last Time

Review Questions‚ What do transformational rules apply to?

‚ What transformational rules have we seen?

‚ How do we draw trees for sentences that we knowhave undergone transformations?

§ Step 1: Figure out what the deep structure must havebeen (undo the transformation)

‚ Answer the question with a complete sentenceDid you finish your homework? – Yes, I finished myhomework.What will you eat? – I will eat some pie.

§ Step 2: Draw the tree for the deep structure

§ Step 3: Apply the transformational rules to the deep

structure

‚ Try it out: When will you see a concert at theHollywood Bowl?

3

Lecture 7: Structure and Meaning Today

Questions for Today‚ What kind of syntax is there across languages?

§ Mostly the same (from UG principles), but with some

controlled variation (from UG parameters)

‚ Why do sentences mean what they mean?

§ They build meaning through rules applied to deep

structures

‚ How is meaning represented?

§ Logical symbols in the brain!

‚ What do sentences mean?

§ Truth values! (mostly)

‚ How can sentences be related in meaning?

§ Entailment and presupposition5

Lecture 7: Structure and Meaning Briefly: Syntactic Typology

(In-)variation‚ So we’ve been talking mostly about English – what

about other languages?

§ Do all languages use the same syntactic rules?

§ Are there any syntactic constants across languages?

§ What kinds of rules do we never see?

‚ The answers to these questions lie in whateverunderpins all languages

§ Universal Grammar!

7


Principles and Parameters‚ Universal Grammar (UG) is, as we’ve said before, a

kind of master-plan blueprint for languages

§ Some things never vary from house-to-house

‚ There are foundations to support the building, walls toseparate rooms, floors to walk on, doors to leave thebuilding, etc.

§ These are principles – like how all languages sharemany properties

‚ What are some principles of language that we’vealready discussed?

‚ But of course, there are many variations too

8


Principles and Parameters‚ Both buildings and languages can vary

§ What are the walls made of? What covers the floor?

What is the building material? Etc.

§ These are parameters

‚ There are a limited number of ways in which you canvalue a parameter

‚ E.g. you can’t build floors out of pine needles

‚ Let’s look at some linguistic varianceTokyo kara densha de ichijikan kurai nishi e ikebaTokyo from train by 1-hour about (the) west towards if you go1 2 3 4 5 6 7 8 9

Kamakura to iu machi-ga aruKamakura called (a) town exists10 11 12 13

§ Now read the gloss backwards!9


Parametric Variation in PSRs‚ To account for this kind of thing, Japanese PSRs must

be different

‚ Verb and Object:

§ Japanese:VP Ñ NP V

§ English: VP Ñ V NP

‚ Pre-/Post-positions:

§ Japanese:PP Ñ NP P

§ English: PP Ñ P NP

‚ No determiners:

§ Japanese:NP Ñ N'

§ English: NP Ñ D N'

‚ These PSRs are the result of different parameter settings

§ Order of constituents (head-final, head-initial)

‚ Japanese XPs are mostly head-final, and English XPs aremostly head-initial (hence opposite orders in somesentences)

§ Presence of constituents 10


Principles in PSRs‚ Different combinations of these (and other)

parameters will give you different languages

§ But which combinations of PSRs make good languages

seems to be constrained

§ remember: no language has the following two rules

VP Ñ V NP & PP Ñ NP P

‚ Though the parameter settings affect the exactformulation of these PSRs...

§ The rules are essentially the same

§ The principle parts of PSRs don’t vary

§ The rules are still hierarchical and binary

‚ These properties are invariant, and likely representsomething about UG

11


Parametric Variation in Transformations‚ In languages like English, Vata (West Africa), Irish

Gaelic, and Dutch, WH phrases move to the front of asentence by a transformational rule

§ Who did Sally kiss ?

§ Vata: (SVO)

yIwhat

KòfíKofi

leeat

lá??

‘What is Kofi eating?’

§ Irish: (VSO)

Cad-aWhat

chonaicsaw

siadthey

??

‘What did they see?’

§ Dutch: (SOV)

WatWhat

gawill

jeyou

eten?eat?

‘What will you eat?’ 12


Parametric Variation in Transformations‚ However, in many languages like Chinese, Tongan

(South Pacific), and Turkish, WH phrases do not move

§ Chinese: (SVO)

LisiLisi

mai-lebought

shenmewhat

ne??

‘What did Lisi buy?’

§ Tongan: (VSO)

Na’epast

fakataataadraw

‘e SioneSione

‘ae haa?what?

‘What did Sione draw?’

§ Turkish: (SOV)

HasanHasan

newhat

yapti?did?

‘What did Hasan do?’

13


Parametric Variation in Transformations‚ There is a parameter whether or not wh-words undergo

a transformational rule and move

§ Languages where they don’t move are called

wh-in-situ languages

‚ There is still variation within this parameter

§ Russian and Serbo-Croatian (unlike English), for

example, allow more than one WH-phrase to move

§ Tongan (unlike Chinese), for example, allows

WH-phrases to move in some cases

14


Principles in Transformation‚ Despite there being variation in the application of

transformational rules, some logically possible uses ofrules are disallowed in all languages

§ These must be due to UG principles!

§ For example...

‚ Transformations are always applied to constituents

‚ WH-words in some contexts are impossible to move, inany language

‚ You can’t passivize an already passive sentence

§ Let’s look at some examples of the last two

15


Principles in Transformation‚ Coordinate Structure Constraint: it is impossible to

apply the WH-phase movement transformation rule toa part of a conjunction

a. You saw Liz and Jack together? (rising intonation)

b. You saw Liz and who together? (rising intonation)

c. Who did you see together? (falling intonation)

d. ˚ Who did you see Liz and together?

(falling/rising intonation)

e. Who did you see Liz with ? (falling intonation)

f. ˚ Who did you see and Jack together?

(falling/rising intonation)

g. Who did you see with Jack? (falling intonation)

16


Principles in Transformation‚ One Advancement Exclusiveness Law (1AEX,

Perlmutter and Postal 1984): it is impossible to apply thepassive transformation rule more than once to thesame sentence

a. Mom cooked beef.

b. Beef was cooked by mom.

c. Mom cooked on the BBQ.

d. The BBQ was cooked on by mom.

e. Mom cooked beef on the BBQ.

f. ˚ The BBQ was cooked on by mom by beef.

17


Typological Review‚ Not all languages have the same PSRs or

transformational rules

§ But the ways in which those rules vary from language tolanguage is constrained – by UG

‚ Absence/presence of a rule, ordering of rules, etc.

‚ This is parametric variation

§ Some aspects of language are universal

‚ PSRs are always binary and hierarchical, transformationsalways target constituents, etc.

‚ These are principles of language

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Lecture 7: Structure and Meaning Introducing... Semantics

Introducing... Semantics‚ Semantics is the linguistic system that determines

meaning for words, phrases, and sentences

§ What does “kick” mean?

§ What does “swiftly kick” mean, and how does that

meaning relate to the meaning of “kick”?

‚ More specifically, semantics relates to thetruth-conditions that determine meaning

§ The questions investigated are more like: “what does it

take for something to be classified as a true ‘swift

kicking’ event?”

§ (There is also non-truth-conditional meaning [inferences

and so on] which we return to shortly)

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Truth Conditional Semantics‚ We will be discussing truth conditional semantics

§ The main idea is that speakers know what needs to bethe case for a sentence to be true – even though theymay not know if it is true or not at a given time

‚ Truth conditions!

§ Just like with syntax, someone can utter a sentence youhave never heard before and you know what it means

‚ Because you know the semantic rules that determinetruth conditions

‚ For example, imagine Angie is someone you havenever heard of before

§ If Devon tells you “Angie has a 41-year-old husband”

you know what it takes to determine if Devon is lying or

not

21


Truth Conditional Semantics‚ If we go to the hall of records and find out that...

§ ...she has a husband born 41.5 years ago the sentence

is true

§ ...she got married to a man born 41.5 years ago, but

they are now divorced, the sentence is false

§ ...she got married to a man born 39 years ago, the

sentence is false

§ ...she has never been married to a man, the sentence is

false

‚ In other words, we intuitively know that...

§ she must have married a man

§ she must still be married to that man currently

§ that man must have been born between 41 and 42

years ago 22


Truth Conditional Semantics‚ The list that we check are the truth conditions

§ These are the things we intuitively know when we know

the meaning of a sentence

§ The ability to calculate the truth conditional semantics

of a sentence does not rely on reality

‚ The result that we find is the truth value

§ This must be investigated by looking at the current state

of affairs in the world and checking that state against

the conditions we laid out

‚ What is the truth conditional semantics of...

§ “Byron has 5 sisters”?

§ or “Some scientists might create a clone of me.”

23


Constant Truth Conditions‚ Some sentences do not depend on situational factors

in the real world in order to be interpreted and to betrue or false

§ For example “Computers are computers” or “All

copyrights are the property of their respective owners”

‚ Are any real-world situations in which these are nottrue?

§ No. These sentences are tautologies: sentences thatare always true solely because of the terms involved

‚ There is no situation in any possible world that wouldmake it false

§ Some more tautologies:

‚ A married man is not single.

‚ When you know, you know.

‚ Cake-lovers love cakes. 24


Constant Truth Conditions‚ Now consider the following:

§ “Books are Computers”, “All green pants are colorless”

‚ Are any real-world situations in which these are true?

§ No. These sentences are contradictions: sentences thatare NEVER true solely because of the terms involved

‚ There is no situation in any possible world that wouldmake it true

§ Some more contradictions:

‚ Few married man have spouses.

‚ If you hate all desserts, then you love pies.

‚ That man is sleepy and not sleepy.

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Types of Sentences‚ Three types of sentences:

§ Situationally true/false

‚ The sentence is true or false based on what is true in theworld

‚ These are the sentences that we are interested in

§ Tautology

‚ The sentence is always true in every possible situation

§ Contradiction

‚ The sentence is always false in every possible situation

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Lecture 7: Structure and Meaning Semantic Compositionality

Syntax and Meaning‚ Syntactic and morphological constituent structures are

reflective of meaning

§ But why?

§ The answer lies in the principle of compositionality

‚ Basic idea:Semantic rules derive the meaning of a phrasefrom the meaning of its parts, based on rules that relyon syntactic constituency

§ That’s why we give different trees for the two meaningsof “Ken heard that his pig died yesterday”

‚ What is the basic constituency difference?

‚ “Ken heard [CP that his pig died yesterday]”“Ken heard [CP that his pig died] yesterday”

‚ This is why sentences with the same syntactic structuremean such similar things

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Syntax and MeaningS

VP

NP

N'

N

N

catsbuffalo

N

TexasBuffalo

V

intimidatebuffalo

NP

N'

CP

S

VP

V

intimidatebuffalo

NP

N'

N

N

catsbuffalo

N

TexasBuffalo

C

thatØ

N'

N

N

dogsbuffalo

N

LABuffalo

29


The Meaning of Words‚ If a phrase’s meaning comes from the meaning of its

parts, what is the meaning of its parts?

§ This brings us to the notion of lexical meaning

§ The basic units of meaning are stored as (arbitrary)form-meaning pairs

‚ That is, the meaning of a (monomorphemic) word israndom and must be memorized

§ Consider another kind of ambiguity: lexical ambiguity

‚ Jack never goes to the bank.bank = financial institution? side of the river?

‚ Form does not dictate lexical meaning – lexicalmeaning is independent from form

§ In fact: language and thought are not co-dependent

(though language does depend on thought)

§ When you get hungry, do you think “I’m hungry”? No! 30


The Meaning of Phrases‚ But form does dictate compositional meaning

§ Why does “dogs chase cats” not mean the same thing

as “cats chase dogs”?

§ They use the same words...

§ But they have different constituencies!

‚ If ‘chase’ and ‘dogs’ forms a constituent first themeaning is entirely different from if ‘chase’ and ‘cats’forms a constituent first

§ In fact, the word order for the meaning “dogs chasecats” in some languages (e.g. Hixkaryana) would be“cats chase dogs”

‚ In all languages, though, to get the meaning wheredogs are doing chasing and cats are being chased,‘chase’ and ‘cats’ forms a constituent first

31


Language and Logic‚ But... why?

§ Semantic rules for how to interpret phrases act upon

syntactic constituents

§ Just like syntactic transformation rules act upon

syntactic constituents

‚ But before we get to that, let’s reminder ourselves thatlanguage is an object that is logical in a mathematicalway

§ Sentences have truth values, remember? Truth is a

logical mathematical object

§ So we use logical terms in describing meaning and

meaningful relationships between constituents

32


Language and Logic: Function Words‚ Conjunction (using ‘and’) and disjunction (using ‘or’)

are mathematical operations: ^

§ [S1 It will rain in two weeks] and [S2 raptors were purple]

‚ Even without knowing whether S1 and S2 are true ornot, I know the truth conditions of the conjoinedsentence because I know the rule on conjunction

§ Same for disjunction:

[S1 It will rain in two weeks] or [S2 raptors were purple]

‚ And Rule:“S1 and S2” “ S1 ^ S2

S1 S2 S1 ^ S2

T T TT F FF T FF F F

Or Rule:“S1 or S2” “ S1 _ S2

S1 S2 S1 _ S2

T T TT F TF T TF F F 33


Language and Logic: Function Words‚ Of course there are many other operations in

language other than “^” and “_”

§ The most complex meanings are often in the little tiny

function words

§ The issue of what the means is still researched

‚ It is interesting to note that whatever meaning(s) it has,the differs in the exact meaning from le in French, el inSpanish, das in German, etc. etc.

§ Many semanticists and syntacticians consider the

meaning of function words to be important, and is yet

to be fully understood

‚ Of course, outside of function words, we have contentwords

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Language and Logic: Adjectives‚ Some content words/phrases denote sets of things

§ For example, nouns and adjectives each denote theset of things that are that specific noun/adjective

‚ “purple” means the set of things that are purple

‚ “raptor” means the set of things that are raptors

§ Let’s add some more rules to that

‚ N'Rule: [N'] = {things that are N'}

‚ Adj Rule: [Adj] = {things that are Adj}

‚ Adj N'Rule: [Adj N'] = [Adj] X [N']

N' Adj

35


Language and Logic: Adjectives‚ Let’s run through examples with the following words

§ [A hungry], [A purple], [N bananas], [N raptors]

‚ ‘hungry purple raptors’

raptors purple

hungry

N'

N'

N'

N

raptors

Adj

purple

Adj

hungry

{ {purple} X {raptors} } X {hungry}

§ What about “purple bananas” and “hungry bananas”?

‚ What’s wrong with “hungry bananas”?

36


Language and Logic: Adjectives‚ Note: the meaning is determined by the constituency

§ This is exactly what we called compositionality

‚ Semantic rules derive the meaning of a phrase from themeaning of its parts, based on rules that rely onsyntactic constituency

§ The rules we relied on here:

‚ N'Rule: [N'] = {things that are N'}

‚ Adj Rule: [Adj] = {things that are Adj}

‚ Adj N'Rule: [Adj N'] = [Adj] X [N']

‚ Puzzle: does “big” meaningfully combine with N's usingthe same rules as “purple”?

§ Why/why not? (Hint: “big mosquito”)

§ Some adjectives are non-intersective

‚ They use a different rule which also relies on acontextually determined ‘standard of comparison’ 37


Language and Logic: Verbs‚ How do we get closer to understanding the meaning

of a sentence?

§ Let’s start by looking at the PSR for sentences:

S Ñ NP VP

§ Whatever the semantic rule is, it should relate the

subject and the predicate

§ A VP denotes the set of things that {do the action/arein the state} described by the VP

‚ “demolish bananas” refers to the set of things thatdemolish bananas

‚ “sing” refers to the set of things that sing

§ More specifically...

‚ VP Rule: [VP] “ {things that VP}

‚ Intrans V Rule: [V] “ {things that V}

‚ Trans V Rule: [VP] “ {things that V {V’ed things}}38


Language and Logic: Sentences‚ So like the other things we’ve seen so far we’re

creating sets

§ [VP sing] = {things that sing}

§ [VP demolish purple bananas]

= {things that demolish [NP purple bananas]}

= {things that demolish [{things that are purple } X

{things that are bananas}]}

‚ Finally, what semantic rule combines NPs and VPs?

§ S Rule: [S] “ True iff {things that are NP} P {things that VP}

‚ The truth conditions of a given sentence are the result ofapplying this rule to the specific sentence

‚ The truth value of this sentence is the result of checkingthe state of affairs against the truth conditions

39


Language and Logic: Sentences‚ S Rule: [S] “ T iff {things that are NP} P {things that VP}

§ [S [NP hungry raptors] [VP sing]]

‚ The truth conditions of “hungry raptors sing” are that it istrue iff [the intersection of the set of things that areraptors and the set of things that are hungry] is amember of the set of things that sing

‚ The truth value of this sentence ... is unknown to me

§ [S [NP raptors] [VP sing]]

‚ The truth conditions of “raptors sing” are that it is true iffthe set of things that are raptors is a member of the setof things that sing

‚ The truth value of this sentence is also unknown to me

‚ But I do know that it is true if “hungry raptors sing” is true

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Lecture 7: Structure and Meaning More On Sentential Logic

Entailment and Presupposition‚ Sometimes sentences are meaningfully related in ways

that might affect grammatical properties

§ The two ways we will investigate now are entailmentand presupposition

§ These relationships only exist between sentences – they

have no meaning as applied to a single sentence

§ Consider the following two sentences

‚ S1 = Liz ate a snack.

‚ S2 = Liz at a cheesy snack.

§ S1 refers to all situations in which Liz was a member ofthe set of things that was eating a snack

‚ Imagine there are six times in her life she’s eaten asnack, and three of them were cheesy snacks

42


Entailment and Presupposition‚ The two sentences:

§ S1 = Liz ate a snack.

§ S2 = Liz at a cheesy snack.

‚ Every time Liz had a cheesy snack, she was having asnack:

S1S2

‚ Since S2 is only true when S1 is also true, S2 entails S1

§ S2 entails S1: if S2 is true, S1 must also be true43


Entailment and Presupposition‚ A sentence X entails another sentence Y if and only if:

every time X is true, Y MUST be true

§ In other words, X entails Y if [the set of situations where X

is true] is a subset of [the set of situations where Y is true]

§ If you can find even one situation where X is true & Y is

false, X does NOT entail Y

Y is trueX is true

X entails YPOSSIBLE IMPOSSIBLE

X=T, Y=T X=T, Y=F

X=F, Y=F

X=F, Y=T

44


Entailment and Presupposition‚ Let’s practice with entailment: does one sentence

entail another? If so, which entails which?

§ S3: Liz read a bookS4: Liz read a dictionary

‚ S4 entails S3 – if Liz is reading a dictionary, she is readinga book

§ S5: Less than 3 guests came to the party.S6: Less than 3 guests came to the party early.

‚ S5 entails S6 – if ă3 people came, it must be that ă3people came early

§ S7: This one guy ate a banana every day.S8: Every guy ate a banana this one day.

‚ There is no entailment – they are entirely independentsituations

45


Entailment and Presupposition‚ When an entailment is false, the original statement is

also false

§ S9: Ken: “Liz read a dictionary.”S10: Jack: “No, she didn’t read a book. She read anewspaper.”

‚ S9 entails that Liz read a book

‚ S10 says that she didn’t read a book, meaning shecould not have read a dictionary

‚ Whenever an entailment is denied, the originalstatement is denied

§ S11: Ken: “Less than 3 guests came to the party.”S12: Jack: “No, 30 guests came early.”

‚ S11 entails that ă 3 guests came early

‚ S12 says 30 guests came late

‚ If Jack is right about S12, Ken must be wrong on S11

46


Entailment and Presupposition‚ Presupposition is a bit like entailment: presuppositions

must be true when the original sentence is true

§ S13: Mike’s dog is sick.S14: Mike has a dog.

‚ Every time S13 is true, S14 must be true

§ However, S13 does not entail S14 – S14 is not anentailment of S13

‚ When S14 is false, it isn’t true that S13 is false

‚ Instead you have to say “hey wait a minute!” to S13

‚ S13 presupposes S14

§ S15: The king of the U.S.A. is fat.S16: The U.S.A. has a king.

‚ Every time S15 is true, S16 must be true

‚ But if someone said S15, you’d say “hey wait a minute!”



Entailment and Presupposition‚ Some words can trigger presuppositions when added

to a sentence – these are called presuppositiontriggers

§ S17: Liz has another boyfriend.S18: Liz had a previous boyfriend.


§ S19: Liz has a boyfriend.

‚ S19 has no presupposition, but S17 does

‚ The word another is a presupposition trigger

§ S20: Jack used to smoke.S21: Jack stopped smoking.

‚ S21 presupposes S20, and stop is the trigger

§ S22: I’m tired.S23: I didn’t sleep for a long time.

‚ No presupposition or entailment48


Entailment and Presupposition‚ Another way to distinguish entailments and

presuppositions – presuppositions remain if you changethe original sentence into a question or negate it

§ S3: Liz read a book

S4: Liz read a dictionary.

S41: Did Liz read a dictionary?S42: Liz didn’t read a dictionary.

‚ S4 entails S3, but i n the question and negation of S4 (S41

and S42), the entailments disappear

§ S13: Mike’s dog is sick.

S14: Mike has a dog.

S141: Is Mike’s dog sick?S142: Mike’s dog isn’t sick.

‚ S13 presupposes S14, and in the question and negationof S13 (S131 and S132), the presuppositions remain

49


Truth Values N/A‚ Truth values can only be evaluated for declarative

sentences

§ Declaratives are basically “statements”

‚ Questions do not have truth values

§ They request information

§ You are familiar with them already

50


Truth Values N/A‚ Sentences like imperatives and performatives do not

have truth values

§ These are speech acts

‚ They elicit action in the world around the speaker

§ Performatives are things like “I hereby declare youguilty”

‚ The act of saying the sentence makes it true

‚ They can usually have “hereby” added

‚ Note that it is funny to say “I am hereby insulting you”,because just saying you are insulting someone does notmean you are insulting them

§ Imperatives are commands like “Go to your room”

‚ The sentence expresses what the speaker wants tomake happen in the world

‚ Imperatives (usually) lack pronounced subjects

51

Lecture 7: Structure and Meaning Random Note

“Semantic Transformation”‚ We have seen that semantics has rules for building up

meaning

§ These are much like PSRs, except the build meaning

based on existing syntactic structure

‚ You may wonder then if there are semantictransformational rules

§ Depending on how you shape your theory there might

be a couple examples

§ Let’s look at one example

53


“Semantic Transformation”‚ To understand this “semantic transformation” you must

first understand the difference between mass nounsand count nouns

§ A mass noun is one that can appear to be singular(and have a singular verb form) but which does notneed to have a determiner in English

‚ Additionally, mass nouns refer to “masses” and notindividual units of something

‚ e.g. Soda is carbonated.

§ A count noun is one that needs to have a determiner inEnglish when it is singular

‚ Additionally, they can be counted

‚ e.g. ˚Brick is red.

54


“Semantic Transformation”‚ However, though these words are semantically

specified as either mass (e.g. soda) or count (e.g.brick), they can be transformed

‚ The first transformational process is called the universalgrinder

§ It takes count nouns and turns them into mass nouns

‚ Brick is all over the room.

§ You can imagine that whatever “brick” refers to has to

be a substance that has been broken up into a mass

‚ Conversely, there is the universal packager

§ It takes mass nouns and turns them into count nouns

‚ A soda would be great.

§ You can imagine that whatever “soda” refers to now is

an individual unit of soda (a can/bottle/glass) 55


Review‚ There are constant principles across languages, as well

as parametric variation

§ Principles: every PSR is hierarchical/binary, every

transformational rule targets a constituent, etc.

§ Parameters: ordering/presence of constituents in the

PSRs, whether or not a transformational rule is used by a

language, etc.

‚ Semantics is the system that determines meaning ofwords/phrases/sentences

§ Knowing meaning of sentences means knowing the

truth conditions

§ Some sentences are always true or always false

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Review‚ Meaning of a phrase is built upon (i) the meaning of its

parts, (ii) syntactic constituency, and (iii) the rules forcombining those parts

‚ Language can be represented with logical expressions

§ _, X, sets, etc.

‚ Sentences and Truth Values

§ A sentence X entails a sentence Y if Y is true whenever

X is true

§ A sentence X presupposes a sentence Y if Y must be

true for X to be true

§ Non-declarative sentences do not have truth values

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