Semantics
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•
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Truth
Semantic Relations
o Entailment
o Mutual entailment
o Tautology
o Contradiction
Set Theory
Truth
•
What does it mean to know the meaning of a sentence?
Consider the following situation:
Based on Situation A, which of the sentences are True and which are False?
1.
2.
3.
4.
John saw Susan.
A person saw another person.
A woman saw a man.
A man saw himself.
Every declarative sentence has a Truth Value.
Truth Value = {True, False}
Now consider the following sentence.
5. John loves Mary.
•
•
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Is (5) True or False?
Can we find conditions in which (1) is True or False?
1!
For some linguists/philosophers, to know the meaning of a sentence is to know its Truth
Conditions. This is called Truth-Conditional Semantics, which was first used by Donald
Davidson.
•
•
Native speakers know the truth conditions of a sentence intuitively.
We will try to derive the truth conditions of sentences compositionally. In other
words, truth conditions of a sentence is determined by its parts.
Semantic Relations
Sentences can have a number of relations among themselves. Given two sentences, we
can detect some relations between the two sentences.
Entailment
• Given two sentences p and q,
o Whenever p is true, if q is also true then p entails q.
6.
p: John loves Mary.
q: A man loves a woman.
In (6), p entails q.
7.
p: John loves Mary.
q: A man loves Mary.
In (7), p entails q.
8.
p: John loves Mary.
q: Mary loves John.
In (8), p does not entail q.
Note that entailment is not a bi-directional relation.
9.
p: John loves Mary.
q: A man loves a woman.
In (9), p entails q, but q does not entail p.
In order to determine whether a sentence entails another sentence, we need to consider
every possible situation. In other words, we should try to find a situation where p is True
but q is False.
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2!
Exercises
Determine whether p sentences entail q sentences in the examples below.
10
p: Patrick is a smart boy.
q: Patrick is a boy.
11.
p: Susan is taller than Mary.
q: Susan is tall.
12.
p: If Sheila works hard, she will be a millionaire.
q: If Sheila is a millionaire, she works hard.
Mutual entailment
In cases where p entails q and q entails p, we get mutual entailment.
13.
p: John loves Mary.
q: John is in love with Mary.
Tautology
Certain sentences are always True, no matter what the situation is. These are usually
general facts, or weird repetitions. There is no way that they can be False.
14. All dinosaurs are dinosaurs.
15. John is either at home or not.
Contradiction
Certain sentences are always False. There is no way they can be True.
16. Mary saw Bill but Mary didn’t see Bill.
17. John is both at home and not at home.
Exercises
Which of the following are tautologies and which of them are contradictions. They may
be neither.
1. The morning star is the morning star.
2. The morning star is the evening star.
3. Mary told John something but Mary didn’t say anything.
4. John is a bachelor and he is an unmarried adult male.
5. John is a bachelor and he is married.
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3!
Set Theory
A set is an unordered collection of individuals.
Sets can be represented by
Braces (curly brackets)
OR
Venn diagram
A={a, b, c}
A
a b!
c!
For any given set A, an individual can be either in the set A or not.
• If an individual x is in the set A, we say: x is a member of A and represent it by
x A.
• If an individual x is not in the set A, we say: x is not a member of A and represent
it by x ∉A.
Set$Relations$
•
Consider two sets A and B. If an individual x is in both sets, then the intersection
of the two sets A and B yields a set that includes x.
o We represent the intersection of two sets as A
B.
Union
• Union of two sets yields a set with all the members of two sets. We represent it
with A B.
b!
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4!
Subset & Superset
• If all the members of set A are also members of set B then A is a subset of B.
o A is a subset of B: A B
•
If A is a subset of B, then B is a superset of A.
o B is a superset of A: B⊇A
•
A set without any members is an empty set.
o Empty set can be represented by {} or Ø.
Cardinality
• Cardinality of a set gives the number of individuals in a set.
• It is represented as | n |
For example, the cardinality of the set A={a, b, c} is |3|.
Exercises
A= {2, 3, 4, 5 6, 9, 11, 33}
B= {2, 3, 0}
C= {2, 5, 11, 33}
Are the following True or False?
• A B
• A C
Give the sets.
• A B=
• C B=
!
5!
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•
Semantic Relations
o Presupposition
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Compositionality & Denotation
Presupposition
Previously, we talked about entailment and some other semantic relations that are related to it.
Now, we will focus on another semantic relation called presupposition.
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In entailment, when sentence p entails sentence q, then q is true whenever p is true.
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In presupposition, sentence p takes for granted the Truth of sentence q. It is more of a
matter of discourse. Consider the following sentence.
1. p. The present king of France is bald.
q. There is a unique present King of France.
Basically, (1)p presupposes (1)q. In other words, it takes for granted the truth of (1)q.
•
However, note that, unlike entailment, presupposition does not require q to be True when
p is True. It is quite possible that (1)q is False. Indeed, it is False. There is no present
king of France because it is no longer a monarchy.
Cases when the sentence taken for granted is False yield to presupposition failure.
P-Family Tests
• Presuppositions are strong. They usually live in different sentences. If we want to
determine whether a given sentence presupposes another one, we need to use P-Family
Tests. Using two or three of those tests will usually suffice to decide whether there is a
presupposition or not.
These tests are:
• Negation: Just add it is not the case that before a sentence to see if presupposition
projects (continues to live). Alternatively, you can negate the sentence but the first option
is more secure.
• Question: Turning a sentence into a question lets the presupposition project.
• Conditional: Presuppositions also project in conditionals.
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Examples:
2.
a) The present king of France is bald.
b) It is not the case that king of the France is bald.
c) I wonder if king of France is bald.
d) If king of France is bald, then king of USA must be bald, too.
e) There is a unique present king of France.
In (2), a-d sentences presuppose (2)e.
3.
a)
b)
c)
d)
e)
It was Lee who got a perfect score on the quiz.
It wasn’t Lee who got a perfect score on the quiz.
Was it Lee who got a perfect score on the quiz?
If it was Lee who got a perfect score on the quiz, why does he look so depressed?
Someone got a perfect score on the quiz.
In (3), a-d sentences presuppose (3)e.
4.
a)
b)
c)
d)
e)
Lee got a perfect score on the quiz.
Lee did not get a perfect score on the quiz.
Did Lee get a perfect score on the quiz.
If Lee got a perfect score on the quiz, then everyone got a perfect score.
Someone got a perfect score on the quiz.
In (4), a-d sentences do not presuppose (4)e.
Exercises
What relationship holds between the sentences in the following examples? Explain why you
think that relation holds. (It could be entailment or presupposition or both).
5.
a) It is false that everyone tried to kill Templeton
b) Someone did not try to kill Templeton.
6.
a) That John left early didn’t bother Mary.
b) John left early.
7.
a) If John discovers that Mary is in New York, he will get angry.
b) Mary is in New York.
8.
a) Seeing is believing.
b) If John sees a riot, he will believe it.
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Compositionality,&,Denotation%
Previously, we talked about meanings of sentences. A sentence like (9) can be either True or
False.
9. The silly boy in the park loves the silly girl.
Nevertheless, we cannot know whether it is True or False, unless given a situation.
On the other hand, as the speakers of the language, we know its truth conditions. We know the
conditions under which sentence (9) would be True or False.
Due to our knowledge of syntax, we also know that a sentence is composed of its parts (namely
phrases). We know that the way a set of words combine yield (un)grammaticality.
Also, the way a set of words combine can yield different meanings. Consider (10)a and (10)b.
10. a. The silly boy in the park loves the silly girl.
b. The silly girl loves the silly boy in the park.
(10) shows that the meaning of a sentence is actually composed of the meaning of its parts and
they way they come together.
We already learnt how the words come together. They come together as phrases to form larger
phrases.
11. a. [TP [DP The [[NP silly boy in the park]] [VP runs]]
Now, we will see the denotation of words and phrases. After we do that, we can work towards
the denotation of sentences.
Let’s now focus on the denotation of NPs in sentence (11).
•
[NP silly boy in the park]
Remember, we talked about set theory. A set is an abstract collection of individuals.
The denotation of nouns, adjectives and PPs are predicates. A predicate is simply a set of
individuals. Linguists use double brackets to show denotation of a word/phrase. ⟦""⟧
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Situation A
⟦John⟧="John"
⟦Billy⟧="Billy"
⟦Tom⟧="Tom"
⟦Mary⟧="Mary"
⟦Susan⟧="Susan"
""Individuals"
⟦boy⟧={John,"Billy,"Tom}"
⟦silly⟧={John,"Billy,"Susan}"
⟦in"the"park⟧={John,"Tom,"Mary}"
""Predicates"
Denotation of names yield individuals. On the other hand, denotation of common nouns,
adjectives and PPs yield sets of individuals.
Now that we know the denotation of each word, we can get the denotation of NPs.
•
Denotation of NPs is merely set intersection of sets.
⟦boy⟧={John,"Billy,"Tom}"
⟦silly boy⟧"="{John,"Billy}"
⟦silly boy in the park⟧"="{John}"
boy
silly
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J%
S%
T%
M%
in the park
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Exercises
⟦John⟧="John"
⟦Billy⟧="Billy"
⟦Tom⟧="Tom"
⟦Mary⟧="Mary"
⟦Susan⟧="Susan"
⟦person⟧={Tom,"Mary,"Billy,"Susan,"John}"
⟦boy⟧={John,"Billy,"Tom}"
⟦silly⟧={John,"Susan}"
⟦in"the"room⟧={Tom,"Mary}"
⟦tall⟧={Mary,"John}"
⟦smart⟧={Tom,"Mary}"
Give the denotations of the following NPs:
1. smart boy in the room
2. tall person in the room
3. smart silly boy
In terms of semantic relations (entailment, tautology, contradiction, presupposition, etc.), what
does smart silly boy remind us?
4. smart boy
5. boy
In terms of set theory, what is the relation between smart boy and boy?
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Sometimes sets can include to a very large number of individuals. It would be hard (even
impossible in some cases) to write every member of the set. Therefore, we give the description of
the set.
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human={x| x is a human} is read as x such that x is a human.
This is the set of human beings. It would be impossible to list all the names of human beings out
there but we can refer to them by simply saying the name of the set.
Consider the following sets in conjunction with the ones above.
Give the denotations of rich and happy by listing their members.
1. ⟦rich⟧"= {x| x is either tall or silly}
2. ⟦happy⟧"= {x| x is both tall and smart and rich}
Consider set theory relations. What do and and or look like?
What is the denotation of
1. Rich person in the room
2. Rich boy in the room
3. Happy person
4. Happy boy
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Semantics
Last time we started working on the denotation (meaning) of the following sentence.
•
The silly boy in the park loves the silly girl.
Denotation of a sentence is its truth conditions. We want to derive the truth conditions of
the sentence above.
By principle of compositionality, denotation of a sentence is derived from its parts. So,
we started with the parts. We started with the following NP.
•
[NP silly boy in the park]
In order to get the denotation of the NP above, we need to know the denotation of silly,
boy and in the park.
Things like John, Mary, Tom, Susan, silly, boy, etc. are words. Things like in the park are
phrases. These refer to actual entities in the world.
At this point we will distinguish two types of words/phrases.
•
Proper nouns (names)
= John, Mary, Susan, Tom, etc.
•
Predicates
= common nouns (boy, girl)
= adjectives (red, silly, tall)
= PPs (in the park, on the table)
Words and phrases are linguistic expressions. They are just a bunch of
sounds/symbols/letters. They are interpreted by semantics. If we want to talk about the
interpretation of a word/phrase, we show it in double brackets !! .
For example:
• The word boy is a linguistic expression. When we put it into double brackets, we
get its denotation (meaning) !"# .
We made a distinction between words/phrases above. We said there are proper nouns
(names) like John and predicates like boy, silly, etc. Semantically, they refer to different
entities.
•
•
Proper nouns are individuals like the person John, the person Mary, etc.
Predicates are sets of individuals.
So the denotations of individuals and predicates are shown in different ways.
!
1!
Individuals
•
•
•
•
•
!"ℎ!
!"##
!"#
!"#$%
!"#$
= John
= Bill
= Tom
= Susan
= Mary
Predicates
•
•
•
•
!"#
!"##$
!"!!ℎ!!!"#$
!"#$
= {John, Billy, Tom}
= {John, Susan}
= {John, Mary}
= {Susan, Mary}
Now, we know the denotations of the words. We need to find out the denotation of the
NPs.
•
[silly boy in the park]
The NP above consists of three predicates silly, boy, and in the park. So, whoever the
silly boy in the park is he must be silly, boy, and in the park. In other words, he must be a
member of the sets denoted by silly, boy and in the park.
Therefore, to get the denotation of [silly boy in the park], we need to intersect the three
sets. The intersection of the three will give us the denotation of [silly boy in the park].
•
!"##$!!"#!!"!!ℎ!!!"#$ != {John}
What is the denotation of [silly girl]?
•
!"##$!!"#$ != { }
Denotation of VPs
Previously, we talked about the denotation of NPs, APs, and PPs. They were all
predicates. In other words, they were sets.
⟦boy⟧& &
={John,&Billy,&Tom}&&
⟦silly⟧& &
={John,&Susan}&
⟦in&the&park⟧& ={John,&Mary}&
!
2!
Similarly, VPs are predicates too. Therefore, they denote sets of individuals.
•
•
•
⟦sleeps⟧&
=&{John,&Mary,&Bill}&&
⟦runs⟧&&
=&{Bill}
⟦loves&Mary⟧& =&{John}&
Denotation of sentences
We already know the meaning of its parts. Now, let’s get the meaning of the whole from
its parts. Let’s start with simple DPs like John, Mary, Susan, etc.
Now we are ready to derive the Truth Conditions of sentences.
Consider the following sentence.
• John sleeps.
Crudely, this sentence consists of a DP and a VP.
• [DP John] [VP sleeps]
Truth condition of a sentence
• A sentence is True iff (if and only is) the denotation of the DP is in the
denotation of the VP.
In other words
• ⟦Sentence⟧= True iff ⟦DP⟧ is in ⟦VP⟧.
Yet another convention
• ⟦Sentence⟧= True iff ⟦DP⟧ ∈ ⟦VP⟧.
Therefore
• ⟦John sleeps⟧= True iff ⟦John⟧ ∈ ⟦sleeps⟧.
In order to get the Truth Value of a sentence, then we test the truth condition against the
situation. In the following scenario, it would be True.
⟦sleeps⟧&
=&{John,&Mary,&Bill}&&
⟦runs⟧&&
=&{Bill}&
⟦loves&Mary⟧& =&{John}&
What are the Truth Conditions and the Truth Value of the following sentence
according to the situation above?
•
!
Bill loves Mary.
3!
Exercises
Give the truth conditions of the sentences below. Additionally, give a situation which
accounts for all the desired Truth Values specified for each sentence.
Situation
Sentences
1. John visited Tom. True
Truth Condition:
2. Mary is brilliant. True
Truth Condition:
3. The brilliant girl visited Tom. False
Truth Condition:
4. Tom is silly. False
Truth Condition:
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Quantifiers
Previously, we worked on the truth conditions of sentences whose subjects were proper
nouns (names).
1. John
slept.
Deriving the truth conditions of sentences like (1) is straightforward. John is an
individual, slept is a predicate. So sentence 1 is true iff John is in the set denoted by slept.
2. ⟦Sentence⟧= True iff ⟦DP⟧ ∈ ⟦VP⟧.
DPs are not always very simple. Consider cases like every student.
Every is a quantifier. Quantifiers establish relations between predicates. In other words,
they take NPs and relate them to VPs.
A sentence like “Every man laughs.” consists of three parts. A quantifier, a NP and a
VP.
!
Let’s give it a template.
“Every
Quantifier
man
A
laughs.”
B
The list below gives the meanings of quantifiers sentences with quantifiers.
Note that | | indicates the cardinality of a set. In other words, it gives us the number of
individuals in the set.
The
The is a little bit different. It has a presupposition. Therefore, we can talk about its felicity
condition.
The A B
is felicitous iff |A|=1
because, when we say the man we assume that there is only one man (not 2 or 3).
!
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Exercises
Give the truth conditions of the following sentences.
1. Exactly three girls are rich.
2. No boy is tall.
3. Most books are cheap.
Consider the following sentence.
•
Two girls are tall.
Two is a quantifier. Can you give a truth condition for the sentence above?
General Exercises on Semantics
1a. If ⟦cat⟧= {Sammy, kitty, princess}, ⟦white⟧= {Sammy} and ⟦cute⟧= {kitty},
then ⟦cute white kitty⟧= {Sammy}or{kitty}.
i. True ii. False
b. “Bill went to NY to see a play” entails “Bill went to NY and saw a play”.
i. True ii. False
2. Suppose there are three individuals, Mary, Suzy and Jane. Only Mary and Suzy are
tall. Jane and Suzy are blond. Mary is lazy. Given this situation, what would be the
semantic values of the following expressions:
a. ⟦lazy and blond⟧ =
b. ⟦Suzy⟧#=
c. Which of the following is a superset of ⟦girls⟧?
i. ⟦girls⟧
ii. ⟦tall and lazy⟧
iii. ⟦blond girl⟧
iv. None of the above.
3. Which of the following contains contradictory sentences?
A. 'John is not tall' - 'John is short'.
B. 'Most girls wished for a white pony' - 'Some girl wished for a black pony'
C. 'All of John's friends are silly.' - 'Some of John's friends are not silly.'
D. 'Every brown dog barks loudly' - 'Some brown dog barks loudly'
E. 'The milk is hot or at room temperature' - 'The milk is cold'
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4. Provide the semantic values for the five items below, such that the sentences in (i) to
(iv) have the given truth values:
⟦dog⟧=
⟦black⟧=
⟦cat⟧=
⟦furry⟧=
⟦white⟧=
i. ⟦Every dog and some cat is white⟧= True
ii. ⟦No dog is black or furry⟧= False
iii. ⟦Two dogs and two cats are white⟧=True
iv. ⟦No cat is furry⟧= True
5. The two following questions are based off the following sentences:
I. Most Ling 201 students studied hard for Midterm 2.
II. John, who is a Ling 201 student, studied hard for Midterm 2.
a) Sentence (II) entails Sentence (I).
A. True
B. False
b) If your answer above is 'True', describe a situation in which Sentence (I) and Sentence
(II) have the same truth values. If your answer above is 'False', describe a situation in
which Sentence (I) and Sentence (II) have different truth values.
6. Consider the following situation:
⟦boy⟧ = {Tom, Paul, John}
⟦girl⟧ = {Mary, Susan }
⟦Tom⟧=Tom ⟦Mary⟧=Mary ⟦Paul⟧=Paul ⟦Susan⟧=Susan ⟦John⟧=John
⟦studied hard for Ling 201 test⟧ = {Tom, Mary}
⟦has missed breakfast⟧ = {Tom, Paul, Mary, Susan}
⟦has work after class⟧= {Mary, Susan}
Answer the following questions based off the situation described above.
a) ⟦has work after class⟧#is a subset of ⟦has missed breakfast⟧
A. True
B. False
!
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b. ⟦studied hard for Ling 201 test⟧#is a superset of the intersection of ⟦ has missed
breakfast⟧#and ⟦ boy⟧
A. True
B. False
c. ⟦Tom⟧#is a member of the set intersection of ⟦ studied hard for Ling 201 test⟧#and ⟦has
work after class⟧.
A. True
B. False
d.⟦Susan⟧#is a member of the set⟦ girl⟧.
A. True
B. False
4. Suppose ‘erc' is an English determiner with the following meaning:
⟦Erc A B⟧= True, if and only if at least 3 individuals in A are not in B.
Situation B:
⟦boy⟧ = {Tom, Paul, John, Ed}
⟦smart⟧= {Tom}
⟦Tom⟧= Tom ⟦John⟧= John ⟦Paul⟧= Paul ⟦Ed⟧= Ed
a. Is ‘Erc boys are smart' True or False in Situation B?
b. Modifying just the set denoted by ⟦smart⟧, how would you make 'Erc boys are smart'
to have the opposite truth value in Situation B to the one you indicated above?
For example, if you had said that Erc boys are smart' is True in Situation B, how would
you modify the denotation of ⟦smart⟧#such that the sentence becomes False in Situation
B.
!
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More on Entailment & Presupposition
Entailment is some sort of an implication relationship between two sentences. However,
it is not any implication relationship.
1.
p: Vincent has a red car.
q: Vincent has a car.
2.
p: Vincent does not have a red car.
q: Vincent has a car.
Entailment relations between two sentences is not preserved when one of them is
negated.
Presupposition is another sort of implication relation between two sentences.
Presupposition does not go away when p is negated.
3.
p: Vincent cleaned his car.
q: Vincent has a car.
4.
p: Vincent did not clean his car.
q: Vincent has a car.
Exercises:
Determine if the p sentences below entail or presuppose the q sentences.
1.
p: Alex is a bachelor.
q: Alex is a man.
2.
p: Alex is a man.
q: Alex is male.
3.
p: Jon Snow knows that Ygritte is dead.
q: Ygritte is dead.
4.
p: Jon Snow does not regret killing Ygritte.
q: Jon Snow killed Ygritte.
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5.
p: Who discovered Pluto?
q: Pluto was discovered.
6.
p: Bill is a better linguist than Mary.
q: Mary is a linguist.
7.
p: John managed to stop smoking.
q: John stopped smoking.
8.
p: Today is sunny.
q: Today is warm.
9.
p: Juan is not aware that Joanna is pregnant.
q: Joanna is pregnant.
10.
p: Oscar and Jenny are rich.
q: Jenny is rich.
11.
p: Oscar and Jenny are young.
q: Jenny is young.
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