Very quick introduction to sentence semantics.

1. Sentence Semantics I

2. Boredom
• Classes
• Meetings
• No problems
• Playing computer games
• Free time

3. Blue
• Sky
• Sea
• Music
• Soccer team shirt

4. Bear
• Forest
• Zoo
• Mountains
• Hair

5. Bad
• Politicians
• Global warming
• Examinations
• Stress

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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
• ?????

12. 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
• ?????

13. Why do this?
• Seems completely pointless
• Maybe it is!

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

15. Words have amazing power

17. Incredibly complex

18. Mysterious

19. Magical

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

22. 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

23. Just individual words

24. Full of information, concepts, meaning

25. That we cannot really explain

26. And another problem

27. Sometimes just word is OK

28. But usually we speak in sentences

29. • 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 ……

30. Even simple sentences …

31. … contain huge amounts of
information

32. And we cannot …

33. … really explain …

34. … at all

35. 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

36. Lots of things we cannot explain …

37. Really … not at all

38. 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

39. Boredom

40. Boredom is good!

41. Interesting is BAD!

42. Boredom feels secure

43. Don’t forget … it’s a mystery

44. Complicated and mysterious

45. So let’s simplify

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

47. Propositional logic: atomic statements

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

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

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

51. • 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!

52. 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

53. 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

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

55. • p (some sentence)
• r (some sentence)
•  (if … then connective)
• p  r
• What does this mean?
• If p is true, then r is also true

56. • If you are human, then you are a mammal
• If … then connective

57. • p  r
• p = you are human
• r = you are a mammal
• If p is true
• Then r is also true

58. • 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

59. • We can have crazy examples if we want

60. • 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

61. • 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

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

63. 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

64. Propositions
• John hit Ben
• Ben was hit by John
• Same proposition
• call it p
• or q
• or r
• etc

65. p = Ben is studying

66. p: Ben is studying
• Ben is studying = True
• p = True
• p = T
• p = 1

67. p = Ben is studying

68. p: Ben is studying
• Ben is studying = False
• p = False
• p = F
• p = 0

69. But maybe he’s studying?!

70. Maybe, but don’t worry too much

71. Keep things simple

72. Imagine simple little worlds

73. Where everything is either T or F

74. p = The Little Prince is standing (= T)

75. q = The Little Prince is gardening (=T)

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

77. Logical Constants or connectives
• ∧
• &
• And
• ∨
• Or
• →
• If … then

78. Connectives
• ¬
• Not
• ~
• Not
• -
• Not

79. Connectives
⇒
→
If… then
p → q
p = you are clever
q = you will study
If you are clever, you will study

80. Connectives
⇒
→
If… then
p → q
p = you are clever
q = you will pass the test
If you are clever, you will pass the test

81. Connectives
⇒
→
If… then
p → q
p = you are studying logic
q = you are bored
If you are studying logic, you are bored

82. Connectives
⇒
→
If… then
p → q
p = you are studying logic
q = you hate your teacher
If you are studying logic, you hate your teacher

83. Connectives
⇒
→
If… then
p → q
p = Eri hates logic
q = Eri studies logic
If Eri hates logic then she studies logic

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

85. Truth tables for p & q

86. p & q
• p = The weather is fine
• q = Tom is sleeping
• Is p & q true?
• If both p & q it is true
• Not otherwise

87. Truth tables for p or q

88. p  q

89. ¬ p

90. p ⇔ q

91. Predicate Logic

92. • P v Q
• P or Q
• - P
• Not P

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

94. Propositional Logic – too simple
• p
• q
• No internal structure

95. 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?

96. Predicate Logic

97. Predicate Logic

98. Simple sentence (grammar)

99. Predicate Logic
• Predicate
• For example …
• Love
• Love (x,y)

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

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

102. Ken is crazy
• Predicate
• Crazy
• One argument
• Crazy (x)
• x = ?
• x = ken
• Crazy (ken)

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

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

105. How about three-place predicate?
• Send
• Ken sent Mary a letter
• Send (ken, mary, letter)

106. Predicate is capital
• Love (x,y)

107. Arguments are small letters
• Love (john, mary)
• Why?
• Why not?
• Who cares?

108. Japan is a country

109. Japan is a country
• Country (x)
• x = japan
• Country (japan)

110. Easy so far?
• Yes
• But what is the point?
• Good question
• Don’t think about that!

111. And it helps with computer
programming

112. Peter fell over
• Predicate?
• Fall_over
• How many arguments?
• One – peter
• Fall_over (peter)

113. Jim donated $100 to the city hospital
• Donated (x,…..n)
• Donated (jim, $100, city_hospital)

114. Ben hates computers
• Hate (ben, computers)

115. Eri gave up
• Gave_up (eri)

116. Eri is a genius
• Genius (eri)

117. Quantifiers

118. The Little Prince is wearing a brown
scarf

119. But the Little Prince is the only person
in this world

120. Everyone is wearing a brown scarf

121. Imagine simple little worlds

122. What do quantifiers mean?

123. All – upside-down A

124. Think of sets

125. • Two sets
• Set A
• Set B

126. • Set A = the set of Linguists
• Linguist (x)
• Set B = the set of crazy people
• Crazy_person (x)

127. Who is in these sets?
• Linguist (x)
• {evans,
• imai,
• ono}

128. Who is in the set of crazy people?
• Crazy_person (x)
• {ken,
• jim,
• ben,
• mary,
• evans}

129. All linguists are crazy
• ∀x (Linguist (x)  Crazy_person (x))
• Is this true?

130. • Set A = the set of Linguists
• Linguist (x)
• Set B = the set of crazy people
• Crazy_person (x)

131. Who is in these sets?
• Linguist (x)
• {evans,
• imai,
• ono}

132. Who is in the set of crazy people?
• Crazy_person (x)
• {ken,
• jim,
• ben,
• mary,
• evans}

133. 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.

134. Some linguists are crazy
• Is this True?

135. 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

136. Who is in the set of crazy people?
• Crazy_person (x)
• {ken,
• jim,
• ben,
• mary,
• evans}

137. 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

138. Predicate
• Mary is a girl
• Girl
• Girl (mary)

139. Predicate
• Mary lives in Tsuru
• Lives_in_Tsuru (mary)

140. Set of girls
• Girl (mary)
• Girl (eri)
• Girl (rie)
• Girl = {mary, eri, rie}

141. 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}

142. A girl lives in Tsuru
• ∃x
• (
• Girl (x)
• & Lives_in_tsuru (x)
• )
• ∃x (Girl (x) & Lives_in_tsuru (x))

143. Set of girls
• Girl (mary)
• Girl (eri)
• Girl (rie)
• Girl = {mary, eri, rie}

144. 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}

145. • ∃x (Girl (x) & Lives_in_tsuru (x))
• Is True!

146. A girl lives in Fujiyoshida
• ∃x
• (
• Girl (x)
• & Lives_in_fujiyoshida (x)
• )
• ∃x (Girl (x) & Lives_in_fujiyoshida (x))

147. Set of girls
• Girl (mary)
• Girl (eri)
• Girl (rie)
• Girl = {mary, eri, rie}

148. 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}

149. • ∃x (Girl (x) & Lives_in_fujiyoshida (x))
• Is False!

150. • 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

151. Any problems with Predicate Logic?
• Yes
• We are not always trying to say things that are
true

152. The sky is blue

153. We don’t say the sky is blue at night

154. Even though it’s true

155. • 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))

156. • 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?

157. New ideas about how we understand
meaning

158. Strong evidence for IMAGES rather
than CODE

159. Strong evidence for ACTION simulation

160. Mental Models as IMAGES

161. Many people say Logical Form cannot
be real

162. 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)

163. • ∀x (Child (x)  Kissed (nurse, x)) on x’s
birthday
• The nurse kissed every child on his birthday

164. But Logic is VERY important in
Linguistics

165. • 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.

166. • 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

167. • 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)

168. • How could you represent “Every Linguist is
crazy” in Predicate Logic?
• ∀x (Linguist (x)  Crazy_person (x))
•

169. • How could you represent “A boy sent Mary
$300” in Predicate Logic?
• ∃x (Boy (x) & Sent (x, mary, $300))

170. • What does an Upside-down A mean?
• Every, All
• What does a backward E mean?
• There is
• Existence

171. • What does an upside-down A say about two
sets?
• One is contained in the other

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

173. • 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