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Sentence semantics
- 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
- 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
- 24. Full of information, concepts, meaning
- 25. That we cannot really explain
- 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!
- 43. Don’t forget … it’s a mystery
- 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
- 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
- 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