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Propositional Logic
• p  q
• NO internal structure
Predicate logic
• Loves (j, m)
• DOES have internal structure
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
It helps with computer programming
We use it to input data
And write code
Data
• You can put in any information you want
Peter fell over
• Predicate?
• Fell_over
• How many arguments?
• One – peter
• Fell_over (peter)
Jim donated $100 to the city hospital
• Donated (x,…..n)
• Donated (jim, $100, city_hospital)
• Donated (x, y, z)
• X = Jim
• Y = $100
• Z= the city hospital
Ben hates computers
• Hate (ben, computers)
Eri gave up
• Gave_up (eri)
Emi is a genius
• Genius (emi)
No limits as long as it’s organized
• Visit (mary, familiymart, sat_15th_april,)
• Computational Linguistics:
• Verb
• Subject
• Complement
Hit
• Hit ( )
• Hit (x, )
• Hit (x, y)
• X = bill
• Y = ken
• Bill hit Ken
Database 1
• “Jun, Ken, and Emi are all Japanese”
Japanese (jun)
Japanese (ken)
Japanese (emi)
Japanese (x)  ………………………
Database 1
Japanese (x)  Human (x)
Japanse (x)  Asian (x)
Database 2
• “Jim is Bob’s father. Fred is Jerry’s father. Bob
is Erica’s father. Will is Mary’s father. Jim, Bob,
Fred, Jerry, and Will are male. Erica and Mary
are female”
Database 2
Father (jim, bob)
Father (fred, jerry)
Father (bob, erica)
Father (will, mary)
Male (jim)
Male (bob)
Male (fred)
Male (jerry)
Male (will)
Female (erica)
Female (mary)
Database 2
Father (x, y) & Male (y)
 ………………………
Father (x, y) & Female (y)
 ……………………
Father (x, y) & Father (y, z)
 ………………….
Father (x, y) & Father (y, z) & Female (z)
 ……………………..
Father (x, y) & Father (y, z) & Male (z)
 ………………………..
Database 2
Father (x, y) & Male (y)
 Son (y)
 Son (y, x)
Father (x, y) & Female (y)
 Daughter (y)
 Daughter (y, x)
Database 2
Father (x, y) & Father (y, z)
 Grandfather (x, z)
Father (x, y) & Father (y, z) & Female (z)
 Granddaughter (z, x)
Father (x, y) & Father (y, z) & Male (z)
 Grandson (z, x)
Quantifiers
The Little Prince is wearing a brown
scarf
• Wearing_a_brown_scarf (p)
• Wearing (p, b_s)
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)
A = Linguists; B = Crazy people
Who is in these sets?
• Linguist (x)
• {evans,
• imai,
• ono}
Computer database
• Linguist (evans)
• Linguist (imai)
• Linguist (ono)
• Set of linguists =
• {evans, imai, ono}
Who is in the set of crazy people?
• Crazy_person (x)
• {ken,
• jim,
• ben,
• mary,
• evans}
Computer database
• Crazy_person (evans)
• Crazy_person (ken)
• Crazy_person (jim)
• Crazy_person (ben)
• Crazy_person (mary)
∀= universal (all)
Linguist (x)  Crazy_person (x)
All linguists are crazy
• Linguist (x)  Crazy_person (x)
• Is this true?
• For all individuals x, if x is in the set of
Linguists, x is also in the set of Crazy persons.
• 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
• 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?
Does this kind of x exist?
• Linguist (x) & Crazy_person (x)
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
Language connects to the body
NOT all in the brain!
Also 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!
• Is it just because computers are important?
Logic is the basis of computer science
∃ (existential)
• M(jen,mary)
• M = mother
• Jen is Mary’s mother
• ∃xM(x,y)
• Someone is the mother of y
• ∀y∃xM(x,y)
• Everyone has a mother
• Likes (ben, emi)
• Ben likes Emi
• Likes (x, y)
• x likes y
X = boys, Y = girls
• Likes (x, y)
• x likes y
• (let’s say x = boys and y = girls)
• ∃y (Likes (x, y)
• There is some girl that x likes
• ∃y (Likes (x, y)
• There is some girl that x likes
• ∀x∃y (Likes (x, y)
• Each boy likes a girl
Each boy likes a girl
Boys Girls
Each boy likes a girl
Boys Girls
Each boy likes a girl
Boys Girls
∀x∃y (Likes (x, y))
Boys (x) Girls (y)
There’s one girl all the boys like
Boys Girls
All the boys like the same one girl
Boys Girls
∃y∀x (Likes (x, y)
Boys (x) Girls (y)
Each girl punched a boy
Girls Boys
∀x∃y (Punched (x, y)
Girls (x) Boys (y)
All the girls punched the same boy
Girls (x) Boys (y)
∃y ∀x (Punched (x, y)
Girls (x) Boys (y)
p → q
• p = Jim is dead
• q = Mary is sad
• p → q =
• If Jim is dead, Mary is sad
p V q
• p = My parrot is clever
• q = Taro fell over
• p V q =
• (either) my parrot is clever or Taro fell over
p → q
• p = I study hard
• q = I will pass the test
• p → q =
• If I study hard, I will pass the test
¬p → ¬q
• P = I study hard
• Q = I will pass the test
• ¬p → ¬q = If I do not study hard, I will not
pass the test
• If NOT(I study hard) then NOT (I will pass the
test)
p → q
• Normal everyday values for p and q
• p = a bear has warm blood
• q = a bear is a mammal
¬p → q
• Normal everyday values for p and q
• p = you study hard
• q = you will fail
• If you DON’T study hard, you will fail
Modern Linguistics
• Studying language helps us understand the
human brain
• Do you think the human brain works like
computer code?
• Do you think studying language tells us
something about the brain?

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Predicate calculus up

  • 1. Propositional Logic • p  q • NO internal structure
  • 2. Predicate logic • Loves (j, m) • DOES have internal structure
  • 3. Japan is a country
  • 4. Japan is a country • Country (x) • x = japan • Country (japan)
  • 5. Easy so far? • Yes • But what is the point? • Good question
  • 6. It helps with computer programming
  • 7. We use it to input data
  • 9. Data • You can put in any information you want
  • 10. Peter fell over • Predicate? • Fell_over • How many arguments? • One – peter • Fell_over (peter)
  • 11. Jim donated $100 to the city hospital • Donated (x,…..n) • Donated (jim, $100, city_hospital) • Donated (x, y, z) • X = Jim • Y = $100 • Z= the city hospital
  • 12. Ben hates computers • Hate (ben, computers)
  • 13. Eri gave up • Gave_up (eri)
  • 14. Emi is a genius • Genius (emi)
  • 15. No limits as long as it’s organized • Visit (mary, familiymart, sat_15th_april,) • Computational Linguistics: • Verb • Subject • Complement
  • 16. Hit • Hit ( ) • Hit (x, ) • Hit (x, y) • X = bill • Y = ken • Bill hit Ken
  • 17. Database 1 • “Jun, Ken, and Emi are all Japanese” Japanese (jun) Japanese (ken) Japanese (emi) Japanese (x)  ………………………
  • 18. Database 1 Japanese (x)  Human (x) Japanse (x)  Asian (x)
  • 19. Database 2 • “Jim is Bob’s father. Fred is Jerry’s father. Bob is Erica’s father. Will is Mary’s father. Jim, Bob, Fred, Jerry, and Will are male. Erica and Mary are female”
  • 20. Database 2 Father (jim, bob) Father (fred, jerry) Father (bob, erica) Father (will, mary) Male (jim) Male (bob) Male (fred) Male (jerry) Male (will) Female (erica) Female (mary)
  • 21. Database 2 Father (x, y) & Male (y)  ……………………… Father (x, y) & Female (y)  …………………… Father (x, y) & Father (y, z)  …………………. Father (x, y) & Father (y, z) & Female (z)  …………………….. Father (x, y) & Father (y, z) & Male (z)  ………………………..
  • 22. Database 2 Father (x, y) & Male (y)  Son (y)  Son (y, x) Father (x, y) & Female (y)  Daughter (y)  Daughter (y, x)
  • 23. Database 2 Father (x, y) & Father (y, z)  Grandfather (x, z) Father (x, y) & Father (y, z) & Female (z)  Granddaughter (z, x) Father (x, y) & Father (y, z) & Male (z)  Grandson (z, x)
  • 25. The Little Prince is wearing a brown scarf
  • 27. But the Little Prince is the only person in this world
  • 28. Everyone is wearing a brown scarf
  • 33. • Two sets • Set A • Set B
  • 34. • Set A = the set of Linguists • Linguist (x) • Set B = the set of crazy people • Crazy_person (x)
  • 35. A = Linguists; B = Crazy people
  • 36. Who is in these sets? • Linguist (x) • {evans, • imai, • ono}
  • 37. Computer database • Linguist (evans) • Linguist (imai) • Linguist (ono) • Set of linguists = • {evans, imai, ono}
  • 38. Who is in the set of crazy people? • Crazy_person (x) • {ken, • jim, • ben, • mary, • evans}
  • 39. Computer database • Crazy_person (evans) • Crazy_person (ken) • Crazy_person (jim) • Crazy_person (ben) • Crazy_person (mary)
  • 40. ∀= universal (all) Linguist (x)  Crazy_person (x)
  • 41. All linguists are crazy • Linguist (x)  Crazy_person (x) • Is this true? • For all individuals x, if x is in the set of Linguists, x is also in the set of Crazy persons.
  • 42. • Set A = the set of Linguists • Linguist (x) • Set B = the set of crazy people • Crazy_person (x)
  • 43. Who is in these sets? • Linguist (x) • {evans, • imai, • ono}
  • 44. Who is in the set of crazy people? • Crazy_person (x) • {ken, • jim, • ben, • mary, • evans}
  • 45. All linguists are crazy • Linguist (x)  Crazy_person (x) • Is Untrue • Because two members of the set of linguists are not in the set of crazy people.
  • 46. Some linguists are crazy • Is this True?
  • 47. Does this kind of x exist? • Linguist (x) & Crazy_person (x)
  • 48. 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
  • 49. Who is in the set of crazy people? • Crazy_person (x) • {ken, • jim, • ben, • mary, • evans}
  • 50. 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
  • 51. Predicate • Mary is a girl • Girl • Girl (mary)
  • 52. Predicate • Mary lives in Tsuru • Lives_in_Tsuru (mary)
  • 53. Set of girls • Girl (mary) • Girl (eri) • Girl (rie) • Girl = {mary, eri, rie}
  • 54. 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}
  • 55. A girl lives in Tsuru • ∃x • ( • Girl (x) • & Lives_in_tsuru (x) • ) • ∃x (Girl (x) & Lives_in_tsuru (x))
  • 56. Set of girls • Girl (mary) • Girl (eri) • Girl (rie) • Girl = {mary, eri, rie}
  • 57. 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}
  • 58. • ∃x (Girl (x) & Lives_in_tsuru (x)) • Is True!
  • 59. A girl lives in Fujiyoshida • ∃x • ( • Girl (x) • & Lives_in_fujiyoshida (x) • ) • ∃x (Girl (x) & Lives_in_fujiyoshida (x))
  • 60. Set of girls • Girl (mary) • Girl (eri) • Girl (rie) • Girl = {mary, eri, rie}
  • 61. 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}
  • 62. • ∃x (Girl (x) & Lives_in_fujiyoshida (x)) • Is False!
  • 63. • 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
  • 64. Any problems with Predicate Logic? • Yes • We are not always trying to say things that are true
  • 65. The sky is blue
  • 66. We don’t say the sky is blue at night
  • 68. • 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))
  • 69. • 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?
  • 70. New ideas about how we understand meaning
  • 72. NOT all in the brain!
  • 73. Also strong evidence for IMAGES rather than CODE
  • 74. Strong evidence for ACTION simulation
  • 76. Many people say Logical Form cannot be real
  • 77. But Logic is VERY important in Linguistics! • Is it just because computers are important?
  • 78. Logic is the basis of computer science
  • 79. ∃ (existential) • M(jen,mary) • M = mother • Jen is Mary’s mother • ∃xM(x,y) • Someone is the mother of y
  • 81. • Likes (ben, emi) • Ben likes Emi • Likes (x, y) • x likes y
  • 82. X = boys, Y = girls • Likes (x, y) • x likes y • (let’s say x = boys and y = girls) • ∃y (Likes (x, y) • There is some girl that x likes
  • 83. • ∃y (Likes (x, y) • There is some girl that x likes • ∀x∃y (Likes (x, y) • Each boy likes a girl
  • 84. Each boy likes a girl Boys Girls
  • 85. Each boy likes a girl Boys Girls
  • 86. Each boy likes a girl Boys Girls
  • 87. ∀x∃y (Likes (x, y)) Boys (x) Girls (y)
  • 88. There’s one girl all the boys like Boys Girls
  • 89. All the boys like the same one girl Boys Girls
  • 90. ∃y∀x (Likes (x, y) Boys (x) Girls (y)
  • 91. Each girl punched a boy Girls Boys
  • 92. ∀x∃y (Punched (x, y) Girls (x) Boys (y)
  • 93. All the girls punched the same boy Girls (x) Boys (y)
  • 94. ∃y ∀x (Punched (x, y) Girls (x) Boys (y)
  • 95. p → q • p = Jim is dead • q = Mary is sad • p → q = • If Jim is dead, Mary is sad
  • 96. p V q • p = My parrot is clever • q = Taro fell over • p V q = • (either) my parrot is clever or Taro fell over
  • 97. p → q • p = I study hard • q = I will pass the test • p → q = • If I study hard, I will pass the test
  • 98. ¬p → ¬q • P = I study hard • Q = I will pass the test • ¬p → ¬q = If I do not study hard, I will not pass the test • If NOT(I study hard) then NOT (I will pass the test)
  • 99. p → q • Normal everyday values for p and q • p = a bear has warm blood • q = a bear is a mammal
  • 100. ¬p → q • Normal everyday values for p and q • p = you study hard • q = you will fail • If you DON’T study hard, you will fail
  • 101. Modern Linguistics • Studying language helps us understand the human brain • Do you think the human brain works like computer code? • Do you think studying language tells us something about the brain?