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Predicate calculus
- 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. Quantifiers
- 18. The Little Prince is wearing a brown scarf
- 19. • Wearing_a_brown_scarf (p) • Wearing (p, b_s)
- 20. But the Little Prince is the only person in this world
- 21. Everyone is wearing a brown scarf
- 22. Imagine simple little worlds
- 23. What do quantifiers mean?
- 24. All – upside-down A
- 25. Think of sets
- 26. • Two sets • Set A • Set B
- 27. • Set A = the set of Linguists • Linguist (x) • Set B = the set of crazy people • Crazy_person (x)
- 28. Who is in these sets? • Linguist (x) • {evans, • imai, • ono}
- 29. Computer database • Linguist (evans) • Linguist (imai) • Linguist (ono) • Set of linguists = • {evans, imai, ono}
- 30. Who is in the set of crazy people? • Crazy_person (x) • {ken, • jim, • ben, • mary, • evans}
- 31. Computer database • Crazy_person (evans) • Crazy_person (ken) • Crazy_person (jim) • Crazy_person (ben) • Crazy_person (mary)
- 32. Linguist (x) Crazy_person (x)
- 33. All linguists are crazy • ∀x (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.
- 34. • Set A = the set of Linguists • Linguist (x) • Set B = the set of crazy people • Crazy_person (x)
- 35. Who is in these sets? • Linguist (x) • {evans, • imai, • ono}
- 36. Who is in the set of crazy people? • Crazy_person (x) • {ken, • jim, • ben, • mary, • evans}
- 37. 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.
- 38. Some linguists are crazy • Is this True?
- 39. • Linguist (x) & Crazy_person (x)
- 40. 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
- 41. Who is in the set of crazy people? • Crazy_person (x) • {ken, • jim, • ben, • mary, • evans}
- 42. 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
- 43. Predicate • Mary is a girl • Girl • Girl (mary)
- 44. Predicate • Mary lives in Tsuru • Lives_in_Tsuru (mary)
- 45. Set of girls • Girl (mary) • Girl (eri) • Girl (rie) • Girl = {mary, eri, rie}
- 46. 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}
- 47. A girl lives in Tsuru • ∃x • ( • Girl (x) • & Lives_in_tsuru (x) • ) • ∃x (Girl (x) & Lives_in_tsuru (x))
- 48. Set of girls • Girl (mary) • Girl (eri) • Girl (rie) • Girl = {mary, eri, rie}
- 49. 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}
- 50. • ∃x (Girl (x) & Lives_in_tsuru (x)) • Is True!
- 51. A girl lives in Fujiyoshida • ∃x • ( • Girl (x) • & Lives_in_fujiyoshida (x) • ) • ∃x (Girl (x) & Lives_in_fujiyoshida (x))
- 52. Set of girls • Girl (mary) • Girl (eri) • Girl (rie) • Girl = {mary, eri, rie}
- 53. 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}
- 54. • ∃x (Girl (x) & Lives_in_fujiyoshida (x)) • Is False!
- 55. • 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
- 56. Any problems with Predicate Logic? • Yes • We are not always trying to say things that are true
- 57. The sky is blue
- 58. We don’t say the sky is blue at night
- 59. Even though it’s true
- 60. • 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))
- 61. • 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?
- 62. New ideas about how we understand meaning
- 63. Language connects to the body
- 64. NOT all in the brain!
- 65. Also strong evidence for IMAGES rather than CODE
- 66. Strong evidence for ACTION simulation
- 67. Mental Models as IMAGES
- 68. Many people say Logical Form cannot be real
- 69. 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)
- 70. • ∀x (Child (x) Kissed (nurse, x)) on x’s birthday • The nurse kissed every child on his birthday
- 71. But Logic is VERY important in Linguistics
- 72. And logic is the basis of computer science
- 73. ∃ (existential) • M(jen,mary) • M = mother • Jen is Mary’s mother • ∃xM(x,y) • Someone is the mother of y
- 74. • ∀y∃xM(x,y) • Everyone has a mother
- 75. • Likes (ben, emi) • Ben likes Emi • Likes (x, y) • x likes y
- 76. 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
- 77. • ∃y (Likes (x, y) • There is some girl that x likes • ∀x∃y (Likes (x, y) • Each boy likes a girl
- 78. Each boy likes a girl Boys Girls
- 79. ∀x∃y (Likes (x, y) Boys Girls
- 80. There’s one girl all the boys like Boys Girls
- 81. All the boys like one girl Boys Girls
- 82. ∃y∀x (Likes (x, y) Boys Girls
- 83. p → q • p = Jim is dead • q = Mary is sad • p → q = • If Jim is dead, Mary is sad
- 84. p V q • p = My parrot is clever • q = Taro fell over • p V q = • (either) my parrot is clever or Taro fell over
- 85. p → q • p = I study hard • q = I will pass the test • p → q = • If I study hard, I will pass the test
- 86. ¬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)
- 87. p → q • Normal everyday values for p and q • p = a bear has warm blood • q = a bear is a mammal
- 88. ¬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
- 89. 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?