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Predicate calculus up

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A simple introduction to Predicate Calculus, Logic, and Computer Science

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Predicate calculus up

  1. 1. Propositional Logic • p  q • NO internal structure
  2. 2. Predicate logic • Loves (j, m) • DOES have internal structure
  3. 3. Japan is a country
  4. 4. Japan is a country • Country (x) • x = japan • Country (japan)
  5. 5. Easy so far? • Yes • But what is the point? • Good question
  6. 6. It helps with computer programming
  7. 7. We use it to input data
  8. 8. And write code
  9. 9. Data • You can put in any information you want
  10. 10. Peter fell over • Predicate? • Fell_over • How many arguments? • One – peter • Fell_over (peter)
  11. 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. 12. Ben hates computers • Hate (ben, computers)
  13. 13. Eri gave up • Gave_up (eri)
  14. 14. Emi is a genius • Genius (emi)
  15. 15. No limits as long as it’s organized • Visit (mary, familiymart, sat_15th_april,) • Computational Linguistics: • Verb • Subject • Complement
  16. 16. Hit • Hit ( ) • Hit (x, ) • Hit (x, y) • X = bill • Y = ken • Bill hit Ken
  17. 17. Database 1 • “Jun, Ken, and Emi are all Japanese” Japanese (jun) Japanese (ken) Japanese (emi) Japanese (x)  ………………………
  18. 18. Database 1 Japanese (x)  Human (x) Japanse (x)  Asian (x)
  19. 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. 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. 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. 22. Database 2 Father (x, y) & Male (y)  Son (y)  Son (y, x) Father (x, y) & Female (y)  Daughter (y)  Daughter (y, x)
  23. 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)
  24. 24. Quantifiers
  25. 25. The Little Prince is wearing a brown scarf
  26. 26. • Wearing_a_brown_scarf (p) • Wearing (p, b_s)
  27. 27. But the Little Prince is the only person in this world
  28. 28. Everyone is wearing a brown scarf
  29. 29. Imagine simple little worlds
  30. 30. What do quantifiers mean?
  31. 31. All – upside-down A
  32. 32. Think of sets
  33. 33. • Two sets • Set A • Set B
  34. 34. • Set A = the set of Linguists • Linguist (x) • Set B = the set of crazy people • Crazy_person (x)
  35. 35. A = Linguists; B = Crazy people
  36. 36. Who is in these sets? • Linguist (x) • {evans, • imai, • ono}
  37. 37. Computer database • Linguist (evans) • Linguist (imai) • Linguist (ono) • Set of linguists = • {evans, imai, ono}
  38. 38. Who is in the set of crazy people? • Crazy_person (x) • {ken, • jim, • ben, • mary, • evans}
  39. 39. Computer database • Crazy_person (evans) • Crazy_person (ken) • Crazy_person (jim) • Crazy_person (ben) • Crazy_person (mary)
  40. 40. ∀= universal (all) Linguist (x)  Crazy_person (x)
  41. 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. 42. • Set A = the set of Linguists • Linguist (x) • Set B = the set of crazy people • Crazy_person (x)
  43. 43. Who is in these sets? • Linguist (x) • {evans, • imai, • ono}
  44. 44. Who is in the set of crazy people? • Crazy_person (x) • {ken, • jim, • ben, • mary, • evans}
  45. 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. 46. Some linguists are crazy • Is this True?
  47. 47. Does this kind of x exist? • Linguist (x) & Crazy_person (x)
  48. 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. 49. Who is in the set of crazy people? • Crazy_person (x) • {ken, • jim, • ben, • mary, • evans}
  50. 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. 51. Predicate • Mary is a girl • Girl • Girl (mary)
  52. 52. Predicate • Mary lives in Tsuru • Lives_in_Tsuru (mary)
  53. 53. Set of girls • Girl (mary) • Girl (eri) • Girl (rie) • Girl = {mary, eri, rie}
  54. 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. 55. A girl lives in Tsuru • ∃x • ( • Girl (x) • & Lives_in_tsuru (x) • ) • ∃x (Girl (x) & Lives_in_tsuru (x))
  56. 56. Set of girls • Girl (mary) • Girl (eri) • Girl (rie) • Girl = {mary, eri, rie}
  57. 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. 58. • ∃x (Girl (x) & Lives_in_tsuru (x)) • Is True!
  59. 59. A girl lives in Fujiyoshida • ∃x • ( • Girl (x) • & Lives_in_fujiyoshida (x) • ) • ∃x (Girl (x) & Lives_in_fujiyoshida (x))
  60. 60. Set of girls • Girl (mary) • Girl (eri) • Girl (rie) • Girl = {mary, eri, rie}
  61. 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. 62. • ∃x (Girl (x) & Lives_in_fujiyoshida (x)) • Is False!
  63. 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. 64. Any problems with Predicate Logic? • Yes • We are not always trying to say things that are true
  65. 65. The sky is blue
  66. 66. We don’t say the sky is blue at night
  67. 67. Even though it’s true
  68. 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. 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. 70. New ideas about how we understand meaning
  71. 71. Language connects to the body
  72. 72. NOT all in the brain!
  73. 73. Also strong evidence for IMAGES rather than CODE
  74. 74. Strong evidence for ACTION simulation
  75. 75. Mental Models as IMAGES
  76. 76. Many people say Logical Form cannot be real
  77. 77. But Logic is VERY important in Linguistics! • Is it just because computers are important?
  78. 78. Logic is the basis of computer science
  79. 79. ∃ (existential) • M(jen,mary) • M = mother • Jen is Mary’s mother • ∃xM(x,y) • Someone is the mother of y
  80. 80. • ∀y∃xM(x,y) • Everyone has a mother
  81. 81. • Likes (ben, emi) • Ben likes Emi • Likes (x, y) • x likes y
  82. 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. 83. • ∃y (Likes (x, y) • There is some girl that x likes • ∀x∃y (Likes (x, y) • Each boy likes a girl
  84. 84. Each boy likes a girl Boys Girls
  85. 85. Each boy likes a girl Boys Girls
  86. 86. Each boy likes a girl Boys Girls
  87. 87. ∀x∃y (Likes (x, y)) Boys (x) Girls (y)
  88. 88. There’s one girl all the boys like Boys Girls
  89. 89. All the boys like the same one girl Boys Girls
  90. 90. ∃y∀x (Likes (x, y) Boys (x) Girls (y)
  91. 91. Each girl punched a boy Girls Boys
  92. 92. ∀x∃y (Punched (x, y) Girls (x) Boys (y)
  93. 93. All the girls punched the same boy Girls (x) Boys (y)
  94. 94. ∃y ∀x (Punched (x, y) Girls (x) Boys (y)
  95. 95. p → q • p = Jim is dead • q = Mary is sad • p → q = • If Jim is dead, Mary is sad
  96. 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. 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. 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. 99. p → q • Normal everyday values for p and q • p = a bear has warm blood • q = a bear is a mammal
  100. 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. 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?

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