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Inductive inference

Inductive inference. Based on Graham Priest’s Logic: A very short introduction, 2000 Translation Petr Hromek (Logika, Dokořán 2007). The Red-Headed League.

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Inductive inference

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  1. Inductive inference Based on Graham Priest’s Logic: A very short introduction, 2000 Translation Petr Hromek (Logika, Dokořán 2007)

  2. The Red-Headed League Sherlock Holmes's quick eye took in my occupation, and he shook his head with a smile as he noticed my questioning glances. "Beyond the obvious facts that he has at some time done manual labor, that he takes snuff, that he is a Freemason, that he has been in China, and that he has done a considerable amount of writing lately, I can deduce nothing else.” Mr. Jabez Wilson started up in his chair, with his forefinger upon the paper, but his eyes upon my companion. How, in the name of good fortune, did you know all that, Mr. Holmes?" he asked. "How did you know, for example, that I did manual labor? It's as true as gospel, for I began as a ship's carpenter." "Your hands, my dear sir. Your right hand is quite a size larger than your left. You have worked with it and the muscles are more developed." "Well, the snuff, then, and the Freemasonry?" "I won't insult your intelligence by telling you how I read that, especially as, rather against the strict rules of your order, you use an arc and compass breastpin." "Ah, of course, I forgot that. But the writing?" "What else can be indicated by that right cuff so very shiny for five inches, and the left one with the smooth patch near the elbow where you rest it upon the desk." "Well, but China?" "The fish which you have tattooed immediately above your wrist could only have been done in China. I have made a small study of tattoo marks, and have even contributed to the literature of the subject. That trick of staining the fishes' scales of a delicate pink is quite peculiar to China. When, in addition, I see a Chinese coin hanging from your watch chain, the matter becomes even more simple." Arthur Conan Doyle: Red-headed League http://www.eastoftheweb.com/short-stories/UBooks/RedHead.shtml

  3. The Red-Headed League Sherlock Holmes's quick eye took in my occupation, and he shook his head with a smile as he noticed my questioning glances. "Beyond the obvious facts that he has at some time done manual labor, that he takes snuff, that he is a Freemason, that he has been in China, and that he has done a considerable amount of writing lately, I can deduce nothing else.” Mr. Jabez Wilson started up in his chair, with his forefinger upon the paper, but his eyes upon my companion. How, in the name of good fortune, did you know all that, Mr. Holmes?" he asked. "How did you know, for example, that I did manual labor? It's as true as gospel, for I began as a ship's carpenter." "Your hands, my dear sir. Your right hand is quite a size larger than your left. You have worked with it and the muscles are more developed." "Well, the snuff, then, and the Freemasonry?" "I won't insult your intelligence by telling you how I read that, especially as, rather against the strict rules of your order, you use an arc and compass breastpin." "Ah, of course, I forgot that. But the writing?" "What else can be indicated by that right cuff so very shiny for five inches, and the left one with the smooth patch near the elbow where you rest it upon the desk." "Well, but China?" "The fish which you have tattooed immediately above your wrist could only have been done in China. I have made a small study of tattoo marks, and have even contributed to the literature of the subject. That trick of staining the fishes' scales of a delicate pink is quite peculiar to China. When, in addition, I see a Chinese coin hanging from your watch chain, the matter becomes even more simple." Arthur Conan Doyle: Red-headed League http://www.eastoftheweb.com/short-stories/UBooks/RedHead.shtml

  4. The Red-Headed League Sherlock Holmes's quick eye took in my occupation, and he shook his head with a smile as he noticed my questioning glances. "Beyond the obvious facts that he has at some time done manual labor, that he takes snuff, that he is a Freemason, that he has been in China, and that he has done a considerable amount of writing lately, I can deduce nothing else.” Mr. Jabez Wilson started up in his chair, with his forefinger upon the paper, but his eyes upon my companion. How, in the name of good fortune, did you know all that, Mr. Holmes?" he asked. "How did you know, for example, that I did manual labor? It's as true as gospel, for I began as a ship's carpenter." "Your hands, my dear sir. Your right hand is quite a size larger than your left. You have worked with it and the muscles are more developed." "Well, the snuff, then, and the Freemasonry?" "I won't insult your intelligence by telling you how I read that, especially as, rather against the strict rules of your order, you use an arc and compass breastpin." "Ah, of course, I forgot that. But the writing?" "What else can be indicated by that right cuff so very shiny for five inches, and the left one with the smooth patch near the elbow where you rest it upon the desk." "Well, but China?" "The fish which you have tattooed immediately above your wrist could only have been done in China. I have made a small study of tattoo marks, and have even contributed to the literature of the subject. That trick of staining the fishes' scales of a delicate pink is quite peculiar to China. When, in addition, I see a Chinese coin hanging from your watch chain, the matter becomes even more simple." Arthur Conan Doyle: Red-headed League http://www.eastoftheweb.com/short-stories/UBooks/RedHead.shtml

  5. Example 1 Guess a number.

  6. Example 1 Guess a number. Given a number, guess the next

  7. Example 1 Guess a number. your guess

  8. Example 1 Guess a number. your guess 1 Next ?

  9. Example 1 Guess a number. your guess 1 Next ? 2

  10. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ?

  11. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ? 3

  12. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ? 3 Yes, 3. Next ?

  13. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ? 3 Yes, 3. Next ? 4

  14. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ? 3 Yes, 3. Next ? 4 No, 5. Next ?

  15. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ? 3 Yes, 3. Next ? 4 No, 5. Next ? 7

  16. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ? 3 Yes, 3. Next ? 4 No, 5. Next ? 7 Yes, 7. Next ?

  17. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ? 3 Yes, 3. Next ? 4 No, 5. Next ? 7 Yes, 7. Next ? 11

  18. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ? 3 Yes, 3. Next ? 4 No, 5. Next ? 7 Yes, 7. Next ? 11 Yes, 11. Next?

  19. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ? 3 Yes, 3. Next ? 4 No, 5. Next ? 7 Yes, 7. Next ? 11 Yes, 11. Next? prime numbers ?

  20. Example 1 Guess a number. your guess 1 Next ? 2 Yes, 2. Next ? 3 Yes, 3. Next ? 4 No, 5. Next ? 7 Yes, 7. Next ? 11 Yes, 11. Next? prime numbers ? Yes.

  21. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an

  22. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an • Can we always find a solution?

  23. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an • Can we always find a solution? • f(i) must be fixed during the process of inference

  24. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an • Can we always find a solution? • f(i) must be fixed during the process of inference • Can we always find a solution?

  25. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an • Can we always find a solution? • f(i) must be fixed during the process of inference • Can we always find a solution? • f(i) must not be dependent on the answer

  26. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an • Can we always find a solution? • f(i) must be fixed during the process of inference • Can we always find a solution? • f(i) must not be dependent on the answer • Can we always find a solution?

  27. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an • Can we always find a solution? • f(i) must be fixed during the process of inference • Can we always find a solution? • f(i) must not be dependent on the answer • Can we always find a solution? • a1, a2, a3, … must be “representative”

  28. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an • Can we always find a solution? • f(i) must be fixed during the process of inference • Can we always find a solution? • f(i) must not be dependent on the answer • Can we always find a solution? • a1, a2, a3, … must be “representative” Example: 0, 0, 0, Next?

  29. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an • Can we always find a solution? • f(i) must be fixed during the process of inference • Can we always find a solution? • f(i) must not be dependent on the answer • Can we always find a solution? • a1, a2, a3, … must be “representative” Example: 0, 0, 0, Next? 0

  30. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an • Can we always find a solution? • f(i) must be fixed during the process of inference • Can we always find a solution? • f(i) must not be dependent on the answer • Can we always find a solution? • a1, a2, a3, … must be “representative” Example: 0, 0, 0, Next? 0 No, 6

  31. Example 1: Guess a number • Knowing a1, a2, a3, … , find a function f(n) = an • Can we always find a solution? • f(i) must be fixed during the process of inference • Can we always find a solution? • f(i) must not be dependent on the answer • Can we always find a solution? • a1, a2, a3, … must be “representative” Example: 0, 0, 0, Next? 0 No, 6 f(n) = (n-1)*(n-2)*(n-3)

  32. Example 2 raining warm Mon Tueyes Wed yes yes Thu yes Fri Sat yes yes Sun yes

  33. Example 2 raining warm Mon P(raining) = Tue yes Wed yes yes Thu yes Fri Sat yes yes Sun yes

  34. Example 2 raining warm Mon P(raining) = 3/7 Tue yes Wed yes yes Thu yes Fri Sat yes yes Sun yes

  35. Example 2 raining warm Mon P(raining) = 3/7 Tue yes P(warm) = Wed yes yes Thu yes Fri Sat yes yes Sun yes

  36. Example 2 raining warm Mon P(raining) = 3/7 Tue yes P(warm) = 4/7 Wed yes yes Thu yes Fri Sat yes yes Sun yes

  37. Example 2 raining warm Mon P(raining) = 3/7 Tue yes P(warm) = 4/7 Wed yes yes P(warm|raining)= Thu yes Fri Sat yes yes Sun yes

  38. Example 2 raining warm Mon P(raining) = 3/7 Tue yes P(warm) = 4/7 Wed yes yes P(warm|raining) = = P(warm & raining)/p(raining) Thu yes Fri Sat yes yes Sun yes

  39. Example 2 raining warm Mon P(raining) = 3/7 Tue yes P(warm) = 4/7 Wed yes yes P(warm|raining) = = P(warm & raining)/p(raining) Thu yes = (2/7)/(3/7) = 2/3 Fri Sat yes yes Sun yes

  40. Example 2 raining warm Mon P(raining) = 3/7 Tue yes P(warm) = 4/7 Wed yes yes P(warm|raining) = ? = P(warm & raining)/p(raining) Thu yes = (2/7)/(3/7) = 2/3 P(¬warm|raining) = Fri Sat yes yes Sun yes

  41. Example 2 raining warm Mon P(raining) = 3/7 Tue yes P(warm) = 4/7 Wed yes yes P(warm|raining) = ? = P(warm & raining)/p(raining) Thu yes = (2/7)/(3/7) = 2/3 P(¬warm|raining) = Fri = P(¬warm & raining)/p(raining) Sat yes yes Sun yes

  42. Example 2 raining warm Mon P(raining) = 3/7 Tue yes P(warm) = 4/7 Wed yes yes P(warm|raining) = = P(warm & raining)/p(raining) Thu yes = (2/7)/(3/7) = 2/3 P(¬warm|raining) = Fri = P(¬warm & raining)/p(raining) = (1/7)/(3/7) = 1/3 Sat yes yes Sun yes

  43. Example 2 raining warm Mon P(raining) = 3/7 Tue yes P(warm) = 4/7 Wed yes yes P(warm|raining) = = P(warm & raining)/p(raining) Thu yes = (2/7)/(3/7) = 2/3 P(¬warm|raining) = Fri = P(¬warm & raining)/p(raining) = (1/7)/(3/7) = 1/3 Sat yes yes P(warm|raining) > P(¬warm|raining) Sun yes

  44. Example 2 raining warm Mon P(raining) = 3/7 Tue yes P(warm) = 4/7 Wed yes yes P(warm|raining) = ? = P(warm & raining)/p(raining) Thu yes = (2/7)/(3/7) = 2/3 P(¬warm|raining) = Fri = P(¬warm & raining)/p(raining) = (1/7)/(3/7) = 1/3 Sat yes yes P(warm|raining) > P(¬warm|raining) => Sun yeswarminductively entails from the premise raining

  45. Inductive entailment P(H|E) > P(¬H|E) => H inductively entails from E E … description of a reference group (database,corpus) E= E+ E- E+ … positive examples, E- … negative examples more than 2 classes: E=  Ei Hi = argmax(i) P(Hi|E)

  46. Back to the Read-headed league H = Jabez Wilson did manual labor ¬H = Jabez Wilson did not manual labor E = London inhabitants ? = rich people ? = Sherlock Holmes’ clients ? = human beings ?

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