Semantic Paradoxes

1. Semantic Paradoxes

2. the barber

3. The Barber Paradox Once upon a time there was a village, and in this village lived a barber named B.

4. The Barber Paradox B shaved all the villagers who did not shave themselves, And B shaved none of the villagers who did shave themselves.

5. The Barber Paradox Question, did B shave B, or not?

6. Suppose B Shaved B 1.B shaved B Assumption 2. B did not shave any villager X where X shaved X Assumption 3. B did not shave B 1,2 Logic

7. Suppose B Did Not Shave B 1.B did not shave B Assumption 2. B shaved every villager X where X did not shave X Assumption 3. B shaved B 1,2 Logic

8. Contradictions with Assumptions We can derive a contradiction from the assumption that B shaved B. We can derive a contradiction from the assumption that B did not shave B.

9. The Law of Excluded Middle Everything is either true or not true. Either P or not-P, for any P. Either B shaved B or B did not shave B, there is not third option.

10. It’s the Law • Either it’s Tuesday or it’s not Tuesday. • Either it’s Wednesday or it’s not Wednesday. • Either killing babies is good or killing babies is not good. • Either this sandwich is good or it is not good.

11. Disjunction Elimination A or B A implies C B implies C Therefore, C

12. Example Either Michael is deador he has no legs If Michael is dead, he can’t run the race. If Michael has no legs, he can’t run the race. Therefore, Michael can’t run the race.

13. Contradiction, No Assumptions B shaves B or B does not shave B [Law of Excluded Middle] If B shaves B, contradiction. If B does not shave B, contradiction. Therefore, contradiction

14. Contradictions Whenever we are confronted with a contradiction, we need to give up something that led us into the contradiction.

15. Give up Logic? For example, we used Logic in the proof that B shaved B if and only if B did not shave B. So we might consider giving up logic. A or B A implies C B implies C Therefore, C

16. No Barber In this instance, however, it makes more sense to give up our initial acquiescence to the story: We assumed that there was a village with a barber who shaved all and only the villagers who did not shave themselves.

17. The Barber Paradox The paradox shows us that there is no such barber, and that there cannot be.

18. Semantic Paradoxes Unfortunately, much of our semantic vocabulary like ‘is true’ and ‘applies to’ leads us into contradictions where it is highly non-obvious what to abandon.

19. The paradox of the liar

20. Disquotation To say P is the same thing as saying ‘P’ is true. This is the “disquotation principle”: P = ‘P’ is true

21. Liar Sentence The liar sentence is a sentence that says that it is false. For example, “This sentence is false,” or “The second example sentence in the powerpoint slide titled ‘Liar Sentence’ is false.”

22. Liar Sentence L = ‘L’ is not true

23. “‘L’ is true” 1. ‘L’ is true Assumption 2. L 1, Disquotation 3. ‘L’ is not true 2, Def of L 1 & 3 form a contradiction

24. “‘L’ is not true” 1. ‘L’ is not true Assumption 2. L 1, Def of L 3. ‘L’ is true 2, Disquotation 1 & 3 form a contradiction

25. Contradiction Thus we can derive a contradiction from the assumption that “‘L’ is true or ‘L’ is not true,” [Law of Excluded Middle] plus the inference rule: A or B A implies C B implies C Therefore, C

26. Contradiction ‘L’ is true or ‘L’ is not true [Law of Excluded Middle] If ‘L’ is true, then ‘L’ is true and not true. If ‘L’ is not true, then ‘L’ is true and not true. Therefore, ‘L’ is true and not true.

27. Solutions • Give up excluded middle • Give up disjunction elimination • Give up disquotation • Disallow self-reference • Accept that some contradictions are true

28. 1. Giving up Excluded Middle The problem with giving up the Law of Excluded Middle is that it seems to collapse into endorsing contradictions: “According to LEM, every sentence is either true or not true. I disagree: I think that some sentences are not true and not not true at the same time.”

29. 2. Give up Disjunction Elimination Basic logical principles are difficult to deny. What would a counterexample to disjunction elimination look like? A or B A implies C B implies C However, not-C

30. 3. Give up Disquotation Principle Giving up the disquotation principle P = ‘P’ is true Involves accepting that sometimes P but ‘P’ is not true or accepting that not-P but ‘P’ is true.

31. 4. Disallow Self-Reference The problem with disallowing self-reference is that self-reference isn’t essential to the paradox. A: ‘B’ is true B: ‘A’ is not true

32. Circular Reference ‘A’ is false. ‘B’ is true. B A

33. Assume ‘A’ Is True ‘A’ is false. ‘B’ is true. B A

34. Then ‘B’ Is Also True ‘A’ is false. ‘B’ is true. B A

35. But Then ‘A’ is False! ‘A’ is false. ‘B’ is true. B A

36. Assume ‘A’ Is False ‘A’ is false. ‘B’ is true. B A

37. Then ‘B’ Is Also False ‘A’ is false. ‘B’ is true. B A

38. But Then ‘A’ Is Also True ‘A’ is false. ‘B’ is true. B A

39. “‘A’ is true” 1. ‘A’ is true Assumption 2. A 1, Disquotation 3. ‘B’ is true 2, Def of A 4. B 3, Disquotation 5. ‘A’ is not true 4, Def of B

40. “‘A’ is not true” 1. ‘A’ is not true Assumption 2. B 1, Def of B 3. ‘B’ is true 2, Disquotation 4. A 3, Def of A 5. ‘A’ is true 4, Disquotation

41. Contradiction, No Assumptions Either ‘A’ is true or ‘A’ is not true. [Law of Excluded Middle] If ‘A’ is true, then ‘A’ is true and not true. If ‘A’ is not true, then ‘A’ is true and not true. Therefore, ‘A’ is true and not true.

42. Disallowing Circular Reference Even circular reference is not essential. Stephen Yablo has shown that non-circular sets of sentences cause paradox too: Let Ai = all sentences ‘Aj’ for j > i are not true. Then {A0, A1, A2,…} are inconsistent.

43. Yablo’s Paradox Set Y1: For all k > 1, Yk is not true. Y2: For all k > 2, Yk is not true. Y3: For all k > 3, Yk is not true. Y4: For all k > 4, Yk is not true. Y5: For all k > 5, Yk is not true. … Yn: For all k > n, Yk is not true. …

44. Yablo’s Paradox Set {A0, A1, A2, A3, A4, A5,A6, A7, A8,…Aj, Aj+1,…}

45. Yablo’s Paradox Set {A0, A1, A2, A3, A4, A5,A6, A7, A8,…Aj, Aj+1,…} All of those guys are false!

46. Yablo’s Paradox Set {A0, A1, A2, A3, A4, A5,A6, A7, A8,…Aj, Aj+1,…} All of those guys are false!

47. Yablo’s Paradox Set {A0, A1, A2, A3, A4, A5,A6, A7, A8,…Aj, Aj+1,…} All of those guys are false!

48. Yablo’s Paradox Set {A0, A1, A2, A3, A4, A5,A6, A7, A8,…Aj, Aj+1,…} All of those guys are false!

49. Yablo’s Paradox Set {A0, A1, A2, A3, A4, A5,A6, A7, A8,…Aj, Aj+1,…} All of those guys are false!

50. Yablo’s Paradox Now consider some number j. Is Yj true or not true? Suppose Yj is true: