1. Paradoxes 2nd Term 2017 Dr. Michael Johnson
The Sorites Paradox
Paradoxes
2nd Term 2017
Dr. Michael Johnson

2. The Sorites Paradox
Premise #1: Some people are poor.

3. The Sorites Paradox
Premise #2: Some people are not poor.

4. The Sorites Paradox
Premise #3: There are no sharp boundaries. A poor man cannot become not-poor simply by finding HKD $0.10. A not-poor man cannot become poor by losing HKD $0.10

5. The Sorites Paradox
Conclusion: Everyone is both poor and not poor.

6. The Reasoning

7. No Sharp Boundaries
Premise #3: There are no sharp boundaries. A poor man cannot become not-poor simply by finding HKD $0.10. A not-poor man cannot become poor by losing HKD $0.10
This logically entails conditionals like the following:
If having $5 is poor, then having $5.10 is also poor.
If having $1M is not poor, then having $1M - $0.10 is not poor.

8. Sorites Series
Premise #1: Some people are poor. (Pick one such person. Suppose he has $0.00.) If having $0.00 is poor, then having $0.10 is poor. If having $0.10 is poor, then having $0.20 is poor. If having $0.20 is poor, then having $0.30 is poor. … If having $0,999, is poor, then having $1M is poor. Conclusion: Someone who has $1M is poor.

9. Reverse Sorites Series
Premise #2: Some people are not poor. (Pick one such person. Suppose he has $1M.) If having $1M is not poor, then having $999, is not poor. If having $999, is not poor, then having $999, is not poor. If having $999, is not poor, then having $999, is not poor. … If having $0.10 is not poor, then having $0.00 is not poor. Conclusion: Someone who has $0.00 is not poor.

10. Vagueness

11. Borderline Cases

12. Sharp Boundaries?

13. YouTube Commenters
Roop Dhillon: “This is fucking stupid.” gedstrom: “This is NOT a paradox! "Heap" is an imprecise number. Removing one grain of sand from an imprecise number leaves an imprecise number. The paradox goes away completely if we define exactly how many grains of sand constitute a heap. It then remains a heap until we reach that defined threshold. There is no ambiguity.”

14. The “Legislative” Solution
Let’s consider the common non- philosopher’s response: the paradox “goes away” if we simply define precisely how much money counts as “rich” or how many grains of sand count as “a heap.”

15. The “Legislative” Solution
If you legislate the precise meanings of ALL the vague terms in the language, then you speak an ideal language, that’s perfectly precise.

16. The “Legislative” Solution
This is what the early founders of analytic philosophy recommended we do: Precisify our messy language and speak a nice ideal language.

17. The “Legislative” Solution
But this doesn’t solve the problem! Just because your language can’t state the paradox doesn’t mean it ceases to exist.

18. Timothy Williamson British philosopher
Wykeham Chair of Logic at Oxford
Knowledge-first epistemology
Solution to sorites: epistemicism

19. Epistemicism
The epistemicist solution says that we already do speak an ideal language with sharp boundaries for each term. This solution explains why we think there are no sharp boundaries as follows: they exist, but it is impossible to know where they are.

20. Epistemicism
The epistemicist solution says that we already do speak an ideal language with sharp boundaries for each term. This solution explains why we think there are no sharp boundaries as follows: they exist, but it is impossible to know where they are.

21. Epistemicism
Basic problem: What determines the boundary if not how we use the words? What determines how we use the words if not what we (can) know?

22. Epistemicism
Further problem: the epistemicist says we can’t know where the sharp boundary is, but that it exists. However, he has to admit that we can:
Guess where the sharp boundary is.
Wonder where the sharp boundary is.
Fear that we are crossing the sharp boundary (e.g. for getting old).
But all these seem silly!

23. New Logic?

24. Many-Valued Logics
Another solution is to introduce a new truth-value: True, False, and Undefined. There’s no sharp boundaries, because there’s no point at which adding one hair moves someone from truly bald to falsely bald.

25. T T T T T T T T T T T U U U U U U U U U U U U F F F F F F F F F F F
Many-Valued Logics
T T T T T T T T T T T U U U U U U U U U U U U F F F F F F F F F F F

26. Higher-Order Vagueness
The problem is that now there are sharp boundaries between being truly bald and undefinedly bald, and between being undefinedly bald, and falsely bald. Intuitively, adding one hair to a truly bald person can’t make them undefinedly bald.

27. T T T T T T T T T T T U U U U U U U U U U U U F F F F F F F F F F F
Two Sharp Boundaries
T T T T T T T T T T T U U U U U U U U U U U U F F F F F F F F F F F

28. Fuzzy Logic
Instead, we might try having infinitely many truth-values: 1 is fully true, 0 is fully false, and any number in between is less than fully true.

29. No Sharp Boundaries …

30. Fuzzy Logic
A fuzzy logician has to explain how to calculate the truth-values of complex expressions from the truth values of their parts. Common rules:
The truth-value of “not-P” is 1 minus the truth-value of P
The truth-value of “P and Q” is the lowest of the truth-values of P and Q.
The truth-value of “P or Q” is the highest of the truth values of P and Q.

31. Problems
“P and not-P” should always be fully false: 0. But if P = 0.5, then “P and not-P” = 0.5