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

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

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

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

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

The Reasoning

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.

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 $0,999,999.90 is poor, then having $1M is poor. Conclusion: Someone who has $1M is poor.

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 $999,999.90 is not poor. If having $999,999.90 is not poor, then having $999,999.80 is not poor. If having $999,999.80 is not poor, then having $999,999.70 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.

Vagueness

Borderline Cases

Sharp Boundaries?

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.”

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.”

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.

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.

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.

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

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.

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.

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?

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!

New Logic?

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.

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

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.

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

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.

No Sharp Boundaries 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25…

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.

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