Sorites • 1 grain of sand is not a heap. • For all numbers n: if n grains of sand are not a heap, then n + 1 grains of sand are not a heap. • Therefore, 200 trillion grains of sand are not a heap.
The Other Way • 200 trillion grains of sand makes a heap. • For all numbers n: if n + 1 grains of sand make a heap, then n grains of sand make a heap. • Therefore 1 grain of sand makes a heap.
Paradox Neither of these sorites arguments results in a contradiction… until you add in the obvious fact that the conclusion of each is false.
Borderline Cases The paradox seems to arise whenever we have a term that admits of borderline cases. There are some people that I don’t know whether they’re rich out of uncertainty: because I don’t know how much money they have. These are not borderline cases.
Borderline Cases The paradox seems to arise whenever we have a term that admits of borderline cases. But there are other people that I don’t know whether they are rich even though I know exactly how much money they have. These are borderline cases.
Borderline Cases Most of our ordinary language admits of borderline cases: Big, tall, short, rich, fast, slow, smart, dumb, funny, long, flat, narrow… Also: mountain, car, tree, horse…
What To Do? Neither of these sorites arguments results in a contradiction… until you add in the obvious fact that the conclusion of each is false. To deny the conclusion, we need to deny either premise 1 or premise 2 or logic.
Denying Premise 1 In the first argument, premise 1 is: 1 grain of sand is not a heap. In the second it’s: 200 trillion grains of sand is a heap.
Denying Premise 2 Premise 2 (Argument 1) says: For all numbers n: if n grains of sand are not a heap, then n + 1 grains of sand are not a heap. The negation of this is: There exists a number n such that: n grains of sand are not a heap, but n + 1 grains of sand are a heap.
Denying Premise 2 Premise 2 (Argument 2) says: For all numbers n: if n + 1 grains of sand make a heap, then n grains of sand make a heap. The negation of this is: There is a number n such that: n + 1 grains of sand make a heap, but n grains of sand do not make a heap.
No Sharp Boundaries Premise 2 in both cases asserts No Sharp Boundaries. It’s never true that one grain of sand makes the difference between a heap and not a heap.
No Sharp Boundaries • One hair doesn’t make the difference between being bald and not bald. • One micrometer doesn’t make the difference between being tall and not tall. • $0.10HKD does not make the difference between being rich and not rich. • One nanosecond does not make the difference between being old and not old.
Solutions • Accept Sharp Boundaries. • Introduce more truth-values.
Epistemicism One solution is to claim that there ARE sharp boundaries, but we can never know where they are. Acquiring $0.10 can make someone go from not rich to rich, but we can’t ever know when this happens.
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!
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.
Many-Valued Logics More hairs → tttttttttttttttttttttttuuuuuuuuuuufffffffffffffffffff
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.
Many-Valued Logics More hairs → tttttttttttttttttttttttuuuuuuuuuuufffffffffffffffffff Two sharp boundaries!
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. More hairs → 1 1 1 1 1 1 .99 .98 .98 .97… .12 .11 .1 .1 0 0 0 0 0
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 “~P” is 1 minus the truth-value of P • The truth-value of “P & 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 & ~P” should always be fully false: 0. But if P = 0.5, then “P & ~P” = 0.5
The OtherDisquotation Principle Our earlier disquotation principle was: P = “P” is true. Here’s a very different principle: If a rational speaker sincerely asserts “P” then the speaker believes P. If a rational speaker won’t sincerely accept an assertion of “P” then the speaker does not believe P.
De Dicto Attitudes Let’s call these your de dictoattitudes. “De dicto” means “concerning what is said.” These are the propositional attitudes you have that match what you would say. For example: Lois Lane believes de dicto that Superman can fly. Lois Lane does not believe de dicto that Clark Kent can fly.
De Re Attitudes Sometimes we use the word “believe” in a way that is not consistent with “believe de dicto.”
De Re Attitudes I once knew a woman who told me the following story. She went to Las Vegas and won lots of money in the slot machines. She got into an elevator with several black men, a number of whom were large and intimidating.
De Re Attitudes The woman thought “these men on the elevator are criminals and they are going to rob me!”
De Re Attitudes She would never have asserted or accepted the sentence “Eddie Murphy is a criminal,” so she did not believe de dicto that Murphy was a criminal. BUT… the man on the elevator was Eddie Murphy (and his bodyguards). He had millions of dollars and no need to rob.
De Re Attitudes This case is quite naturally described in the following way: The woman believed that Eddie Murphy was a criminal.
Eddie Murphy She believes X is a criminal That Guy
Eddie Murphy She believes X is a criminal That Guy
De Re Attitudes “De re” means “concerning the thing.” They’re a way we have of reporting propositional attitudes that people have about things, regardless of what people would say about those things. Another example: Dr. Baker says “that guy wanted to pee on my car”– the guy didn’t know it was Baker’s car, or he wouldn’t have tried to pee on it!
Frege Cases We saw before a class of cases where people: • Had an attitude de dicto that X was F • Did not have an attitude de dicto that Y was F • Even though X = Y
Frege Cases • John believes Benjamin Franklin liked Belgian waffles. • Mary discovered that Benjamin Franklin liked potato salad. • Sam doubts that Benjamin Franklin liked deep dish pizza
Frege Cases • John believes that the inventor of bifocals liked Belgian waffles. • Mary discovered that the first postmaster general liked potato salad. • Sam doubts that the author of Poor Richard’s Almanac liked deep dish pizza.
Factual Errors In cases where our de dicto beliefs correspond to contradictory de re beliefs, it seems reasonable to say that this is because we are lacking some factual information. There are things regarding how the world is that we are simply ignorant of.
Two Possibilities Clark Kent Bruce Wayne
The Messy Shopper “I once followed a trail of sugar on a supermarket floor, pushing my cart down the aisle on one side of a tall counter and back the aisle on the other, seeking the shopper with a torn sack to tell him he was making a mess…”
The Messy Shopper “…With each trip around the counter, the trail became thicker. But I seemed unable to catch up. Finally it dawned on me. I was the shopper I was trying to catch.” – John Perry, “The Essential Indexical”
De Se Cases Suppose Ada sees herself in the mirror, unaware that it’s her who is in the mirror. She believes what she’d express by saying “I am pretty.” But she also believes what she’d express by saying “she is not pretty.”
De Se Attitudes “De se” means “concerning the self.” These are attitudes we have about ourselves… but only a special sort of attitudes. (NOT: that person in the mirror/ that person on TV/ etc… attitudes about me.)
Rudolf Lingens Perry imagines another case: Rudolf Lingens has lost his memory. He doesn’t know who he is or where he is or why he’s there. As a matter of fact, he’s in the library at Stanford Univeristy.
Rudolf Lingens In the library, Lingens finds a biography of himself. He reads the entire life story of Rudolf Lingens. The biography even says, “Lingens is currently wandering around the Stanford Library without his memory.” Lingens knows that Lingens did X, Y, and Z. But he does not know that he did X, Y, and Z. (He does not know what he’d express by saying “I know that I did X, Y, and Z.”)
Non-Factual Problem Lingens seems to know all the relevant facts. He knows everything there is to know about LingensEXCEPT that he is Lingens. Is this a further fact? Or are there things you can know that are non-factual?
Two Possibilities Lingens