Monday, March 20, 2006

A heaping contradiction

The paradox of heap (or Sorites Paradox) starts with the following premises.

(1) A million grains of sand make a heap.
(2) If N grains of sand make a heap, then so do N-1 grains of sand.
(3) A single grain of sand does not make a heap.

Many people believe this leads to paradox because they intuitively affirm (1), (2), and (3) despite the fact that one can easily deduce a contradiction from (1), (2), and (3). (Consider the minimum N for which N grains of sand make a heap.) I don't think much of this paradox because I think premise (2) is obviously false.

"Do you agree that a million grains of sand make a heap?"
"Yes."
"Do you agree that one grain of sand makes a heap?"
"No."
"Do you agree that a 500,000 grains of sand make a heap?"
"Yes."
"Do you agree that a 250,000 grains of sand make a heap?"
"Yes."
"Do you agree that a 125,000 grains of sand make a heap?"
"Yes."
"Do you agree that a 62,500 grains of sand make a heap?"
"Well, uh, no; at least that's not how I'd use the word 'heap.'"
"Do you agree that a 93,750 grains of sand make a heap?"
"I don't like where this is going. I'm not sure."
"So you don't agree?"
"Well, I partially agree."
"But you don't fully agree?"
"No, I don't fully agree."
"Do you fully agree that 125,000 grains of sand make a heap?"
"Yes."
"Do you fully agree that 109,375 grains of sand make a heap?"
"Yes."
"Do you fully agree that 101,563 grains of sand make a heap?"
"Uh, yes."
"Do you fully agree that 97,656 grains of sand make a heap?"
"I don't know."
"So you don't fully agree?"
"I don't know!"
"Do you knowingly fully agree that 101,562 grains of sand make a heap?"
"Yes."
"Do you knowingly fully agree that 99,609 grains of sand make a heap?"
"Uh, yes."
"Do you knowingly fully agree that 98,632 grains of sand make a heap?"
"No, I don't know"
"Do you knowingly fully agree that 99,121 grains of sand make a heap?"
"No."
"Do you knowingly fully agree that 99,364 grains of sand make a heap?"
"No."
"Do you knowingly fully agree that 99,487 grains of sand make a heap?"
"Look, I know where this is going. Let's just say that I knowingly fully agree that 99,600 grains of sand make a heap, but I don't knowingly fully agree that 99,599 grains of sand make a heap."
"Are you ready to renounce premise (2)?"

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