Haiming added a topic in ParadoxesAchilles and the TortoiseZeno's second paradox of motion, of Achilles and the tortoise, is probably the best known of his four paradoxes of motion. In this problem, the fleet Greek warrior runs a race against a slow-moving tortoise. Assume Achilles runs at ten times the speed of the tortoise (1 meter per second to 0.1 meter per second). The tortoise is given a 100-meter handicap in a race that is 1,000 meters. By the time Achilles reaches the tortoise's starting point T0, the tortoise will have moved on to point T1. Soon, Achilles will reach point T1, but by then the tortoise would have moved on to T2, and so on, ad infinitum. Every time Achilles reaches a point where the tortoise has just been, the tortoise has moved on a bit. Although the distances between the two runners will diminish rapidly, Achilles can never catch up with the tortoise, or so it would seem.
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Haiming added a topic in ParadoxesHempel's ravens (The Confirmation Paradox)While taking a group of benefactors on a tour through the new aviary they had just helped to build, a noted ornithologist commented, "And here we have two of the finest examples of ravens that I have ever seen. Notice the lustrous black plumage for which all ravens are famous." The ornithologist continued his lecture, commenting on the corvine feeding and nesting habits as well as on the birds' legendary role as harbingers of ill fortune.
When the ornithologist had finished, a young man said, "Sir, excuse me, but did you say that 'All ravens are black'?"
"I don't know if I said exactly that, but it's true. All ravens are black."
"But, how do you know that - for certain, I mean?" asked the young man.
"Well, I've seen a few hundred ravens in my day and every one of them has been black."
"Yes, but a few hundred are not all. How many ravens are there, anyway?"
"I would guess several million. As for your question, many other scientists, and non-scientists for that matter, have observed ravens over thousands of years and so far the birds have all been black. At least, I don't know of a single instance in which someone has produced a non-black raven."
"That's true, but it's still not all - just most."
"True, but there is other evidence. For example, take all these lovely multicolored birds we have seen today - the parrots, toucans, the peacocks -"
"They're lovely, but what do they have to do with your claim that all ravens are black?"
"Don't you see?" asked the ornithologist.
"No, I don't see. Please explain."
"Well, you accept the idea that every new instance of another black raven that is observed adds to the support of the generalization that all ravens are black?"
"Yes, of course."
"Well then, the statement 'All ravens are black' is logically equivalent to the statement 'All non-black things are non-ravens.' This being so and because whatever confirms a statement also confirms any logically equivalent statement, it's clear that any non-black non-raven supports the generalization 'All ravens are black.' Hence, all these colorful, non-black non-ravens also support the generalization."
"That's ridiculous," chided the young man. "In that case you might as well say that your blue jacket and gray pants also confirm the statement 'All ravens are black.' After all, they're also non-black non-ravens."
"That's correct," said the ornithologist. "Now you're beginning to think like a true scientist."
Who is reasoning correctly, the ornithologist or the young man?
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Haiming added a topic in ParadoxesThe Unexpected HangingA man condemned to be hanged was sentenced on Saturday. "The hanging will take place at noon," said the judge to the prisoner, "on one of the seven days of next week. But you will not know which day it is until you are so informed on the morning of the day of the hanging."
The judge was known to be a man who always kept his word. The prisoner, accompanied by his lawyer, went back to his cell. As soon as the two men were alone, the lawyer broke into a grin. "Don't you see?" he exclaimed. "The judge's sentence cannot possibly be carried out."
"I don't see," said the prisoner.
"Let me explain They obviously can't hang you next Saturday. Saturday is the last day of the week. On Friday afternoon you would still be alive and you would know with absolute certainty that the hanging would be on Saturday. You would know this before you were told so on Saturday morning. That would violate the judge's decree."
"True," said the prisoner.
"Saturday, then is positively ruled out," continued the lawyer. "This leaves Friday as the last day they can hang you. But they can't hang you on Friday because by Thursday only two days would remain: Friday and Saturday. Since Saturday is not a possible day, the hanging would have to be on Friday. Your knowledge of that fact would violate the judge's decree again. So Friday is out. This leaves Thursday as the last possible day. But Thursday is out because if you're alive Wednesday afternoon, you'll know that Thursday is to be the day."
"I get it," said the prisoner, who was beginning to feel much better. "In exactly the same way I can rule out Wednesday, Tuesday and Monday. That leaves only tomorrow. But they can't hang me tomorrow because I know it today!"
... He is convinced, by what appears to be unimpeachable logic, that he cannot be hanged without contradicting the conditions specified in his sentence. Then on Thursday morning, to his great surprise, the hangman arrives. Clearly he did not expect him. What is more surprising, the judge's decree is now seen to be perfectly correctly. The sentence can be carried out exactly as stated.
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