[busy signaL]
[Friday, March 25, 2005]
i. the hempel's ravens paradox
The statement "all ravens are black" is logically equivalent to the statement "all non-black-things are non-ravens". If we observe a red apple, that is consistent with that statement. A red apple is a non-black-thing, and when we examine it, we observe that it is a non-raven. So by the principle of induction, observing a red apple should increase our belief that all ravens are black!
ii. the horse paradox
A farmer has 11 horses, and he dies. His will says that his eldest son is to receive 1/2 of the horses in his stable, his middle son is to receive 1/4 of the horses, and the youngest son is to receive 1/6 of the horses. How can the horses be divided?
The lawyer rides to the farm, bringing his own horse. Now there are 12 horses in the stable. The eldest son receives 6 horses, the middle son receives 3 horses, and the youngest son receives 2 horses. The lawyer then takes back his horse.
This paradox works because the farmer, in giving 1/2, 1/4, and 1/6 of his horses, has not given away all of his horses but instead only 11/12 of them. Therefore without the 12th horse, the will gives away 10.083333 horses and leaves 0.916667 horses remaining. Adding the final horse evens out the fractions and allows the 1 remaining horse that the farmer did not give away to return to the lawyer.
iii. the unexpected hanging paradox
A judge makes two statements to a condemned prisoner:
1. You will be hanged at noon one day next week, Monday through Friday.
2. The choice of day will be a surprise to you, in that you won't know the day of the hanging until the executioner knocks on your cell door at noon that day.
The prisoner reflects on these statements, and then smiles. If the hanging were on Friday, then it wouldn't be a surprise in the sense announced. For he would know by Thursday night that he was going to be hanged on Friday, since no hanging had yet occurred and only one day was left. So the hanging can't be on Friday.
But then the hanging can't be on Thursday either. If it were, then it wouldn't be a surprise either. For he would know on Wednesday night that he was going to be hanged on Thursday, since no hanging had yet occurred and only two days were left, one of which (Friday) he already knows is impossible. So a Thursday hanging is impossible too. Similar reasoning shows that the hanging can't be on Wednesday, Tuesday or even Monday! He returns to his cell confident in his safety.
The next week, the executioner knocks on his door at noon on Wednesday - an utter surprise. Everything the judge said has come true, but where is the flaw in the prisoner's reasoning?
By believing himself safe, the executioner's arrival became a surprise.
iv. the george washington's axe and tin woodsman paradoxes
George Washington is supposed, in an apocryphal story, to have cut down his father's cherry tree. The axe is supposedly on display in an (unverifiable) American museum, although, having had both its handle and its head replaced several times, no part of the original axe remains.
Oz' Tin Man wasn't always made of tin. A witch cast a spell on his axe, and it cut off his leg. He proceeded to get the leg replaced with a tin leg, only to have the enchanted axe chop off the other. He then replaced that missing leg with a tin prosthetic, only to have the process repeat itself with both his arms, torso, and eventually head. By the end of the ordeal, he was entirely made out of tin replacements, yet still living, and had become the Tin Man. In a later book, the paradox is further satirized when the Tin Man meets his old flesh body parts, which have been re-assembled with magic glue. His former lover is not sure whom to marry.
v. the omnipotence paradox
The omnipotence paradox is an argument that an omnipotent being cannot exist due to the following lines of argument. An omnipotent being is one that can do anything. Assuming there is such a being, one can then pose a question like the following: "Can an omnipotent being create a stone that is too heavy for him to lift? If he can create it, then there is one thing he can't do: lift the stone. If he can't create it, then there is one thing he can't do: create such a stone. Either way, there is something he can't do, which contradicts the assumption that he is omnipotent.
vi. the grand hotel paradox
In a hotel with a finite number of rooms, once it is full, no more guests can be accommodated. Now imagine a hotel with an infinite number of rooms. You might assume that the same problem will arise when all the rooms are taken. However, there is a way to solve this: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1.
vii. the thinking cap paradox
click me.
sige ingat.ΓΌ
posted by stani at 12:52 AM
The statement "all ravens are black" is logically equivalent to the statement "all non-black-things are non-ravens". If we observe a red apple, that is consistent with that statement. A red apple is a non-black-thing, and when we examine it, we observe that it is a non-raven. So by the principle of induction, observing a red apple should increase our belief that all ravens are black!
ii. the horse paradox
A farmer has 11 horses, and he dies. His will says that his eldest son is to receive 1/2 of the horses in his stable, his middle son is to receive 1/4 of the horses, and the youngest son is to receive 1/6 of the horses. How can the horses be divided?
The lawyer rides to the farm, bringing his own horse. Now there are 12 horses in the stable. The eldest son receives 6 horses, the middle son receives 3 horses, and the youngest son receives 2 horses. The lawyer then takes back his horse.
This paradox works because the farmer, in giving 1/2, 1/4, and 1/6 of his horses, has not given away all of his horses but instead only 11/12 of them. Therefore without the 12th horse, the will gives away 10.083333 horses and leaves 0.916667 horses remaining. Adding the final horse evens out the fractions and allows the 1 remaining horse that the farmer did not give away to return to the lawyer.
iii. the unexpected hanging paradox
A judge makes two statements to a condemned prisoner:
1. You will be hanged at noon one day next week, Monday through Friday.
2. The choice of day will be a surprise to you, in that you won't know the day of the hanging until the executioner knocks on your cell door at noon that day.
The prisoner reflects on these statements, and then smiles. If the hanging were on Friday, then it wouldn't be a surprise in the sense announced. For he would know by Thursday night that he was going to be hanged on Friday, since no hanging had yet occurred and only one day was left. So the hanging can't be on Friday.
But then the hanging can't be on Thursday either. If it were, then it wouldn't be a surprise either. For he would know on Wednesday night that he was going to be hanged on Thursday, since no hanging had yet occurred and only two days were left, one of which (Friday) he already knows is impossible. So a Thursday hanging is impossible too. Similar reasoning shows that the hanging can't be on Wednesday, Tuesday or even Monday! He returns to his cell confident in his safety.
The next week, the executioner knocks on his door at noon on Wednesday - an utter surprise. Everything the judge said has come true, but where is the flaw in the prisoner's reasoning?
By believing himself safe, the executioner's arrival became a surprise.
iv. the george washington's axe and tin woodsman paradoxes
George Washington is supposed, in an apocryphal story, to have cut down his father's cherry tree. The axe is supposedly on display in an (unverifiable) American museum, although, having had both its handle and its head replaced several times, no part of the original axe remains.
Oz' Tin Man wasn't always made of tin. A witch cast a spell on his axe, and it cut off his leg. He proceeded to get the leg replaced with a tin leg, only to have the enchanted axe chop off the other. He then replaced that missing leg with a tin prosthetic, only to have the process repeat itself with both his arms, torso, and eventually head. By the end of the ordeal, he was entirely made out of tin replacements, yet still living, and had become the Tin Man. In a later book, the paradox is further satirized when the Tin Man meets his old flesh body parts, which have been re-assembled with magic glue. His former lover is not sure whom to marry.
v. the omnipotence paradox
The omnipotence paradox is an argument that an omnipotent being cannot exist due to the following lines of argument. An omnipotent being is one that can do anything. Assuming there is such a being, one can then pose a question like the following: "Can an omnipotent being create a stone that is too heavy for him to lift? If he can create it, then there is one thing he can't do: lift the stone. If he can't create it, then there is one thing he can't do: create such a stone. Either way, there is something he can't do, which contradicts the assumption that he is omnipotent.
vi. the grand hotel paradox
In a hotel with a finite number of rooms, once it is full, no more guests can be accommodated. Now imagine a hotel with an infinite number of rooms. You might assume that the same problem will arise when all the rooms are taken. However, there is a way to solve this: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1.
vii. the thinking cap paradox
click me.
sige ingat.ΓΌ
posted by stani at 12:52 AM
Anonymous at 10:01 AM